LMFDB - Newform orbit 3762.2.g.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(2089,3762)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3762.2089");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$30.0397212404$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} + 2 x^{18} + 199 x^{16} - 414 x^{15} + 430 x^{14} + 184 x^{13} + 6939 x^{12} + \cdots + 64 $ x^20 - 2x^19 + 2x^18 + 199x^16 - 414x^15 + 430x^14 + 184x^13 + 6939x^12 - 14914x^11 + 16078x^10 + 10764x^9 + 7501x^8 - 14446x^7 + 21922x^6 + 21836x^5 + 10900x^4 - 80x^3 + 128x^2 + 128x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 1254)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + q^{2} + q^{4} - \beta_{5} q^{5} - \beta_{4} q^{7} + q^{8} - \beta_{5} q^{10} + \beta_{8} q^{11} + \beta_{10} q^{13} - \beta_{4} q^{14} + q^{16} + \beta_{18} q^{17} - \beta_{17} q^{19} - \beta_{5} q^{20}+ \cdots + (\beta_{17} - 3 \beta_{10} + \cdots + \beta_1) q^{98}+O(q^{100}) $ q + q^2 + q^4 - b5 * q^5 - b4 * q^7 + q^8 - b5 * q^10 + b8 * q^11 + b10 * q^13 - b4 * q^14 + q^16 + b18 * q^17 - b17 * q^19 - b5 * q^20 + b8 * q^22 + (-b19 - b15 - b10 - b9 - b5 + 1) * q^23 + (-b19 + b7 + 2) * q^25 + b10 * q^26 - b4 * q^28 + (b15 - b12 - b7 - b5) * q^29 + (b16 - b12 + b11 + b9 + b4) * q^31 + q^32 + b18 * q^34 + (-b18 + b16 - b14 + b13 + b11 + 2b4 + b2) q^35 + (b18 - b16 - b11 - b4) * q^37 - b17 * q^38 - b5 * q^40 + (b19 + b17 + b15 - b12 - b10 + b8 - b7 - b6 + b5 + b3 + b1 + 2) * q^41 + (-b16 + b12 - 2b11 - b9) q^43 + b8 * q^44 + (-b19 - b15 - b10 - b9 - b5 + 1) * q^46 + (-b19 + b15 - b12 - b10 + b8 - b5 + b1 + 1) * q^47 + (b17 - 3b10 + b8 + b7 + b6 - 2b5 + b3 + b1) * q^49 + (-b19 + b7 + 2) * q^50 + b10 * q^52 + (-b16 + b13 + b12 - b11 - b9) * q^53 + (-b19 - b17 + b16 - b15 - b14 - b8 - b3 + b2) * q^55 - b4 * q^56 + (b15 - b12 - b7 - b5) * q^58 + (b16 + b13 + b11 - b8 + b4 + b1) * q^59 + (2b16 + b13 + b2) q^61 + (b16 - b12 + b11 + b9 + b4) * q^62 + q^64 + (-b19 - b15 - b9 - b8 - b5 - b1 + 1) * q^65 + (b18 - 2b16 - 2b11 - b8 - 2b2 + b1) q^67 + b18 * q^68 + (-b18 + b16 - b14 + b13 + b11 + 2b4 + b2) q^70 + (-2b18 - b17 + b11 + 2b8 + b4 + b3 + b2 - 2b1) q^71 + (-b16 + b13 + b8 + b4 + b2 - b1) * q^73 + (b18 - b16 - b11 - b4) * q^74 - b17 * q^76 + (-b18 + b17 - b14 + b13 - 3b10 + b8 + b4 + b3 + b2 + b1) q^77 + (b19 + b17 - b15 - b10 - b9 - b8 - b5 + b3 - b1 + 1) * q^79 - b5 * q^80 + (b19 + b17 + b15 - b12 - b10 + b8 - b7 - b6 + b5 + b3 + b1 + 2) * q^82 + (-b18 - 2b17 + b16 - b14 + b13 + 2b12 - b11 - 2b9 + b8 + 2b3 + b2 - b1) * q^83 + (-b17 + b16 - 2b14 - b12 + b9 + b3) q^85 + (-b16 + b12 - 2b11 - b9) q^86 + b8 * q^88 + (2b17 + b16 - 2b12 + 3b11 + 2b9 - b8 + 2b4 - 2b3 - b2 + b1) * q^89 + (-b18 + b16 - b14 + b13 - b12 + b11 + b9 + b8 + b2 - b1) * q^91 + (-b19 - b15 - b10 - b9 - b5 + 1) * q^92 + (-b19 + b15 - b12 - b10 + b8 - b5 + b1 + 1) * q^94 + (b19 + b18 + b17 - b16 + b14 + b12 - b10 + b8 + b5 - b3 - b2 + 2b1 - 1) q^95 + (-b18 + 3b16 - b14 + b13 - b12 + b11 + b9 - b8 + b2 + b1) q^97 + (b17 - 3b10 + b8 + b7 + b6 - 2b5 + b3 + b1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{2} + 20 q^{4} - 4 q^{5} + 20 q^{8} - 4 q^{10} - 6 q^{11} + 4 q^{13} + 20 q^{16} + 2 q^{19} - 4 q^{20} - 6 q^{22} + 8 q^{23} + 32 q^{25} + 4 q^{26} - 4 q^{29} + 20 q^{32} + 2 q^{38} - 4 q^{40}+ \cdots - 36 q^{98}+O(q^{100}) $ 20 * q + 20 * q^2 + 20 * q^4 - 4 * q^5 + 20 * q^8 - 4 * q^10 - 6 * q^11 + 4 * q^13 + 20 * q^16 + 2 * q^19 - 4 * q^20 - 6 * q^22 + 8 * q^23 + 32 * q^25 + 4 * q^26 - 4 * q^29 + 20 * q^32 + 2 * q^38 - 4 * q^40 + 24 * q^41 - 6 * q^44 + 8 * q^46 - 8 * q^47 - 36 * q^49 + 32 * q^50 + 4 * q^52 + 8 * q^55 - 4 * q^58 + 20 * q^64 + 24 * q^65 + 2 * q^76 - 28 * q^77 + 24 * q^79 - 4 * q^80 + 24 * q^82 - 6 * q^88 + 8 * q^92 - 8 * q^94 - 32 * q^95 - 36 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{20} - 2 x^{19} + 2 x^{18} + 199 x^{16} - 414 x^{15} + 430 x^{14} + 184 x^{13} + 6939 x^{12} + \cdots + 64 $ x^20 - 2*x^19 + 2*x^18 + 199*x^16 - 414*x^15 + 430*x^14 + 184*x^13 + 6939*x^12 - 14914*x^11 + 16078*x^10 + 10764*x^9 + 7501*x^8 - 14446*x^7 + 21922*x^6 + 21836*x^5 + 10900*x^4 - 80*x^3 + 128*x^2 + 128*x + 64 :

$\beta_{1}$	$=$	$ ( 83\!\cdots\!97 \nu^{19} + \cdots - 85\!\cdots\!48 ) / 25\!\cdots\!32 $ (831512139114535450548703521487597v^19 - 10666407926939662759125896203236036v^18 + 22860895175813138956587432928926594v^17 - 25760806681975940984402432923331300v^16 + 174405512164447914468271208302239747v^15 - 2138258713474058491600609115265851196v^14 + 4719112230040428175476850958338672750v^13 - 5311520213026849338708156643981644964v^12 + 6014483044728094757135059070542162503v^11 - 74793485893463825882507228662915714264v^10 + 169357074766720013170857688442325215622v^9 - 192735071031044728645462500359147842320v^8 - 20415302169180015618628115953831668823v^7 - 64076190866093345393030504416161641336v^6 + 151024951581155030110311448035773178570v^5 - 229654424057454985505867939559364250904v^4 - 106103989325303755262205656976726854924v^3 - 52836856026315397821040882549692910520v^2 - 1416872117330124046362828554101352720*v - 8551213805621052352120142111949060448) / 2586557172353689357723696874231052032
$\beta_{2}$	$=$	$ ( 58\!\cdots\!68 \nu^{19} + \cdots - 31\!\cdots\!40 ) / 16\!\cdots\!52 $ (58968872310765640035833969525368v^19 - 432932192584308517788799632195651v^18 + 745123419287860098564203915040150v^17 - 569891016501074563469650044511270v^16 + 11565287226998413835139557507812624v^15 - 86852874034747412771397361300406677v^14 + 155042908950589299105085459531940754v^13 - 112536614361184787594763749765083034v^12 + 316522335182877188666846404921801728v^11 - 3014539672431770094441212781509999105v^10 + 5601781906410811606288060906619313654v^9 - 4007836601928422920532995608009353690v^8 - 4184234081547719413449201565852673444v^7 - 1377888766061996769075665589444393111v^6 + 5124266521759516864281634283525597426v^5 - 5308937466486005621203518725174724358v^4 - 7289077822537706286655440947460340884v^3 - 1343564938878091479633620051436172220v^2 - 527606395635819039866839703713900272*v - 31064523958690418954325864694808640) / 161659823272105584857731054639440752
$\beta_{3}$	$=$	$ ( 12\!\cdots\!51 \nu^{19} + \cdots - 48\!\cdots\!44 ) / 12\!\cdots\!16 $ (1221058225521071066382552515384751v^19 - 10922146641137419692730694649184900v^18 + 21701018437526849628623054648177926v^17 - 21538018721546088330010316856972316v^16 + 247454834540315782443085557140390689v^15 - 2192544333300649994670123111440200956v^14 + 4492307451549938067240783371285688714v^13 - 4368753483902038832692127535546637884v^12 + 7873696923471455810490300492037456973v^11 - 76528768185508387615366305926562524048v^10 + 161861810739920293792643190162006246962v^9 - 157196687988746723570089630866062379744v^8 - 46165337918939855703216075622032198237v^7 - 52842486772560456164804292579394828832v^6 + 159522253204079716056915258156340530942v^5 - 187194930679889936167729874254044572760v^4 - 123488228508948335855018769276590691748v^3 - 37786599731303499822234313174651094408v^2 + 15486203157475804862093837799228054224*v - 482504086620460202132735031865875744) / 1293278586176844678861848437115526016
$\beta_{4}$	$=$	$ ( 82\!\cdots\!31 \nu^{19} + \cdots + 10\!\cdots\!68 ) / 85\!\cdots\!44 $ (8219799132211139472376890131v^19 - 16233416365423891897265888796v^18 + 16326477034319462952356570366v^17 - 471622439071943244033777372v^16 + 1637076972115028739231189750845v^15 - 3362795186327553364091475559076v^14 + 3508981978880631016427921415186v^13 + 1420277124815268337022088856996v^12 + 57355834049581047374211409931833v^11 - 121281224458000421756413201538760v^10 + 131166438763969750182749287242362v^9 + 85355885443356451179259934323280v^8 + 74232070475493958453250308739351v^7 - 120553803535436370793020962411304v^6 + 178607841082818966928704078799798v^5 + 178535678548482930049770654178456v^4 + 111990664831243016102375228945932v^3 - 685616995259350144663758090440v^2 + 9605389887803950644425763403280*v + 1047964158565003474000541737568) / 8562547330006453160189410927744
$\beta_{5}$	$=$	$ ( 73\!\cdots\!17 \nu^{19} + \cdots - 63\!\cdots\!24 ) / 25\!\cdots\!32 $ (7352022686928822650700141022969517v^19 - 16509623872869331751879311782797060v^18 + 20787222866119439824074790169560578v^17 - 10062708840826055295155266714870692v^16 + 1471176521307216195008147178081020611v^15 - 3406142403280512203249189727315405948v^14 + 4400446762812428904657219663875227758v^13 - 746039552667763294136463093378497764v^12 + 52402873610360935452436782054707418631v^11 - 122388720471366872695793671408220527384v^10 + 161938686056779731023867529311299996422v^9 + 3016722403064478125663234349731612592v^8 + 99026562971885097263836974047369380841v^7 - 118106060749704335739288349277651286072v^6 + 187110227248062446591972571110736191434v^5 + 83047957752232608436799204765960830248v^4 + 116745411472638109001223646930694701172v^3 - 670337657467895864164417423963907128v^2 - 7824631763825627339874357295381928720*v - 6325000853694545625878349382727184224) / 2586557172353689357723696874231052032
$\beta_{6}$	$=$	$ ( 21\!\cdots\!05 \nu^{19} + \cdots - 18\!\cdots\!28 ) / 64\!\cdots\!08 $ (2199097885898303239446423967014905v^19 - 4800427959168052281083189344139284v^18 + 5764377964740736782447612175908730v^17 - 2251911993901795533472919769561140v^16 + 439450560515521087997515696736473847v^15 - 991195931571010359391538159285940652v^14 + 1223965627885863390401311752600035382v^13 - 65535867247078162661468640620027572v^12 + 15572961137556029028427508580210883179v^11 - 35640740469909170423457313390061699480v^10 + 45205100433877059841984586143536258158v^9 + 6555958025332338430389426832477449872v^8 + 26466817933953010120885329854575842693v^7 - 34552000293547676358476401305384031928v^6 + 55150059904187034842859775929786435234v^5 + 28320394392850505948012713314274979560v^4 + 34072418364956031639545177116016588324v^3 - 196140432480182869599937077744979544v^2 - 2281473313051752608199056286940500176*v - 1847763776516549269101319447525839328) / 646639293088422339430924218557763008
$\beta_{7}$	$=$	$ ( 53\!\cdots\!05 \nu^{19} + \cdots + 63\!\cdots\!92 ) / 12\!\cdots\!16 $ (5383105619836592703591631911267005v^19 - 11044966372091234690519992982840164v^18 + 12134508868471636394925559714063074v^17 - 2829885892088287498071577443144676v^16 + 1074312228141139351599174731751719667v^15 - 2285343878841805384769808236335883676v^14 + 2590719517818359765195907736286961230v^13 + 407115319403250637938888656475000668v^12 + 37948876008941508076837488685144995511v^11 - 82343468569437308713789910031994366200v^10 + 96060850735060096362518819740665234982v^9 + 37229606927284618359353189285171380528v^8 + 60954580913292341982888505550028089465v^7 - 81481296021019131459709085601705954904v^6 + 115951407306861964811330692464407869674v^5 + 109084169832188853888628606489474461608v^4 + 74919200004860443931492448687148690228v^3 - 463446702707583604063751024262561464v^2 - 5046525265040546394721439595935135248*v + 6310275698736577220462928477865647392) / 1293278586176844678861848437115526016
$\beta_{8}$	$=$	$ ( - 10\!\cdots\!23 \nu^{19} + \cdots - 86\!\cdots\!56 ) / 25\!\cdots\!32 $ (-10821892343525956121989901322820323v^19 + 23717312543902070671725845302061500v^18 - 22318750491688401608531615796636062v^17 - 4679123926434180422441836655890852v^16 - 2142641755375541815623800107483870509v^15 + 4888899539594549451016863387018421316v^14 - 4820789003484492082405914910017284978v^13 - 2910885900008975532756418917952283236v^12 - 72398867785744345932984182400751543849v^11 + 175565154651482768384705593069350018920v^10 - 181145265855005250381257467286718324442v^9 - 147809938991784111883961898166857488656v^8 + 26391219348099857452338574919776956185v^7 + 177038586373421333168754413173948238152v^6 - 260206566745069700387500177627106634710v^5 - 247606398512949410047160525003554090392v^4 + 30112928464856238701331669549098401716v^3 + 53484640475719300167322944109673761992v^2 + 6612991148098370241018370792226572272*v - 8660178046473441287217785653446562656) / 2586557172353689357723696874231052032
$\beta_{9}$	$=$	$ ( 15\!\cdots\!15 \nu^{19} + \cdots + 49\!\cdots\!08 ) / 25\!\cdots\!32 $ (15620909516103631150329463849240415v^19 - 40199255713362636462958499772458188v^18 + 48564003857607124081896916002408054v^17 - 15290334869165388841157075268503628v^16 + 3103573697688304052516825232861331953v^15 - 8244451061505100790915481613201496244v^14 + 10304885947014042207666592994431769562v^13 - 444510259318638472361484641373961356v^12 + 105710678603045514905184944501975387933v^11 - 294152021176091896483296396767751377832v^10 + 380354456846855737466530786661397836770v^9 + 43022753590340609727609727641834931728v^8 - 16616379493428426131544481829807973069v^7 - 259414900600791849498639847860291972936v^6 + 462887880618737349937110342325616788878v^5 + 162639746883305545311278034575006746424v^4 - 63791483500086382643125766740907360708v^3 - 61662910087623333121457116468162318312v^2 + 13403443827655780282620887595542393424*v + 4985912705522407379710179656216368608) / 2586557172353689357723696874231052032
$\beta_{10}$	$=$	$ ( - 97\!\cdots\!25 \nu^{19} + \cdots - 38\!\cdots\!04 ) / 12\!\cdots\!16 $ (-9700552370857735000074863292719725v^19 + 16662875163802437416451127082162308v^18 - 11749842928698825157599510834192834v^17 - 10557207027332910953966000776960924v^16 - 1924415962325504736906872278843581891v^15 + 3469502747033429158769406843622370428v^14 - 2605087040082340629951949237809832046v^13 - 4007537373579416599801726189893505500v^12 - 66542546151253907016930255635551838407v^11 + 125725491890562209332288497048142507544v^10 - 100220285543603233803196687595932338502v^9 - 185772364060790445738058215955217855984v^8 - 55201838613177233467517313633470783593v^7 + 129984192436692274023501011229878641848v^6 - 166228618980673632720153382420788219082v^5 - 301393263396418638700439795261124910824v^4 - 108522727472368663458958129402994629236v^3 + 741451603094371887870588525620607032v^2 + 7333864341705461703139360709159133968*v - 3894532451092790065181257108724870304) / 1293278586176844678861848437115526016
$\beta_{11}$	$=$	$ ( 51\!\cdots\!33 \nu^{19} + \cdots + 19\!\cdots\!40 ) / 64\!\cdots\!08 $ (5182233568789923801067330362493133v^19 - 14107465411691033142539853714558144v^18 + 17776967785079681292647089784662010v^17 - 7211433595577452983600568241065628v^16 + 1030728873500969016755704367826170051v^15 - 2889728678653953856317602499295530272v^14 + 3762932054302962195135684220350737174v^13 - 600002829024419938229553659188823020v^12 + 35159833581397232276728024789137304903v^11 - 103154078726118312553612339155645477548v^10 + 138571062550421532860062014040951923454v^9 - 2411243528413731480530644993643803992v^8 - 5418439534468101544954406076496620455v^7 - 99390519266478208544331633442665178628v^6 + 165347779771241907597076530504218271858v^5 + 32819425740425451680113197944792031920v^4 - 29172708927658664164124843582891890204v^3 - 39599884452768342635981704174025268456v^2 - 1270181937667323947629348804409534160*v + 193855944172884117977818952117934240) / 646639293088422339430924218557763008
$\beta_{12}$	$=$	$ ( - 29\!\cdots\!71 \nu^{19} + \cdots + 17\!\cdots\!96 ) / 25\!\cdots\!32 $ (-29383342024216997287188492151284471v^19 + 69208483447703882420375827460451500v^18 - 81485254775716380671131300832084070v^17 + 24582916371072519917591225148706220v^16 - 5850425814915515707485846694918739161v^15 + 14240887676380051558717683758214378708v^14 - 17320720974990102281632867295345024106v^13 - 151415218312942134414557911633203732v^12 - 202654394429328318276309047401595418597v^11 + 509987762499376280983793224516849262312v^10 - 640419849560248885225878209618739325874v^9 - 120814856017586423567824399852760282896v^8 - 134098643012576142315278731115226740459v^7 + 470456205503737650965373825648391880072v^6 - 791395119072717065092327587476133290302v^5 - 383784725863844822190478127177945927800v^4 - 140158022061522199551401550542217485980v^3 + 62861691438931972545856342212030105128v^2 + 265524180909903975536433534471783728*v + 1772008566361791457856826839884856096) / 2586557172353689357723696874231052032
$\beta_{13}$	$=$	$ ( 16\!\cdots\!83 \nu^{19} + \cdots + 13\!\cdots\!32 ) / 12\!\cdots\!16 $ (16705866266073313106409589134315183v^19 - 39398930830861539023868468432326012v^18 + 45317194259002148811414894776052726v^17 - 12550115886716729626173781089807788v^16 + 3326583624655663066926696505706982689v^15 - 8111242148307516662733533094665670852v^14 + 9651176844154386735144764428380871706v^13 + 385023504790464541506636246843127316v^12 + 115250174836960412205104203068169856909v^11 - 291453061187297514642949937236406887672v^10 + 357845008004452706498504291524339163810v^9 + 79594250231906976288663050063231358768v^8 + 75945796691704316394224625289179904643v^7 - 313943603199006459692403010683455003064v^6 + 465179220974545562702059613590911383438v^5 + 230770992806048748964442702136053824728v^4 + 57936551602525694936709987833926446012v^3 - 92481513943530817685700140464494477992v^2 + 7131385693089671596768718913429792848*v + 1356655861654772471183901415557374432) / 1293278586176844678861848437115526016
$\beta_{14}$	$=$	$ ( - 40\!\cdots\!97 \nu^{19} + \cdots - 54\!\cdots\!72 ) / 25\!\cdots\!32 $ (-40581671258688668660044122696480997v^19 + 76903625187692962119313033062320580v^18 - 76750592314150880260308186460826226v^17 - 1488863413054186347608631200845820v^16 - 8080579656028191912551466093403680267v^15 + 15948271476169475912574862905691759036v^14 - 16500853143036865706339572570181404286v^13 - 7834739200513177204505034742866827068v^12 - 283453518031464838806423095834095185199v^11 + 573902382795884509421044549106012415736v^10 - 617025893874164127558572517898551270102v^9 - 452259955406060304798890607811437816624v^8 - 391730653689279223685330279435248822849v^7 + 473116689805677281733017623241215651672v^6 - 848430693363426019045438665492070605178v^5 - 924803750429799503534426050321089082728v^4 - 608596778381215262803129620515735853140v^3 - 161234423125531722111781548592712809608v^2 - 51490108944920232692581728730960323184*v - 5427704590810226097313408690237709472) / 2586557172353689357723696874231052032
$\beta_{15}$	$=$	$ ( - 12\!\cdots\!39 \nu^{19} + \cdots + 25\!\cdots\!04 ) / 64\!\cdots\!08 $ (-12035516946903983645924003908251439v^19 + 25727310763510197276603384207098092v^18 - 27225559067462219599952574592340534v^17 + 2925848952845467261076114059757932v^16 - 2394363317082558324340372205356579745v^15 + 5311562200606596060429492792757489652v^14 - 5828873019462720725229904709416798938v^13 - 1581848338455212451214491207214989620v^12 - 83063343322246312552568604394154890317v^11 + 190866022239526653291762967581955813128v^10 - 216986364662577567027943133448234714594v^9 - 105730493874871659967440159309662710512v^8 - 67170708615322332990520762989511759427v^7 + 182427073597359407015344770773942880648v^6 - 282208431399527155484899599578414500750v^5 - 226983502035819423486081092165185618488v^4 - 90281059897200722657467010830993350716v^3 + 16084894218356334291478850890658051304v^2 + 3798758572134686849954648503685557872*v + 2516386656793101208056355911131291104) / 646639293088422339430924218557763008
$\beta_{16}$	$=$	$ ( - 25\!\cdots\!11 \nu^{19} + \cdots - 18\!\cdots\!08 ) / 12\!\cdots\!16 $ (-25335525903379485767829777006484011v^19 + 61222436025920087569674449002378132v^18 - 72586452643889303741471091907998622v^17 + 21914375505591888762688981857047916v^16 - 5040967009472340995557813167301537125v^15 + 12585130469000795527156096417658996396v^14 - 15421850953462993416048304303360548594v^13 + 49088218928556787526558731425282892v^12 - 173715298237845275949475903206800584513v^11 + 450185085558257747057939737006146794640v^10 - 569987139194755822620809471995578387962v^9 - 96613956660626030070966267896984181632v^8 - 71943019021371291726840430880444167663v^7 + 409618166713453734396560962025148720480v^6 - 703970428376140917763914752727334773942v^5 - 313971709008337115573156547066932878696v^4 - 50335208443039430836533465511400077900v^3 + 74412973689344994540970029305697021288v^2 - 7944700305899930581190333213332649232*v - 1846524452507148131571888834557088608) / 1293278586176844678861848437115526016
$\beta_{17}$	$=$	$ ( - 27\!\cdots\!13 \nu^{19} + \cdots - 26\!\cdots\!16 ) / 12\!\cdots\!16 $ (-27694520924907307500139835036938513v^19 + 58487346806674537037239796863616876v^18 - 60240128260669338382466725014200666v^17 + 2761225595626649613079359957401412v^16 - 5506477392416031332221841831323749407v^15 + 12079612950888476214253658505973288980v^14 - 12922775491278493108737198712670613398v^13 - 4468482206268483963821070312101109532v^12 - 190605689929814751101596176658651228787v^11 + 434256130691239264015177500026467479360v^10 - 482104034680711095970098018963240921646v^9 - 273373961009431580990315212867276146240v^8 - 137836728297013512180980792085775498717v^7 + 418313826593893821680885184731742165904v^6 - 643046136977379272849406737677605449506v^5 - 554266456295563646512706464955620509048v^4 - 190411516912181442296614586626150691940v^3 + 39869645398529780433386711334641063096v^2 + 5700910774936590737200110454847007184*v - 2641109547635419631578118955230896416) / 1293278586176844678861848437115526016
$\beta_{18}$	$=$	$ ( 61\!\cdots\!99 \nu^{19} + \cdots + 65\!\cdots\!76 ) / 25\!\cdots\!32 $ (61772359811664697394146213609667299v^19 - 132346571607950063167205505257903852v^18 + 144150364616436924377368953655358078v^17 - 23728187850999709221319258211349820v^16 + 12299313911622367980236525422343772333v^15 - 27326419020280907220905591719606831252v^14 + 30804866634630660243125655879291178130v^13 + 6313917926056944344421208954895266628v^12 + 428432606161462043064489801901476496041v^11 - 982024593939352288708490278528419924088v^10 + 1145546064648328266442169827309381352890v^9 + 477732636142725785929740250709362706416v^8 + 421125339128595797208819847057282200615v^7 - 934286467909813103370877040555372901816v^6 + 1513077777503585669862401843786540094454v^5 + 1110764876647909879372637720991644706488v^4 + 547306782740524018740189912792085191436v^3 - 38299512210320504979915567534405285512v^2 + 50658103551315032450202228173348203024*v + 6523540048031629687897541423861370976) / 2586557172353689357723696874231052032
$\beta_{19}$	$=$	$ ( 31\!\cdots\!67 \nu^{19} + \cdots + 85\!\cdots\!04 ) / 12\!\cdots\!16 $ (31768412599442519361033905478107167v^19 - 57687677869832334255904301603393660v^18 + 48119499811400342389345901708628022v^17 + 19827828528341268545967907481936756v^16 + 6312920234455297954356615664825262321v^15 - 11986452586130237841123225058238434628v^14 + 10498561731528801537584486789493010522v^13 + 10037554200329361892078225510997137332v^12 + 219597564563071043518749519704296170845v^11 - 433530034208164804386773143035213615224v^10 + 397810185089276533887900313275403498850v^9 + 496179353044696249280315270791576348624v^8 + 229975445389102298648249054312637953523v^7 - 441658621112658576717016342319905239416v^6 + 589578583850144289742894265003578488910v^5 + 877291176025403274389430890695864752248v^4 + 382026690847735964145900787666695566716v^3 - 2516782724206445540331589160235927144v^2 - 25776422550651075919652896393921848752*v + 8539630336415312152634333628477425504) / 1293278586176844678861848437115526016

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{12} + \beta_{9} + \beta_{6} + 2\beta_{4} ) / 4 $ (b12 + b9 + b6 + 2*b4) / 4
$\nu^{2}$	$=$	$ ( 2\beta_{17} - 2\beta_{16} + 2\beta_{13} - 3\beta_{11} + 2\beta_{4} - 2\beta_{3} + 5\beta_{2} ) / 4 $ (2b17 - 2b16 + 2b13 - 3b11 + 2b4 - 2b3 + 5*b2) / 4
$\nu^{3}$	$=$	$ ( 4 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} - 7 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - 11 \beta_{9} + \cdots + 4 ) / 4 $ (4b16 - 2b15 - 2b14 - 7b12 + 4b11 + 2b10 - 11b9 - 2b8 - 11b6 - 4b5 + 18b4 + 4b2 + 4*b1 + 4) / 4
$\nu^{4}$	$=$	$ ( - 4 \beta_{19} + 13 \beta_{17} - 6 \beta_{15} + 4 \beta_{12} - 42 \beta_{10} - 2 \beta_{9} + 11 \beta_{8} + \cdots - 58 ) / 2 $ (-4b19 + 13b17 - 6b15 + 4b12 - 42b10 - 2b9 + 11b8 + 14b7 + 14b6 - 24b5 + 13b3 + 11b1 - 58) / 2
$\nu^{5}$	$=$	$ ( 14 \beta_{19} - 14 \beta_{18} - 56 \beta_{16} - 24 \beta_{15} + 24 \beta_{14} + 10 \beta_{13} + \cdots + 72 ) / 4 $ (14b19 - 14b18 - 56b16 - 24b15 + 24b14 + 10b13 - 97b12 - 62b11 + 26b10 - 99b9 + 74b8 - 10b7 - 129b6 - 50b5 - 186b4 + 20b3 - 44b2 - 32b1 + 72) / 4
$\nu^{6}$	$=$	$ ( 142 \beta_{18} - 326 \beta_{17} + 70 \beta_{16} + 210 \beta_{14} - 362 \beta_{13} + 90 \beta_{12} + \cdots - 124 \beta_1 ) / 4 $ (142b18 - 326b17 + 70b16 + 210b14 - 362b13 + 90b12 + 161b11 - 90b9 + 124b8 - 284b4 + 326b3 - 633b2 - 124*b1) / 4
$\nu^{7}$	$=$	$ ( - 282 \beta_{19} - 282 \beta_{18} - 380 \beta_{17} - 690 \beta_{16} + 254 \beta_{15} + 254 \beta_{14} + \cdots - 990 ) / 4 $ (-282b19 - 282b18 - 380b17 - 690b16 + 254b15 + 254b14 + 216b13 + 889b12 - 808b11 - 338b10 + 1391b9 + 434b8 + 216b7 + 1571b6 + 574b5 - 2064b4 - 456b2 - 1062b1 - 990) / 4
$\nu^{8}$	$=$	$ ( 1019 \beta_{19} - 2015 \beta_{17} + 1463 \beta_{15} - 1034 \beta_{12} + 6744 \beta_{10} + 429 \beta_{9} + \cdots + 6749 ) / 2 $ (1019b19 - 2015b17 + 1463b15 - 1034b12 + 6744b10 + 429b9 - 1326b8 - 2286b7 - 2140b6 + 3121b5 - 2015b3 - 1326b1 + 6749) / 2
$\nu^{9}$	$=$	$ ( - 4276 \beta_{19} + 4276 \beta_{18} + 8338 \beta_{16} + 2680 \beta_{15} - 2680 \beta_{14} + \cdots - 12894 ) / 4 $ (-4276b19 + 4276b18 + 8338b16 + 2680b15 - 2680b14 - 3418b13 + 13835b12 + 10108b11 - 4582b10 + 13471b9 - 13952b8 + 3418b7 + 19391b6 + 6584b5 + 23888b4 - 5352b3 + 4814b2 + 5556b1 - 12894) / 4
$\nu^{10}$	$=$	$ ( - 27256 \beta_{18} + 49494 \beta_{17} - 3858 \beta_{16} - 37968 \beta_{14} + 57242 \beta_{13} + \cdots + 23700 \beta_1 ) / 4 $ (-27256b18 + 49494b17 - 3858b16 - 37968b14 + 57242b13 - 18708b12 - 19467b11 + 18708b9 - 23700b8 + 39626b4 - 49494b3 + 89165b2 + 23700*b1) / 4
$\nu^{11}$	$=$	$ ( 58720 \beta_{19} + 58720 \beta_{18} + 67752 \beta_{17} + 101036 \beta_{16} - 28742 \beta_{15} + \cdots + 165956 ) / 4 $ (58720b19 + 58720b18 + 67752b17 + 101036b16 - 28742b15 - 28742b14 - 48544b13 - 132155b12 + 125300b11 + 63262b10 - 199759b9 - 69538b8 - 48544b7 - 240555b6 - 76116b5 + 283370b4 + 51852b2 + 176152b1 + 165956) / 4
$\nu^{12}$	$=$	$ ( - 176628 \beta_{19} + 303173 \beta_{17} - 239246 \beta_{15} + 182010 \beta_{12} - 1042960 \beta_{10} + \cdots - 965092 ) / 2 $ (-176628b19 + 303173b17 - 239246b15 + 182010b12 - 1042960b10 - 57236b9 + 182085b8 + 356956b7 + 338388b6 - 440196b5 + 303173b3 + 182085b1 - 965092) / 2
$\nu^{13}$	$=$	$ ( 772178 \beta_{19} - 772178 \beta_{18} - 1231612 \beta_{16} - 312660 \beta_{15} + 312660 \beta_{14} + \cdots + 2131180 ) / 4 $ (772178b19 - 772178b18 - 1231612b16 - 312660b15 + 312660b14 + 657706b13 - 2128093b12 - 1551582b11 + 875006b10 - 1938451b9 + 2181226b8 - 657706b7 - 2990893b6 - 884590b5 - 3408838b4 + 816124b3 - 564620b2 - 862800b1 + 2131180) / 4
$\nu^{14}$	$=$	$ ( 4507906 \beta_{18} - 7423230 \beta_{17} + 534738 \beta_{16} + 5949350 \beta_{14} - 8887110 \beta_{13} + \cdots - 3547448 \beta_1 ) / 4 $ (4507906b18 - 7423230b17 + 534738b16 + 5949350b14 - 8887110b13 + 3069558b12 + 3040273b11 - 3069558b9 + 3547448b8 - 5410332b4 + 7423230b3 - 13141025b2 - 3547448*b1) / 4
$\nu^{15}$	$=$	$ ( - 9945318 \beta_{19} - 9945318 \beta_{18} - 9567468 \beta_{17} - 15085666 \beta_{16} + 3427262 \beta_{15} + \cdots - 27359034 ) / 4 $ (-9945318b19 - 9945318b18 - 9567468b17 - 15085666b16 + 3427262b15 + 3427262b14 + 8709032b13 + 20052317b12 - 19227140b11 - 12036254b10 + 29986103b9 + 10672414b8 + 8709032b7 + 37231255b6 + 10301474b5 - 41329388b4 - 6160000b2 - 26746406b1 - 27359034) / 4
$\nu^{16}$	$=$	$ ( 28533035 \beta_{19} - 45442251 \beta_{17} + 36744031 \beta_{15} - 30333120 \beta_{12} + 159902054 \beta_{10} + \cdots + 146079151 ) / 2 $ (28533035b19 - 45442251b17 + 36744031b15 - 30333120b12 + 159902054b10 + 6410911b9 - 26180376b8 - 55265876b7 - 54624074b6 + 64580653b5 - 45442251b3 - 26180376b1 + 146079151) / 2
$\nu^{17}$	$=$	$ ( - 126759560 \beta_{19} + 126759560 \beta_{18} + 185384902 \beta_{16} + 37613004 \beta_{15} + \cdots - 351132370 ) / 4 $ (-126759560b19 + 126759560b18 + 185384902b16 + 37613004b15 - 37613004b14 - 113937738b13 + 331907043b12 + 238481948b11 - 164282626b10 + 285325755b9 - 326346616b8 + 113937738b7 + 463810535b6 + 119947588b5 + 503295060b4 - 110248832b3 + 66850542b2 + 131903492b1 - 351132370) / 4
$\nu^{18}$	$=$	$ ( - 719370636 \beta_{18} + 1113095910 \beta_{17} - 106969886 \beta_{16} - 904845476 \beta_{14} + \cdots + 500642640 \beta_1 ) / 4 $ (-719370636b18 + 1113095910b17 - 106969886b16 - 904845476b14 + 1374216438b13 - 481206616b12 - 501089447b11 + 481206616b9 - 500642640b8 + 722393466b4 - 1113095910b3 + 1967430849b2 + 500642640*b1) / 4
$\nu^{19}$	$=$	$ ( 1606937508 \beta_{19} + 1606937508 \beta_{18} + 1254307784 \beta_{17} + 2282997988 \beta_{16} + \cdots + 4504244008 ) / 4 $ (1606937508b19 + 1606937508b18 + 1254307784b17 + 2282997988b16 - 411003878b15 - 411003878b14 - 1480426256b13 - 3063248155b12 + 2960309264b11 + 2224623834b10 - 4561713083b9 - 1630309606b8 - 1480426256b7 - 5781018811b6 - 1394517872b5 + 6144534982b4 + 717206596b2 + 3972078440b1 + 4504244008) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3762\mathbb{Z}\right)^\times$.

$n$	$343$	$2377$	$2927$
$\chi(n)$	$-1$	$-1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

2089.1

−1.82753	+	1.82753i
−1.82753	−	1.82753i
−0.407475	+	0.407475i
−0.407475	−	0.407475i
2.45556	+	2.45556i
2.45556	−	2.45556i
1.75638	+	1.75638i
1.75638	−	1.75638i
−0.684824	+	0.684824i
−0.684824	−	0.684824i
−2.50724	+	2.50724i
−2.50724	−	2.50724i
−0.196118	+	0.196118i
−0.196118	−	0.196118i
1.32386	+	1.32386i
1.32386	−	1.32386i
0.891508	+	0.891508i
0.891508	−	0.891508i
0.195859	+	0.195859i
0.195859	−	0.195859i

1.00000

−4.00532

−

3.65505i

1.00000

−4.00532

2089.2

1.00000

−4.00532

3.65505i

1.00000

−4.00532

2089.3

1.00000

−3.77178

−

0.814950i

1.00000

−3.77178

2089.4

1.00000

−3.77178

0.814950i

1.00000

−3.77178

2089.5

1.00000

−2.33584

−

4.91113i

1.00000

−2.33584

2089.6

1.00000

−2.33584

4.91113i

1.00000

−2.33584

2089.7

1.00000

−1.24031

−

3.51276i

1.00000

−1.24031

2089.8

1.00000

−1.24031

3.51276i

1.00000

−1.24031

2089.9

1.00000

−0.845289

−

1.36965i

1.00000

−0.845289

2089.10

1.00000

−0.845289

1.36965i

1.00000

−0.845289

2089.11

1.00000

0.397906

−

5.01447i

1.00000

0.397906

2089.12

1.00000

0.397906

5.01447i

1.00000

0.397906

2089.13

1.00000

1.28096

−

0.392237i

1.00000

1.28096

2089.14

1.00000

1.28096

0.392237i

1.00000

1.28096

2089.15

1.00000

2.09100

−

2.64773i

1.00000

2.09100

2089.16

1.00000

2.09100

2.64773i

1.00000

2.09100

2089.17

1.00000

2.44406

−

1.78302i

1.00000

2.44406

2089.18

1.00000

2.44406

1.78302i

1.00000

2.44406

2089.19

1.00000

3.98461

−

0.391718i

1.00000

3.98461

2089.20

1.00000

3.98461

0.391718i

1.00000

3.98461

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
209.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3762.2.g.l		20
3.b	odd	2	1		1254.2.g.a	✓	20
11.b	odd	2	1		3762.2.g.k		20
19.b	odd	2	1		3762.2.g.k		20
33.d	even	2	1		1254.2.g.b	yes	20
57.d	even	2	1		1254.2.g.b	yes	20
209.d	even	2	1	inner	3762.2.g.l		20
627.b	odd	2	1		1254.2.g.a	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1254.2.g.a	✓	20	3.b	odd	2	1
1254.2.g.a	✓	20	627.b	odd	2	1
1254.2.g.b	yes	20	33.d	even	2	1
1254.2.g.b	yes	20	57.d	even	2	1
3762.2.g.k		20	11.b	odd	2	1
3762.2.g.k		20	19.b	odd	2	1
3762.2.g.l		20	1.a	even	1	1	trivial
3762.2.g.l		20	209.d	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3762, [\chi])$:

$ T_{5}^{10} + 2 T_{5}^{9} - 31 T_{5}^{8} - 50 T_{5}^{7} + 308 T_{5}^{6} + 336 T_{5}^{5} - 1168 T_{5}^{4} + \cdots - 384 $

$ T_{13}^{10} - 2 T_{13}^{9} - 48 T_{13}^{8} + 40 T_{13}^{7} + 768 T_{13}^{6} + 64 T_{13}^{5} + \cdots - 4608 $

$ T_{23}^{10} - 4 T_{23}^{9} - 148 T_{23}^{8} + 704 T_{23}^{7} + 6960 T_{23}^{6} - 41056 T_{23}^{5} + \cdots + 4633088 $

$ T_{29}^{10} + 2 T_{29}^{9} - 207 T_{29}^{8} - 540 T_{29}^{7} + 14824 T_{29}^{6} + 49360 T_{29}^{5} + \cdots + 24032000 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T - 1)^{20} $ (T - 1)^20
$3$	$ T^{20} $ T^20
$5$	$ (T^{10} + 2 T^{9} + \cdots - 384)^{2} $ (T^10 + 2T^9 - 31T^8 - 50T^7 + 308T^6 + 336T^5 - 1168T^4 - 768T^3 + 1440T^2 + 576*T - 384)^2
$7$	$ T^{20} + 88 T^{18} + \cdots + 65536 $ T^20 + 88T^18 + 3068T^16 + 54352T^14 + 524144T^12 + 2744704T^10 + 7447104T^8 + 9688832T^6 + 5264384T^4 + 1032192*T^2 + 65536
$11$	$ T^{20} + \cdots + 25937424601 $ T^20 + 6T^19 + 18T^18 + 18T^17 - 91T^16 + 24T^15 + 3416T^14 + 19432T^13 + 45266T^12 - 8972T^11 - 234004T^10 - 98692T^9 + 5477186T^8 + 25863992T^7 + 50013656T^6 + 3865224T^5 - 161212051T^4 + 350769078T^3 + 3858459858T^2 + 14147686146*T + 25937424601
$13$	$ (T^{10} - 2 T^{9} + \cdots - 4608)^{2} $ (T^10 - 2T^9 - 48T^8 + 40T^7 + 768T^6 + 64T^5 - 4256T^4 - 1728T^3 + 7680T^2 + 2304*T - 4608)^2
$17$	$ T^{20} + \cdots + 59395538944 $ T^20 + 222T^18 + 20713T^16 + 1057116T^14 + 32140176T^12 + 593815040T^10 + 6517181440T^8 + 39756218368T^6 + 118740680704T^4 + 147593363456*T^2 + 59395538944
$19$	$ T^{20} + \cdots + 6131066257801 $ T^20 - 2T^19 - 2T^18 + 178T^17 - 139T^16 - 1600T^15 + 16424T^14 - 26608T^13 - 75022T^12 + 264204T^11 - 1741964T^10 + 5019876T^9 - 27082942T^8 - 182504272T^7 + 2140392104T^6 - 3961758400T^5 - 6539377459T^4 + 159109169542T^3 - 33967126082T^2 - 645375395558*T + 6131066257801
$23$	$ (T^{10} - 4 T^{9} + \cdots + 4633088)^{2} $ (T^10 - 4T^9 - 148T^8 + 704T^7 + 6960T^6 - 41056T^5 - 88608T^4 + 845312T^3 - 785152T^2 - 2944512*T + 4633088)^2
$29$	$ (T^{10} + 2 T^{9} + \cdots + 24032000)^{2} $ (T^10 + 2T^9 - 207T^8 - 540T^7 + 14824T^6 + 49360T^5 - 419904T^4 - 1783616T^3 + 3172480T^2 + 21796096*T + 24032000)^2
$31$	$ T^{20} + 224 T^{18} + \cdots + 80281600 $ T^20 + 224T^18 + 18924T^16 + 792848T^14 + 18255408T^12 + 238865920T^10 + 1750229056T^8 + 6693340928T^6 + 10844148736T^4 + 2639314944*T^2 + 80281600
$37$	$ T^{20} + 258 T^{18} + \cdots + 1024 $ T^20 + 258T^18 + 24717T^16 + 1061864T^14 + 19109916T^12 + 100894896T^10 + 221523056T^8 + 215497600T^6 + 78946368T^4 + 3869440*T^2 + 1024
$41$	$ (T^{10} - 12 T^{9} + \cdots + 32157696)^{2} $ (T^10 - 12T^9 - 224T^8 + 2736T^7 + 15920T^6 - 205952T^5 - 341248T^4 + 5672448T^3 - 764928T^2 - 41902080*T + 32157696)^2
$43$	$ T^{20} + 314 T^{18} + \cdots + 51380224 $ T^20 + 314T^18 + 35433T^16 + 1846412T^14 + 49497360T^12 + 697289728T^10 + 4915111936T^8 + 15558139904T^6 + 20068630528T^4 + 6902513664*T^2 + 51380224
$47$	$ (T^{10} + 4 T^{9} + \cdots + 594432)^{2} $ (T^10 + 4T^9 - 276T^8 - 864T^7 + 23728T^6 + 28192T^5 - 793760T^4 + 621312T^3 + 5545728T^2 - 4631040*T + 594432)^2
$53$	$ T^{20} + \cdots + 14880096256 $ T^20 + 368T^18 + 51244T^16 + 3433264T^14 + 119193264T^12 + 2288886528T^10 + 24881008448T^8 + 149009379584T^6 + 443740430336T^4 + 484855255040*T^2 + 14880096256
$59$	$ T^{20} + \cdots + 2415919104 $ T^20 + 520T^18 + 111888T^16 + 12922880T^14 + 870008832T^12 + 34674745344T^10 + 789061697536T^8 + 9139075940352T^6 + 38735774220288T^4 + 3122877431808*T^2 + 2415919104
$61$	$ T^{20} + \cdots + 3226240000 $ T^20 + 570T^18 + 121389T^16 + 12620492T^14 + 704009340T^12 + 21724656320T^10 + 364426271920T^8 + 3041124273472T^6 + 9638272773440T^4 + 463467830784*T^2 + 3226240000
$67$	$ T^{20} + \cdots + 31021072384 $ T^20 + 762T^18 + 223049T^16 + 31916336T^14 + 2363023104T^12 + 86902090240T^10 + 1349697560576T^8 + 6215719714816T^6 + 6265929531392T^4 + 1172030095360*T^2 + 31021072384
$71$	$ T^{20} + \cdots + 10\!\cdots\!00 $ T^20 + 972T^18 + 396988T^16 + 88646464T^14 + 11788236336T^12 + 952810866496T^10 + 45814513738304T^8 + 1230279472224256T^6 + 16341621818959872T^4 + 84498163091062784*T^2 + 104777443099033600
$73$	$ T^{20} + \cdots + 2578054119424 $ T^20 + 656T^18 + 151232T^16 + 14951936T^14 + 657551360T^12 + 13711966208T^10 + 146470141952T^8 + 847422423040T^6 + 2660719656960T^4 + 4205779615744*T^2 + 2578054119424
$79$	$ (T^{10} - 12 T^{9} + \cdots - 70604800)^{2} $ (T^10 - 12T^9 - 408T^8 + 4592T^7 + 44432T^6 - 445088T^5 - 1259744T^4 + 12166784T^3 + 8676992T^2 - 63419392*T - 70604800)^2
$83$	$ T^{20} + \cdots + 48\!\cdots\!44 $ T^20 + 1412T^18 + 867200T^16 + 304335936T^14 + 67400525056T^12 + 9812862902272T^10 + 948001115324416T^8 + 59807993094733824T^6 + 2351361359124365312T^4 + 51933250393902940160*T^2 + 489255871611607711744
$89$	$ T^{20} + \cdots + 31\!\cdots\!56 $ T^20 + 1170T^18 + 559937T^16 + 142395068T^14 + 21037573776T^12 + 1871380905472T^10 + 101639085991936T^8 + 3367328390307840T^6 + 66093036796968960T^4 + 703601258187718656*T^2 + 3121764519883309056
$97$	$ T^{20} + \cdots + 43\!\cdots\!76 $ T^20 + 712T^18 + 199120T^16 + 28768512T^14 + 2390082560T^12 + 119212367872T^10 + 3579152695296T^8 + 62533801607168T^6 + 589913473941504T^4 + 2664487047921664*T^2 + 4372624599678976
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + q^{2} + q^{4} - \beta_{5} q^{5} - \beta_{4} q^{7} + q^{8} - \beta_{5} q^{10} + \beta_{8} q^{11} + \beta_{10} q^{13} - \beta_{4} q^{14} + q^{16} + \beta_{18} q^{17} - \beta_{17} q^{19} - \beta_{5} q^{20}+ \cdots + (\beta_{17} - 3 \beta_{10} + \cdots + \beta_1) q^{98}+O(q^{100}) \) q + q^2 + q^4 - b5 * q^5 - b4 * q^7 + q^8 - b5 * q^10 + b8 * q^11 + b10 * q^13 - b4 * q^14 + q^16 + b18 * q^17 - b17 * q^19 - b5 * q^20 + b8 * q^22 + (-b19 - b15 - b10 - b9 - b5 + 1) * q^23 + (-b19 + b7 + 2) * q^25 + b10 * q^26 - b4 * q^28 + (b15 - b12 - b7 - b5) * q^29 + (b16 - b12 + b11 + b9 + b4) * q^31 + q^32 + b18 * q^34 + (-b18 + b16 - b14 + b13 + b11 + 2b4 + b2) q^35 + (b18 - b16 - b11 - b4) * q^37 - b17 * q^38 - b5 * q^40 + (b19 + b17 + b15 - b12 - b10 + b8 - b7 - b6 + b5 + b3 + b1 + 2) * q^41 + (-b16 + b12 - 2b11 - b9) q^43 + b8 * q^44 + (-b19 - b15 - b10 - b9 - b5 + 1) * q^46 + (-b19 + b15 - b12 - b10 + b8 - b5 + b1 + 1) * q^47 + (b17 - 3b10 + b8 + b7 + b6 - 2b5 + b3 + b1) * q^49 + (-b19 + b7 + 2) * q^50 + b10 * q^52 + (-b16 + b13 + b12 - b11 - b9) * q^53 + (-b19 - b17 + b16 - b15 - b14 - b8 - b3 + b2) * q^55 - b4 * q^56 + (b15 - b12 - b7 - b5) * q^58 + (b16 + b13 + b11 - b8 + b4 + b1) * q^59 + (2b16 + b13 + b2) q^61 + (b16 - b12 + b11 + b9 + b4) * q^62 + q^64 + (-b19 - b15 - b9 - b8 - b5 - b1 + 1) * q^65 + (b18 - 2b16 - 2b11 - b8 - 2b2 + b1) q^67 + b18 * q^68 + (-b18 + b16 - b14 + b13 + b11 + 2b4 + b2) q^70 + (-2b18 - b17 + b11 + 2b8 + b4 + b3 + b2 - 2b1) q^71 + (-b16 + b13 + b8 + b4 + b2 - b1) * q^73 + (b18 - b16 - b11 - b4) * q^74 - b17 * q^76 + (-b18 + b17 - b14 + b13 - 3b10 + b8 + b4 + b3 + b2 + b1) q^77 + (b19 + b17 - b15 - b10 - b9 - b8 - b5 + b3 - b1 + 1) * q^79 - b5 * q^80 + (b19 + b17 + b15 - b12 - b10 + b8 - b7 - b6 + b5 + b3 + b1 + 2) * q^82 + (-b18 - 2b17 + b16 - b14 + b13 + 2b12 - b11 - 2b9 + b8 + 2b3 + b2 - b1) * q^83 + (-b17 + b16 - 2b14 - b12 + b9 + b3) q^85 + (-b16 + b12 - 2b11 - b9) q^86 + b8 * q^88 + (2b17 + b16 - 2b12 + 3b11 + 2b9 - b8 + 2b4 - 2b3 - b2 + b1) * q^89 + (-b18 + b16 - b14 + b13 - b12 + b11 + b9 + b8 + b2 - b1) * q^91 + (-b19 - b15 - b10 - b9 - b5 + 1) * q^92 + (-b19 + b15 - b12 - b10 + b8 - b5 + b1 + 1) * q^94 + (b19 + b18 + b17 - b16 + b14 + b12 - b10 + b8 + b5 - b3 - b2 + 2b1 - 1) q^95 + (-b18 + 3b16 - b14 + b13 - b12 + b11 + b9 - b8 + b2 + b1) q^97 + (b17 - 3b10 + b8 + b7 + b6 - 2b5 + b3 + b1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{2} + 20 q^{4} - 4 q^{5} + 20 q^{8} - 4 q^{10} - 6 q^{11} + 4 q^{13} + 20 q^{16} + 2 q^{19} - 4 q^{20} - 6 q^{22} + 8 q^{23} + 32 q^{25} + 4 q^{26} - 4 q^{29} + 20 q^{32} + 2 q^{38} - 4 q^{40}+ \cdots - 36 q^{98}+O(q^{100}) \) 20 * q + 20 * q^2 + 20 * q^4 - 4 * q^5 + 20 * q^8 - 4 * q^10 - 6 * q^11 + 4 * q^13 + 20 * q^16 + 2 * q^19 - 4 * q^20 - 6 * q^22 + 8 * q^23 + 32 * q^25 + 4 * q^26 - 4 * q^29 + 20 * q^32 + 2 * q^38 - 4 * q^40 + 24 * q^41 - 6 * q^44 + 8 * q^46 - 8 * q^47 - 36 * q^49 + 32 * q^50 + 4 * q^52 + 8 * q^55 - 4 * q^58 + 20 * q^64 + 24 * q^65 + 2 * q^76 - 28 * q^77 + 24 * q^79 - 4 * q^80 + 24 * q^82 - 6 * q^88 + 8 * q^92 - 8 * q^94 - 32 * q^95 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3762.2.g.l

Level	$3762$
Weight	$2$
Character orbit	3762.g
Analytic conductor	$30.040$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(30.0397212404\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - 2 x^{19} + 2 x^{18} + 199 x^{16} - 414 x^{15} + 430 x^{14} + 184 x^{13} + 6939 x^{12} + \cdots + 64 \) x^20 - 2x^19 + 2x^18 + 199x^16 - 414x^15 + 430x^14 + 184x^13 + 6939x^12 - 14914x^11 + 16078x^10 + 10764x^9 + 7501x^8 - 14446x^7 + 21922x^6 + 21836x^5 + 10900x^4 - 80x^3 + 128x^2 + 128x + 64
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{18} \)
Twist minimal:	no (minimal twist has level 1254)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( 83\!\cdots\!97 \nu^{19} + \cdots - 85\!\cdots\!48 ) / 25\!\cdots\!32 \) (831512139114535450548703521487597v^19 - 10666407926939662759125896203236036v^18 + 22860895175813138956587432928926594v^17 - 25760806681975940984402432923331300v^16 + 174405512164447914468271208302239747v^15 - 2138258713474058491600609115265851196v^14 + 4719112230040428175476850958338672750v^13 - 5311520213026849338708156643981644964v^12 + 6014483044728094757135059070542162503v^11 - 74793485893463825882507228662915714264v^10 + 169357074766720013170857688442325215622v^9 - 192735071031044728645462500359147842320v^8 - 20415302169180015618628115953831668823v^7 - 64076190866093345393030504416161641336v^6 + 151024951581155030110311448035773178570v^5 - 229654424057454985505867939559364250904v^4 - 106103989325303755262205656976726854924v^3 - 52836856026315397821040882549692910520v^2 - 1416872117330124046362828554101352720*v - 8551213805621052352120142111949060448) / 2586557172353689357723696874231052032
\(\beta_{2}\)	\(=\)	\( ( 58\!\cdots\!68 \nu^{19} + \cdots - 31\!\cdots\!40 ) / 16\!\cdots\!52 \) (58968872310765640035833969525368v^19 - 432932192584308517788799632195651v^18 + 745123419287860098564203915040150v^17 - 569891016501074563469650044511270v^16 + 11565287226998413835139557507812624v^15 - 86852874034747412771397361300406677v^14 + 155042908950589299105085459531940754v^13 - 112536614361184787594763749765083034v^12 + 316522335182877188666846404921801728v^11 - 3014539672431770094441212781509999105v^10 + 5601781906410811606288060906619313654v^9 - 4007836601928422920532995608009353690v^8 - 4184234081547719413449201565852673444v^7 - 1377888766061996769075665589444393111v^6 + 5124266521759516864281634283525597426v^5 - 5308937466486005621203518725174724358v^4 - 7289077822537706286655440947460340884v^3 - 1343564938878091479633620051436172220v^2 - 527606395635819039866839703713900272*v - 31064523958690418954325864694808640) / 161659823272105584857731054639440752
\(\beta_{3}\)	\(=\)	\( ( 12\!\cdots\!51 \nu^{19} + \cdots - 48\!\cdots\!44 ) / 12\!\cdots\!16 \) (1221058225521071066382552515384751v^19 - 10922146641137419692730694649184900v^18 + 21701018437526849628623054648177926v^17 - 21538018721546088330010316856972316v^16 + 247454834540315782443085557140390689v^15 - 2192544333300649994670123111440200956v^14 + 4492307451549938067240783371285688714v^13 - 4368753483902038832692127535546637884v^12 + 7873696923471455810490300492037456973v^11 - 76528768185508387615366305926562524048v^10 + 161861810739920293792643190162006246962v^9 - 157196687988746723570089630866062379744v^8 - 46165337918939855703216075622032198237v^7 - 52842486772560456164804292579394828832v^6 + 159522253204079716056915258156340530942v^5 - 187194930679889936167729874254044572760v^4 - 123488228508948335855018769276590691748v^3 - 37786599731303499822234313174651094408v^2 + 15486203157475804862093837799228054224*v - 482504086620460202132735031865875744) / 1293278586176844678861848437115526016
\(\beta_{4}\)	\(=\)	\( ( 82\!\cdots\!31 \nu^{19} + \cdots + 10\!\cdots\!68 ) / 85\!\cdots\!44 \) (8219799132211139472376890131v^19 - 16233416365423891897265888796v^18 + 16326477034319462952356570366v^17 - 471622439071943244033777372v^16 + 1637076972115028739231189750845v^15 - 3362795186327553364091475559076v^14 + 3508981978880631016427921415186v^13 + 1420277124815268337022088856996v^12 + 57355834049581047374211409931833v^11 - 121281224458000421756413201538760v^10 + 131166438763969750182749287242362v^9 + 85355885443356451179259934323280v^8 + 74232070475493958453250308739351v^7 - 120553803535436370793020962411304v^6 + 178607841082818966928704078799798v^5 + 178535678548482930049770654178456v^4 + 111990664831243016102375228945932v^3 - 685616995259350144663758090440v^2 + 9605389887803950644425763403280*v + 1047964158565003474000541737568) / 8562547330006453160189410927744
\(\beta_{5}\)	\(=\)	\( ( 73\!\cdots\!17 \nu^{19} + \cdots - 63\!\cdots\!24 ) / 25\!\cdots\!32 \) (7352022686928822650700141022969517v^19 - 16509623872869331751879311782797060v^18 + 20787222866119439824074790169560578v^17 - 10062708840826055295155266714870692v^16 + 1471176521307216195008147178081020611v^15 - 3406142403280512203249189727315405948v^14 + 4400446762812428904657219663875227758v^13 - 746039552667763294136463093378497764v^12 + 52402873610360935452436782054707418631v^11 - 122388720471366872695793671408220527384v^10 + 161938686056779731023867529311299996422v^9 + 3016722403064478125663234349731612592v^8 + 99026562971885097263836974047369380841v^7 - 118106060749704335739288349277651286072v^6 + 187110227248062446591972571110736191434v^5 + 83047957752232608436799204765960830248v^4 + 116745411472638109001223646930694701172v^3 - 670337657467895864164417423963907128v^2 - 7824631763825627339874357295381928720*v - 6325000853694545625878349382727184224) / 2586557172353689357723696874231052032
\(\beta_{6}\)	\(=\)	\( ( 21\!\cdots\!05 \nu^{19} + \cdots - 18\!\cdots\!28 ) / 64\!\cdots\!08 \) (2199097885898303239446423967014905v^19 - 4800427959168052281083189344139284v^18 + 5764377964740736782447612175908730v^17 - 2251911993901795533472919769561140v^16 + 439450560515521087997515696736473847v^15 - 991195931571010359391538159285940652v^14 + 1223965627885863390401311752600035382v^13 - 65535867247078162661468640620027572v^12 + 15572961137556029028427508580210883179v^11 - 35640740469909170423457313390061699480v^10 + 45205100433877059841984586143536258158v^9 + 6555958025332338430389426832477449872v^8 + 26466817933953010120885329854575842693v^7 - 34552000293547676358476401305384031928v^6 + 55150059904187034842859775929786435234v^5 + 28320394392850505948012713314274979560v^4 + 34072418364956031639545177116016588324v^3 - 196140432480182869599937077744979544v^2 - 2281473313051752608199056286940500176*v - 1847763776516549269101319447525839328) / 646639293088422339430924218557763008
\(\beta_{7}\)	\(=\)	\( ( 53\!\cdots\!05 \nu^{19} + \cdots + 63\!\cdots\!92 ) / 12\!\cdots\!16 \) (5383105619836592703591631911267005v^19 - 11044966372091234690519992982840164v^18 + 12134508868471636394925559714063074v^17 - 2829885892088287498071577443144676v^16 + 1074312228141139351599174731751719667v^15 - 2285343878841805384769808236335883676v^14 + 2590719517818359765195907736286961230v^13 + 407115319403250637938888656475000668v^12 + 37948876008941508076837488685144995511v^11 - 82343468569437308713789910031994366200v^10 + 96060850735060096362518819740665234982v^9 + 37229606927284618359353189285171380528v^8 + 60954580913292341982888505550028089465v^7 - 81481296021019131459709085601705954904v^6 + 115951407306861964811330692464407869674v^5 + 109084169832188853888628606489474461608v^4 + 74919200004860443931492448687148690228v^3 - 463446702707583604063751024262561464v^2 - 5046525265040546394721439595935135248*v + 6310275698736577220462928477865647392) / 1293278586176844678861848437115526016
\(\beta_{8}\)	\(=\)	\( ( - 10\!\cdots\!23 \nu^{19} + \cdots - 86\!\cdots\!56 ) / 25\!\cdots\!32 \) (-10821892343525956121989901322820323v^19 + 23717312543902070671725845302061500v^18 - 22318750491688401608531615796636062v^17 - 4679123926434180422441836655890852v^16 - 2142641755375541815623800107483870509v^15 + 4888899539594549451016863387018421316v^14 - 4820789003484492082405914910017284978v^13 - 2910885900008975532756418917952283236v^12 - 72398867785744345932984182400751543849v^11 + 175565154651482768384705593069350018920v^10 - 181145265855005250381257467286718324442v^9 - 147809938991784111883961898166857488656v^8 + 26391219348099857452338574919776956185v^7 + 177038586373421333168754413173948238152v^6 - 260206566745069700387500177627106634710v^5 - 247606398512949410047160525003554090392v^4 + 30112928464856238701331669549098401716v^3 + 53484640475719300167322944109673761992v^2 + 6612991148098370241018370792226572272*v - 8660178046473441287217785653446562656) / 2586557172353689357723696874231052032
\(\beta_{9}\)	\(=\)	\( ( 15\!\cdots\!15 \nu^{19} + \cdots + 49\!\cdots\!08 ) / 25\!\cdots\!32 \) (15620909516103631150329463849240415v^19 - 40199255713362636462958499772458188v^18 + 48564003857607124081896916002408054v^17 - 15290334869165388841157075268503628v^16 + 3103573697688304052516825232861331953v^15 - 8244451061505100790915481613201496244v^14 + 10304885947014042207666592994431769562v^13 - 444510259318638472361484641373961356v^12 + 105710678603045514905184944501975387933v^11 - 294152021176091896483296396767751377832v^10 + 380354456846855737466530786661397836770v^9 + 43022753590340609727609727641834931728v^8 - 16616379493428426131544481829807973069v^7 - 259414900600791849498639847860291972936v^6 + 462887880618737349937110342325616788878v^5 + 162639746883305545311278034575006746424v^4 - 63791483500086382643125766740907360708v^3 - 61662910087623333121457116468162318312v^2 + 13403443827655780282620887595542393424*v + 4985912705522407379710179656216368608) / 2586557172353689357723696874231052032
\(\beta_{10}\)	\(=\)	\( ( - 97\!\cdots\!25 \nu^{19} + \cdots - 38\!\cdots\!04 ) / 12\!\cdots\!16 \) (-9700552370857735000074863292719725v^19 + 16662875163802437416451127082162308v^18 - 11749842928698825157599510834192834v^17 - 10557207027332910953966000776960924v^16 - 1924415962325504736906872278843581891v^15 + 3469502747033429158769406843622370428v^14 - 2605087040082340629951949237809832046v^13 - 4007537373579416599801726189893505500v^12 - 66542546151253907016930255635551838407v^11 + 125725491890562209332288497048142507544v^10 - 100220285543603233803196687595932338502v^9 - 185772364060790445738058215955217855984v^8 - 55201838613177233467517313633470783593v^7 + 129984192436692274023501011229878641848v^6 - 166228618980673632720153382420788219082v^5 - 301393263396418638700439795261124910824v^4 - 108522727472368663458958129402994629236v^3 + 741451603094371887870588525620607032v^2 + 7333864341705461703139360709159133968*v - 3894532451092790065181257108724870304) / 1293278586176844678861848437115526016
\(\beta_{11}\)	\(=\)	\( ( 51\!\cdots\!33 \nu^{19} + \cdots + 19\!\cdots\!40 ) / 64\!\cdots\!08 \) (5182233568789923801067330362493133v^19 - 14107465411691033142539853714558144v^18 + 17776967785079681292647089784662010v^17 - 7211433595577452983600568241065628v^16 + 1030728873500969016755704367826170051v^15 - 2889728678653953856317602499295530272v^14 + 3762932054302962195135684220350737174v^13 - 600002829024419938229553659188823020v^12 + 35159833581397232276728024789137304903v^11 - 103154078726118312553612339155645477548v^10 + 138571062550421532860062014040951923454v^9 - 2411243528413731480530644993643803992v^8 - 5418439534468101544954406076496620455v^7 - 99390519266478208544331633442665178628v^6 + 165347779771241907597076530504218271858v^5 + 32819425740425451680113197944792031920v^4 - 29172708927658664164124843582891890204v^3 - 39599884452768342635981704174025268456v^2 - 1270181937667323947629348804409534160*v + 193855944172884117977818952117934240) / 646639293088422339430924218557763008
\(\beta_{12}\)	\(=\)	\( ( - 29\!\cdots\!71 \nu^{19} + \cdots + 17\!\cdots\!96 ) / 25\!\cdots\!32 \) (-29383342024216997287188492151284471v^19 + 69208483447703882420375827460451500v^18 - 81485254775716380671131300832084070v^17 + 24582916371072519917591225148706220v^16 - 5850425814915515707485846694918739161v^15 + 14240887676380051558717683758214378708v^14 - 17320720974990102281632867295345024106v^13 - 151415218312942134414557911633203732v^12 - 202654394429328318276309047401595418597v^11 + 509987762499376280983793224516849262312v^10 - 640419849560248885225878209618739325874v^9 - 120814856017586423567824399852760282896v^8 - 134098643012576142315278731115226740459v^7 + 470456205503737650965373825648391880072v^6 - 791395119072717065092327587476133290302v^5 - 383784725863844822190478127177945927800v^4 - 140158022061522199551401550542217485980v^3 + 62861691438931972545856342212030105128v^2 + 265524180909903975536433534471783728*v + 1772008566361791457856826839884856096) / 2586557172353689357723696874231052032
\(\beta_{13}\)	\(=\)	\( ( 16\!\cdots\!83 \nu^{19} + \cdots + 13\!\cdots\!32 ) / 12\!\cdots\!16 \) (16705866266073313106409589134315183v^19 - 39398930830861539023868468432326012v^18 + 45317194259002148811414894776052726v^17 - 12550115886716729626173781089807788v^16 + 3326583624655663066926696505706982689v^15 - 8111242148307516662733533094665670852v^14 + 9651176844154386735144764428380871706v^13 + 385023504790464541506636246843127316v^12 + 115250174836960412205104203068169856909v^11 - 291453061187297514642949937236406887672v^10 + 357845008004452706498504291524339163810v^9 + 79594250231906976288663050063231358768v^8 + 75945796691704316394224625289179904643v^7 - 313943603199006459692403010683455003064v^6 + 465179220974545562702059613590911383438v^5 + 230770992806048748964442702136053824728v^4 + 57936551602525694936709987833926446012v^3 - 92481513943530817685700140464494477992v^2 + 7131385693089671596768718913429792848*v + 1356655861654772471183901415557374432) / 1293278586176844678861848437115526016
\(\beta_{14}\)	\(=\)	\( ( - 40\!\cdots\!97 \nu^{19} + \cdots - 54\!\cdots\!72 ) / 25\!\cdots\!32 \) (-40581671258688668660044122696480997v^19 + 76903625187692962119313033062320580v^18 - 76750592314150880260308186460826226v^17 - 1488863413054186347608631200845820v^16 - 8080579656028191912551466093403680267v^15 + 15948271476169475912574862905691759036v^14 - 16500853143036865706339572570181404286v^13 - 7834739200513177204505034742866827068v^12 - 283453518031464838806423095834095185199v^11 + 573902382795884509421044549106012415736v^10 - 617025893874164127558572517898551270102v^9 - 452259955406060304798890607811437816624v^8 - 391730653689279223685330279435248822849v^7 + 473116689805677281733017623241215651672v^6 - 848430693363426019045438665492070605178v^5 - 924803750429799503534426050321089082728v^4 - 608596778381215262803129620515735853140v^3 - 161234423125531722111781548592712809608v^2 - 51490108944920232692581728730960323184*v - 5427704590810226097313408690237709472) / 2586557172353689357723696874231052032
\(\beta_{15}\)	\(=\)	\( ( - 12\!\cdots\!39 \nu^{19} + \cdots + 25\!\cdots\!04 ) / 64\!\cdots\!08 \) (-12035516946903983645924003908251439v^19 + 25727310763510197276603384207098092v^18 - 27225559067462219599952574592340534v^17 + 2925848952845467261076114059757932v^16 - 2394363317082558324340372205356579745v^15 + 5311562200606596060429492792757489652v^14 - 5828873019462720725229904709416798938v^13 - 1581848338455212451214491207214989620v^12 - 83063343322246312552568604394154890317v^11 + 190866022239526653291762967581955813128v^10 - 216986364662577567027943133448234714594v^9 - 105730493874871659967440159309662710512v^8 - 67170708615322332990520762989511759427v^7 + 182427073597359407015344770773942880648v^6 - 282208431399527155484899599578414500750v^5 - 226983502035819423486081092165185618488v^4 - 90281059897200722657467010830993350716v^3 + 16084894218356334291478850890658051304v^2 + 3798758572134686849954648503685557872*v + 2516386656793101208056355911131291104) / 646639293088422339430924218557763008
\(\beta_{16}\)	\(=\)	\( ( - 25\!\cdots\!11 \nu^{19} + \cdots - 18\!\cdots\!08 ) / 12\!\cdots\!16 \) (-25335525903379485767829777006484011v^19 + 61222436025920087569674449002378132v^18 - 72586452643889303741471091907998622v^17 + 21914375505591888762688981857047916v^16 - 5040967009472340995557813167301537125v^15 + 12585130469000795527156096417658996396v^14 - 15421850953462993416048304303360548594v^13 + 49088218928556787526558731425282892v^12 - 173715298237845275949475903206800584513v^11 + 450185085558257747057939737006146794640v^10 - 569987139194755822620809471995578387962v^9 - 96613956660626030070966267896984181632v^8 - 71943019021371291726840430880444167663v^7 + 409618166713453734396560962025148720480v^6 - 703970428376140917763914752727334773942v^5 - 313971709008337115573156547066932878696v^4 - 50335208443039430836533465511400077900v^3 + 74412973689344994540970029305697021288v^2 - 7944700305899930581190333213332649232*v - 1846524452507148131571888834557088608) / 1293278586176844678861848437115526016
\(\beta_{17}\)	\(=\)	\( ( - 27\!\cdots\!13 \nu^{19} + \cdots - 26\!\cdots\!16 ) / 12\!\cdots\!16 \) (-27694520924907307500139835036938513v^19 + 58487346806674537037239796863616876v^18 - 60240128260669338382466725014200666v^17 + 2761225595626649613079359957401412v^16 - 5506477392416031332221841831323749407v^15 + 12079612950888476214253658505973288980v^14 - 12922775491278493108737198712670613398v^13 - 4468482206268483963821070312101109532v^12 - 190605689929814751101596176658651228787v^11 + 434256130691239264015177500026467479360v^10 - 482104034680711095970098018963240921646v^9 - 273373961009431580990315212867276146240v^8 - 137836728297013512180980792085775498717v^7 + 418313826593893821680885184731742165904v^6 - 643046136977379272849406737677605449506v^5 - 554266456295563646512706464955620509048v^4 - 190411516912181442296614586626150691940v^3 + 39869645398529780433386711334641063096v^2 + 5700910774936590737200110454847007184*v - 2641109547635419631578118955230896416) / 1293278586176844678861848437115526016
\(\beta_{18}\)	\(=\)	\( ( 61\!\cdots\!99 \nu^{19} + \cdots + 65\!\cdots\!76 ) / 25\!\cdots\!32 \) (61772359811664697394146213609667299v^19 - 132346571607950063167205505257903852v^18 + 144150364616436924377368953655358078v^17 - 23728187850999709221319258211349820v^16 + 12299313911622367980236525422343772333v^15 - 27326419020280907220905591719606831252v^14 + 30804866634630660243125655879291178130v^13 + 6313917926056944344421208954895266628v^12 + 428432606161462043064489801901476496041v^11 - 982024593939352288708490278528419924088v^10 + 1145546064648328266442169827309381352890v^9 + 477732636142725785929740250709362706416v^8 + 421125339128595797208819847057282200615v^7 - 934286467909813103370877040555372901816v^6 + 1513077777503585669862401843786540094454v^5 + 1110764876647909879372637720991644706488v^4 + 547306782740524018740189912792085191436v^3 - 38299512210320504979915567534405285512v^2 + 50658103551315032450202228173348203024*v + 6523540048031629687897541423861370976) / 2586557172353689357723696874231052032
\(\beta_{19}\)	\(=\)	\( ( 31\!\cdots\!67 \nu^{19} + \cdots + 85\!\cdots\!04 ) / 12\!\cdots\!16 \) (31768412599442519361033905478107167v^19 - 57687677869832334255904301603393660v^18 + 48119499811400342389345901708628022v^17 + 19827828528341268545967907481936756v^16 + 6312920234455297954356615664825262321v^15 - 11986452586130237841123225058238434628v^14 + 10498561731528801537584486789493010522v^13 + 10037554200329361892078225510997137332v^12 + 219597564563071043518749519704296170845v^11 - 433530034208164804386773143035213615224v^10 + 397810185089276533887900313275403498850v^9 + 496179353044696249280315270791576348624v^8 + 229975445389102298648249054312637953523v^7 - 441658621112658576717016342319905239416v^6 + 589578583850144289742894265003578488910v^5 + 877291176025403274389430890695864752248v^4 + 382026690847735964145900787666695566716v^3 - 2516782724206445540331589160235927144v^2 - 25776422550651075919652896393921848752*v + 8539630336415312152634333628477425504) / 1293278586176844678861848437115526016

\(\nu\)	\(=\)	\( ( \beta_{12} + \beta_{9} + \beta_{6} + 2\beta_{4} ) / 4 \) (b12 + b9 + b6 + 2*b4) / 4
\(\nu^{2}\)	\(=\)	\( ( 2\beta_{17} - 2\beta_{16} + 2\beta_{13} - 3\beta_{11} + 2\beta_{4} - 2\beta_{3} + 5\beta_{2} ) / 4 \) (2b17 - 2b16 + 2b13 - 3b11 + 2b4 - 2b3 + 5*b2) / 4
\(\nu^{3}\)	\(=\)	\( ( 4 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} - 7 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} - 11 \beta_{9} + \cdots + 4 ) / 4 \) (4b16 - 2b15 - 2b14 - 7b12 + 4b11 + 2b10 - 11b9 - 2b8 - 11b6 - 4b5 + 18b4 + 4b2 + 4*b1 + 4) / 4
\(\nu^{4}\)	\(=\)	\( ( - 4 \beta_{19} + 13 \beta_{17} - 6 \beta_{15} + 4 \beta_{12} - 42 \beta_{10} - 2 \beta_{9} + 11 \beta_{8} + \cdots - 58 ) / 2 \) (-4b19 + 13b17 - 6b15 + 4b12 - 42b10 - 2b9 + 11b8 + 14b7 + 14b6 - 24b5 + 13b3 + 11b1 - 58) / 2
\(\nu^{5}\)	\(=\)	\( ( 14 \beta_{19} - 14 \beta_{18} - 56 \beta_{16} - 24 \beta_{15} + 24 \beta_{14} + 10 \beta_{13} + \cdots + 72 ) / 4 \) (14b19 - 14b18 - 56b16 - 24b15 + 24b14 + 10b13 - 97b12 - 62b11 + 26b10 - 99b9 + 74b8 - 10b7 - 129b6 - 50b5 - 186b4 + 20b3 - 44b2 - 32b1 + 72) / 4
\(\nu^{6}\)	\(=\)	\( ( 142 \beta_{18} - 326 \beta_{17} + 70 \beta_{16} + 210 \beta_{14} - 362 \beta_{13} + 90 \beta_{12} + \cdots - 124 \beta_1 ) / 4 \) (142b18 - 326b17 + 70b16 + 210b14 - 362b13 + 90b12 + 161b11 - 90b9 + 124b8 - 284b4 + 326b3 - 633b2 - 124*b1) / 4
\(\nu^{7}\)	\(=\)	\( ( - 282 \beta_{19} - 282 \beta_{18} - 380 \beta_{17} - 690 \beta_{16} + 254 \beta_{15} + 254 \beta_{14} + \cdots - 990 ) / 4 \) (-282b19 - 282b18 - 380b17 - 690b16 + 254b15 + 254b14 + 216b13 + 889b12 - 808b11 - 338b10 + 1391b9 + 434b8 + 216b7 + 1571b6 + 574b5 - 2064b4 - 456b2 - 1062b1 - 990) / 4
\(\nu^{8}\)	\(=\)	\( ( 1019 \beta_{19} - 2015 \beta_{17} + 1463 \beta_{15} - 1034 \beta_{12} + 6744 \beta_{10} + 429 \beta_{9} + \cdots + 6749 ) / 2 \) (1019b19 - 2015b17 + 1463b15 - 1034b12 + 6744b10 + 429b9 - 1326b8 - 2286b7 - 2140b6 + 3121b5 - 2015b3 - 1326b1 + 6749) / 2
\(\nu^{9}\)	\(=\)	\( ( - 4276 \beta_{19} + 4276 \beta_{18} + 8338 \beta_{16} + 2680 \beta_{15} - 2680 \beta_{14} + \cdots - 12894 ) / 4 \) (-4276b19 + 4276b18 + 8338b16 + 2680b15 - 2680b14 - 3418b13 + 13835b12 + 10108b11 - 4582b10 + 13471b9 - 13952b8 + 3418b7 + 19391b6 + 6584b5 + 23888b4 - 5352b3 + 4814b2 + 5556b1 - 12894) / 4
\(\nu^{10}\)	\(=\)	\( ( - 27256 \beta_{18} + 49494 \beta_{17} - 3858 \beta_{16} - 37968 \beta_{14} + 57242 \beta_{13} + \cdots + 23700 \beta_1 ) / 4 \) (-27256b18 + 49494b17 - 3858b16 - 37968b14 + 57242b13 - 18708b12 - 19467b11 + 18708b9 - 23700b8 + 39626b4 - 49494b3 + 89165b2 + 23700*b1) / 4
\(\nu^{11}\)	\(=\)	\( ( 58720 \beta_{19} + 58720 \beta_{18} + 67752 \beta_{17} + 101036 \beta_{16} - 28742 \beta_{15} + \cdots + 165956 ) / 4 \) (58720b19 + 58720b18 + 67752b17 + 101036b16 - 28742b15 - 28742b14 - 48544b13 - 132155b12 + 125300b11 + 63262b10 - 199759b9 - 69538b8 - 48544b7 - 240555b6 - 76116b5 + 283370b4 + 51852b2 + 176152b1 + 165956) / 4
\(\nu^{12}\)	\(=\)	\( ( - 176628 \beta_{19} + 303173 \beta_{17} - 239246 \beta_{15} + 182010 \beta_{12} - 1042960 \beta_{10} + \cdots - 965092 ) / 2 \) (-176628b19 + 303173b17 - 239246b15 + 182010b12 - 1042960b10 - 57236b9 + 182085b8 + 356956b7 + 338388b6 - 440196b5 + 303173b3 + 182085b1 - 965092) / 2
\(\nu^{13}\)	\(=\)	\( ( 772178 \beta_{19} - 772178 \beta_{18} - 1231612 \beta_{16} - 312660 \beta_{15} + 312660 \beta_{14} + \cdots + 2131180 ) / 4 \) (772178b19 - 772178b18 - 1231612b16 - 312660b15 + 312660b14 + 657706b13 - 2128093b12 - 1551582b11 + 875006b10 - 1938451b9 + 2181226b8 - 657706b7 - 2990893b6 - 884590b5 - 3408838b4 + 816124b3 - 564620b2 - 862800b1 + 2131180) / 4
\(\nu^{14}\)	\(=\)	\( ( 4507906 \beta_{18} - 7423230 \beta_{17} + 534738 \beta_{16} + 5949350 \beta_{14} - 8887110 \beta_{13} + \cdots - 3547448 \beta_1 ) / 4 \) (4507906b18 - 7423230b17 + 534738b16 + 5949350b14 - 8887110b13 + 3069558b12 + 3040273b11 - 3069558b9 + 3547448b8 - 5410332b4 + 7423230b3 - 13141025b2 - 3547448*b1) / 4
\(\nu^{15}\)	\(=\)	\( ( - 9945318 \beta_{19} - 9945318 \beta_{18} - 9567468 \beta_{17} - 15085666 \beta_{16} + 3427262 \beta_{15} + \cdots - 27359034 ) / 4 \) (-9945318b19 - 9945318b18 - 9567468b17 - 15085666b16 + 3427262b15 + 3427262b14 + 8709032b13 + 20052317b12 - 19227140b11 - 12036254b10 + 29986103b9 + 10672414b8 + 8709032b7 + 37231255b6 + 10301474b5 - 41329388b4 - 6160000b2 - 26746406b1 - 27359034) / 4
\(\nu^{16}\)	\(=\)	\( ( 28533035 \beta_{19} - 45442251 \beta_{17} + 36744031 \beta_{15} - 30333120 \beta_{12} + 159902054 \beta_{10} + \cdots + 146079151 ) / 2 \) (28533035b19 - 45442251b17 + 36744031b15 - 30333120b12 + 159902054b10 + 6410911b9 - 26180376b8 - 55265876b7 - 54624074b6 + 64580653b5 - 45442251b3 - 26180376b1 + 146079151) / 2
\(\nu^{17}\)	\(=\)	\( ( - 126759560 \beta_{19} + 126759560 \beta_{18} + 185384902 \beta_{16} + 37613004 \beta_{15} + \cdots - 351132370 ) / 4 \) (-126759560b19 + 126759560b18 + 185384902b16 + 37613004b15 - 37613004b14 - 113937738b13 + 331907043b12 + 238481948b11 - 164282626b10 + 285325755b9 - 326346616b8 + 113937738b7 + 463810535b6 + 119947588b5 + 503295060b4 - 110248832b3 + 66850542b2 + 131903492b1 - 351132370) / 4
\(\nu^{18}\)	\(=\)	\( ( - 719370636 \beta_{18} + 1113095910 \beta_{17} - 106969886 \beta_{16} - 904845476 \beta_{14} + \cdots + 500642640 \beta_1 ) / 4 \) (-719370636b18 + 1113095910b17 - 106969886b16 - 904845476b14 + 1374216438b13 - 481206616b12 - 501089447b11 + 481206616b9 - 500642640b8 + 722393466b4 - 1113095910b3 + 1967430849b2 + 500642640*b1) / 4
\(\nu^{19}\)	\(=\)	\( ( 1606937508 \beta_{19} + 1606937508 \beta_{18} + 1254307784 \beta_{17} + 2282997988 \beta_{16} + \cdots + 4504244008 ) / 4 \) (1606937508b19 + 1606937508b18 + 1254307784b17 + 2282997988b16 - 411003878b15 - 411003878b14 - 1480426256b13 - 3063248155b12 + 2960309264b11 + 2224623834b10 - 4561713083b9 - 1630309606b8 - 1480426256b7 - 5781018811b6 - 1394517872b5 + 6144534982b4 + 717206596b2 + 3972078440b1 + 4504244008) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2089.20
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T - 1)^{20} \) (T - 1)^20
$3$	\( T^{20} \) T^20
$5$	\( (T^{10} + 2 T^{9} + \cdots - 384)^{2} \) (T^10 + 2T^9 - 31T^8 - 50T^7 + 308T^6 + 336T^5 - 1168T^4 - 768T^3 + 1440T^2 + 576*T - 384)^2
$7$	\( T^{20} + 88 T^{18} + \cdots + 65536 \) T^20 + 88T^18 + 3068T^16 + 54352T^14 + 524144T^12 + 2744704T^10 + 7447104T^8 + 9688832T^6 + 5264384T^4 + 1032192*T^2 + 65536
$11$	\( T^{20} + \cdots + 25937424601 \) T^20 + 6T^19 + 18T^18 + 18T^17 - 91T^16 + 24T^15 + 3416T^14 + 19432T^13 + 45266T^12 - 8972T^11 - 234004T^10 - 98692T^9 + 5477186T^8 + 25863992T^7 + 50013656T^6 + 3865224T^5 - 161212051T^4 + 350769078T^3 + 3858459858T^2 + 14147686146*T + 25937424601
$13$	\( (T^{10} - 2 T^{9} + \cdots - 4608)^{2} \) (T^10 - 2T^9 - 48T^8 + 40T^7 + 768T^6 + 64T^5 - 4256T^4 - 1728T^3 + 7680T^2 + 2304*T - 4608)^2
$17$	\( T^{20} + \cdots + 59395538944 \) T^20 + 222T^18 + 20713T^16 + 1057116T^14 + 32140176T^12 + 593815040T^10 + 6517181440T^8 + 39756218368T^6 + 118740680704T^4 + 147593363456*T^2 + 59395538944
$19$	\( T^{20} + \cdots + 6131066257801 \) T^20 - 2T^19 - 2T^18 + 178T^17 - 139T^16 - 1600T^15 + 16424T^14 - 26608T^13 - 75022T^12 + 264204T^11 - 1741964T^10 + 5019876T^9 - 27082942T^8 - 182504272T^7 + 2140392104T^6 - 3961758400T^5 - 6539377459T^4 + 159109169542T^3 - 33967126082T^2 - 645375395558*T + 6131066257801
$23$	\( (T^{10} - 4 T^{9} + \cdots + 4633088)^{2} \) (T^10 - 4T^9 - 148T^8 + 704T^7 + 6960T^6 - 41056T^5 - 88608T^4 + 845312T^3 - 785152T^2 - 2944512*T + 4633088)^2
$29$	\( (T^{10} + 2 T^{9} + \cdots + 24032000)^{2} \) (T^10 + 2T^9 - 207T^8 - 540T^7 + 14824T^6 + 49360T^5 - 419904T^4 - 1783616T^3 + 3172480T^2 + 21796096*T + 24032000)^2
$31$	\( T^{20} + 224 T^{18} + \cdots + 80281600 \) T^20 + 224T^18 + 18924T^16 + 792848T^14 + 18255408T^12 + 238865920T^10 + 1750229056T^8 + 6693340928T^6 + 10844148736T^4 + 2639314944*T^2 + 80281600
$37$	\( T^{20} + 258 T^{18} + \cdots + 1024 \) T^20 + 258T^18 + 24717T^16 + 1061864T^14 + 19109916T^12 + 100894896T^10 + 221523056T^8 + 215497600T^6 + 78946368T^4 + 3869440*T^2 + 1024
$41$	\( (T^{10} - 12 T^{9} + \cdots + 32157696)^{2} \) (T^10 - 12T^9 - 224T^8 + 2736T^7 + 15920T^6 - 205952T^5 - 341248T^4 + 5672448T^3 - 764928T^2 - 41902080*T + 32157696)^2
$43$	\( T^{20} + 314 T^{18} + \cdots + 51380224 \) T^20 + 314T^18 + 35433T^16 + 1846412T^14 + 49497360T^12 + 697289728T^10 + 4915111936T^8 + 15558139904T^6 + 20068630528T^4 + 6902513664*T^2 + 51380224
$47$	\( (T^{10} + 4 T^{9} + \cdots + 594432)^{2} \) (T^10 + 4T^9 - 276T^8 - 864T^7 + 23728T^6 + 28192T^5 - 793760T^4 + 621312T^3 + 5545728T^2 - 4631040*T + 594432)^2
$53$	\( T^{20} + \cdots + 14880096256 \) T^20 + 368T^18 + 51244T^16 + 3433264T^14 + 119193264T^12 + 2288886528T^10 + 24881008448T^8 + 149009379584T^6 + 443740430336T^4 + 484855255040*T^2 + 14880096256
$59$	\( T^{20} + \cdots + 2415919104 \) T^20 + 520T^18 + 111888T^16 + 12922880T^14 + 870008832T^12 + 34674745344T^10 + 789061697536T^8 + 9139075940352T^6 + 38735774220288T^4 + 3122877431808*T^2 + 2415919104
$61$	\( T^{20} + \cdots + 3226240000 \) T^20 + 570T^18 + 121389T^16 + 12620492T^14 + 704009340T^12 + 21724656320T^10 + 364426271920T^8 + 3041124273472T^6 + 9638272773440T^4 + 463467830784*T^2 + 3226240000
$67$	\( T^{20} + \cdots + 31021072384 \) T^20 + 762T^18 + 223049T^16 + 31916336T^14 + 2363023104T^12 + 86902090240T^10 + 1349697560576T^8 + 6215719714816T^6 + 6265929531392T^4 + 1172030095360*T^2 + 31021072384
$71$	\( T^{20} + \cdots + 10\!\cdots\!00 \) T^20 + 972T^18 + 396988T^16 + 88646464T^14 + 11788236336T^12 + 952810866496T^10 + 45814513738304T^8 + 1230279472224256T^6 + 16341621818959872T^4 + 84498163091062784*T^2 + 104777443099033600
$73$	\( T^{20} + \cdots + 2578054119424 \) T^20 + 656T^18 + 151232T^16 + 14951936T^14 + 657551360T^12 + 13711966208T^10 + 146470141952T^8 + 847422423040T^6 + 2660719656960T^4 + 4205779615744*T^2 + 2578054119424
$79$	\( (T^{10} - 12 T^{9} + \cdots - 70604800)^{2} \) (T^10 - 12T^9 - 408T^8 + 4592T^7 + 44432T^6 - 445088T^5 - 1259744T^4 + 12166784T^3 + 8676992T^2 - 63419392*T - 70604800)^2
$83$	\( T^{20} + \cdots + 48\!\cdots\!44 \) T^20 + 1412T^18 + 867200T^16 + 304335936T^14 + 67400525056T^12 + 9812862902272T^10 + 948001115324416T^8 + 59807993094733824T^6 + 2351361359124365312T^4 + 51933250393902940160*T^2 + 489255871611607711744
$89$	\( T^{20} + \cdots + 31\!\cdots\!56 \) T^20 + 1170T^18 + 559937T^16 + 142395068T^14 + 21037573776T^12 + 1871380905472T^10 + 101639085991936T^8 + 3367328390307840T^6 + 66093036796968960T^4 + 703601258187718656*T^2 + 3121764519883309056
$97$	\( T^{20} + \cdots + 43\!\cdots\!76 \) T^20 + 712T^18 + 199120T^16 + 28768512T^14 + 2390082560T^12 + 119212367872T^10 + 3579152695296T^8 + 62533801607168T^6 + 589913473941504T^4 + 2664487047921664*T^2 + 4372624599678976
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\(n\)	\(343\)	\(2377\)	\(2927\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)