LMFDB - Newform orbit 1710.2.n.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1710,2,Mod(647,1710)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1710, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("1710.647");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1710.n (of order $4$ , degree $2$ , 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:	$13.6544187456$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 4 x^{19} - 10 x^{18} + 56 x^{17} + 50 x^{16} - 336 x^{15} + 672 x^{14} - 776 x^{13} + 626 x^{12} + \cdots + 32$ x^20 - 4x^19 - 10x^18 + 56x^17 + 50x^16 - 336x^15 + 672x^14 - 776x^13 + 626x^12 - 748x^11 + 514x^10 + 1340x^9 - 648x^8 - 2216x^7 + 3348x^6 - 2684x^5 + 1761x^4 - 728x^3 + 408x^2 - 96*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{10}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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.

$f(q)$	$=$	$q - \beta_{7} q^{2} - \beta_{10} q^{4} + \beta_{14} q^{5} + \beta_{17} q^{7} - \beta_{5} q^{8} + \beta_{16} q^{10} + (\beta_{12} - \beta_{11} + \cdots - \beta_{5}) q^{11} + (\beta_{10} + \beta_{8} - 1) q^{13}+ \cdots + ( - \beta_{19} + \beta_{14} + \cdots - 2 \beta_{2}) q^{98}+O(q^{100})$ q - b7 * q^2 - b10 * q^4 + b14 * q^5 + b17 * q^7 - b5 * q^8 + b16 * q^10 + (b12 - b11 + b9 - b7 - b5) * q^11 + (b10 + b8 - 1) * q^13 + b6 * q^14 - q^16 + (b14 - b11 + 2b7) q^17 + b10 * q^19 + b12 * q^20 + (-b15 - b10 - b3 - b1 - 1) * q^22 + (b19 - b14 - b11 - b6) * q^23 + (-b17 - b16 - b15 - b10 + b8) * q^25 + (-b9 + b7 + b5) * q^26 + b18 * q^28 + (-b14 - b13 - 2b12 - 2b11 - 2b7 + 2b5 - b2) * q^29 + (-b16 - b15 - b8 + b4 + b3 + b1 + 2) * q^31 + b7 * q^32 + (b16 - b15 + 2b10) q^34 + (b13 - 3b11 + b9 - 3b7 + b5 - 2b2) q^35 + (-b10 - b3 - 1) * q^37 + b5 * q^38 - b1 * q^40 + (-b19 + b14 - b13 - b12 + b11 - 2b9 - b7 - b5) q^41 + (-b16 - 2b15 - b10 - b8 - b4 + 2b1 + 1) * q^43 + (-b14 - b13 + b7 - b5 + b2) * q^44 + (-b18 - b17 - b16 - b15) * q^46 + (b13 + b12 - b9 + 8b7 + b2) q^47 + (-b18 + b17 - b16 + b15 + 7b10 + 2b8 - b4 + 2b3 + b1) q^49 + (-b13 - b12 - b9 - b6 - b5) * q^50 + (b10 + b3 + 1) * q^52 + (-3b14 - b13 + b12 - 3b11 + b9 + 2b5 + b2) q^53 + (-b18 - b17 + b16 + b15 - 4b10 - b8) q^55 + b19 * q^56 + (-b16 - 2b15 - 2b10 - b8 - b4 + 2b1 + 2) q^58 + (b12 + b11 + b7 + b6 - b5 + b2) * q^59 + (b18 + b17 + 3b16 + 3b15 - b4 - b1 - 2) * q^61 + (b14 - b13 - b12 - b11 + b9 - 2b7 - b2) q^62 + b10 * q^64 + (-b19 - b14 - b13 - b12 - b11 - b9 - b7 - b5 + b2) * q^65 + (-b13 + b12 + 2b5) q^68 + (-3b15 - 3b10 - 2b8 + b4 - b3 + 1) q^70 + (-2b19 + 2b12 - 2b11 - 2b7 - 2b5) q^71 + (2b18 + 2b16 + 2b15 - b10 - 2b8 + 2b4 - 2b1 + 1) * q^73 + (b7 - b5 + b2) * q^74 + q^76 + (b19 + 2b14 - 4b13 - 4b12 - 2b11 + b6) * q^77 + (2b16 - 2b15 - 3b8 - 3b3) * q^79 - b14 * q^80 + (b17 + b16 + b15 - b10 - b4 + 2b3 + b1 - 1) q^82 + (b19 - 2b14 - b13 + b12 - 2b11 + 2b9 - b6 - 2b5 + 2b2) q^83 + (-b17 - 3b16 - b15 - 6b10 + b8) * q^85 + (2b14 - 2b13 - b12 + b11 + b9 - b7 - b5) * q^86 + (-b16 + b10 + b8 - b4 - 1) * q^88 + (-b14 - b13 - 3b12 - 3b11 - b7 - b6 + b5 + 2b2) q^89 + (-b18 - b17 + b16 + b15 - b8 - b4 + b3 - b1 + 2) * q^91 + (-b19 - b13 - b12 - b6) * q^92 + (8b10 + b8 + b4 + b3 - b1) q^94 - b12 * q^95 + (-2b17 - b16 - 2b15 + b4 + b3 - 2b1) q^97 + (-b19 + b14 + b13 - b12 + b11 - 2b9 + b6 + 7b5 - 2b2) q^98
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 4 q^{10} - 12 q^{13} - 20 q^{16} - 16 q^{22} + 16 q^{31} - 12 q^{37} - 8 q^{46} + 12 q^{52} + 20 q^{58} - 16 q^{61} + 20 q^{73} + 20 q^{76} - 28 q^{82} - 8 q^{85} - 16 q^{88} + 32 q^{91} - 20 q^{97}+O(q^{100})$ 20 * q + 4 * q^10 - 12 * q^13 - 20 * q^16 - 16 * q^22 + 16 * q^31 - 12 * q^37 - 8 * q^46 + 12 * q^52 + 20 * q^58 - 16 * q^61 + 20 * q^73 + 20 * q^76 - 28 * q^82 - 8 * q^85 - 16 * q^88 + 32 * q^91 - 20 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} - 4 x^{19} - 10 x^{18} + 56 x^{17} + 50 x^{16} - 336 x^{15} + 672 x^{14} - 776 x^{13} + 626 x^{12} + \cdots + 32$ x^20 - 4*x^19 - 10*x^18 + 56*x^17 + 50*x^16 - 336*x^15 + 672*x^14 - 776*x^13 + 626*x^12 - 748*x^11 + 514*x^10 + 1340*x^9 - 648*x^8 - 2216*x^7 + 3348*x^6 - 2684*x^5 + 1761*x^4 - 728*x^3 + 408*x^2 - 96*x + 32 :

$\beta_{1}$	$=$	$( 72\!\cdots\!24 \nu^{19} + \cdots - 12\!\cdots\!06 ) / 27\!\cdots\!18$ (72457492633883920232309142352324v^19 - 1131814671577947269293754429396609v^18 + 2429000990175842151535236670212456v^17 + 13474015064891651140599614656705574v^16 - 41816240657060140443345965796947048v^15 - 80305574887278206324797525238670552v^14 + 327253667248014951190915769913269500v^13 - 536342364840570679979797536406096584v^12 + 516596701662388156583356731224555116v^11 - 362722418322941195260717582377715296v^10 + 493171526397907299062508327492642732v^9 - 157048702414771725814952709996107116v^8 - 1345662996408246640015723639964676136v^7 + 92947353117994614368894432002700480v^6 + 2478516477772987131222880368172050680v^5 - 2438497602296799034853106361174673622v^4 + 1342784331905519484786420616942664720v^3 - 819277042824411810443074061797315225v^2 + 163288755642445759897543613824784252*v - 123711349576345973206336873321967106) / 27737445045072193221814949390731618
$\beta_{2}$	$=$	$( 17\!\cdots\!69 \nu^{19} + \cdots - 23\!\cdots\!40 ) / 55\!\cdots\!36$ (171748347985777524381796606710569v^19 - 65605975789733787962814078574390v^18 - 3862240212038111914265116478029226v^17 + 1798010245427062630140150336764908v^16 + 40618588946399611431349810651488782v^15 - 3879965883508449907812263263700360v^14 - 86260727937723465950970368713111080v^13 + 139641009382846059183874589811054304v^12 - 85649163042651305109535177205154714v^11 - 55811977299073071271791254486978832v^10 - 165203195735102898973448126691681238v^9 + 320640056814953993293342015515636200v^8 + 957969023577541605911005676282027724v^7 - 276243401455094378728668600795489932v^6 - 1459505985891408759334770036938701044v^5 + 508916612127364215996144447691579260v^4 + 449422777070133950855760110467788997v^3 + 174477686617134013726228160633534318v^2 - 5536243191678422874201489467239000*v - 23210863153210119826792446911731840) / 55474890090144386443629898781463236
$\beta_{3}$	$=$	$( - 15\!\cdots\!44 \nu^{19} + \cdots - 11\!\cdots\!40 ) / 27\!\cdots\!18$ (-150110213610710052025999689146744v^19 + 807014360887961564413347146917455v^18 + 1019281103231285810663868646828720v^17 - 11778878370030898315787042263044598v^16 + 216512672630399942063156240154368v^15 + 79733191033887073884860257231193456v^14 - 147558038850684446972874174107889076v^13 + 137359853562172067889671116165628492v^12 - 57563387107276889910704784940256028v^11 + 57354923307200218839421818152658118v^10 - 118323433647881522373568066848074880v^9 - 273594628405299122916914883045482050v^8 + 483927576365404014931699424667222444v^7 + 746061407822155663687147618153884226v^6 - 1099491143154375962880321059164524384v^5 + 85831018271186531238736539417253176v^4 + 107009436871462449994681081518898304v^3 + 100738285046315557876971413613830067v^2 + 44826451601683856068709991059597996*v - 11093429850263343878415521735051240) / 27737445045072193221814949390731618
$\beta_{4}$	$=$	$( 19\!\cdots\!80 \nu^{19} + \cdots - 14\!\cdots\!86 ) / 27\!\cdots\!18$ (197802295312886259066098497704580v^19 - 1604882779798492498973253476011963v^18 + 1120583909961695756954175242119852v^17 + 19986368135105591212664589541487980v^16 - 34608583108516549152246192258101888v^15 - 117768031993291566733421568639753540v^14 + 405767428668804947811467591921157712v^13 - 634690301091987084413849969489902390v^12 + 606295460705111644911159207598661500v^11 - 472199043770281413412203707079239512v^10 + 557984931802137973924492761608971616v^9 - 8386503494045907323903843033950966v^8 - 1366281360726345661379013360682933816v^7 - 109950056773121845472799781793015500v^6 + 2806556434701042876430990362521131300v^5 - 2846937197786100483513526571037646224v^4 + 1612092365446877365654274384666875744v^3 - 945249195282527632161227208442503159v^2 + 275099474810232675277925094008781124*v - 141623317405427906329192619651646586) / 27737445045072193221814949390731618
$\beta_{5}$	$=$	$( 29\!\cdots\!52 \nu^{19} + \cdots - 22\!\cdots\!30 ) / 27\!\cdots\!18$ (294959931887167913939990807767852v^19 - 1184499377938513312148535948048036v^18 - 2979261838661350892628589173962025v^17 + 16756288842584584139656144873369328v^16 + 14975876469166329077346151583066808v^15 - 102036548073916061558351210269857812v^14 + 197305132888246952860730155394614182v^13 - 215824504277172394475730151208391592v^12 + 155975152794863502437962317020261384v^11 - 186806284359242832548435110404285876v^10 + 128471238162158908645981599181321216v^9 + 423815816642987845051614046495900184v^8 - 216319484408338939120339612962924608v^7 - 718702192277079375884786129791200828v^6 + 1029305762975328277987264926502791266v^5 - 702934152685622611606218207307403736v^4 + 365356921828015439455795782966431088v^3 - 64895479410913185320481702148362240v^2 + 65654133781701367906371954014631095*v - 22865845254507924516488547156255330) / 27737445045072193221814949390731618
$\beta_{6}$	$=$	$( - 65\!\cdots\!87 \nu^{19} + \cdots - 26\!\cdots\!04 ) / 55\!\cdots\!36$ (-650781232929161600654553245763787v^19 + 1156034855113177863017494235940330v^18 + 11385200286758140874797977276786446v^17 - 18398792629385681580640913719982564v^16 - 103989106103630144016733660966826626v^15 + 95222971010417649753300537899077840v^14 - 2601371524979460898376376854997664v^13 - 154006376378212167509616663110773856v^12 + 137428487871316485846836156084097006v^11 + 175823028189934658836384837031039984v^10 + 362686319625345477050789233376079826v^9 - 1115963544108301800723199617773422984v^8 - 1826755337447871929005541169518509332v^7 + 1007126048015168115293494319200491852v^6 + 1606963165889363029420789304504532180v^5 - 756698342795103774187920261368612212v^4 - 301475302909284712287142226472457399v^3 - 296043074932904392189971375100770234v^2 + 34969502262106265244958976059550920*v - 261542611753322558399072763374552704) / 55474890090144386443629898781463236
$\beta_{7}$	$=$	$( 14\!\cdots\!11 \nu^{19} + \cdots + 58\!\cdots\!44 ) / 11\!\cdots\!72$ (1459336207727512200150968729390711v^19 - 5557146150577058063897013058941698v^18 - 15642256225663168521739695756069602v^17 + 78838768613801816645225675664677700v^16 + 87229139008851095430026372154444838v^15 - 474773178758606695674286891674975500v^14 + 900383626567401927957666495055975992v^13 - 952062530415141860131599317297767944v^12 + 682695545490908272986474289057392758v^11 - 869932152965971972890376125045643272v^10 + 491626098087921599290789839364169334v^9 + 2107895747918301639386496664769477192v^8 - 643526377895698024080253823114854504v^7 - 3265698835515318071402040997985282488v^6 + 4520670791843298844846023636179336164v^5 - 3101079408134802617548302669293826732v^4 + 1573059710271377110297233648792825095v^3 - 334842193966706656379022525592229222v^2 + 279429727651015312148498382550749332*v + 58637206181740924723472281374161144) / 110949780180288772887259797562926472
$\beta_{8}$	$=$	$( - 37\!\cdots\!76 \nu^{19} + \cdots + 78\!\cdots\!96 ) / 27\!\cdots\!18$ (-372774897585154434836748407000376v^19 + 1153563384542439068432642350636071v^18 + 5089016463412321875669397408710872v^17 - 17368628238799133637563813127902338v^16 - 38283018784746326161123260467767976v^15 + 106891941447808771737223285281192980v^14 - 127254950481597555347356187802089972v^13 + 72405433482912551544608958223652440v^12 - 21825677032679670025741921812134092v^11 + 146075693708159727476534928976048134v^10 - 5439978195965488291491600638419648v^9 - 642846544755178572897419685721590382v^8 - 288914261909860719675965529277746716v^7 + 1093876320663038851867954425858132110v^6 - 222474946212682027919149946932369560v^5 - 164533632115963687187305006005592152v^4 - 253232644869342801825954058427790016v^3 + 5604125878144221600114140822585267v^2 + 42417620582894765370858098522874284*v + 7858232311678793060444470136752896) / 27737445045072193221814949390731618
$\beta_{9}$	$=$	$( - 81\!\cdots\!29 \nu^{19} + \cdots - 44\!\cdots\!84 ) / 55\!\cdots\!36$ (-814638404523488813207698506395129v^19 + 3058684969353963253720306837828822v^18 + 9481714359277340015504373624156226v^17 - 45351105025199605911884570237530916v^16 - 58932692834629388769074059533285750v^15 + 289429977055492087951768456570580928v^14 - 429829878487307032112471929891276096v^13 + 340519435260871780555232847081888936v^12 - 141491083530392914071044409313342598v^11 + 333036642526714947691167647213042960v^10 - 202326194816977123218907847232112130v^9 - 1416393331096604482670392656391664928v^8 + 296494599280016735048699529489109340v^7 + 2825608693549595459590170624599223684v^6 - 1990519732341096994367754395809595836v^5 + 55553542123318833474125347541226660v^4 - 243406711791657141140654509351790149v^3 + 192667351804777093805353553644071570v^2 + 116455390410336329535221302019323824*v - 441921396309713551039618889243984) / 55474890090144386443629898781463236
$\beta_{10}$	$=$	$( - 22\!\cdots\!16 \nu^{19} + \cdots + 33\!\cdots\!04 ) / 12\!\cdots\!36$ (-2281117129166267659316v^19 + 8888682866514118302081v^18 + 23931099814635490097632v^17 - 126118037099035899679778v^16 - 128838713612465455060384v^15 + 764654675981355690017634v^14 - 1447706349878076279720552v^13 + 1553699390955339799449568v^12 - 1105256308250397968052432v^11 + 1389120874679654629430038v^10 - 855865919912609727324628v^9 - 3314140701130007225460990v^8 + 1254902467081578214594604v^7 + 5416939767225036335791752v^6 - 7312057529387927415633680v^5 + 5131452678311051999959636v^4 - 2558159736876131214020192v^3 + 555869402145844274135661v^2 - 457281223288902782372396*v + 33470096506401333103604) / 129160249464651768357236
$\beta_{11}$	$=$	$( - 39\!\cdots\!85 \nu^{19} + \cdots - 26\!\cdots\!92 ) / 11\!\cdots\!72$ (-3975723947085516111119353016282385v^19 + 12234867880937588162098054042650630v^18 + 54269546602623106596702140656845062v^17 - 184696283191742314649323700037925980v^16 - 405141825716257255426272608203718186v^15 + 1137125413865316887895281324394881700v^14 - 1410768534189972190506451041046934000v^13 + 706422091126278137091687666772808728v^12 + 23205013063564704716051316140513102v^11 + 1210597650592063703060268617867359240v^10 + 174585096710155390457817139372693942v^9 - 6799340171754054076644944917914409792v^8 - 2790416472384091334869399364342343896v^7 + 11269009517966550855904498193947656288v^6 - 4130243211209349192844542725008633108v^5 - 1716101716971013416907332815507858132v^4 + 479231216951563829904209927646298791v^3 - 1063687083522993714576106777026951270v^2 - 232611353123671194183888743183898724*v - 268913828201123519443349839286921592) / 110949780180288772887259797562926472
$\beta_{12}$	$=$	$( 21\!\cdots\!23 \nu^{19} + \cdots - 28\!\cdots\!60 ) / 55\!\cdots\!36$ (2141104324356571727200922703117523v^19 - 9590786738512666717535352440708458v^18 - 18067835373444396138223666123831584v^17 + 133031842702455798994099775675225220v^16 + 58051752368991629801272654939329342v^15 - 811878827058430478209679094400165096v^14 + 1733510763253301670927222903661332064v^13 - 2099800306433296836528986155395237048v^12 + 1700896667697200567896734418938253962v^11 - 1800799628319048644010831488070697700v^10 + 1560929822411281872176185933401006454v^9 + 2801580134224371917162470594826204300v^8 - 3032862993195777531025512976290595088v^7 - 5222220147595341830623523359861094224v^6 + 9656784997005206240526389315685003804v^5 - 7207709686126283450216677903453286912v^4 + 4580575613984929812274376215690834687v^3 - 2188340910001194281169890253116926646v^2 + 807195063144771618652005169251431942*v - 283153079281704177662775218639651460) / 55474890090144386443629898781463236
$\beta_{13}$	$=$	$( - 21\!\cdots\!69 \nu^{19} + \cdots + 25\!\cdots\!96 ) / 55\!\cdots\!36$ (-2189252000890924411794496718940669v^19 + 9574894591300935765491638424506222v^18 + 19289190859567693748356755002326956v^17 - 133357945685271670751061316811872864v^16 - 70920268149052420693568904701380578v^15 + 813612452512170486708440828991051472v^14 - 1704607995724049657904860813877133192v^13 + 2019274296210963945215025710199161892v^12 - 1607855501256525675748965895504380070v^11 + 1747766422797006282817972622164886540v^10 - 1469410209509524452148310176206359122v^9 - 2907121015276466685346849052549644272v^8 + 2763152130890269409791361829398983456v^7 + 5348731803509881320664257410474517608v^6 - 9461530838556790600132198244411640028v^5 + 6771391773783303399094347649340533548v^4 - 4170172708149356051758224966020084961v^3 + 2033617976473719739322103636648601450v^2 - 871455654286427875829465004284200222*v + 256273030989263208837575901926510196) / 55474890090144386443629898781463236
$\beta_{14}$	$=$	$( 44\!\cdots\!85 \nu^{19} + \cdots + 12\!\cdots\!60 ) / 11\!\cdots\!72$ (4439825954682792215345928701878385v^19 - 14364440472137143106645498999380086v^18 - 57594953668254244502893109262258550v^17 + 212568734471061541980356489134449268v^16 + 410532368231810480794557311480222506v^15 - 1294238773008809411985326837402272548v^14 + 1829338802811291430922280054034179328v^13 - 1329171952663972648335672828247343696v^12 + 606179980646199385224582699804451794v^11 - 1833609487004187768619602524764293800v^10 + 313224588220230031966331509788449626v^9 + 7192505379073324303511335829528182296v^8 + 2187375149390773020666108483248186808v^7 - 11746359869394295878341869432362338048v^6 + 6152309474917860728946719668337043844v^5 - 1150196810788490433674572260046082548v^4 + 1796683725237428141789857990503617433v^3 + 122282326264425299854339826008979366v^2 + 723456695448100417479419227035553908*v + 121756214321680800983289207179633560) / 110949780180288772887259797562926472
$\beta_{15}$	$=$	$( 30\!\cdots\!00 \nu^{19} + \cdots - 90\!\cdots\!64 ) / 55\!\cdots\!36$ (3012399520356226133000489324284500v^19 - 11378492053635809815327333167810431v^18 - 33355685771833401252379040942719440v^17 + 163681817565598869816782811223527662v^16 + 195553173247069828687137086873582024v^15 - 1004017160334425878140974578379033502v^14 + 1744739334982839498126281244519805128v^13 - 1731687675633449707534120617175173808v^12 + 1159299097454014229952600890266941456v^11 - 1714173540523890491117558567431192018v^10 + 1017337540715408869776729198738175852v^9 + 4572443376521715321924325731325001318v^8 - 1054267910168286687528140430434240116v^7 - 8003789036696756369850470513923249996v^6 + 8243911944103212161268881878894378952v^5 - 4365527365279632039539790008340811080v^4 + 2810141246385444467246587125902985088v^3 - 1108681337564700814851345553946782143v^2 + 705255258370790393066953303533664676*v - 90849534405896255781341269324581764) / 55474890090144386443629898781463236
$\beta_{16}$	$=$	$( - 32\!\cdots\!00 \nu^{19} + \cdots + 16\!\cdots\!80 ) / 55\!\cdots\!36$ (-3244450524154864185113777167082500v^19 + 12443278349235587287601055646175159v^18 + 35018389304648970205474525245426184v^17 - 177618043205258483482299205771789306v^16 - 198248444504846441371279438511834184v^15 + 1082573839906172140185997334882728926v^14 - 1954024469293499118334195751013427792v^13 + 2043062606402296963156113197912441292v^12 - 1473991594308896274922917898239423904v^11 + 2025679458729952523897225520879659298v^10 - 1261242383180601580988803523318747636v^9 - 4769025980181350435357521187131887570v^8 + 1355788571664945844629785870981318660v^7 + 8242464212410628881069156133130590876v^6 - 9254945075957467929319970350558584320v^5 + 5798676629159383964830742546117781420v^4 - 3948098717479940453093621084977943200v^3 + 1579383716193985022212229029455768095v^2 - 839728149352716231827458747896565796*v + 164428341345617615011371585378225780) / 55474890090144386443629898781463236
$\beta_{17}$	$=$	$( - 19\!\cdots\!96 \nu^{19} + \cdots + 12\!\cdots\!12 ) / 27\!\cdots\!18$ (-1977057182897076177127299067496296v^19 + 8017562012075905566861351388896617v^18 + 19884265120083987988959677795499788v^17 - 113944388218019311830132844430550826v^16 - 98581671882768991041534644677174032v^15 + 700231274138543366829143919329302412v^14 - 1333286478821114573363259771312151104v^13 + 1421643280583585499961969920902796112v^12 - 978183069674395659872717753953379436v^11 + 1192874281133482492565518815854161306v^10 - 852308058858855118335976058034654860v^9 - 2909410033879316536636750314823724702v^8 + 1626858205064140808843220020149697044v^7 + 5129419155159288356894296550032750970v^6 - 7169853307136981245476498236590830628v^5 + 4294773953740990285139802782745417740v^4 - 2030305991674237402489408968877763904v^3 + 661213156236860175556169783207777169v^2 - 338109859476849901123030675780860060*v + 123826058477644642842095134985707912) / 27737445045072193221814949390731618
$\beta_{18}$	$=$	$( 21\!\cdots\!80 \nu^{19} + \cdots + 66\!\cdots\!68 ) / 27\!\cdots\!18$ (2195692115636330266982140092718280v^19 - 8131212944373579891185094702043977v^18 - 24704670384693271499559462095476492v^17 + 116797842545897309911106655050440054v^16 + 148369054569774512811631767421649320v^15 - 710390113854158299104098410243439584v^14 + 1239543972864908252335915617687295592v^13 - 1237618185437313171403331533325873128v^12 + 833318712976540286119441671762419676v^11 - 1214712007514733126029857248493754426v^10 + 626270522555292977723555107332911156v^9 + 3338570685361833111911419535208367818v^8 - 513508047596620909447898843881483372v^7 - 5512948429047714274913457789255398246v^6 + 5693271733492812074151651032845058580v^5 - 3514113296031395347499394002233659008v^4 + 2148944910465606477956433936917452448v^3 - 387829816313888797269562365645013233v^2 + 299089025768558590272007128072766492*v + 66914811089156630529760813674362368) / 27737445045072193221814949390731618
$\beta_{19}$	$=$	$( - 57\!\cdots\!19 \nu^{19} + \cdots + 71\!\cdots\!96 ) / 55\!\cdots\!36$ (-5747925141321489570790981240651819v^19 + 22251287068065603546467877694559338v^18 + 61288785189850722872561599426969454v^17 - 317710542107981465735445491081299164v^16 - 338166446448455842188258840457451786v^15 + 1940663853245722105510123017528674776v^14 - 3559846235693991809172991066959285600v^13 + 3678630349880420359573528015624459824v^12 - 2527703961936698540479911646813131538v^11 + 3326453078179494101524583605828472352v^10 - 2057959560363489543740740789348162446v^9 - 8570649192342200161673276330338836000v^8 + 3018736976919800942926189807439856244v^7 + 14516044952175022611483264008673637332v^6 - 17890436773984930857967550199901382724v^5 + 10656678962260241747259507185145120068v^4 - 5820837817050175875727169644729271607v^3 + 1386335832709054737492249854091035526v^2 - 884832999330514305239413989822024304*v + 71095334748728964894555038075671696) / 55474890090144386443629898781463236

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

$\nu$	$=$	$( \beta_{16} + \beta_{15} + \beta_{14} + \beta_{11} ) / 2$ (b16 + b15 + b14 + b11) / 2
$\nu^{2}$	$=$	$( \beta_{19} - \beta_{17} + 4\beta_{10} - \beta_{9} - \beta_{6} + 8\beta_{5} + \beta_{3} - \beta_{2} + 4 ) / 2$ (b19 - b17 + 4b10 - b9 - b6 + 8b5 + b3 - b2 + 4) / 2
$\nu^{3}$	$=$	$( 2 \beta_{16} + 5 \beta_{15} + 10 \beta_{14} - 2 \beta_{13} - 5 \beta_{12} + 11 \beta_{11} + 3 \beta_{10} + \cdots + 1 ) / 2$ (2b16 + 5b15 + 10b14 - 2b13 - 5b12 + 11b11 + 3b10 - 4b9 + 2b8 + b7 + 3b5 - 10b4 + 4b3 - 2b2 + 11b1 + 1) / 2
$\nu^{4}$	$=$	$( 16 \beta_{19} + 12 \beta_{18} - 12 \beta_{17} + 4 \beta_{14} - 4 \beta_{13} - 4 \beta_{12} + \cdots + 6 \beta_1 ) / 2$ (16b19 + 12b18 - 12b17 + 4b14 - 4b13 - 4b12 + 4b11 + 84b10 - 20b9 + 15b8 + 56b7 + 56b5 - 6b4 + 15b3 + 6*b1) / 2
$\nu^{5}$	$=$	$( 7 \beta_{19} + 7 \beta_{18} - 3 \beta_{17} - 74 \beta_{16} - 34 \beta_{15} + 34 \beta_{14} - 127 \beta_{13} + \cdots - 32 ) / 2$ (7b19 + 7b18 - 3b17 - 74b16 - 34b15 + 34b14 - 127b13 - 143b12 + 74b11 + 80b10 - 70b9 + 70b8 + 80b7 + 3b6 + 32b5 - 143b4 + 30b3 + 30b2 + 127*b1 - 32) / 2
$\nu^{6}$	$=$	$( 155 \beta_{19} + 217 \beta_{18} - 119 \beta_{16} - 105 \beta_{15} - 10 \beta_{14} - 160 \beta_{13} + \cdots - 770 ) / 2$ (155b19 + 217b18 - 119b16 - 105b15 - 10b14 - 160b13 - 160b12 + 10b11 + 770b10 - 225b9 + 315b8 + 1100b7 + 155b6 - 119b4 + 225b2 + 105b1 - 770) / 2
$\nu^{7}$	$=$	$( 85 \beta_{19} + 204 \beta_{18} + 85 \beta_{17} - 1936 \beta_{16} - 1719 \beta_{15} - 1016 \beta_{14} + \cdots - 1615 ) / 2$ (85b19 + 204b18 + 85b17 - 1936b16 - 1719b15 - 1016b14 - 1936b13 - 1719b12 - 489b11 + 665b10 - 455b9 + 1092b8 + 1615b7 + 204b6 - 665b5 - 1016b4 - 455b3 + 1092b2 + 489*b1 - 1615) / 2
$\nu^{8}$	$=$	$( 2091 \beta_{18} + 2091 \beta_{17} - 3213 \beta_{16} - 3213 \beta_{15} - 2472 \beta_{14} - 2472 \beta_{13} + \cdots - 15198 ) / 2$ (2091b18 + 2091b17 - 3213b16 - 3213b15 - 2472b14 - 2472b13 - 2064b12 - 2064b11 + 3400b8 + 10728b7 + 2952b6 - 10728b5 - 289b4 - 3400b3 + 4800b2 - 289b1 - 15198) / 2
$\nu^{9}$	$=$	$( - 1751 \beta_{19} + 1751 \beta_{18} + 4223 \beta_{17} - 23928 \beta_{16} - 26880 \beta_{15} + \cdots - 29058 ) / 2$ (-1751b19 + 1751b18 + 4223b17 - 23928b16 - 26880b15 - 26880b14 - 14079b13 - 6945b12 - 23928b11 - 12024b10 + 6953b9 + 6953b8 + 12024b7 + 4223b6 - 29058b5 + 6945b4 - 16769b3 + 16769b2 - 14079*b1 - 29058) / 2
$\nu^{10}$	$=$	$( - 28971 \beta_{19} + 40959 \beta_{17} - 36490 \beta_{16} - 44936 \beta_{15} - 57594 \beta_{14} + \cdots - 152110 ) / 2$ (-28971b19 + 40959b17 - 36490b16 - 44936b15 - 57594b14 - 5974b13 + 5974b12 - 57594b11 - 152110b10 + 51533b9 + 28971b6 - 215180b5 + 36490b4 - 72857b3 + 51533b2 - 44936b1 - 152110) / 2
$\nu^{11}$	$=$	$( - 76720 \beta_{19} - 31784 \beta_{18} + 76720 \beta_{17} - 99769 \beta_{16} - 198670 \beta_{15} + \cdots - 203811 ) / 2$ (-76720b19 - 31784b18 + 76720b17 - 99769b16 - 198670b15 - 339831b14 + 99769b13 + 198670b12 - 380790b11 - 492129b10 + 256620b9 - 106314b8 - 203811b7 + 31784b6 - 492129b5 + 339831b4 - 256620b3 + 106314b2 - 380790*b1 - 203811) / 2
$\nu^{12}$	$=$	$( - 579460 \beta_{19} - 409761 \beta_{18} + 409761 \beta_{17} + 108504 \beta_{16} - 108504 \beta_{15} + \cdots - 973170 \beta_1 ) / 2$ (-579460b19 - 409761b18 + 409761b17 + 108504b16 - 108504b15 - 611380b14 + 611380b13 + 764820b12 - 764820b11 - 3099492b10 + 1106280b9 - 782298b8 - 2191560b7 - 2191560b5 + 973170b4 - 782298b3 - 973170*b1) / 2
$\nu^{13}$	$=$	$( - 1305657 \beta_{19} - 1305657 \beta_{18} + 540837 \beta_{17} + 2851151 \beta_{16} + 1452169 \beta_{15} + \cdots + 3331860 ) / 2$ (-1305657b19 - 1305657b18 + 540837b17 + 2851151b16 + 1452169b15 - 1452169b14 + 4904992b13 + 5484452b12 - 2851151b11 - 8044062b10 + 3921512b9 - 3921512b8 - 8044062b7 - 540837b6 - 3331860b5 + 5484452b4 - 1624392b3 - 1624392b2 - 4904992*b1 + 3331860) / 2
$\nu^{14}$	$=$	$( - 5894213 \beta_{19} - 8335603 \beta_{18} + 12537223 \beta_{16} + 9925909 \beta_{15} + 1846494 \beta_{14} + \cdots + 31996842 ) / 2$ (-5894213b19 - 8335603b18 + 12537223b16 + 9925909b15 + 1846494b14 + 15883972b13 + 15883972b12 - 1846494b11 - 31996842b10 + 11887629b9 - 16811499b8 - 45250764b7 - 5894213b6 + 12537223b4 - 11887629b2 - 9925909b1 + 31996842) / 2
$\nu^{15}$	$=$	$( - 8865233 \beta_{19} - 21402456 \beta_{18} - 8865233 \beta_{17} + 80056861 \beta_{16} + \cdots + 128586181 ) / 2$ (-8865233b19 - 21402456b18 - 8865233b17 + 80056861b16 + 71721258b15 + 41496131b14 + 80056861b13 + 71721258b12 + 21372102b11 - 53261867b10 + 24800581b9 - 59873592b8 - 128586181b7 - 21402456b6 + 53261867b5 + 41496131b4 + 24800581b3 - 59873592b2 - 21372102*b1 + 128586181) / 2
$\nu^{16}$	$=$	$( - 85951074 \beta_{18} - 85951074 \beta_{17} + 253671703 \beta_{16} + 253671703 \beta_{15} + \cdots + 667647854 ) / 2$ (-85951074b18 - 85951074b17 + 253671703b16 + 253671703b15 + 200775168b14 + 200775168b13 + 157970256b12 + 157970256b11 - 180768907b8 - 472097616b7 - 121552992b6 + 472097616b5 + 30267689b4 + 180768907b3 - 255645456b2 + 30267689b1 + 667647854) / 2
$\nu^{17}$	$=$	$( 141969696 \beta_{19} - 141969696 \beta_{18} - 342744864 \beta_{17} + 1059472869 \beta_{16} + \cdots + 2025632800 ) / 2$ (141969696b19 - 141969696b18 - 342744864b17 + 1059472869b16 + 1181025861b15 + 1181025861b14 + 610749642b13 + 317294502b12 + 1059472869b11 + 839043840b10 - 378421988b9 - 378421988b8 - 839043840b7 - 342744864b6 + 2025632800b5 - 317294502b4 + 913590692b3 - 913590692b2 + 610749642*b1 + 2025632800) / 2
$\nu^{18}$	$=$	$( 1266976935 \beta_{19} - 1791775503 \beta_{17} + 2481169810 \beta_{16} + 3166659538 \beta_{15} + \cdots + 7022575476 ) / 2$ (1266976935b19 - 1791775503b17 + 2481169810b16 + 3166659538b15 + 3993619460b14 + 484714560b13 - 484714560b12 + 3993619460b11 + 7022575476b10 - 2750249035b9 - 1266976935b6 + 9931424040b5 - 2481169810b4 + 3889438483b3 - 2750249035b2 + 3166659538b1 + 7022575476) / 2
$\nu^{19}$	$=$	$( 5405826548 \beta_{19} + 2239167010 \beta_{18} - 5405826548 \beta_{17} + 4742172932 \beta_{16} + \cdots + 13087434199 ) / 2$ (5405826548b19 + 2239167010b18 - 5405826548b17 + 4742172932b16 + 9067902305b15 + 15774350232b14 - 4742172932b13 - 9067902305b12 + 17566125735b11 + 31595865909b10 - 13934062216b9 + 5771678420b8 + 13087434199b7 - 2239167010b6 + 31595865909b5 - 15774350232b4 + 13934062216b3 - 5771678420b2 + 17566125735*b1 + 13087434199) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$ .

$n$	$191$	$1027$	$1351$
$\chi(n)$	$-1$	$\beta_{10}$	$1$

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}

647.1

0.232147	−	0.560453i
−0.185128	+	0.446939i
0.618697	−	1.49367i
0.153903	−	0.371555i
−0.526726	+	1.27163i
−3.06999	−	1.27163i
0.897013	+	0.371555i
3.60603	+	1.49367i
−1.07901	−	0.446939i
1.35305	+	0.560453i
0.232147	+	0.560453i
−0.185128	−	0.446939i
0.618697	+	1.49367i
0.153903	+	0.371555i
−0.526726	−	1.27163i
−3.06999	+	1.27163i
0.897013	−	0.371555i
3.60603	−	1.49367i
−1.07901	+	0.446939i
1.35305	−	0.560453i

−0.707107

0.707107i

−

1.00000i

−2.03893

0.918023i

−1.07805

−

1.07805i

0.707107

0.707107i

0.792601

−

2.09088i

647.2

−0.707107

0.707107i

−

1.00000i

−1.06972

1.96359i

3.58826

3.58826i

0.707107

0.707107i

−0.632067

−

2.14488i

647.3

−0.707107

0.707107i

−

1.00000i

−0.975056

−

2.01228i

−3.02518

−

3.02518i

0.707107

0.707107i

2.11236

0.733428i

647.4

−0.707107

0.707107i

−

1.00000i

1.16531

−

1.90842i

2.34049

2.34049i

0.707107

0.707107i

0.525458

2.17345i

647.5

−0.707107

0.707107i

−

1.00000i

2.21129

0.331972i

−1.82552

−

1.82552i

0.707107

0.707107i

−1.79836

1.32888i

647.6

0.707107

−

0.707107i

−

1.00000i

−2.21129

−

0.331972i

−1.82552

−

1.82552i

−0.707107

−

0.707107i

−1.79836

1.32888i

647.7

0.707107

−

0.707107i

−

1.00000i

−1.16531

1.90842i

2.34049

2.34049i

−0.707107

−

0.707107i

0.525458

2.17345i

647.8

0.707107

−

0.707107i

−

1.00000i

0.975056

2.01228i

−3.02518

−

3.02518i

−0.707107

−

0.707107i

2.11236

0.733428i

647.9

0.707107

−

0.707107i

−

1.00000i

1.06972

−

1.96359i

3.58826

3.58826i

−0.707107

−

0.707107i

−0.632067

−

2.14488i

647.10

0.707107

−

0.707107i

−

1.00000i

2.03893

−

0.918023i

−1.07805

−

1.07805i

−0.707107

−

0.707107i

0.792601

−

2.09088i

1673.1

−0.707107

−

0.707107i

1.00000i

−2.03893

−

0.918023i

−1.07805

1.07805i

0.707107

−

0.707107i

0.792601

2.09088i

1673.2

−0.707107

−

0.707107i

1.00000i

−1.06972

−

1.96359i

3.58826

−

3.58826i

0.707107

−

0.707107i

−0.632067

2.14488i

1673.3

−0.707107

−

0.707107i

1.00000i

−0.975056

2.01228i

−3.02518

3.02518i

0.707107

−

0.707107i

2.11236

−

0.733428i

1673.4

−0.707107

−

0.707107i

1.00000i

1.16531

1.90842i

2.34049

−

2.34049i

0.707107

−

0.707107i

0.525458

−

2.17345i

1673.5

−0.707107

−

0.707107i

1.00000i

2.21129

−

0.331972i

−1.82552

1.82552i

0.707107

−

0.707107i

−1.79836

−

1.32888i

1673.6

0.707107

0.707107i

1.00000i

−2.21129

0.331972i

−1.82552

1.82552i

−0.707107

0.707107i

−1.79836

−

1.32888i

1673.7

0.707107

0.707107i

1.00000i

−1.16531

−

1.90842i

2.34049

−

2.34049i

−0.707107

0.707107i

0.525458

−

2.17345i

1673.8

0.707107

0.707107i

1.00000i

0.975056

−

2.01228i

−3.02518

3.02518i

−0.707107

0.707107i

2.11236

−

0.733428i

1673.9

0.707107

0.707107i

1.00000i

1.06972

1.96359i

3.58826

−

3.58826i

−0.707107

0.707107i

−0.632067

2.14488i

1673.10

0.707107

0.707107i

1.00000i

2.03893

0.918023i

−1.07805

1.07805i

−0.707107

0.707107i

0.792601

2.09088i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1710.2.n.i	✓	20
3.b	odd	2	1	inner	1710.2.n.i	✓	20
5.c	odd	4	1	inner	1710.2.n.i	✓	20
15.e	even	4	1	inner	1710.2.n.i	✓	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1710.2.n.i	✓	20	1.a	even	1	1	trivial
1710.2.n.i	✓	20	3.b	odd	2	1	inner
1710.2.n.i	✓	20	5.c	odd	4	1	inner
1710.2.n.i	✓	20	15.e	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1710, [\chi])$ :

T_{7}^{10} + 32T_{7}^{7} + 584T_{7}^{6} + 624T_{7}^{5} + 512T_{7}^{4} + 5760T_{7}^{3} + 48400T_{7}^{2} + 88000T_{7} + 80000

T_{17}^{20} + 2288T_{17}^{16} + 1179008T_{17}^{12} + 165189632T_{17}^{8} + 6196826112T_{17}^{4} + 68719476736

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} + 1)^{5}$ (T^4 + 1)^5
$3$	$T^{20}$ T^20
$5$	$T^{20} + 13 T^{16} + \cdots + 9765625$ T^20 + 13T^16 - 224T^14 + 858T^12 - 1472T^10 + 21450T^8 - 140000T^6 + 203125*T^4 + 9765625
$7$	$(T^{10} + 32 T^{7} + \cdots + 80000)^{2}$ (T^10 + 32T^7 + 584T^6 + 624T^5 + 512T^4 + 5760T^3 + 48400T^2 + 88000*T + 80000)^2
$11$	$(T^{10} + 64 T^{8} + \cdots + 128)^{2}$ (T^10 + 64T^8 + 1208T^6 + 7968T^4 + 16784T^2 + 128)^2
$13$	$(T^{10} + 6 T^{9} + \cdots + 128)^{2}$ (T^10 + 6T^9 + 18T^8 + 24T^7 + 272T^6 + 1520T^5 + 4512T^4 + 6336T^3 + 4096T^2 - 1024*T + 128)^2
$17$	$T^{20} + \cdots + 68719476736$ T^20 + 2288T^16 + 1179008T^12 + 165189632T^8 + 6196826112T^4 + 68719476736
$19$	$(T^{2} + 1)^{10}$ (T^2 + 1)^10
$23$	$T^{20} + \cdots + 4294967296$ T^20 + 5072T^16 + 7726208T^12 + 3725701120T^8 + 40827162624T^4 + 4294967296
$29$	$(T^{10} - 218 T^{8} + \cdots - 21727232)^{2}$ (T^10 - 218T^8 + 15920T^6 - 452864T^4 + 5358592T^2 - 21727232)^2
$31$	$(T^{5} - 4 T^{4} - 56 T^{3} + \cdots - 64)^{4}$ (T^5 - 4T^4 - 56T^3 + 96T^2 + 592T - 64)^4
$37$	$(T^{10} + 6 T^{9} + \cdots + 128)^{2}$ (T^10 + 6T^9 + 18T^8 + 24T^7 + 272T^6 + 1520T^5 + 4512T^4 + 6336T^3 + 4096T^2 - 1024*T + 128)^2
$41$	$(T^{10} + 226 T^{8} + \cdots + 123008)^{2}$ (T^10 + 226T^8 + 14832T^6 + 337920T^4 + 1988224T^2 + 123008)^2
$43$	$(T^{10} + 248 T^{7} + \cdots + 76880000)^{2}$ (T^10 + 248T^7 + 5096T^6 + 13392T^5 + 30752T^4 + 580320T^3 + 8179600T^2 + 35464000*T + 76880000)^2
$47$	$T^{20} + \cdots + 10\!\cdots\!56$ T^20 + 41488T^16 + 424347776T^12 + 1123749838848T^8 + 464794501386240T^4 + 1052640129581056
$53$	$T^{20} + \cdots + 68719476736$ T^20 + 69696T^16 + 1209894400T^12 + 985648873472T^8 + 521427550208T^4 + 68719476736
$59$	$(T^{10} - 104 T^{8} + \cdots - 8192)^{2}$ (T^10 - 104T^8 + 3344T^6 - 34592T^4 + 33856T^2 - 8192)^2
$61$	$(T^{5} + 4 T^{4} + \cdots + 8896)^{4}$ (T^5 + 4T^4 - 168T^3 - 224T^2 + 3856T + 8896)^4
$67$	$T^{20}$ T^20
$71$	$(T^{10} + 512 T^{8} + \cdots + 22151168)^{2}$ (T^10 + 512T^8 + 81792T^6 + 4057088T^4 + 33329152T^2 + 22151168)^2
$73$	$(T^{10} - 10 T^{9} + \cdots + 4921113632)^{2}$ (T^10 - 10T^9 + 50T^8 - 160T^7 + 23784T^6 - 260432T^5 + 1427920T^4 - 623232T^3 + 79495056T^2 - 884538528*T + 4921113632)^2
$79$	$(T^{10} + 640 T^{8} + \cdots + 904806400)^{2}$ (T^10 + 640T^8 + 144992T^6 + 13618944T^4 + 453054720T^2 + 904806400)^2
$83$	$T^{20} + 47248 T^{16} + \cdots + 16777216$ T^20 + 47248T^16 + 715708032T^12 + 4178736967680T^8 + 8144623323385856T^4 + 16777216
$89$	$(T^{10} - 578 T^{8} + \cdots - 4739484800)^{2}$ (T^10 - 578T^8 + 117936T^6 - 10300992T^4 + 384759680T^2 - 4739484800)^2
$97$	$(T^{10} + 10 T^{9} + \cdots + 1073512448)^{2}$ (T^10 + 10T^9 + 50T^8 + 1008T^7 + 47552T^6 + 642432T^5 + 4554752T^4 + 11427840T^3 + 200704T^2 - 20758528*T + 1073512448)^2
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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 $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 647.10
Significant digits:
Format:

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	1710.2.n.i

Level	$1710$
Weight	$2$
Character orbit	1710.n
Analytic conductor	$13.654$
Analytic rank	$0$
Dimension	$20$
Inner twists	$4$