LMFDB - Newform orbit 616.2.r.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [616,2,Mod(113,616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(616, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 0, 8]))

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

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

chi := DirichletCharacter("616.113");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$616 = 2^{3} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	616.r (of order $5$ , degree $4$ , 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:	$4.91878476451$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$5$ over $\Q(\zeta_{5})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - 4 x^{19} + 21 x^{18} - 58 x^{17} + 225 x^{16} - 348 x^{15} + 1296 x^{14} - 755 x^{13} + \cdots + 10000$ x^20 - 4x^19 + 21x^18 - 58x^17 + 225x^16 - 348x^15 + 1296x^14 - 755x^13 + 6557x^12 - 5753x^11 + 29289x^10 - 14174x^9 + 85487x^8 + 13430x^7 + 191109x^6 + 166407x^5 + 212971x^4 - 179670x^3 + 72900x^2 - 3000*x + 10000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$ -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.

$f(q)$	$=$	$q - \beta_{3} q^{3} + ( - \beta_{11} - \beta_{6} - \beta_{4} + \cdots - 1) q^{5} - \beta_{11} q^{7} + (\beta_{19} - \beta_{18} + \cdots - \beta_1) q^{9} + ( - \beta_{17} - \beta_{15} + \beta_{6}) q^{11}+ \cdots + (2 \beta_{19} - 2 \beta_{18} + \cdots - 7) q^{99}+O(q^{100})$ q - b3 * q^3 + (-b11 - b6 - b4 - b2 - 1) * q^5 - b11 * q^7 + (b19 - b18 - b16 - b12 - b7 + b4 + b3 - b1) * q^9 + (-b17 - b15 + b6) * q^11 + (b19 - b18 + b15 + b13 + b12 - b11 - 2b6 - 2b4 - 2b2 - b1 - 1) q^13 + (b19 - b18 + b17 + b16 + b13 - b12 - 2b11 + 2b10 + b9 - b8 - b6 - 2b4 - b2 - 1) q^15 + (b19 - b16 + b15 + b11 - b10 - b7 + b6 + b4 + 1) * q^17 + (-b18 - b17 - b15 - 2b13 - b12 - b8 + b6 + 2b4 + b2 + 2) * q^19 + (b7 + b5 - b3 + b1) * q^21 + (b19 + b17 + b16 + b15 - b14 + b13 - 2b11 + b10 - b9 - 2b6 - b4 - 2b2 - 2) q^23 + (b18 + b15 + b13 + 2b12 + b8 - b7 - b5 - b2) q^25 + (-b19 - 2b17 + b16 - 2b15 - 2b14 + 3b11 - b10 - b9 + b8 + 2b7 + 4b6 + b5 + 3b4 + 2b2 + b1 + 4) * q^27 + (b19 - b17 - b16 + b13 + b12 - b11 - b9 + b8 - 2b6 - b4 + b3 - b2 - b1 - 1) q^29 + (b16 - b14 - b11 - b10 + b9 - b4 + b1 - 1) * q^31 + (-b19 + 2b18 + b17 + b16 - b14 - b11 + b9 + b6 - b5 + 2b1 - 1) * q^33 + (b9 - b4) * q^35 + (b19 + b13 + b12 + b11 + b10 - b9 + b8 - 3b6 - b5 - 2b4 + b3 - b2 - b1 - 1) * q^37 + (-b19 - b17 + 2b16 - 3b15 - b14 + 4b11 + b10 + b9 - b8 + b7 + 4b6 + b5 + 4b4 + 2b2 + b1 + 4) * q^39 + (b18 + b17 + b15 - b14 + 2b13 - b12 - 2b6 - 2b4 + b3 - 2) q^41 + (b19 - b18 + b17 + b15 + b12 - b11 + b10 + b8 - 2b6 - b5 - b4 + b3 - b2 - 3) q^43 + (-b19 + b16 - b15 - b14 - b13 - 3b12 - 3b11 + b9 - 3b8 + 2b7 + 2b5 + b4 - 2b3 + 2b2 + 2b1) * q^45 + (b12 - 2b6 - 2b3) * q^47 + (-b11 - b6 - b4 - 1) * q^49 + (b19 - b18 + b13 + 2b12 - 2b11 + 2b10 - 2b9 + b8 - 2b6 - 2b5 + 3b3 - b2 - 3b1 - 1) * q^51 + (-b19 + b18 - b16 - 2b7 + b6 + 5b4 + 2b3 + b2 - b1) q^53 + (-b18 - b17 - b14 + b13 - 2b10 + b8 - b7 - 2b5 - 3b4 + b3 - b1 - 1) q^55 + (-b19 + b18 - b15 - b14 - b13 + b12 - b10 - b9 + 2b7 + 2b4 - 2b3) q^57 + (-b19 - b18 - b17 - b16 - b13 - b12 - b11 - 2b10 + b9 - b6 - 2b4 - b3 + b2 + b1 + 1) * q^59 + (b19 - b16 - b15 + 5b11 - b10 + 5b6 + b5 + 5b4 + 2b2 + b1 + 5) * q^61 + (b17 - b13 - b12 - b8 - b7 - b5 + b4 + b2 + 1) * q^63 + (b19 + b18 + 2b17 - b16 + b15 + b14 + 2b13 + 2b12 + 2b11 + 2b10 - 2b9 + 2b8 + b7 - b4 - 3b2 + b1 - 3) * q^65 + (-b19 - b18 - 2b17 - b15 - 2b13 - b11 - 2b10 - 2b9 + b7 - b6 + 2b5 + b4 - 2b3 - b2 + b1 - 1) * q^67 + (-2b18 - b17 - 2b15 - 4b12 - 3b8 + b7 + 2b6 + b5 + 3b4 - b3 + 3b2 + 3) q^69 + (b19 - 2b16 + b15 - 2b10 - 2b9 + 2b8 + 2b6 + 2b5 + 2b1 + 2) q^71 + (-2b18 - b17 - b16 - b12 - 3b11 - b10 + b9 + b6 - b5 + b3 - b1) * q^73 + (b19 - b18 + b15 + 3b14 + b13 + 2b12 - 7b11 + 3b10 + b9 - b7 - 3b6 - 5b4 + b3 - 3b2 - b1 - 7) q^75 + (b19 - b18 - b16 + b14 - b11 - b6 - b4 - b2 - 2) * q^77 + (2b19 - 2b18 - b16 - b15 + 2b14 - b13 + 2b11 + 2b10 - 2b7 - 2b6 - 2b4 + 2b3 - 2b2 - 4b1 + 2) q^79 + (-4b19 + 3b18 - 4b13 - b12 + 4b11 - 3b10 + b9 + b8 + 4b6 - 3b5 + 3b4 - 3b3 + 4b2 + 3b1 + 4) q^81 + (-b19 - b16 + 3b15 + 5b11 - b10 - 3b7 + 5b6 - b5 + 5b4 - b1 + 5) q^83 + (b17 - 2b12 + 3b7 - 5b6 + 3b5 - 4b4 - b3 - 4) q^85 + (-3b19 + 3b18 - 2b17 + 2b16 - 3b15 - 2b14 + 3b12 + 7b11 - 2b10 - 2b9 + 3b8 + 3b7 + 4b6 + b5 + 3b4 - b3 + b2 + 3b1 + 2) q^87 + (b19 + b18 + b17 - 2b16 + b15 + 2b14 + 2b13 + b12 - b11 + b10 + b9 + b8 - b7 - 2b6 + 2b5 - b4 - 2b3 - b1 - 3) * q^89 + (b18 + b15 + b12 + b8 - b3 - b2) * q^91 + (2b19 + 3b17 + b16 + 3b15 + 3b14 - 6b11 + 4b10 + 2b9 - 2b8 - b7 - 3b6 - b5 - 6b4 - b2 - b1 - 3) * q^93 + (-b18 + b17 + b16 - 3b12 - 2b11 - b10 + 3b9 - 4b8 - 3b6 - b5 - 6b4 + 2b3 - 2b1) * q^95 + (-2b19 + 2b18 + 3b16 + b15 + b13 + b12 + 4b11 + b9 - 2b7 + b6 + 2b4 + 2b3 + b2 - 2b1 + 4) * q^97 + (2b19 - 2b18 + 2b17 + 3b15 + 3b14 - b13 - 2b12 - 8b11 + 4b10 + 2b9 - 3b8 - 3b7 - 4b6 - 2b5 - 8b4 - 2b2 - 3b1 - 7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$20 q - q^{3} + 5 q^{7} - 6 q^{9} - 3 q^{11} + 2 q^{13} + 8 q^{15} + 21 q^{19} + 6 q^{21} - 6 q^{23} - 7 q^{25} + 32 q^{27} + 6 q^{29} - 18 q^{31} - 16 q^{33} + 5 q^{37} + 12 q^{39} - 12 q^{41} - 42 q^{43}+ \cdots - 57 q^{99}+O(q^{100})$ 20 * q - q^3 + 5 * q^7 - 6 * q^9 - 3 * q^11 + 2 * q^13 + 8 * q^15 + 21 * q^19 + 6 * q^21 - 6 * q^23 - 7 * q^25 + 32 * q^27 + 6 * q^29 - 18 * q^31 - 16 * q^33 + 5 * q^37 + 12 * q^39 - 12 * q^41 - 42 * q^43 + 18 * q^45 + 3 * q^47 - 5 * q^49 - q^51 - 38 * q^53 - 15 * q^55 - 3 * q^57 + 36 * q^59 + 23 * q^61 + 11 * q^63 - 30 * q^65 - 2 * q^67 + 43 * q^69 + 48 * q^71 + 6 * q^73 - 67 * q^75 - 17 * q^77 + 43 * q^79 + 16 * q^81 + 5 * q^83 - 16 * q^85 - 16 * q^87 - 46 * q^89 - 2 * q^91 + q^93 + 44 * q^95 + 9 * q^97 - 57 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} - 4 x^{19} + 21 x^{18} - 58 x^{17} + 225 x^{16} - 348 x^{15} + 1296 x^{14} - 755 x^{13} + \cdots + 10000$ x^20 - 4*x^19 + 21*x^18 - 58*x^17 + 225*x^16 - 348*x^15 + 1296*x^14 - 755*x^13 + 6557*x^12 - 5753*x^11 + 29289*x^10 - 14174*x^9 + 85487*x^8 + 13430*x^7 + 191109*x^6 + 166407*x^5 + 212971*x^4 - 179670*x^3 + 72900*x^2 - 3000*x + 10000 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 25\!\cdots\!87 \nu^{19} + \cdots + 17\!\cdots\!00 ) / 87\!\cdots\!00$ (-25060573608933462657955518830847082527188576887v^19 + 11556442089774960139224361438460250617765053048v^18 - 190567307848127805599532771514907190485318493627v^17 - 400236933711633807288152121944892641123824963554v^16 - 579625933682813863459725427594993125186098463075v^15 - 11644973493713756475882111663051294613422464464824v^14 - 1294440809575935460602061016335148798592222151552v^13 - 105839972730725981139715584231581748548167601122315v^12 - 91365977834847082232697972415726927199607743085559v^11 - 513677935359061164862839579155176626629076488056089v^10 - 245990707904268698777713225549845955066795576520843v^9 - 2541942148839484894924181036938661343951383152960662v^8 - 733424827251535299177335007020074163444350020681969v^7 - 9399738001633894959794191625594323270867524128110410v^6 - 5618227724087166678700556947594847475164662325680683v^5 - 25152060303860130754446861719603191366061957047446509v^4 - 21081759418369782765196352467160404447136266390351277v^3 - 23585087514321117937515010600407090612184494400833210v^2 + 13236687730948644939134821309052808423680122934742700*v + 1783076289543920291440878202489416150266305614961000) / 8707082140645302621003645575223925946567069737049000
$\beta_{3}$	$=$	$( - 33\!\cdots\!79 \nu^{19} + \cdots - 18\!\cdots\!00 ) / 87\!\cdots\!00$ (-3340848245555479824188330065995668980891926779v^19 + 7800403317664773619346831010079927077957419146v^18 - 49141735600815370350951791751890203106930165879v^17 + 81879626869219017043247232198588426757663861612v^16 - 455884221603558586962586528209590372400466497615v^15 - 12655485234427808918462936324711283988749954958v^14 - 2696479225358001098593031027013713388714759917324v^13 - 4165049723425059505246933640012811826655802262175v^12 - 19653491961212449902860648139558289235237795387153v^11 - 15520289066395173609332689590301852295636289567903v^10 - 76207488081940049045462196026521535464103583449721v^9 - 104649300854135381012086024729679421988119885524384v^8 - 252423189972991065764623037468723757899221158204593v^7 - 486557760457899971113544033990338605092833866564360v^6 - 861120761896744353236703228325469389539580527689811v^5 - 1562020809282300253195725278757259053248133163645983v^4 - 2229724731227042239323281369366135151143546759956399v^3 - 633341014125456869461892004379333959867344432093940v^2 + 324705493859510185857335224870876097724705555723000*v - 182849891132521437523243104465239297728192935938000) / 870708214064530262100364557522392594656706973704900
$\beta_{4}$	$=$	$( - 15\!\cdots\!89 \nu^{19} + \cdots + 39\!\cdots\!00 ) / 21\!\cdots\!50$ (-15457877268253702872777756547634619841814202989v^19 + 78162940646109855109360390214388021848629725526v^18 - 397822763705756950145412191327844101465919864424v^17 + 1267515248793514189766962646071207556708207649082v^16 - 4582801700315572708563057165047953441011872403960v^15 + 9459565097253158232164526207540108521670557061672v^14 - 27375019139773785623108858827010735840297841456229v^13 + 34974849808598461948003597811853967782800598534165v^12 - 123350006231563402527254815730322354214469935354723v^11 + 197521986114230518914284934742029889366955534321832v^10 - 597277554006319170973504992656680769187879910701406v^9 + 714299702785728523204421353420821802649523932700191v^8 - 1775322947006757322204843985422233775768306724259948v^7 + 1215599703107617360549789270958311125182442568644320v^6 - 3393890915352644933403723684332976740049079414525576v^5 - 13493507594077624312817292328164860668531450216643v^4 - 2076402961174605863190595682140918423835977358842954v^3 + 3632024369867409331873051023590152629084450455113975v^2 - 6300778734566586951811279433405288231684477915377875*v + 393302673413656877773181179741542160178897325011500) / 2176770535161325655250911393805981486641767434262250
$\beta_{5}$	$=$	$( 32\!\cdots\!14 \nu^{19} + \cdots + 30\!\cdots\!00 ) / 43\!\cdots\!50$ (3266286314619008723649872804769908496274582714v^19 - 14641468214485837963415860765503416957564320331v^18 + 74191673446959884629170553261679921176596775144v^17 - 220955862991697912437612388366032795320735346287v^16 + 816044761580173926487573385792652163343842884300v^15 - 1468322040023397339997777268255253705061326876497v^14 + 4660830494213383255811278323677965960446175055494v^13 - 4398540996724774558090213209496430382338841271170v^12 + 21718563637993393257438900264697584283399684905223v^11 - 28906357339287293506543456226602077728196743871317v^10 + 99039950077100107737133886422929740202329883906821v^9 - 90775078595110604943938382286918605870227190667861v^8 + 284639798964052918026238908278608813931601462957318v^7 - 87950289698789606218007881654214435339961779100155v^6 + 511761094996843261927502168298813864669648925314776v^5 + 243135323704530698265551181513074839699007053185473v^4 + 170941512216053307344214300935328096421138520816069v^3 - 1034779896342178402477156196216544889043244503495955v^2 + 69385808321779153830969582019727660130690943208900*v + 30915754536507405745555513095269239683628405978000) / 435354107032265131050182278761196297328353486852450
$\beta_{6}$	$=$	$( - 84\!\cdots\!79 \nu^{19} + \cdots - 62\!\cdots\!00 ) / 87\!\cdots\!00$ (-84389927043024962888440410026201801016307334279v^19 + 215171777843052387305954217586652389175126327706v^18 - 1309128993930824990924996008562277545403032685819v^17 + 2418512858055555828249954401755206106787575019972v^16 - 12355359532597037766157060964754000861574535111595v^15 + 2937184239109838285581562226209254315484692913642v^14 - 71180486381788520449018030337148536956741447345904v^13 - 90793360719226876975251744453326068701882787595915v^12 - 482261694394936105047689147169187462443235784189453v^11 - 326194530068066969318034344201722922007389962280283v^10 - 1886673766590842224243587474565062962232139470028101v^9 - 2342120351440637817618597461991600519958641758762144v^8 - 5969870353942009606009694146918946876780292624700933v^7 - 11877195541335593445631114692284306366991355559984240v^6 - 18916640990098732571200027945945796236252677651272911v^5 - 39601379373686859265704880554803630643492496207918243v^4 - 41229347081383797020041542965065591292237468418188179v^3 - 18166707729384372675696793338232281506777652871572380v^2 + 12148896967344095916896905983735713677962243944202000*v - 6205240826316547383849115115569010147164246472736000) / 8707082140645302621003645575223925946567069737049000
$\beta_{7}$	$=$	$( - 12\!\cdots\!41 \nu^{19} + \cdots + 84\!\cdots\!00 ) / 87\!\cdots\!00$ (-12238793032904746424780742251815481489010300941v^19 + 46305947397269922973225260198796027593942133404v^18 - 247610291043989201927958937976449835215825036821v^17 + 663237405208357888374203129114140436709461510118v^16 - 2643051037186284879959570046290897243819025941545v^15 + 3818885906597183145440074105680899716039285787968v^14 - 15450775563671072395602425402310842846919482497656v^13 + 7108305722617857661181462137261774682069140667795v^12 - 81168978034658958081523202308246188325420605638737v^11 + 58502280657231591379594369469236158773448604366953v^10 - 353826317734847364159935183370298484756378191483269v^9 + 124437133918506539643441118499096648670077246080794v^8 - 1074383882114776819403935998563241617934234806061727v^7 - 278896642283327493855306962624839624582719930454750v^6 - 2555830478423820426632817724357346754177184163255269v^5 - 2325673992910372764872750040137536752799347912945527v^4 - 3332904592120466775786288180763995909537759162148031v^3 + 1830092264878061571146421187464582497205104861314110v^2 - 645841060744562227251443634564761555021316847557300*v + 84389927043024962888440410026201801016307334279000) / 870708214064530262100364557522392594656706973704900
$\beta_{8}$	$=$	$( 77\!\cdots\!91 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 43\!\cdots\!00$ (77322277328288115209309533597837928023297402991v^19 - 293046094669233164060297728758909440506905639329v^18 + 1584887107144719563858100815929468019734927280171v^17 - 4252743212444503685069793214023769477676942105493v^16 + 17021075130392516433549584654140163573637579458295v^15 - 24754024837711736292474049710257915437550678255193v^14 + 100222002758571124785704812452616198520552325276356v^13 - 45238710521513009712630108235373153365698124538695v^12 + 523707079217194418877689111050030822793186736817712v^11 - 341069026893637726000146193863305961816659193152528v^10 + 2307401956152244718238615010694856968493017673828294v^9 - 679146738057718102138970287068819810115372558628619v^8 + 7055695287856703476851322511536733174716809342209977v^7 + 2509854606069625093580423123846857367781299716399025v^6 + 16375353936378160459469489149004819179597873381985119v^5 + 18188484963601129011707664974897188996258096339842802v^4 + 22068552675229140132128403368890539153700819829236306v^3 - 2766146639405223043950170061161806008545597456519085v^2 + 5931494311529377003709636811018503436660866984378600*v + 2134883202520273344189445546205847219854984410272000) / 4353541070322651310501822787611962973283534868524500
$\beta_{9}$	$=$	$( - 39\!\cdots\!98 \nu^{19} + \cdots + 12\!\cdots\!00 ) / 21\!\cdots\!50$ (-39257564716844729757689182883839599689738457198v^19 + 161207450328707056276520770619574303759650404432v^18 - 828000262163279525105798477159968073872644205293v^17 + 2318425557060177191683949332223123720377039686724v^16 - 8819616928150748331389335499598472681770848419495v^15 + 13949794055689374886514208556408190838522900627754v^14 - 49626841357381213711228309327805486177700362802203v^13 + 31560080259468841378304923211799298092629082260905v^12 - 244493000902141788869681973649543878733441255043561v^11 + 249850658950122183736036401697355316051604418113949v^10 - 1088130710212655767242143602665505999219275174365167v^9 + 648684812185456087700161858028415516472093888259012v^8 - 3061208986253013851340239034540439991415039635604336v^7 - 122298679259044227095697855696007296879048901802285v^6 - 6385074100963835631549473460202172092335696749138407v^5 - 4921028869070229443041508649155822219064614924095326v^4 - 4920019192824581934758178296424849180115537830897828v^3 + 11875186490936259896130870867892364190452329039535325v^2 - 2422428155409051117891973728498786137668573962984575*v + 1293454562459380730839232623105110195846870768230500) / 2176770535161325655250911393805981486641767434262250
$\beta_{10}$	$=$	$( - 15\!\cdots\!89 \nu^{19} + \cdots + 41\!\cdots\!00 ) / 87\!\cdots\!00$ (-15856505306933082916358739556149927139572905389v^19 + 93684903618395517196495888593754629402467957184v^18 - 434529290140954352184972451904235712410932516796v^17 + 1485403481284676162883885704908636961883432332960v^16 - 4938874717492846239795214779951667819092349816439v^15 + 11345512820931316163711641531930599310364936963692v^14 - 27025415645299238237813868934151987549513873912163v^13 + 45835292318171277010142184611668719074914836057628v^12 - 102621553994664022673891953816121864555460542629803v^11 + 282724611299252328210263613152615439019687485539613v^10 - 504067945445123200036306667034759959048567139599310v^9 + 1042046663685412777646429490893686596846894560898573v^8 - 1193712496461909810275413878924576252171694913295800v^7 + 2283174945052359071806579884469341055924193006551326v^6 - 818651705452203322195972676549691018898000146652341v^5 + 3862927644945229691674574869749271583805122229106154v^4 + 6541560719289013989246243390725564067942285519385217v^3 + 13968384092766773831619282236229410105367611168276763v^2 + 1410327319925679664040919271298377258709355355667700*v + 419521031630154489024516728411480243892621837504000) / 870708214064530262100364557522392594656706973704900
$\beta_{11}$	$=$	$( - 91\!\cdots\!69 \nu^{19} + \cdots - 13\!\cdots\!00 ) / 43\!\cdots\!00$ (-91424945566260718761621552232619648864096467969v^19 + 382404023492820274167427859260456940360845505771v^18 - 1958925873479798962090786751935412261535812923079v^17 + 5548355520847198539928808988251390649652245971597v^16 - 20980010886754756806581085413332363128210024601085v^15 + 34095302165076523063857232817999589666707903341287v^14 - 118423452027701752470469217011851508507925272713034v^13 + 82508230029316848157989427070696401835966632903215v^12 - 578648119460846235393717849789222978468601529161858v^11 + 624235171648760164549912030692052286091335957161422v^10 - 2600143785358234323762470195389687634102340002504526v^9 + 1676894618865879672954534861477758580320221254241211v^8 - 7292397817352253159714311512061558812504415329643983v^7 + 34278930910073875854537740859536905251290226199295v^6 - 15039341119933019846247013055672015449304442564265821v^5 - 10908147107361025660781641500746190960829798306868328v^4 - 11660758035780610269402677206746943971994823061595984v^3 + 27574943626025274536516951136465448067129946199772225v^2 - 3498173461153122050412751135861302602855910354470400*v - 1349252632598768773001811467656521542031238374708000) / 4353541070322651310501822787611962973283534868524500
$\beta_{12}$	$=$	$( - 38\!\cdots\!87 \nu^{19} + \cdots - 37\!\cdots\!00 ) / 16\!\cdots\!50$ (-381161769028387769547693121692558323509571287v^19 + 1237009647416582184917211898604665564824305173v^18 - 6996183711905421178852280497456813009808387452v^17 + 16575386609251665164031927204411768542908188971v^16 - 71722650615034035645823869777261056931565606100v^15 + 74422045776805231131285538291672678505788017726v^14 - 419906169435058657865852543255157448449326763652v^13 - 57730086315902943392568048997944493670686857190v^12 - 2412947588243947858046175814940602299081547699809v^11 + 264524567070514862381828328873662991916755558661v^10 - 10184807591779839915151422975576052523333396634543v^9 - 2929915385018587143897615369012425476036862008862v^8 - 31368535457929394525815510740785030589591715127169v^7 - 30932465592600430787709899959505262807198603743435v^6 - 84196618339716835666859403537739517264936145989808v^5 - 130451474041819638962537825611573695840180670028459v^4 - 140769735929074692272558796554075994792011394097802v^3 - 36201515015823772859258225200604205684693861477960v^2 - 907862648820782756503797181195580394584349744500*v - 3739006700209433474618996107440549847067957327000) / 16616568970697142406495506822946423562150896444750
$\beta_{13}$	$=$	$( - 48\!\cdots\!31 \nu^{19} + \cdots + 42\!\cdots\!00 ) / 17\!\cdots\!00$ (-48493393082305699960712917872814437589491575531v^19 + 195756273086294877802346966847751422993053067616v^18 - 1054355194835558625117078870681706830957772243179v^17 + 2955574734367908799559568774403565226645575245570v^16 - 11582993415510117069855300716891387911058479949421v^15 + 18753708147423379140993706361964193007309627733268v^14 - 69464092007042621217196723138624812151171327375942v^13 + 47103790244097573923786141528797866808205958093517v^12 - 354716369887774526894637967556332916384044663262187v^11 + 303809230899453480159736429289948322870759983682837v^10 - 1624222962102889416117762590092823701235091478626455v^9 + 866793341593294821236122514521780018956090568418112v^8 - 5008087294007400085393313162417330387613080147662095v^7 - 284184892269856369100556582228220711129404216375486v^6 - 11693462779487214262277606954773006630971626002078939v^5 - 8410237547834177523651990666896839933170554090145689v^4 - 16385593979765758940722287207291460229289263757695397v^3 + 4033830461780697922580549812545209186547516439336732v^2 - 11190062413984356456041820565389452414011939875375200*v + 4289823191355676675504729371777289746631613994194400) / 1741416428129060524200729115044785189313413947409800
$\beta_{14}$	$=$	$( - 15\!\cdots\!87 \nu^{19} + \cdots + 52\!\cdots\!00 ) / 43\!\cdots\!00$ (-153022519123500551381685465511237574188122090387v^19 + 533554764262391197884800007807292740870461842568v^18 - 2962636855202344273917635703496533474195599381257v^17 + 7455692780236016906420492784556050131350998211941v^16 - 31168029547431767527062302153987467122833897184160v^15 + 38989510054501763986580813667511621216977355659576v^14 - 184926410489323168480807701942384216000130943025012v^13 + 33636425192554066111481386603031030530987006791505v^12 - 1029709546531936741370508638185114914730182802655834v^11 + 401337402437971148884262569259570680906436911909676v^10 - 4490315929317460169476114788428114354364462956717003v^9 + 143782031938045141476004768123558787903963944446393v^8 - 13987959688031798763667390587360143423752617278832774v^7 - 8081382518686113172121777276250369823390258135349595v^6 - 36429907931056728365645425164715233481511191519642808v^5 - 42186798150418624286536670376137479647760023483629204v^4 - 62447117029821460056487561767006994080600283854848337v^3 - 2733251266656803800189172934887073170292322766968390v^2 - 16081877741223855069908634792930106448560463915250250*v + 5206156551689989235199245215500162165631039255104500) / 4353541070322651310501822787611962973283534868524500
$\beta_{15}$	$=$	$( - 30\!\cdots\!93 \nu^{19} + \cdots - 51\!\cdots\!00 ) / 87\!\cdots\!00$ (-306749490661164544707269263465618851383818076893v^19 + 1552748684907054334075543774527012106209661525852v^18 - 7627686886977445229231745920044323555904512225823v^17 + 24084238726567022761019944721806747261066012797724v^16 - 85101806463020052101611119684190741375802390826365v^15 + 171497645065524035509793792958253568304232433200364v^14 - 479520266238159410286167150863427185332967210802268v^13 + 594216867265710463184095292012211327099440960593945v^12 - 2073842347292909260832463756080183022443952851690751v^11 + 3707073978833969691110174953874304290675951778303389v^10 - 10006525370252883645095280782431479509985402240600567v^9 + 12703152135323633726607465888917405410887030161326652v^8 - 26819023360073057138088196238707720903918391225833161v^7 + 19667428995765991189487260320701285260511762354512420v^6 - 43155839562245048941086136382429883059972475165542087v^5 + 6365994916670167717703858374032893953298544320687069v^4 + 9475044738117295410897106057870438454604895717768057v^3 + 129191520504770097489052765194700340972233551646892290v^2 - 65188513771161106046681596978738322133095837883873500*v - 5198576200473072898824464073404884604295789629829000) / 8707082140645302621003645575223925946567069737049000
$\beta_{16}$	$=$	$( 15\!\cdots\!64 \nu^{19} + \cdots - 52\!\cdots\!00 ) / 43\!\cdots\!00$ (159926191827731375304466674479192467705092512764v^19 - 634803979264182105349054295798044509158431557301v^18 + 3290420586816783177413796989838943339042066254674v^17 - 9070510858869036487311323115478432593857760451057v^16 + 35003865882373538462690510011813981941844407526760v^15 - 53445933738050171788705046639122613375765758155822v^14 + 198971924971285294096080079204052720661281885023529v^13 - 114911419863918274294561162942449303286442898093040v^12 + 1002474407681433781808633479057847955588339808077423v^11 - 946278777767880688356106490142438220924216152981007v^10 + 4341487331414836175527771731091614859903583788035956v^9 - 2342719707317499335352443467721640096725747288508566v^8 + 12393499328899176361902664114222072102083185715145298v^7 + 1201796647140363111441875366620888376317974146019480v^6 + 26397613425639677059795508000438314894201208307972851v^5 + 20958533147747373213883677234467460335494571728090118v^4 + 19446268819986189048989862820289416134888757418110604v^3 - 48257832545666989459021163149015981394449822707594600v^2 - 21897823894341084498305526514150452000172309870508850*v - 5250072133715239473168653583319572833927161219224000) / 4353541070322651310501822787611962973283534868524500
$\beta_{17}$	$=$	$( - 95\!\cdots\!37 \nu^{19} + \cdots - 96\!\cdots\!00 ) / 17\!\cdots\!00$ (-95872924217885060708534795697359544017929321137v^19 + 284085250615601669788485208300276580224801832492v^18 - 1655114990134580051911875438364171010041765134873v^17 + 3626850863866867013648779094288005611210055033450v^16 - 16617451738285823989568014589817204027961119114347v^15 + 13228771705386185658401416056688174649070804318036v^14 - 98459095649486135207080964170461561063976132393314v^13 - 42913358376493553080282507074245792471023973463081v^12 - 605516314365811817282819441410863779785686334673749v^11 - 68010553965005852599983211217068074060875108182181v^10 - 2507191980103606206133626039599413506586156840767565v^9 - 1316129198993109100008995262940591291666308276839636v^8 - 7971153092922062480161137652169425350642574407879105v^7 - 9200586909054292349700686713138700050324286334421702v^6 - 23341379368549133776545817182786659979090738238262853v^5 - 35105854488365187729848595269675373686113376003959663v^4 - 45361313429856163508268051438749485684178779145164299v^3 - 9759202797856854313186652742817926742295756177817876v^2 - 406982415222491130524390377505179087520807361731400*v - 960795425560021232271384185853172324589343382589200) / 1741416428129060524200729115044785189313413947409800
$\beta_{18}$	$=$	$( 67\!\cdots\!58 \nu^{19} + \cdots + 17\!\cdots\!00 ) / 87\!\cdots\!00$ (67216138265765641349547124933114531351703988958v^19 - 253910999274503376562323345229650458611424321078v^18 + 1352806453429920366570947328827280154822234550507v^17 - 3578709905222524676816195719389821571656617642950v^16 + 14244577722673561911824026616851343908900139748798v^15 - 19915263477167394900108733547278533979783935383924v^14 + 81759510996684390613844247626414918029369887705251v^13 - 30131096626966562704677058524495440322292589392571v^12 + 429612913417282392266076175158735610783550415375816v^11 - 281746662217313777609059086007927277106531360316196v^10 + 1895730615927451104079457450106089806101177622874385v^9 - 476386298488193663753866090720698862965012515653401v^8 + 5564479915643009249802584836530765284126050097827495v^7 + 2346069176576058144340217392066442474296419218485218v^6 + 13299927392065125988605572156025763768715294329547452v^5 + 14444000045504617651064463616074164410802201824299717v^4 + 17797771158820277439355006448048274882642970717080416v^3 - 7621738770181676779940903396628073122838743223367041v^2 + 4775111433029129587477234966561711927583672402849600*v + 1714931839301611010961651955736905070478220735442000) / 870708214064530262100364557522392594656706973704900
$\beta_{19}$	$=$	$( 47\!\cdots\!62 \nu^{19} + \cdots + 41\!\cdots\!00 ) / 43\!\cdots\!00$ (476510198789462892340586485950346317205130340162v^19 - 2010976286639818399320252272286798676796959754083v^18 + 10400419717950898126863715649345380358729761808667v^17 - 29740299407773405857241924634698539729166567554631v^16 + 112803260555402873041038656259657071988840612732080v^15 - 188206923784360757404777718567978237933719862459576v^14 + 649530647623249932606711819276924653096804291253832v^13 - 491281139068469463910462393517722772319425181401645v^12 + 3184559981036015275670904867371252338353463077458809v^11 - 3444471970778751392879925564929460561004361375044281v^10 + 14471799642725031104026660553541968523937464301232423v^9 - 9808891856389838262084159474574822676515944642177853v^8 + 41772485298179714349839402459812949166190368139723809v^7 - 2762937395384201355821650559736492345027985445991410v^6 + 88934155378666237544825170205159321343538403470407033v^5 + 56577635131807009888881928116881292669998650193721769v^4 + 82804342832243163592154684785247924087795179843782232v^3 - 113647196426794515399436978641901397321631996573281875v^2 + 50037035376777365894110234710095248179568951405971150*v + 416476403000858533497764020926396895044999952605000) / 4353541070322651310501822787611962973283534868524500

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{18} + \beta_{12} + 5\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{4} - \beta_{3} + \beta_1$ b18 + b12 + 5*b11 - b10 - b9 + b8 + b4 - b3 + b1
$\nu^{3}$	$=$	$\beta_{18} + 2 \beta_{17} + \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{8} - \beta_{7} + \cdots - 1$ b18 + 2b17 + b15 + b14 - b13 - 2b12 - b8 - b7 - b6 - b5 - b4 - 7*b3 + b2 - 1
$\nu^{4}$	$=$	$4 \beta_{19} + 12 \beta_{17} + 12 \beta_{15} + 12 \beta_{14} - 40 \beta_{11} + 12 \beta_{10} + 8 \beta_{9} + \cdots - 32$ 4b19 + 12b17 + 12b15 + 12b14 - 40b11 + 12b10 + 8b9 - 8b8 - 15b7 - 32b6 - 12b5 - 40b4 - 6b2 - 12b1 - 32
$\nu^{5}$	$=$	$33 \beta_{19} - 18 \beta_{18} + 14 \beta_{17} - 29 \beta_{16} + 33 \beta_{15} + 29 \beta_{14} + \cdots - 77$ 33b19 - 18b18 + 14b17 - 29b16 + 33b15 + 29b14 + 15b13 + 25b12 - 44b11 + 14b10 - 12b9 + 25b8 - 77b7 - 69b6 - 58b5 - 33b4 + 58b3 - 45b2 - 77*b1 - 77
$\nu^{6}$	$=$	$138 \beta_{19} - 138 \beta_{18} - 134 \beta_{16} + 63 \beta_{15} + 11 \beta_{14} + 63 \beta_{13} + \cdots - 87$ 138b19 - 138b18 - 134b16 + 63b15 + 11b14 + 63b13 - 65b12 - 87b11 + 11b10 - 31b9 - 133b7 - 73b6 + 158b4 + 133b3 - 73b2 - 190b1 - 87
$\nu^{7}$	$=$	$238 \beta_{19} - 428 \beta_{18} - 170 \beta_{17} - 170 \beta_{16} + 238 \beta_{13} - 138 \beta_{12} + \cdots - 238$ 238b19 - 428b18 - 170b17 - 170b16 + 238b13 - 138b12 - 652b11 + 190b10 + 138b9 - 260b8 + 22b6 + 531b5 - 116b4 + 278b3 - 238b2 - 278b1 - 238
$\nu^{8}$	$=$	$- 808 \beta_{18} - 1477 \beta_{17} - 808 \beta_{15} - 260 \beta_{14} + 773 \beta_{13} + 909 \beta_{12} + \cdots - 229$ -808b18 - 1477b17 - 808b15 - 260b14 + 773b13 + 909b12 + 553b8 + 1477b7 + 1487b6 + 1477b5 - 229b4 + 812b3 - 553*b2 - 229
$\nu^{9}$	$=$	$- 2901 \beta_{19} - 4266 \beta_{17} + 2005 \beta_{16} - 5150 \beta_{15} - 4266 \beta_{14} + 7643 \beta_{11} + \cdots + 9136$ -2901b19 - 4266b17 + 2005b16 - 5150b15 - 4266b14 + 7643b11 - 2261b10 - 1357b9 + 1357b8 + 8864b7 + 9136b6 + 3681b5 + 7643b4 + 4106b2 + 3681*b1 + 9136
$\nu^{10}$	$=$	$- 18128 \beta_{19} + 9792 \beta_{18} - 4272 \beta_{17} + 16332 \beta_{16} - 18128 \beta_{15} + \cdots + 42984$ -18128b19 + 9792b18 - 4272b17 + 16332b16 - 18128b15 - 16332b14 - 8336b13 - 8578b12 + 18580b11 - 4272b10 + 3722b9 - 8578b8 + 26991b7 + 27158b6 + 10371b5 + 18128b4 - 10371b3 + 21850b2 + 26991*b1 + 42984
$\nu^{11}$	$=$	$- 60317 \beta_{19} + 60317 \beta_{18} + 49633 \beta_{16} - 34406 \beta_{15} - 23582 \beta_{14} + \cdots + 79456$ -60317b19 + 60317b18 + 49633b16 - 34406b15 - 23582b14 - 34406b13 + 12881b12 + 79456b11 - 23582b10 + 11884b9 + 46363b7 + 47436b6 + 8132b4 - 46363b3 + 47436b2 + 99205b1 + 79456
$\nu^{12}$	$=$	$- 116235 \beta_{19} + 208158 \beta_{18} + 60679 \beta_{17} + 60679 \beta_{16} - 116235 \beta_{13} + \cdots + 116235$ -116235b19 + 208158b18 + 60679b17 + 60679b16 - 116235b13 + 43341b12 + 350615b11 - 121279b10 - 43341b9 + 81000b8 + 40733b6 - 125745b5 + 84074b4 - 189213b3 + 116235b2 + 189213b1 + 116235
$\nu^{13}$	$=$	$403786 \beta_{18} + 572872 \beta_{17} + 403786 \beta_{15} + 278074 \beta_{14} - 295606 \beta_{13} + \cdots - 246870$ 403786b18 + 572872b17 + 403786b15 + 278074b14 - 295606b13 - 237708b12 - 119866b8 - 567374b7 - 572936b6 - 567374b5 - 246870b4 - 555795b3 + 119866*b2 - 246870
$\nu^{14}$	$=$	$1367076 \beta_{19} + 2042737 \beta_{17} - 800204 \beta_{16} + 2393409 \beta_{15} + 2042737 \beta_{14} + \cdots - 4283224$ 1367076b19 + 2042737b17 - 800204b16 + 2393409b15 + 2042737b14 - 3674901b11 + 1242533b10 + 388909b9 - 388909b8 - 3655609b7 - 4283224b6 - 2171765b5 - 3674901b4 - 1744672b2 - 2171765*b1 - 4283224
$\nu^{15}$	$=$	$8080442 \beta_{19} - 4716641 \beta_{18} + 3285217 \beta_{17} - 6589394 \beta_{16} + 8080442 \beta_{15} + \cdots - 17276877$ 8080442b19 - 4716641b18 + 3285217b17 - 6589394b16 + 8080442b15 + 6589394b14 + 3363801b13 + 2276898b12 - 11231883b11 + 3285217b10 - 1177425b9 + 2276898b8 - 12802072b7 - 13508781b6 - 5981951b5 - 8080442b4 + 5981951b3 - 9257867b2 - 12802072*b1 - 17276877
$\nu^{16}$	$=$	$27549860 \beta_{19} - 27549860 \beta_{18} - 23080828 \beta_{16} + 15999404 \beta_{15} + 10111728 \beta_{14} + \cdots - 36677924$ 27549860b19 - 27549860b18 - 23080828b16 + 15999404b15 + 10111728b14 + 15999404b13 - 3482584b12 - 36677924b11 + 10111728b10 - 3791218b9 - 25054844b7 - 24067276b6 - 4350112b4 + 25054844b3 - 24067276b2 - 42311039b1 - 36677924
$\nu^{17}$	$=$	$54957194 \beta_{19} - 93250361 \beta_{18} - 38818014 \beta_{17} - 38818014 \beta_{16} + 54957194 \beta_{13} + \cdots - 54957194$ 54957194b19 - 93250361b18 - 38818014b17 - 38818014b16 + 54957194b13 - 9949237b12 - 119685355b11 + 36877959b10 + 9949237b9 - 21814513b8 - 5368506b6 + 65500346b5 - 15317743b4 + 81032819b3 - 54957194b2 - 81032819b1 - 54957194
$\nu^{18}$	$=$	$- 186673303 \beta_{18} - 262122886 \beta_{17} - 186673303 \beta_{15} - 124440195 \beta_{14} + \cdots + 124242485$ -186673303b18 - 262122886b17 - 186673303b15 - 124440195b14 + 130722235b13 + 69248416b12 + 30942457b8 + 289897061b7 + 292468661b6 + 289897061b5 + 124242485b4 + 199117593b3 - 30942457*b2 + 124242485
$\nu^{19}$	$=$	$- 639316518 \beta_{19} - 869376568 \beta_{17} + 458111114 \beta_{16} - 1075896580 \beta_{15} + \cdots + 1825417254$ -639316518b19 - 869376568b17 + 458111114b16 - 1075896580b15 - 869376568b14 + 1334000596b11 - 411265454b10 - 88653162b9 + 88653162b8 + 1681841257b7 + 1825417254b6 + 955313094b5 + 1334000596b4 + 759990560b2 + 955313094*b1 + 1825417254

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/616\mathbb{Z}\right)^\times$ .

$n$	$57$	$309$	$353$	$463$
$\chi(n)$	$\beta_{11}$	$1$	$1$	$1$

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}

113.1

2.75053	−	1.99838i
1.68999	−	1.22785i
0.425479	−	0.309129i
−1.00586	+	0.730797i
−1.74211	+	1.26571i
2.75053	+	1.99838i
1.68999	+	1.22785i
0.425479	+	0.309129i
−1.00586	−	0.730797i
−1.74211	−	1.26571i
0.847039	+	2.60692i
0.641063	+	1.97299i
−0.0976679	−	0.300591i
−0.560457	−	1.72491i
−0.948011	−	2.91768i
0.847039	−	2.60692i
0.641063	−	1.97299i
−0.0976679	+	0.300591i
−0.560457	+	1.72491i
−0.948011	+	2.91768i

−1.05061

−

3.23344i

−0.556182

−

0.404090i

−0.309017

0.951057i

−6.92433

5.03082i

113.2

−0.645517

−

1.98670i

1.70961

1.24210i

−0.309017

0.951057i

−1.10322

0.801539i

113.3

−0.162519

−

0.500181i

−1.76076

−

1.27927i

−0.309017

0.951057i

2.20328

−

1.60078i

113.4

0.384203

1.18245i

−2.71639

−

1.97357i

−0.309017

0.951057i

1.17646

−

0.854751i

113.5

0.665425

2.04797i

2.20569

1.60253i

−0.309017

0.951057i

−1.32433

0.962184i

169.1

−1.05061

3.23344i

−0.556182

0.404090i

−0.309017

−

0.951057i

−6.92433

−

5.03082i

169.2

−0.645517

1.98670i

1.70961

−

1.24210i

−0.309017

−

0.951057i

−1.10322

−

0.801539i

169.3

−0.162519

0.500181i

−1.76076

1.27927i

−0.309017

−

0.951057i

2.20328

1.60078i

169.4

0.384203

−

1.18245i

−2.71639

1.97357i

−0.309017

−

0.951057i

1.17646

0.854751i

169.5

0.665425

−

2.04797i

2.20569

−

1.60253i

−0.309017

−

0.951057i

−1.32433

−

0.962184i

225.1

−2.21758

−

1.61116i

−0.395619

1.21759i

0.809017

−

0.587785i

1.39475

4.29259i

225.2

−1.67832

−

1.21937i

0.914896

−

2.81576i

0.809017

−

0.587785i

0.402849

1.23984i

225.3

0.255698

0.185775i

0.102849

−

0.316537i

0.809017

−

0.587785i

−0.896182

−

2.75816i

225.4

1.46730

1.06605i

−0.699852

2.15392i

0.809017

−

0.587785i

0.0894373

0.275260i

225.5

2.48193

1.80322i

1.19576

−

3.68017i

0.809017

−

0.587785i

1.98128

6.09777i

449.1

−2.21758

1.61116i

−0.395619

−

1.21759i

0.809017

0.587785i

1.39475

−

4.29259i

449.2

−1.67832

1.21937i

0.914896

2.81576i

0.809017

0.587785i

0.402849

−

1.23984i

449.3

0.255698

−

0.185775i

0.102849

0.316537i

0.809017

0.587785i

−0.896182

2.75816i

449.4

1.46730

−

1.06605i

−0.699852

−

2.15392i

0.809017

0.587785i

0.0894373

−

0.275260i

449.5

2.48193

−

1.80322i

1.19576

3.68017i

0.809017

0.587785i

1.98128

−

6.09777i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	616.2.r.f	✓	20
11.c	even	5	1	inner	616.2.r.f	✓	20
11.c	even	5	1		6776.2.a.bm		10
11.d	odd	10	1		6776.2.a.bn		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
616.2.r.f	✓	20	1.a	even	1	1	trivial
616.2.r.f	✓	20	11.c	even	5	1	inner
6776.2.a.bm		10	11.c	even	5	1
6776.2.a.bn		10	11.d	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{20} + T_{3}^{19} + 11 T_{3}^{18} - 8 T_{3}^{17} + 85 T_{3}^{16} + 112 T_{3}^{15} + 1081 T_{3}^{14} + \cdots + 10000$ T3^20 + T3^19 + 11*T3^18 - 8*T3^17 + 85*T3^16 + 112*T3^15 + 1081*T3^14 + 395*T3^13 + 6377*T3^12 + 1027*T3^11 + 22864*T3^10 - 6414*T3^9 + 75177*T3^8 - 12610*T3^7 + 296139*T3^6 - 188178*T3^5 + 446621*T3^4 - 125070*T3^3 + 97900*T3^2 - 43000*T3 + 10000 acting on $S_{2}^{\mathrm{new}}(616, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20}$ T^20
$3$	$T^{20} + T^{19} + \cdots + 10000$ T^20 + T^19 + 11T^18 - 8T^17 + 85T^16 + 112T^15 + 1081T^14 + 395T^13 + 6377T^12 + 1027T^11 + 22864T^10 - 6414T^9 + 75177T^8 - 12610T^7 + 296139T^6 - 188178T^5 + 446621T^4 - 125070T^3 + 97900T^2 - 43000T + 10000
$5$	$T^{20} + 16 T^{18} + \cdots + 102400$ T^20 + 16T^18 + 34T^17 + 168T^16 + 18T^15 + 2021T^14 + 358T^13 + 11889T^12 + 10088T^11 + 78257T^10 + 64910T^9 + 241677T^8 - 24366T^7 + 1104436T^6 + 1532968T^5 + 2845136T^4 + 1969280T^3 + 770560T^2 + 76800T + 102400
$7$	$(T^{4} - T^{3} + T^{2} + \cdots + 1)^{5}$ (T^4 - T^3 + T^2 - T + 1)^5
$11$	$T^{20} + \cdots + 25937424601$ T^20 + 3T^19 + 17T^18 + 68T^17 + 297T^16 + 516T^15 + 2061T^14 + 5244T^13 + 2191T^12 - 7971T^11 + 23066T^10 - 87681T^9 + 265111T^8 + 6979764T^7 + 30175101T^6 + 83102316T^5 + 526153617T^4 + 1325127628T^3 + 3644100977T^2 + 7073843073*T + 25937424601
$13$	$T^{20} + \cdots + 5609410816$ T^20 - 2T^19 + 15T^18 - 60T^17 + 1180T^16 + 778T^15 + 30946T^14 - 64419T^13 + 767545T^12 - 451785T^11 + 9328712T^10 - 8375841T^9 + 246089003T^8 - 359548218T^7 + 1972880480T^6 - 2267000120T^5 + 6525679312T^4 - 7359439904T^3 + 7218488448T^2 - 3629160576*T + 5609410816
$17$	$T^{20} + 39 T^{18} + \cdots + 51552400$ T^20 + 39T^18 - 207T^17 + 1413T^16 + 5649T^15 + 70309T^14 + 25826T^13 + 675374T^12 - 1735546T^11 + 12068323T^10 - 15603720T^9 + 78490677T^8 - 243602703T^7 + 1447867529T^6 - 4024673589T^5 + 8452230051T^4 - 10216671990T^3 + 7337652140T^2 + 322166600T + 51552400
$19$	$T^{20} + \cdots + 2296709776$ T^20 - 21T^19 + 251T^18 - 1945T^17 + 12057T^16 - 63382T^15 + 378499T^14 - 2150646T^13 + 11732500T^12 - 61242570T^11 + 344207209T^10 - 1359912083T^9 + 3936933637T^8 - 8797634243T^7 + 20147279617T^6 - 28716041476T^5 + 36517727941T^4 - 24979532146T^3 + 11106808448T^2 - 3226818768*T + 2296709776
$23$	$(T^{10} + 3 T^{9} + \cdots - 368404)^{2}$ (T^10 + 3T^9 - 147T^8 - 290T^7 + 6598T^6 + 6646T^5 - 87923T^4 - 79653T^3 + 328317T^2 + 184756*T - 368404)^2
$29$	$T^{20} + \cdots + 638362648576$ T^20 - 6T^19 + 108T^18 - 648T^17 + 6687T^16 - 30024T^15 + 258600T^14 - 1050443T^13 + 11097184T^12 - 50234476T^11 + 320965203T^10 - 1210371232T^9 + 4921734960T^8 - 14049748055T^7 + 33307393364T^6 - 36862808535T^5 + 45618349009T^4 - 44302494896T^3 + 287370264064T^2 - 604153692160*T + 638362648576
$31$	$T^{20} + \cdots + 101572239616$ T^20 + 18T^19 + 170T^18 + 849T^17 + 5024T^16 + 32614T^15 + 348717T^14 + 2317870T^13 + 14685758T^12 + 61461003T^11 + 349783230T^10 + 987520284T^9 + 4589940337T^8 + 13579709584T^7 + 80157374040T^6 + 307082777216T^5 + 982498038576T^4 + 1263810725568T^3 + 1786567587392T^2 - 200383227776*T + 101572239616
$37$	$T^{20} + \cdots + 5836348816$ T^20 - 5T^19 + 77T^18 + 119T^17 + 2138T^16 - 5757T^15 + 301330T^14 - 799257T^13 + 7304327T^12 + 24873754T^11 + 89725165T^10 + 106938308T^9 + 4397438345T^8 - 489444799T^7 - 10470834190T^6 + 68490789750T^5 + 612652478765T^4 - 79647777746T^3 + 135476238872T^2 + 9690679808*T + 5836348816
$41$	$T^{20} + 12 T^{19} + \cdots + 20647936$ T^20 + 12T^19 + 78T^18 - 541T^17 + 531T^16 - 24256T^15 + 950867T^14 - 2620898T^13 - 6779843T^12 + 20493317T^11 + 391962329T^10 - 1102424425T^9 + 1684030885T^8 - 2159455836T^7 + 11836565544T^6 + 6146256728T^5 + 18252462672T^4 + 2958417280T^3 + 1920426496T^2 - 82809856*T + 20647936
$43$	$(T^{10} + 21 T^{9} + \cdots - 398345)^{2}$ (T^10 + 21T^9 - 8T^8 - 2795T^7 - 16727T^6 + 28236T^5 + 391755T^4 + 328507T^3 - 2243476T^2 - 3662065*T - 398345)^2
$47$	$T^{20} + \cdots + 6845245696$ T^20 - 3T^19 + 90T^18 - 736T^17 + 6035T^16 - 14990T^15 + 294777T^14 - 349099T^13 + 2327287T^12 - 11427731T^11 + 157513497T^10 - 425338636T^9 + 1360404041T^8 + 1933283666T^7 + 188783540T^6 - 12723268128T^5 + 38705401872T^4 + 26515893056T^3 + 50919555840T^2 + 13225846016*T + 6845245696
$53$	$T^{20} + \cdots + 17188652646400$ T^20 + 38T^19 + 735T^18 + 9405T^17 + 108645T^16 + 1078829T^15 + 8130067T^14 + 40056665T^13 + 231254765T^12 + 1065946095T^11 + 3662270247T^10 - 4928690649T^9 + 140398203795T^8 - 633087485645T^7 + 2610687748155T^6 - 8154030653268T^5 + 24092407419601T^4 - 38794044456880T^3 + 32669238826240T^2 - 3589040025600*T + 17188652646400
$59$	$T^{20} + \cdots + 23809724416$ T^20 - 36T^19 + 709T^18 - 9200T^17 + 95420T^16 - 779204T^15 + 5437927T^14 - 28872626T^13 + 130307549T^12 - 245537298T^11 + 840302657T^10 + 4223815076T^9 + 21963345292T^8 + 32719619598T^7 + 159342106359T^6 + 258248741586T^5 + 430747367449T^4 + 477578123296T^3 + 1099155919744T^2 - 264544949760*T + 23809724416
$61$	$T^{20} + \cdots + 27\!\cdots\!00$ T^20 - 23T^19 + 398T^18 - 5538T^17 + 85111T^16 - 704040T^15 + 7083117T^14 - 49437195T^13 + 374979207T^12 - 856000785T^11 + 11633329517T^10 - 12410563240T^9 + 494050363761T^8 + 82467743448T^7 + 13691684294288T^6 + 49740433669608T^5 + 571351168510896T^4 + 2555518588344640T^3 + 9558707749749760T^2 + 20136058292326400*T + 27414987903078400
$67$	$(T^{10} + T^{9} + \cdots + 78764149)^{2}$ (T^10 + T^9 - 355T^8 + 549T^7 + 42627T^6 - 149476T^5 - 1761143T^4 + 8362063T^3 + 16151023T^2 - 108229455T + 78764149)^2
$71$	$T^{20} + \cdots + 12037569030400$ T^20 - 48T^19 + 1431T^18 - 29118T^17 + 457632T^16 - 5702260T^15 + 61141965T^14 - 555048195T^13 + 4419939598T^12 - 29953918559T^11 + 174424034781T^10 - 844785670563T^9 + 3433712384279T^8 - 9291480100773T^7 + 11152674215578T^6 + 13693382604721T^5 + 12052406236641T^4 - 57699227975640T^3 + 68296842794240T^2 - 29589523758400*T + 12037569030400
$73$	$T^{20} + \cdots + 129431214182656$ T^20 - 6T^19 + 175T^18 - 2160T^17 + 26648T^16 - 37018T^15 + 2229829T^14 - 962314T^13 + 20451419T^12 - 41254902T^11 + 1468872093T^10 - 376732904T^9 + 24151834452T^8 - 3101983546T^7 + 288633040217T^6 + 215200820918T^5 + 2809725085161T^4 + 5135914685804T^3 + 40685711458880T^2 + 50129659860608*T + 129431214182656
$79$	$T^{20} + \cdots + 907134334096$ T^20 - 43T^19 + 965T^18 - 12857T^17 + 137731T^16 - 2044440T^15 + 48704411T^14 - 922675503T^13 + 12557181066T^12 - 126214206434T^11 + 1057234834777T^10 - 7354009616532T^9 + 42821325303419T^8 - 195635076105898T^7 + 714399701214538T^6 - 1774343645147743T^5 + 4277431035963589T^4 - 8145851124348052T^3 + 9618454075081848T^2 - 58351618242888*T + 907134334096
$83$	$T^{20} + \cdots + 15\!\cdots\!76$ T^20 - 5T^19 + 305T^18 - 4278T^17 + 58489T^16 - 389172T^15 + 6497239T^14 - 84055484T^13 + 1300714428T^12 - 11816529808T^11 + 102683861277T^10 - 675698172713T^9 + 4906996273659T^8 - 16118333932509T^7 + 72408526831408T^6 - 179725000808055T^5 + 1809180872491009T^4 + 7829505131887072T^3 + 41656090995201920T^2 + 98776944638218752*T + 156264442834456576
$89$	$(T^{10} + 23 T^{9} + \cdots + 455935796)^{2}$ (T^10 + 23T^9 - 215T^8 - 7723T^7 - 9646T^6 + 704119T^5 + 2746413T^4 - 18353276T^3 - 77054387T^2 + 127123014*T + 455935796)^2
$97$	$T^{20} + \cdots + 17\!\cdots\!16$ T^20 - 9T^19 + 455T^18 + 730T^17 + 89538T^16 + 344155T^15 + 29868303T^14 + 160245431T^13 + 6288658864T^12 + 48665443024T^11 + 1108228538363T^10 + 7494592433463T^9 + 96376117586383T^8 + 606724308117230T^7 + 16858088594030180T^6 + 143710238700181296T^5 + 1160616594653343760T^4 + 4722577672640610752T^3 + 12042891733844361088T^2 + 17512060068523806208*T + 17801104401113334016
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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 113.5
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	616.2.r.f

Level	$616$
Weight	$2$
Character orbit	616.r
Analytic conductor	$4.919$
Analytic rank	$0$
Dimension	$20$
Inner twists	$2$