Properties

Label 1.44.a
Level 1
Weight 44
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 1
Sturm bound 3
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 44 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(3\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{44}(\Gamma_0(1))\).

Total New Old
Modular forms 4 4 0
Cusp forms 3 3 0
Eisenstein series 1 1 0

Trace form

\( 3q - 2209944q^{2} + 24401437812q^{3} + 9822618421824q^{4} + 535205380774170q^{5} - 91793974758443424q^{6} + 301971425665478856q^{7} - 15830863913787348480q^{8} + 477482125171066743231q^{9} + O(q^{10}) \) \( 3q - 2209944q^{2} + 24401437812q^{3} + 9822618421824q^{4} + 535205380774170q^{5} - 91793974758443424q^{6} + 301971425665478856q^{7} - 15830863913787348480q^{8} + 477482125171066743231q^{9} + 424411733447436982320q^{10} + 26665053285822824598396q^{11} + 591067459530801007163136q^{12} + 2675160683143344673595682q^{13} + 14021968617857741064904128q^{14} + 35619426649405115420082840q^{15} - 79484355831457348071124992q^{16} - 408654222932442443796841194q^{17} - 4107998354973953853903433848q^{18} + 1577331357975519701176464900q^{19} + 14558304937461364568422335360q^{20} + 39305230121617849136195725536q^{21} + 139626647360508967807005852192q^{22} - 1215339635928561142484811048q^{23} - 1116055600560337571570968012800q^{24} - 2310484082897201607166539950475q^{25} - 1738302877111230473099097494544q^{26} + 12636088863973677226324883265480q^{27} + 16881555000174595924306194912768q^{28} + 57118989018136587596191009705650q^{29} - 73851642261923451351367502727360q^{30} - 255565972342397832812135282963424q^{31} - 144698056390774688063435943542784q^{32} - 274896238871162964163796672607216q^{33} + 460541119788760022406607937310288q^{34} + 2398213530857285851552147980593520q^{35} + 8176018057545637654201506848470848q^{36} - 2320645576165979428293572256842694q^{37} - 30659398199572233844432077267455520q^{38} - 3966989860384498209469308423541128q^{39} - 32703585947405178164040064943232000q^{40} + 25742376604739002042016577514061166q^{41} + 142089722887598025677946391121586432q^{42} + 240714339689128182512406300123419292q^{43} - 103743099769332204870397943787455232q^{44} + 257230496455952342677786043056557090q^{45} - 1897545066442972080018141849284700864q^{46} + 308713553347399687962182796433367856q^{47} - 2551621376300181083415582718933352448q^{48} + 3434980292250886821160584750829904379q^{49} + 1991781591031845842073119994352349400q^{50} + 13352566627975131351422398848669478056q^{51} - 1723278372049486147633277645433304704q^{52} + 15000497558784669132889458695112796362q^{53} - 84497407838510132201868358325218843200q^{54} + 3996571389110251928625895283213506440q^{55} - 64188456562605926675684864956824883200q^{56} + 194498189404429212580032012092952587760q^{57} - 103653602500512241322054964115851073680q^{58} + 226996261182733939538282061249836505900q^{59} + 172179025249459513233108554074130526720q^{60} - 9393429368478033721028929889113591854q^{61} - 208603081649583360642384119938663534848q^{62} - 967475285671148246145956691206790781848q^{63} - 1119010168436659689243622515721507700736q^{64} - 270218921102297342477494789001032451460q^{65} + 2927960709398652565212160365298168924032q^{66} - 733015616887151283726086660162187354444q^{67} + 5475599618745313926138424496016357744768q^{68} + 1967042253746083561325124578559972227232q^{69} + 5947953535225896394715748735968145697920q^{70} - 18523878066158899993345833814569898287864q^{71} - 9888656801008664055042728388175401192960q^{72} - 20251106025908798058937016734369224029298q^{73} + 16747422277652165725566811926177721225008q^{74} - 21273380043530396090723723962888745519700q^{75} + 77507609539490133886259513865936229344000q^{76} + 59055613160181778232413942942559747685792q^{77} - 19617648418411520686241571637167142696896q^{78} + 15233568018599764739559826813006999294800q^{79} - 100962488695267720109813470111835940986880q^{80} + 73114835779397486920654645241710822062363q^{81} - 320808727539069549327137360482471983405168q^{82} - 89206610196719670053085531955706819434428q^{83} - 345724182543475881552276196164469715392512q^{84} + 438725373228516358990025164547060204838420q^{85} + 251282426033382455934203370485070655838496q^{86} + 1720273148596604243726575204332689696886840q^{87} - 25677613216166441601684725395949811087360q^{88} + 203691422331055117225246388044508249665950q^{89} - 2339725556415524057280625761410723956665360q^{90} - 1168954967602248602369466051364648836229584q^{91} + 688199578839743838535125048241381149958656q^{92} - 4949189234312030019905814954548842124935296q^{93} - 1324027207656692012199753325229034662658432q^{94} + 3125746305909018830165692627580442029979000q^{95} + 10923904714711076554208778658628955663630336q^{96} - 3881062021784775072625368160679539917597594q^{97} + 24120041445786820251814869563279238673540008q^{98} - 14255275025472043026078170466415579823927508q^{99} + O(q^{100}) \)

Decomposition of \(S_{44}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.44.a.a \(3\) \(11.711\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-2209944\) \(24401437812\) \(53\!\cdots\!70\) \(30\!\cdots\!56\) \(+\) \(q+(-736648-\beta _{1})q^{2}+(8133812604+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2209944 T + 10724756563968 T^{2} + 18900761561310167040 T^{3} + \)\(94\!\cdots\!44\)\( T^{4} + \)\(17\!\cdots\!16\)\( T^{5} + \)\(68\!\cdots\!12\)\( T^{6} \)
$3$ \( 1 - 24401437812 T + \)\(55\!\cdots\!97\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!19\)\( T^{4} - \)\(26\!\cdots\!48\)\( T^{5} + \)\(35\!\cdots\!83\)\( T^{6} \)
$5$ \( 1 - 535205380774170 T + \)\(30\!\cdots\!75\)\( T^{2} - \)\(99\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!75\)\( T^{4} - \)\(69\!\cdots\!50\)\( T^{5} + \)\(14\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 301971425665478856 T + \)\(16\!\cdots\!93\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!99\)\( T^{4} - \)\(14\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 - \)\(26\!\cdots\!96\)\( T + \)\(16\!\cdots\!65\)\( T^{2} - \)\(28\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(96\!\cdots\!56\)\( T^{5} + \)\(21\!\cdots\!91\)\( T^{6} \)
$13$ \( 1 - \)\(26\!\cdots\!82\)\( T + \)\(44\!\cdots\!87\)\( T^{2} - \)\(47\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!39\)\( T^{4} - \)\(16\!\cdots\!38\)\( T^{5} + \)\(49\!\cdots\!73\)\( T^{6} \)
$17$ \( 1 + \)\(40\!\cdots\!94\)\( T + \)\(16\!\cdots\!43\)\( T^{2} + \)\(69\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!59\)\( T^{4} + \)\(26\!\cdots\!86\)\( T^{5} + \)\(53\!\cdots\!97\)\( T^{6} \)
$19$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(10\!\cdots\!77\)\( T^{2} - \)\(52\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!43\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{5} + \)\(91\!\cdots\!79\)\( T^{6} \)
$23$ \( 1 + \)\(12\!\cdots\!48\)\( T + \)\(53\!\cdots\!77\)\( T^{2} + \)\(37\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!59\)\( T^{4} + \)\(15\!\cdots\!72\)\( T^{5} + \)\(46\!\cdots\!63\)\( T^{6} \)
$29$ \( 1 - \)\(57\!\cdots\!50\)\( T + \)\(27\!\cdots\!67\)\( T^{2} - \)\(79\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!63\)\( T^{4} - \)\(33\!\cdots\!50\)\( T^{5} + \)\(44\!\cdots\!69\)\( T^{6} \)
$31$ \( 1 + \)\(25\!\cdots\!24\)\( T + \)\(53\!\cdots\!65\)\( T^{2} + \)\(66\!\cdots\!80\)\( T^{3} + \)\(72\!\cdots\!15\)\( T^{4} + \)\(46\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!71\)\( T^{6} \)
$37$ \( 1 + \)\(23\!\cdots\!94\)\( T + \)\(78\!\cdots\!43\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!79\)\( T^{4} + \)\(17\!\cdots\!46\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \)
$41$ \( 1 - \)\(25\!\cdots\!66\)\( T + \)\(26\!\cdots\!15\)\( T^{2} - \)\(88\!\cdots\!20\)\( T^{3} + \)\(59\!\cdots\!15\)\( T^{4} - \)\(12\!\cdots\!06\)\( T^{5} + \)\(11\!\cdots\!61\)\( T^{6} \)
$43$ \( 1 - \)\(24\!\cdots\!92\)\( T + \)\(66\!\cdots\!57\)\( T^{2} - \)\(83\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!99\)\( T^{4} - \)\(72\!\cdots\!08\)\( T^{5} + \)\(52\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(30\!\cdots\!56\)\( T + \)\(20\!\cdots\!93\)\( T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!39\)\( T^{4} - \)\(19\!\cdots\!24\)\( T^{5} + \)\(50\!\cdots\!67\)\( T^{6} \)
$53$ \( 1 - \)\(15\!\cdots\!62\)\( T + \)\(48\!\cdots\!47\)\( T^{2} - \)\(42\!\cdots\!40\)\( T^{3} + \)\(67\!\cdots\!19\)\( T^{4} - \)\(29\!\cdots\!98\)\( T^{5} + \)\(27\!\cdots\!33\)\( T^{6} \)
$59$ \( 1 - \)\(22\!\cdots\!00\)\( T + \)\(50\!\cdots\!37\)\( T^{2} - \)\(62\!\cdots\!00\)\( T^{3} + \)\(71\!\cdots\!23\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{5} + \)\(27\!\cdots\!39\)\( T^{6} \)
$61$ \( 1 + \)\(93\!\cdots\!54\)\( T + \)\(89\!\cdots\!15\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(52\!\cdots\!15\)\( T^{4} + \)\(32\!\cdots\!94\)\( T^{5} + \)\(20\!\cdots\!41\)\( T^{6} \)
$67$ \( 1 + \)\(73\!\cdots\!44\)\( T + \)\(79\!\cdots\!93\)\( T^{2} + \)\(52\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!59\)\( T^{4} + \)\(80\!\cdots\!36\)\( T^{5} + \)\(36\!\cdots\!47\)\( T^{6} \)
$71$ \( 1 + \)\(18\!\cdots\!64\)\( T + \)\(22\!\cdots\!65\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!15\)\( T^{4} + \)\(29\!\cdots\!44\)\( T^{5} + \)\(64\!\cdots\!31\)\( T^{6} \)
$73$ \( 1 + \)\(20\!\cdots\!98\)\( T + \)\(46\!\cdots\!27\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(61\!\cdots\!59\)\( T^{4} + \)\(35\!\cdots\!22\)\( T^{5} + \)\(23\!\cdots\!13\)\( T^{6} \)
$79$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(11\!\cdots\!17\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(43\!\cdots\!63\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{5} + \)\(62\!\cdots\!19\)\( T^{6} \)
$83$ \( 1 + \)\(89\!\cdots\!28\)\( T + \)\(68\!\cdots\!17\)\( T^{2} + \)\(57\!\cdots\!40\)\( T^{3} + \)\(22\!\cdots\!79\)\( T^{4} + \)\(97\!\cdots\!32\)\( T^{5} + \)\(36\!\cdots\!03\)\( T^{6} \)
$89$ \( 1 - \)\(20\!\cdots\!50\)\( T + \)\(14\!\cdots\!07\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(98\!\cdots\!83\)\( T^{4} - \)\(90\!\cdots\!50\)\( T^{5} + \)\(29\!\cdots\!09\)\( T^{6} \)
$97$ \( 1 + \)\(38\!\cdots\!94\)\( T + \)\(63\!\cdots\!43\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!39\)\( T^{4} + \)\(28\!\cdots\!26\)\( T^{5} + \)\(19\!\cdots\!17\)\( T^{6} \)
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