Properties

Label 1.54
Level 1
Weight 54
Dimension 4
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 54 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{54}(\Gamma_1(1))\).

Total New Old
Modular forms 5 5 0
Cusp forms 4 4 0
Eisenstein series 1 1 0

Trace form

\( 4q - 68476320q^{2} - 1048411007280q^{3} + 7829639419798528q^{4} - 4563895793294313000q^{5} + 365243285516994476928q^{6} - 224841770748445429600q^{7} + 139144349902380051824640q^{8} + 11374560253581116395684212q^{9} + O(q^{10}) \) \( 4q - 68476320q^{2} - 1048411007280q^{3} + 7829639419798528q^{4} - 4563895793294313000q^{5} + 365243285516994476928q^{6} - 224841770748445429600q^{7} + 139144349902380051824640q^{8} + 11374560253581116395684212q^{9} + 239691034935725362703784000q^{10} + 2482225436399137396212828528q^{11} - 36954074065409878778429091840q^{12} + 38846785417115118418757458040q^{13} - 2318389217831653006768429089024q^{14} + 10885572770546350531014554412000q^{15} - 158675761303426594387641091751936q^{16} - 869470789827683332623267869375160q^{17} - 6763457530054882037574097250222880q^{18} - 25573430606481293737377136094590960q^{19} - 137124186244908431721319713321216000q^{20} - 456430072728043514442750337944104832q^{21} - 930239272324579210970924539473171840q^{22} + 692283403499940921426611559815316960q^{23} + 8087484160806183674287697801240248320q^{24} + 36481255519181820020535795184467437500q^{25} + 88199592302469577023283184091414497088q^{26} + 85289145712610214901983766257673297440q^{27} - 5693966184427848677134093343909109760q^{28} - 1403448065249545071892827367228561632840q^{29} - 5799093432823921615312476874120821216000q^{30} - 3964172885133396823441211087460727887232q^{31} + 3478175004061909463146496356345215713280q^{32} + 26488618539649572306469337360654123586240q^{33} + 149997773608319010417738024908146242937536q^{34} + 61365948170559642529868826584708753304000q^{35} + 74373890656883085418751005021443502838784q^{36} - 283737807523680188930907412534612515795880q^{37} - 2149858349100743242412726210681586741479040q^{38} - 3695759405159081384443679869760359201600416q^{39} - 530824655589569872286083924732862423040000q^{40} + 816268439352205816307560932682632884757288q^{41} + 34473336675516302350057635130637873086909440q^{42} + 45680017888765970285202819125021276867644400q^{43} + 36450257868828743991533939202628376274333696q^{44} - 52785302554790432865174172783106255610789000q^{45} - 286598377899888278123152881845505043619242752q^{46} - 450331332464449690190061856166374876708223040q^{47} - 307327868606734954261884348178650744005591040q^{48} + 61256589204322440136838991964906594854806628q^{49} + 521470775015683964723409941128468604170500000q^{50} + 4823104417223604238685131462744139603812556448q^{51} + 6480873745173555872757727943119261362842777600q^{52} + 3894630980633224571473003534722579109607488920q^{53} - 9094643349894867160783307676594158361928776960q^{54} - 25220087960941681699071033843031097345513916000q^{55} - 41708301235969326798480993683317763265335132160q^{56} - 78794438935790744156133212572387836678302205120q^{57} + 23153390818000102952173078969301145284748805440q^{58} + 161352440713620380366889082872890081870684067120q^{59} + 406305479737925709888509415210821103661353984000q^{60} + 65647964430669998286897420448040378258467279928q^{61} + 397783228528983419561799088374192602352940661760q^{62} - 302986559547633524951279897150694105228921738720q^{63} - 1033735951572319175631379922707833053066697900032q^{64} - 3420620126550512806970332375686936386825214798000q^{65} - 658347942888276364671353073747068107649649237504q^{66} - 1114206317023626852397290701407230552884994982960q^{67} + 2848472333882807118405541680155589548651324344320q^{68} + 775291276168249766930956334518267852448076104064q^{69} + 33179725699094578918821931009364962909309028928000q^{70} + 3568565136657140046536067623675232742954247775648q^{71} + 26402495617891679740981292232213348882933665464320q^{72} - 70697396612705842206652346649767546611106446705240q^{73} - 2477665041456578897595255396512213772982320606144q^{74} - 214147644415917420957952976584023331338227237250000q^{75} + 45189859364823457604586927205450004182295006842880q^{76} - 161023821416166064076421008340115759390173471196800q^{77} + 512081797195935502825354508308769411884974336211200q^{78} - 156417020870005797258130007144678737115075518344640q^{79} + 1135588155590291950317639086884448883835074772992000q^{80} - 498743339483968532647094896032944674900788125971356q^{81} + 1595344511843776298585510858680959074676233895092160q^{82} - 2616078236644862619515249170129703738064518568891120q^{83} - 769036580140824400081375315496419857132807898497024q^{84} - 2322985242765977498040783878693923846942012787106000q^{85} - 771473689440649042704510533443081756253708267624832q^{86} - 4553516071908987502170780484180858967295593004331680q^{87} + 5676678682320308097974311975312643702936459537940480q^{88} - 377838445322199523132380916699165405584669879895320q^{89} + 20963916598038401334962666668658272688885372614152000q^{90} + 12668228612190081366491587480915595422042752342094528q^{91} + 8910978937854086377493208678167975007758313267978240q^{92} - 9941509412818307372855389204673554966584205049372160q^{93} - 37159704993639903541096652460057479885138542709712384q^{94} - 17088387525725080516970439013001314242158750454180000q^{95} - 115890361493294838169648124263288994270902966699425792q^{96} + 10362822344952442405677621761316902058987153293434760q^{97} - 88214006056362955165305640276151180685109758813947040q^{98} + 200357011548750056817277275156798035565052172823063984q^{99} + O(q^{100}) \)

Decomposition of \(S_{54}^{\mathrm{new}}(\Gamma_1(1))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1.54.a \(\chi_{1}(1, \cdot)\) 1.54.a.a 4 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 68476320 T + 16444081999953920 T^{2} + \)\(76\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!28\)\( T^{4} + \)\(69\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(50\!\cdots\!60\)\( T^{7} + \)\(65\!\cdots\!96\)\( T^{8} \)
$3$ \( 1 + 1048411007280 T + \)\(33\!\cdots\!40\)\( T^{2} - \)\(33\!\cdots\!60\)\( T^{3} + \)\(70\!\cdots\!58\)\( T^{4} - \)\(64\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!60\)\( T^{6} + \)\(76\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!41\)\( T^{8} \)
$5$ \( 1 + 4563895793294313000 T + \)\(14\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!50\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{5} + \)\(17\!\cdots\!00\)\( T^{6} + \)\(62\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 + \)\(22\!\cdots\!00\)\( T + \)\(12\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!98\)\( T^{4} + \)\(20\!\cdots\!00\)\( T^{5} + \)\(45\!\cdots\!00\)\( T^{6} + \)\(52\!\cdots\!00\)\( T^{7} + \)\(14\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - \)\(24\!\cdots\!28\)\( T + \)\(45\!\cdots\!68\)\( T^{2} - \)\(92\!\cdots\!76\)\( T^{3} + \)\(96\!\cdots\!70\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} - \)\(94\!\cdots\!48\)\( T^{7} + \)\(59\!\cdots\!21\)\( T^{8} \)
$13$ \( 1 - \)\(38\!\cdots\!40\)\( T + \)\(15\!\cdots\!80\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!18\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} + \)\(19\!\cdots\!20\)\( T^{6} - \)\(50\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} \)
$17$ \( 1 + \)\(86\!\cdots\!60\)\( T + \)\(53\!\cdots\!60\)\( T^{2} + \)\(20\!\cdots\!20\)\( T^{3} + \)\(89\!\cdots\!38\)\( T^{4} + \)\(33\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!40\)\( T^{6} + \)\(38\!\cdots\!80\)\( T^{7} + \)\(71\!\cdots\!61\)\( T^{8} \)
$19$ \( 1 + \)\(25\!\cdots\!60\)\( T + \)\(37\!\cdots\!36\)\( T^{2} + \)\(37\!\cdots\!20\)\( T^{3} + \)\(31\!\cdots\!86\)\( T^{4} + \)\(22\!\cdots\!80\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} + \)\(53\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$23$ \( 1 - \)\(69\!\cdots\!60\)\( T + \)\(50\!\cdots\!20\)\( T^{2} - \)\(27\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(40\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} - \)\(22\!\cdots\!20\)\( T^{7} + \)\(48\!\cdots\!21\)\( T^{8} \)
$29$ \( 1 + \)\(14\!\cdots\!40\)\( T + \)\(18\!\cdots\!56\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!26\)\( T^{4} + \)\(46\!\cdots\!20\)\( T^{5} + \)\(19\!\cdots\!76\)\( T^{6} + \)\(46\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \)
$31$ \( 1 + \)\(39\!\cdots\!32\)\( T + \)\(33\!\cdots\!48\)\( T^{2} + \)\(99\!\cdots\!84\)\( T^{3} + \)\(52\!\cdots\!70\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(40\!\cdots\!88\)\( T^{6} + \)\(53\!\cdots\!72\)\( T^{7} + \)\(14\!\cdots\!61\)\( T^{8} \)
$37$ \( 1 + \)\(28\!\cdots\!80\)\( T + \)\(36\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(60\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!20\)\( T^{5} + \)\(61\!\cdots\!20\)\( T^{6} + \)\(62\!\cdots\!40\)\( T^{7} + \)\(28\!\cdots\!81\)\( T^{8} \)
$41$ \( 1 - \)\(81\!\cdots\!88\)\( T + \)\(68\!\cdots\!88\)\( T^{2} + \)\(84\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!70\)\( T^{4} + \)\(25\!\cdots\!44\)\( T^{5} + \)\(61\!\cdots\!08\)\( T^{6} - \)\(22\!\cdots\!68\)\( T^{7} + \)\(81\!\cdots\!81\)\( T^{8} \)
$43$ \( 1 - \)\(45\!\cdots\!00\)\( T + \)\(22\!\cdots\!00\)\( T^{2} - \)\(55\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!98\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!00\)\( T^{6} - \)\(24\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + \)\(45\!\cdots\!40\)\( T + \)\(21\!\cdots\!40\)\( T^{2} + \)\(57\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!58\)\( T^{4} + \)\(24\!\cdots\!60\)\( T^{5} + \)\(38\!\cdots\!60\)\( T^{6} + \)\(32\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 - \)\(38\!\cdots\!20\)\( T + \)\(69\!\cdots\!40\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(21\!\cdots\!58\)\( T^{4} - \)\(40\!\cdots\!80\)\( T^{5} + \)\(41\!\cdots\!60\)\( T^{6} - \)\(56\!\cdots\!40\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} \)
$59$ \( 1 - \)\(16\!\cdots\!20\)\( T + \)\(23\!\cdots\!16\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!46\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{5} + \)\(12\!\cdots\!56\)\( T^{6} - \)\(59\!\cdots\!80\)\( T^{7} + \)\(26\!\cdots\!81\)\( T^{8} \)
$61$ \( 1 - \)\(65\!\cdots\!28\)\( T + \)\(79\!\cdots\!68\)\( T^{2} - \)\(10\!\cdots\!76\)\( T^{3} + \)\(33\!\cdots\!70\)\( T^{4} - \)\(42\!\cdots\!56\)\( T^{5} + \)\(14\!\cdots\!48\)\( T^{6} - \)\(48\!\cdots\!48\)\( T^{7} + \)\(30\!\cdots\!21\)\( T^{8} \)
$67$ \( 1 + \)\(11\!\cdots\!60\)\( T + \)\(16\!\cdots\!60\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!38\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} + \)\(60\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!61\)\( T^{8} \)
$71$ \( 1 - \)\(35\!\cdots\!48\)\( T + \)\(37\!\cdots\!08\)\( T^{2} - \)\(14\!\cdots\!96\)\( T^{3} + \)\(65\!\cdots\!70\)\( T^{4} - \)\(19\!\cdots\!56\)\( T^{5} + \)\(64\!\cdots\!68\)\( T^{6} - \)\(79\!\cdots\!88\)\( T^{7} + \)\(29\!\cdots\!41\)\( T^{8} \)
$73$ \( 1 + \)\(70\!\cdots\!40\)\( T + \)\(40\!\cdots\!20\)\( T^{2} + \)\(14\!\cdots\!20\)\( T^{3} + \)\(41\!\cdots\!78\)\( T^{4} + \)\(80\!\cdots\!60\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!21\)\( T^{8} \)
$79$ \( 1 + \)\(15\!\cdots\!40\)\( T + \)\(10\!\cdots\!56\)\( T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(51\!\cdots\!26\)\( T^{4} + \)\(44\!\cdots\!20\)\( T^{5} + \)\(14\!\cdots\!76\)\( T^{6} + \)\(82\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!41\)\( T^{8} \)
$83$ \( 1 + \)\(26\!\cdots\!20\)\( T + \)\(41\!\cdots\!60\)\( T^{2} + \)\(45\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!38\)\( T^{4} + \)\(23\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} + \)\(35\!\cdots\!40\)\( T^{7} + \)\(69\!\cdots\!61\)\( T^{8} \)
$89$ \( 1 + \)\(37\!\cdots\!20\)\( T + \)\(51\!\cdots\!76\)\( T^{2} + \)\(97\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!66\)\( T^{4} + \)\(20\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!36\)\( T^{6} + \)\(33\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!21\)\( T^{8} \)
$97$ \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(42\!\cdots\!40\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(28\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!60\)\( T^{6} - \)\(81\!\cdots\!80\)\( T^{7} + \)\(15\!\cdots\!41\)\( T^{8} \)
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