Properties

Label 4.36.a
Level 4
Weight 36
Character orbit a
Rep. character \(\chi_{4}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 1
Sturm bound 18
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 4.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{36}(\Gamma_0(4))\).

Total New Old
Modular forms 19 3 16
Cusp forms 16 3 13
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim.
\(-\)\(3\)

Trace form

\( 3q + 50908884q^{3} + 280720890q^{5} - 5549296289016q^{7} - 4115694118097313q^{9} + O(q^{10}) \) \( 3q + 50908884q^{3} + 280720890q^{5} - 5549296289016q^{7} - 4115694118097313q^{9} - 420463958869151460q^{11} - 7370701315414736574q^{13} + 35737802623493622360q^{15} + 616802274448822103958q^{17} - 3523177807465210008348q^{19} - 35614415089970552016672q^{21} - 97200569155131383968872q^{23} - 1397444430311587979767275q^{25} - 13813263585892101497770872q^{27} - 72104888547394509330484398q^{29} - 305306760334252700869662432q^{31} - 1206118755141042924734246640q^{33} - 4139993873456897291989313040q^{35} - 10467805589252853184477978086q^{37} - 21306312046147482048067771848q^{39} - 28614809221544804827249675602q^{41} - 20661331953141752160177135300q^{43} + 109445534106618058172074201410q^{45} + 431671100729347757519313060528q^{47} + 1202271820749265164710734860219q^{49} + 1812312130969213433267629787496q^{51} + 2094590682902326557681205190442q^{53} - 424018895462656971242660883000q^{55} - 7056738639317962169249569687824q^{57} - 21898322952557962385462881418676q^{59} - 46553596324122391351852871731854q^{61} - 60534256794589553738854966512984q^{63} - 47301026213190020793582172528260q^{65} + 75982094975394264303461043768084q^{67} + 266168516453886472772566159456416q^{69} + 591751031973420639599491172643144q^{71} + 840249720714294658198657086588366q^{73} + 844328642366224046122007927037900q^{75} + 366867399909875520359123617491360q^{77} - 1512021839065233136089826558058544q^{79} - 4715323858342810015899488694567333q^{81} - 8944825848970974146478286242709212q^{83} - 11518175153366806194794571426747180q^{85} - 9434346897547249340232797055364104q^{87} + 1625382197094823895014829864559198q^{89} + 28933989451866564710063788057094832q^{91} + 43351982414907351295090224432607104q^{93} + 96155354088435090124008638354927160q^{95} + 83007021412015685382745763811847014q^{97} + 91112511762343802272300511308928460q^{99} + O(q^{100}) \)

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(4))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
4.36.a.a \(3\) \(31.038\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(50908884\) \(280720890\) \(-5\!\cdots\!16\) \(-\) \(q+(16969628+\beta _{1})q^{3}+(93573630+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{36}^{\mathrm{old}}(\Gamma_0(4))\) into lower level spaces

\( S_{36}^{\mathrm{old}}(\Gamma_0(4)) \cong \) \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{36}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 50908884 T + 78401021942610945 T^{2} + \)\(15\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!15\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$5$ \( 1 - 280720890 T + \)\(50\!\cdots\!75\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!75\)\( T^{4} - \)\(23\!\cdots\!50\)\( T^{5} + \)\(24\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + 5549296289016 T - \)\(32\!\cdots\!67\)\( T^{2} - \)\(26\!\cdots\!60\)\( T^{3} - \)\(12\!\cdots\!81\)\( T^{4} + \)\(79\!\cdots\!84\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 + 420463958869151460 T + \)\(35\!\cdots\!53\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(99\!\cdots\!03\)\( T^{4} + \)\(33\!\cdots\!60\)\( T^{5} + \)\(22\!\cdots\!51\)\( T^{6} \)
$13$ \( 1 + 7370701315414736574 T + \)\(12\!\cdots\!15\)\( T^{2} + \)\(10\!\cdots\!92\)\( T^{3} + \)\(12\!\cdots\!55\)\( T^{4} + \)\(69\!\cdots\!26\)\( T^{5} + \)\(92\!\cdots\!93\)\( T^{6} \)
$17$ \( 1 - \)\(61\!\cdots\!58\)\( T + \)\(14\!\cdots\!35\)\( T^{2} - \)\(66\!\cdots\!36\)\( T^{3} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(83\!\cdots\!42\)\( T^{5} + \)\(15\!\cdots\!57\)\( T^{6} \)
$19$ \( 1 + \)\(35\!\cdots\!48\)\( T + \)\(89\!\cdots\!65\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(50\!\cdots\!35\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} + \)\(18\!\cdots\!99\)\( T^{6} \)
$23$ \( 1 + \)\(97\!\cdots\!72\)\( T + \)\(80\!\cdots\!97\)\( T^{2} - \)\(91\!\cdots\!20\)\( T^{3} + \)\(36\!\cdots\!79\)\( T^{4} + \)\(20\!\cdots\!28\)\( T^{5} + \)\(95\!\cdots\!43\)\( T^{6} \)
$29$ \( 1 + \)\(72\!\cdots\!98\)\( T + \)\(60\!\cdots\!15\)\( T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(92\!\cdots\!35\)\( T^{4} + \)\(16\!\cdots\!98\)\( T^{5} + \)\(35\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 + \)\(30\!\cdots\!32\)\( T + \)\(68\!\cdots\!61\)\( T^{2} + \)\(96\!\cdots\!48\)\( T^{3} + \)\(10\!\cdots\!11\)\( T^{4} + \)\(75\!\cdots\!32\)\( T^{5} + \)\(39\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 + \)\(10\!\cdots\!86\)\( T + \)\(59\!\cdots\!23\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!39\)\( T^{4} + \)\(62\!\cdots\!14\)\( T^{5} + \)\(45\!\cdots\!57\)\( T^{6} \)
$41$ \( 1 + \)\(28\!\cdots\!02\)\( T + \)\(31\!\cdots\!71\)\( T^{2} + \)\(13\!\cdots\!08\)\( T^{3} + \)\(88\!\cdots\!71\)\( T^{4} + \)\(22\!\cdots\!02\)\( T^{5} + \)\(21\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(15\!\cdots\!21\)\( T^{2} + \)\(95\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!47\)\( T^{4} + \)\(45\!\cdots\!00\)\( T^{5} + \)\(32\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(43\!\cdots\!28\)\( T + \)\(93\!\cdots\!45\)\( T^{2} - \)\(14\!\cdots\!56\)\( T^{3} + \)\(31\!\cdots\!35\)\( T^{4} - \)\(48\!\cdots\!72\)\( T^{5} + \)\(37\!\cdots\!07\)\( T^{6} \)
$53$ \( 1 - \)\(20\!\cdots\!42\)\( T + \)\(66\!\cdots\!87\)\( T^{2} - \)\(80\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!59\)\( T^{4} - \)\(10\!\cdots\!58\)\( T^{5} + \)\(11\!\cdots\!93\)\( T^{6} \)
$59$ \( 1 + \)\(21\!\cdots\!76\)\( T + \)\(40\!\cdots\!89\)\( T^{2} + \)\(43\!\cdots\!36\)\( T^{3} + \)\(38\!\cdots\!11\)\( T^{4} + \)\(19\!\cdots\!76\)\( T^{5} + \)\(86\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 + \)\(46\!\cdots\!54\)\( T + \)\(14\!\cdots\!75\)\( T^{2} + \)\(30\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!75\)\( T^{4} + \)\(43\!\cdots\!54\)\( T^{5} + \)\(28\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 - \)\(75\!\cdots\!84\)\( T + \)\(14\!\cdots\!13\)\( T^{2} - \)\(46\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!59\)\( T^{4} - \)\(50\!\cdots\!16\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$71$ \( 1 - \)\(59\!\cdots\!44\)\( T + \)\(29\!\cdots\!65\)\( T^{2} - \)\(80\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} - \)\(22\!\cdots\!44\)\( T^{5} + \)\(24\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 - \)\(84\!\cdots\!66\)\( T + \)\(49\!\cdots\!35\)\( T^{2} - \)\(19\!\cdots\!48\)\( T^{3} + \)\(81\!\cdots\!95\)\( T^{4} - \)\(22\!\cdots\!34\)\( T^{5} + \)\(44\!\cdots\!93\)\( T^{6} \)
$79$ \( 1 + \)\(15\!\cdots\!44\)\( T + \)\(80\!\cdots\!09\)\( T^{2} + \)\(76\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!91\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 + \)\(89\!\cdots\!12\)\( T + \)\(61\!\cdots\!77\)\( T^{2} + \)\(24\!\cdots\!40\)\( T^{3} + \)\(89\!\cdots\!39\)\( T^{4} + \)\(19\!\cdots\!88\)\( T^{5} + \)\(31\!\cdots\!43\)\( T^{6} \)
$89$ \( 1 - \)\(16\!\cdots\!98\)\( T + \)\(16\!\cdots\!15\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(28\!\cdots\!35\)\( T^{4} - \)\(46\!\cdots\!98\)\( T^{5} + \)\(48\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 - \)\(83\!\cdots\!14\)\( T + \)\(11\!\cdots\!03\)\( T^{2} - \)\(57\!\cdots\!40\)\( T^{3} + \)\(39\!\cdots\!79\)\( T^{4} - \)\(98\!\cdots\!86\)\( T^{5} + \)\(40\!\cdots\!57\)\( T^{6} \)
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