Properties

Label 1.36.a
Level 1
Weight 36
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: \( k \) \(=\) \( 36 \)
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 \(M_{36}(\Gamma_0(1))\).

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

Trace form

\( 3q + 139656q^{2} - 104875308q^{3} + 34841262144q^{4} + 892652054010q^{5} - 4786530564384q^{6} + 878422149346056q^{7} + 22336009925337600q^{8} + 150091978876243551q^{9} + O(q^{10}) \) \( 3q + 139656q^{2} - 104875308q^{3} + 34841262144q^{4} + 892652054010q^{5} - 4786530564384q^{6} + 878422149346056q^{7} + 22336009925337600q^{8} + 150091978876243551q^{9} + 1019870812729298160q^{10} - 1157945428549987044q^{11} - 22548891526776486144q^{12} - 62139610550998650558q^{13} + 135909343255071438528q^{14} + 706062479352366674520q^{15} + 2155694017285818421248q^{16} - 3932636930193139724394q^{17} - 23007400170989698594008q^{18} - 32303396242934620550940q^{19} + 147257059647928607018880q^{20} + 222122458753667398807776q^{21} + 22658809266113247402912q^{22} - 514822069421375217224808q^{23} - 3760729014923614708254720q^{24} + 646885156836409574920725q^{25} + 4284481859934908169468336q^{26} + 27373531144823515220747400q^{27} + 10448754713452503022622208q^{28} - 38460399143984437598248110q^{29} - 234613011392444285921119680q^{30} + 103731594563724689730029856q^{31} - 9222369125611166314758144q^{32} + 1184565114402567148957069584q^{33} - 554136735348503783026707312q^{34} + 1596564453000564139676248560q^{35} - 6052204716130590310531484352q^{36} + 2464613172948888017021063706q^{37} - 12543374691225905672793885600q^{38} + 18611981954062877742706387512q^{39} + 16458708947711983604007091200q^{40} + 23522986584532768129037087406q^{41} - 33931371616975894880367698688q^{42} - 47516297135443858184325274308q^{43} - 80954820698705302013923898112q^{44} - 110576300682246572226345828030q^{45} + 310121697020384938950505889856q^{46} + 164346768841318947159729902256q^{47} + 445342931723856383878644744192q^{48} - 595089038084925731622037450821q^{49} + 373669468267473207632799996600q^{50} - 1803044074850794727352259927704q^{51} - 511632617456400883939009686144q^{52} - 1670480551111723484744458809558q^{53} + 4424231404769003900494105901760q^{54} + 3067178595313878889174894808520q^{55} + 5670920217413056567252590243840q^{56} + 4048712721984947152977307522800q^{57} - 23757672063982901326871265032400q^{58} + 4370528583092646699845243631180q^{59} - 40771862895691439845448715210240q^{60} + 23537078276516028947219080488306q^{61} + 29401231764447267309574710842112q^{62} + 45614247892704690123948892506792q^{63} + 93510280736634320631093264384q^{64} + 75762337609865790449417835274620q^{65} - 75640417841065837797640948042368q^{66} - 188845033384156918977315211921644q^{67} + 21990159186015794982823459732608q^{68} - 325983212396274250308653315934048q^{69} + 223263426310223328568197933240960q^{70} + 348774602295358210538621162006856q^{71} + 314101495165322886769992842611200q^{72} - 285174964854822786615792218987058q^{73} + 2106337383176566865790045395789808q^{74} - 1577819189216049218913461709237300q^{75} - 2729074287668324197849535448894720q^{76} + 691501421964810422051915307575712q^{77} - 2213086042571083132613626950273216q^{78} - 426212862591061661901736630874160q^{79} + 6831119561623806264802413342351360q^{80} + 1867221150796978455595408934723163q^{81} - 4061181362173286538873115152234288q^{82} + 14867187331664065317008883105766692q^{83} - 13024788508690842242825242575869952q^{84} - 8752040931978292430566755304420140q^{85} + 148490750656470812540405098190496q^{86} - 7833119385476156993771606256673800q^{87} - 18777218849949122679898044025804800q^{88} + 30583590846193751335205992418034270q^{89} + 2860720282233327924484406809049520q^{90} + 1003391702519487362566122156997296q^{91} + 83662070125024358323030039384121856q^{92} - 40914650682166972019906140161446016q^{93} + 19076193270418512575422277142000768q^{94} - 84521392849966457182141665888160200q^{95} - 32392027865954425966997293222723584q^{96} - 106165667044630951063328618701091994q^{97} - 13202051562017808504422115585262392q^{98} + 30288686341597731928441860752034252q^{99} + O(q^{100}) \)

Decomposition of \(S_{36}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.36.a.a \(3\) \(7.760\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(139656\) \(-104875308\) \(892652054010\) \(87\!\cdots\!56\) \(+\) \(q+(46552+\beta _{1})q^{2}+(-34958436+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 139656 T + 43870875648 T^{2} - 11064712290631680 T^{3} + \)\(15\!\cdots\!64\)\( T^{4} - \)\(16\!\cdots\!44\)\( T^{5} + \)\(40\!\cdots\!32\)\( T^{6} \)
$3$ \( 1 + 104875308 T + 5500743324425217 T^{2} - \)\(10\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!19\)\( T^{4} + \)\(26\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!43\)\( T^{6} \)
$5$ \( 1 - 892652054010 T + \)\(44\!\cdots\!75\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!75\)\( T^{4} - \)\(75\!\cdots\!50\)\( T^{5} + \)\(24\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 878422149346056 T + \)\(12\!\cdots\!93\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(47\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 + 1157945428549987044 T + \)\(66\!\cdots\!65\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} + \)\(91\!\cdots\!44\)\( T^{5} + \)\(22\!\cdots\!51\)\( T^{6} \)
$13$ \( 1 + 62139610550998650558 T + \)\(35\!\cdots\!27\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(34\!\cdots\!39\)\( T^{4} + \)\(58\!\cdots\!42\)\( T^{5} + \)\(92\!\cdots\!93\)\( T^{6} \)
$17$ \( 1 + \)\(39\!\cdots\!94\)\( T + \)\(33\!\cdots\!83\)\( T^{2} + \)\(84\!\cdots\!60\)\( T^{3} + \)\(38\!\cdots\!19\)\( T^{4} + \)\(53\!\cdots\!06\)\( T^{5} + \)\(15\!\cdots\!57\)\( T^{6} \)
$19$ \( 1 + \)\(32\!\cdots\!40\)\( T + \)\(15\!\cdots\!97\)\( T^{2} + \)\(36\!\cdots\!20\)\( T^{3} + \)\(90\!\cdots\!03\)\( T^{4} + \)\(10\!\cdots\!40\)\( T^{5} + \)\(18\!\cdots\!99\)\( T^{6} \)
$23$ \( 1 + \)\(51\!\cdots\!08\)\( T + \)\(83\!\cdots\!37\)\( T^{2} + \)\(26\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!59\)\( T^{4} + \)\(10\!\cdots\!92\)\( T^{5} + \)\(95\!\cdots\!43\)\( T^{6} \)
$29$ \( 1 + \)\(38\!\cdots\!10\)\( T + \)\(30\!\cdots\!47\)\( T^{2} + \)\(92\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!03\)\( T^{4} + \)\(89\!\cdots\!10\)\( T^{5} + \)\(35\!\cdots\!49\)\( T^{6} \)
$31$ \( 1 - \)\(10\!\cdots\!56\)\( T + \)\(46\!\cdots\!65\)\( T^{2} - \)\(31\!\cdots\!20\)\( T^{3} + \)\(72\!\cdots\!15\)\( T^{4} - \)\(25\!\cdots\!56\)\( T^{5} + \)\(39\!\cdots\!51\)\( T^{6} \)
$37$ \( 1 - \)\(24\!\cdots\!06\)\( T + \)\(58\!\cdots\!63\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(45\!\cdots\!59\)\( T^{4} - \)\(14\!\cdots\!94\)\( T^{5} + \)\(45\!\cdots\!57\)\( T^{6} \)
$41$ \( 1 - \)\(23\!\cdots\!06\)\( T + \)\(83\!\cdots\!15\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!15\)\( T^{4} - \)\(18\!\cdots\!06\)\( T^{5} + \)\(21\!\cdots\!01\)\( T^{6} \)
$43$ \( 1 + \)\(47\!\cdots\!08\)\( T + \)\(45\!\cdots\!57\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(67\!\cdots\!99\)\( T^{4} + \)\(10\!\cdots\!92\)\( T^{5} + \)\(32\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(16\!\cdots\!56\)\( T + \)\(69\!\cdots\!53\)\( T^{2} - \)\(62\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!79\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{5} + \)\(37\!\cdots\!07\)\( T^{6} \)
$53$ \( 1 + \)\(16\!\cdots\!58\)\( T + \)\(28\!\cdots\!67\)\( T^{2} + \)\(29\!\cdots\!80\)\( T^{3} + \)\(64\!\cdots\!19\)\( T^{4} + \)\(83\!\cdots\!42\)\( T^{5} + \)\(11\!\cdots\!93\)\( T^{6} \)
$59$ \( 1 - \)\(43\!\cdots\!80\)\( T + \)\(26\!\cdots\!97\)\( T^{2} - \)\(72\!\cdots\!40\)\( T^{3} + \)\(24\!\cdots\!03\)\( T^{4} - \)\(39\!\cdots\!80\)\( T^{5} + \)\(86\!\cdots\!99\)\( T^{6} \)
$61$ \( 1 - \)\(23\!\cdots\!06\)\( T + \)\(92\!\cdots\!15\)\( T^{2} - \)\(14\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!15\)\( T^{4} - \)\(22\!\cdots\!06\)\( T^{5} + \)\(28\!\cdots\!01\)\( T^{6} \)
$67$ \( 1 + \)\(18\!\cdots\!44\)\( T + \)\(27\!\cdots\!33\)\( T^{2} + \)\(29\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!19\)\( T^{4} + \)\(12\!\cdots\!56\)\( T^{5} + \)\(54\!\cdots\!07\)\( T^{6} \)
$71$ \( 1 - \)\(34\!\cdots\!56\)\( T + \)\(16\!\cdots\!65\)\( T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{5} + \)\(24\!\cdots\!51\)\( T^{6} \)
$73$ \( 1 + \)\(28\!\cdots\!58\)\( T + \)\(22\!\cdots\!87\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(37\!\cdots\!59\)\( T^{4} + \)\(77\!\cdots\!42\)\( T^{5} + \)\(44\!\cdots\!93\)\( T^{6} \)
$79$ \( 1 + \)\(42\!\cdots\!60\)\( T + \)\(17\!\cdots\!97\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!03\)\( T^{4} + \)\(29\!\cdots\!60\)\( T^{5} + \)\(17\!\cdots\!99\)\( T^{6} \)
$83$ \( 1 - \)\(14\!\cdots\!92\)\( T + \)\(11\!\cdots\!97\)\( T^{2} - \)\(54\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!79\)\( T^{4} - \)\(32\!\cdots\!08\)\( T^{5} + \)\(31\!\cdots\!43\)\( T^{6} \)
$89$ \( 1 - \)\(30\!\cdots\!70\)\( T + \)\(80\!\cdots\!47\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!03\)\( T^{4} - \)\(87\!\cdots\!70\)\( T^{5} + \)\(48\!\cdots\!49\)\( T^{6} \)
$97$ \( 1 + \)\(10\!\cdots\!94\)\( T + \)\(12\!\cdots\!03\)\( T^{2} + \)\(72\!\cdots\!40\)\( T^{3} + \)\(42\!\cdots\!79\)\( T^{4} + \)\(12\!\cdots\!06\)\( T^{5} + \)\(40\!\cdots\!57\)\( T^{6} \)
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