Properties

Label 3.34
Level 3
Weight 34
Dimension 6
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 22
Trace bound 0

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Defining parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 34 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(22\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{34}(\Gamma_1(3))\).

Total New Old
Modular forms 12 6 6
Cusp forms 10 6 4
Eisenstein series 2 0 2

Trace form

\( 6q + 177822q^{2} + 13482087828q^{4} - 209226212244q^{5} - 4107432024378q^{6} + 86874432908088q^{7} + 2960501398298568q^{8} + 11118121133111046q^{9} + O(q^{10}) \) \( 6q + 177822q^{2} + 13482087828q^{4} - 209226212244q^{5} - 4107432024378q^{6} + 86874432908088q^{7} + 2960501398298568q^{8} + 11118121133111046q^{9} - 62509461110728236q^{10} - 449006118086241936q^{11} - 675705385624963212q^{12} + 2645460304339642212q^{13} - 623424763998893808q^{14} + 13419809246242181304q^{15} + 7934564529906055440q^{16} - 258653397384591512580q^{17} + 329507756022012070302q^{18} + 2328317563074002616432q^{19} - 2832186239254640505096q^{20} + 2813233869418186771704q^{21} - 36961884613614549287160q^{22} + 17355959020293329080944q^{23} - 28075187719107417563064q^{24} + 309760212790824676319706q^{25} + 240939296120424419983332q^{26} - 3179954796066109814583264q^{28} - 87520055623899111320436q^{29} + 2168520208582488630910596q^{30} + 9404979446089190524964376q^{31} - 1377800974112414254997472q^{32} + 10415525222258975937185496q^{33} - 70047252666116601527703204q^{34} + 42741570963318342266667600q^{35} + 24982580933157666841491348q^{36} - 24398623144056563705950332q^{37} + 138515143387720884389889336q^{38} + 51026072063587382678815728q^{39} - 666603972770909171495825232q^{40} - 82495185163465057480504404q^{41} + 788179982212326927140369424q^{42} + 689701393818295549314642768q^{43} - 174033782712048399977072016q^{44} - 387700395325132247736141204q^{45} + 3471243809650395496715385936q^{46} - 6836926096450491478798863024q^{47} - 4574194001448167622481425648q^{48} + 2153650856362724309507673750q^{49} - 7402582325439675829830417246q^{50} + 9674788827085671124615387248q^{51} + 42398780957106823522699650840q^{52} - 3852173940864945775224631236q^{53} - 7611154465509021146142179898q^{54} + 115421563110144076811045862336q^{55} - 390089298585719034008857160640q^{56} + 126232097217675503608179371016q^{57} - 72582753964845845996239927068q^{58} - 117982592922362455434893493312q^{59} + 696925242983005702549805489976q^{60} + 398824389169005399434159376180q^{61} - 500551338110073204948731499744q^{62} + 160980078073741816283402590008q^{63} + 582951333761644192897747140672q^{64} - 4825025736195151172098980215352q^{65} + 3801619691249157714568461812136q^{66} + 984965068868360104472930212800q^{67} - 6650821662680525716356005131416q^{68} + 6100220380573604686558690163232q^{69} + 14479638597204604630826115408480q^{70} - 19813073594379731833841112469968q^{71} + 5485868860171351827118834463688q^{72} + 15416367676240187982354626307324q^{73} - 52680237352801562465269545820428q^{74} + 23093952125795679596533694675424q^{75} + 29988221603153600309366590357008q^{76} - 17485051106419525012295737625184q^{77} + 34043746928968522497668334156180q^{78} + 49260232851840949380842264008440q^{79} - 148847362062960976344714747351456q^{80} + 20602102921755074907947094535686q^{81} + 44575651200347810159192872953516q^{82} - 146226750147659319601601214665424q^{83} + 140529829321164494053048180550688q^{84} + 31684625191014759932677029007992q^{85} + 119550971068579657891314147831624q^{86} + 104703420491450418708009240541992q^{87} - 151795518860587545649068868831392q^{88} - 19279840944590570783081053457028q^{89} - 115831293432428446551652219282476q^{90} - 391502509912830777018467946307824q^{91} + 386075363713040990861985630295008q^{92} - 553807765870392878194908982534488q^{93} - 214603113669067969123982552129952q^{94} + 1749705275522762958515615329497984q^{95} - 1264214847656649287350127672987616q^{96} - 60064883721192870252886280710740q^{97} + 2994672129981768309774723023624238q^{98} - 832017401731800053096906385004176q^{99} + O(q^{100}) \)

Decomposition of \(S_{34}^{\mathrm{new}}(\Gamma_1(3))\)

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
3.34.a \(\chi_{3}(1, \cdot)\) 3.34.a.a 3 1
3.34.a.b 3

Decomposition of \(S_{34}^{\mathrm{old}}(\Gamma_1(3))\) into lower level spaces

\( S_{34}^{\mathrm{old}}(\Gamma_1(3)) \cong \) \(S_{34}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 41202 T + 14287438176 T^{2} - 832098411159552 T^{3} + \)\(12\!\cdots\!92\)\( T^{4} - \)\(30\!\cdots\!28\)\( T^{5} + \)\(63\!\cdots\!88\)\( T^{6} \))(\( 1 - 136620 T + 14922636288 T^{2} - 1399657087107072 T^{3} + \)\(12\!\cdots\!96\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{5} + \)\(63\!\cdots\!88\)\( T^{6} \))
$3$ (\( ( 1 - 43046721 T )^{3} \))(\( ( 1 + 43046721 T )^{3} \))
$5$ (\( 1 - 51261823890 T - \)\(35\!\cdots\!25\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} - \)\(41\!\cdots\!25\)\( T^{4} - \)\(69\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!25\)\( T^{6} \))(\( 1 + 260488036134 T + \)\(26\!\cdots\!75\)\( T^{2} + \)\(57\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!75\)\( T^{4} + \)\(35\!\cdots\!50\)\( T^{5} + \)\(15\!\cdots\!25\)\( T^{6} \))
$7$ (\( 1 - 76113734015256 T + \)\(22\!\cdots\!65\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(17\!\cdots\!55\)\( T^{4} - \)\(45\!\cdots\!44\)\( T^{5} + \)\(46\!\cdots\!43\)\( T^{6} \))(\( 1 - 10760698892832 T + \)\(25\!\cdots\!61\)\( T^{2} - \)\(59\!\cdots\!48\)\( T^{3} + \)\(19\!\cdots\!27\)\( T^{4} - \)\(64\!\cdots\!68\)\( T^{5} + \)\(46\!\cdots\!43\)\( T^{6} \))
$11$ (\( 1 + 103523748885455580 T + \)\(52\!\cdots\!61\)\( T^{2} + \)\(32\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} + \)\(55\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} \))(\( 1 + 345482369200786356 T + \)\(65\!\cdots\!53\)\( T^{2} + \)\(91\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!43\)\( T^{4} + \)\(18\!\cdots\!16\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} \))
$13$ (\( 1 - 1915412601357848490 T + \)\(65\!\cdots\!31\)\( T^{2} - \)\(25\!\cdots\!52\)\( T^{3} + \)\(37\!\cdots\!43\)\( T^{4} - \)\(63\!\cdots\!10\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \))(\( 1 - 730047702981793722 T + \)\(58\!\cdots\!79\)\( T^{2} + \)\(55\!\cdots\!04\)\( T^{3} + \)\(33\!\cdots\!87\)\( T^{4} - \)\(24\!\cdots\!98\)\( T^{5} + \)\(19\!\cdots\!77\)\( T^{6} \))
$17$ (\( 1 + 16951254031996646346 T + \)\(10\!\cdots\!83\)\( T^{2} + \)\(11\!\cdots\!72\)\( T^{3} + \)\(44\!\cdots\!71\)\( T^{4} + \)\(27\!\cdots\!74\)\( T^{5} + \)\(65\!\cdots\!53\)\( T^{6} \))(\( 1 + \)\(24\!\cdots\!34\)\( T + \)\(13\!\cdots\!63\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{3} + \)\(52\!\cdots\!31\)\( T^{4} + \)\(39\!\cdots\!46\)\( T^{5} + \)\(65\!\cdots\!53\)\( T^{6} \))
$19$ (\( 1 - \)\(26\!\cdots\!64\)\( T + \)\(44\!\cdots\!37\)\( T^{2} - \)\(50\!\cdots\!52\)\( T^{3} + \)\(70\!\cdots\!83\)\( T^{4} - \)\(65\!\cdots\!84\)\( T^{5} + \)\(39\!\cdots\!79\)\( T^{6} \))(\( 1 + \)\(30\!\cdots\!32\)\( T + \)\(46\!\cdots\!13\)\( T^{2} + \)\(94\!\cdots\!56\)\( T^{3} + \)\(73\!\cdots\!67\)\( T^{4} + \)\(75\!\cdots\!92\)\( T^{5} + \)\(39\!\cdots\!79\)\( T^{6} \))
$23$ (\( 1 - \)\(79\!\cdots\!68\)\( T + \)\(45\!\cdots\!85\)\( T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(39\!\cdots\!55\)\( T^{4} - \)\(59\!\cdots\!52\)\( T^{5} + \)\(64\!\cdots\!87\)\( T^{6} \))(\( 1 + \)\(62\!\cdots\!24\)\( T + \)\(34\!\cdots\!69\)\( T^{2} + \)\(10\!\cdots\!84\)\( T^{3} + \)\(29\!\cdots\!27\)\( T^{4} + \)\(46\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!87\)\( T^{6} \))
$29$ (\( 1 - \)\(11\!\cdots\!58\)\( T + \)\(39\!\cdots\!87\)\( T^{2} - \)\(24\!\cdots\!24\)\( T^{3} + \)\(70\!\cdots\!43\)\( T^{4} - \)\(38\!\cdots\!18\)\( T^{5} + \)\(59\!\cdots\!69\)\( T^{6} \))(\( 1 + \)\(12\!\cdots\!94\)\( T + \)\(46\!\cdots\!51\)\( T^{2} + \)\(46\!\cdots\!32\)\( T^{3} + \)\(85\!\cdots\!39\)\( T^{4} + \)\(41\!\cdots\!74\)\( T^{5} + \)\(59\!\cdots\!69\)\( T^{6} \))
$31$ (\( 1 + \)\(17\!\cdots\!76\)\( T + \)\(28\!\cdots\!17\)\( T^{2} + \)\(47\!\cdots\!32\)\( T^{3} + \)\(46\!\cdots\!47\)\( T^{4} + \)\(46\!\cdots\!56\)\( T^{5} + \)\(44\!\cdots\!71\)\( T^{6} \))(\( 1 - \)\(11\!\cdots\!52\)\( T + \)\(86\!\cdots\!73\)\( T^{2} - \)\(40\!\cdots\!64\)\( T^{3} + \)\(14\!\cdots\!43\)\( T^{4} - \)\(29\!\cdots\!12\)\( T^{5} + \)\(44\!\cdots\!71\)\( T^{6} \))
$37$ (\( 1 - \)\(23\!\cdots\!86\)\( T + \)\(11\!\cdots\!95\)\( T^{2} - \)\(65\!\cdots\!24\)\( T^{3} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(73\!\cdots\!74\)\( T^{5} + \)\(17\!\cdots\!73\)\( T^{6} \))(\( 1 + \)\(47\!\cdots\!18\)\( T + \)\(52\!\cdots\!51\)\( T^{2} - \)\(22\!\cdots\!08\)\( T^{3} + \)\(29\!\cdots\!47\)\( T^{4} + \)\(15\!\cdots\!62\)\( T^{5} + \)\(17\!\cdots\!73\)\( T^{6} \))
$41$ (\( 1 - \)\(42\!\cdots\!14\)\( T + \)\(39\!\cdots\!27\)\( T^{2} - \)\(10\!\cdots\!68\)\( T^{3} + \)\(65\!\cdots\!67\)\( T^{4} - \)\(11\!\cdots\!74\)\( T^{5} + \)\(46\!\cdots\!61\)\( T^{6} \))(\( 1 + \)\(51\!\cdots\!18\)\( T + \)\(18\!\cdots\!83\)\( T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(30\!\cdots\!43\)\( T^{4} + \)\(14\!\cdots\!38\)\( T^{5} + \)\(46\!\cdots\!61\)\( T^{6} \))
$43$ (\( 1 - \)\(93\!\cdots\!44\)\( T + \)\(19\!\cdots\!53\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!79\)\( T^{4} - \)\(60\!\cdots\!56\)\( T^{5} + \)\(51\!\cdots\!07\)\( T^{6} \))(\( 1 + \)\(24\!\cdots\!76\)\( T + \)\(13\!\cdots\!33\)\( T^{2} + \)\(42\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} + \)\(15\!\cdots\!24\)\( T^{5} + \)\(51\!\cdots\!07\)\( T^{6} \))
$47$ (\( 1 + \)\(87\!\cdots\!52\)\( T + \)\(64\!\cdots\!57\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{3} + \)\(97\!\cdots\!39\)\( T^{4} + \)\(19\!\cdots\!08\)\( T^{5} + \)\(34\!\cdots\!83\)\( T^{6} \))(\( 1 - \)\(19\!\cdots\!28\)\( T + \)\(37\!\cdots\!17\)\( T^{2} - \)\(43\!\cdots\!20\)\( T^{3} + \)\(57\!\cdots\!59\)\( T^{4} - \)\(43\!\cdots\!12\)\( T^{5} + \)\(34\!\cdots\!83\)\( T^{6} \))
$53$ (\( 1 + \)\(11\!\cdots\!02\)\( T + \)\(18\!\cdots\!15\)\( T^{2} + \)\(16\!\cdots\!12\)\( T^{3} + \)\(14\!\cdots\!95\)\( T^{4} + \)\(75\!\cdots\!58\)\( T^{5} + \)\(50\!\cdots\!17\)\( T^{6} \))(\( 1 - \)\(80\!\cdots\!66\)\( T + \)\(17\!\cdots\!19\)\( T^{2} - \)\(16\!\cdots\!36\)\( T^{3} + \)\(14\!\cdots\!87\)\( T^{4} - \)\(50\!\cdots\!14\)\( T^{5} + \)\(50\!\cdots\!17\)\( T^{6} \))
$59$ (\( 1 - \)\(12\!\cdots\!56\)\( T + \)\(82\!\cdots\!37\)\( T^{2} - \)\(68\!\cdots\!48\)\( T^{3} + \)\(22\!\cdots\!23\)\( T^{4} - \)\(97\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!39\)\( T^{6} \))(\( 1 + \)\(24\!\cdots\!68\)\( T + \)\(92\!\cdots\!73\)\( T^{2} + \)\(13\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!67\)\( T^{4} + \)\(18\!\cdots\!88\)\( T^{5} + \)\(20\!\cdots\!39\)\( T^{6} \))
$61$ (\( 1 - \)\(36\!\cdots\!62\)\( T + \)\(12\!\cdots\!59\)\( T^{2} - \)\(22\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!79\)\( T^{4} - \)\(24\!\cdots\!82\)\( T^{5} + \)\(55\!\cdots\!41\)\( T^{6} \))(\( 1 - \)\(37\!\cdots\!18\)\( T + \)\(22\!\cdots\!99\)\( T^{2} - \)\(75\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!19\)\( T^{4} - \)\(25\!\cdots\!98\)\( T^{5} + \)\(55\!\cdots\!41\)\( T^{6} \))
$67$ (\( 1 + \)\(15\!\cdots\!96\)\( T + \)\(11\!\cdots\!33\)\( T^{2} + \)\(29\!\cdots\!72\)\( T^{3} + \)\(20\!\cdots\!71\)\( T^{4} + \)\(52\!\cdots\!24\)\( T^{5} + \)\(60\!\cdots\!03\)\( T^{6} \))(\( 1 - \)\(10\!\cdots\!96\)\( T + \)\(24\!\cdots\!33\)\( T^{2} - \)\(23\!\cdots\!72\)\( T^{3} + \)\(45\!\cdots\!71\)\( T^{4} - \)\(33\!\cdots\!24\)\( T^{5} + \)\(60\!\cdots\!03\)\( T^{6} \))
$71$ (\( 1 + \)\(66\!\cdots\!84\)\( T + \)\(51\!\cdots\!85\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!35\)\( T^{4} + \)\(10\!\cdots\!64\)\( T^{5} + \)\(18\!\cdots\!31\)\( T^{6} \))(\( 1 + \)\(13\!\cdots\!84\)\( T + \)\(94\!\cdots\!85\)\( T^{2} + \)\(40\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!35\)\( T^{4} + \)\(20\!\cdots\!64\)\( T^{5} + \)\(18\!\cdots\!31\)\( T^{6} \))
$73$ (\( 1 - \)\(17\!\cdots\!18\)\( T + \)\(16\!\cdots\!55\)\( T^{2} - \)\(10\!\cdots\!88\)\( T^{3} + \)\(52\!\cdots\!15\)\( T^{4} - \)\(16\!\cdots\!02\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \))(\( 1 + \)\(21\!\cdots\!94\)\( T + \)\(76\!\cdots\!79\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(23\!\cdots\!07\)\( T^{4} + \)\(20\!\cdots\!66\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \))
$79$ (\( 1 - \)\(47\!\cdots\!40\)\( T + \)\(18\!\cdots\!17\)\( T^{2} - \)\(41\!\cdots\!20\)\( T^{3} + \)\(79\!\cdots\!63\)\( T^{4} - \)\(82\!\cdots\!40\)\( T^{5} + \)\(73\!\cdots\!19\)\( T^{6} \))(\( 1 - \)\(22\!\cdots\!00\)\( T + \)\(70\!\cdots\!17\)\( T^{2} - \)\(23\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!63\)\( T^{4} - \)\(38\!\cdots\!00\)\( T^{5} + \)\(73\!\cdots\!19\)\( T^{6} \))
$83$ (\( 1 + \)\(10\!\cdots\!64\)\( T + \)\(92\!\cdots\!69\)\( T^{2} + \)\(46\!\cdots\!56\)\( T^{3} + \)\(19\!\cdots\!47\)\( T^{4} + \)\(50\!\cdots\!16\)\( T^{5} + \)\(97\!\cdots\!47\)\( T^{6} \))(\( 1 + \)\(36\!\cdots\!60\)\( T + \)\(97\!\cdots\!57\)\( T^{2} - \)\(13\!\cdots\!24\)\( T^{3} + \)\(20\!\cdots\!91\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} + \)\(97\!\cdots\!47\)\( T^{6} \))
$89$ (\( 1 - \)\(28\!\cdots\!94\)\( T + \)\(81\!\cdots\!67\)\( T^{2} - \)\(12\!\cdots\!72\)\( T^{3} + \)\(17\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!34\)\( T^{5} + \)\(97\!\cdots\!09\)\( T^{6} \))(\( 1 + \)\(30\!\cdots\!22\)\( T + \)\(93\!\cdots\!23\)\( T^{2} + \)\(13\!\cdots\!56\)\( T^{3} + \)\(20\!\cdots\!87\)\( T^{4} + \)\(13\!\cdots\!42\)\( T^{5} + \)\(97\!\cdots\!09\)\( T^{6} \))
$97$ (\( 1 - \)\(81\!\cdots\!54\)\( T + \)\(65\!\cdots\!03\)\( T^{2} - \)\(25\!\cdots\!48\)\( T^{3} + \)\(23\!\cdots\!31\)\( T^{4} - \)\(10\!\cdots\!66\)\( T^{5} + \)\(49\!\cdots\!33\)\( T^{6} \))(\( 1 + \)\(87\!\cdots\!94\)\( T + \)\(67\!\cdots\!43\)\( T^{2} + \)\(41\!\cdots\!68\)\( T^{3} + \)\(24\!\cdots\!11\)\( T^{4} + \)\(11\!\cdots\!26\)\( T^{5} + \)\(49\!\cdots\!33\)\( T^{6} \))
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