Properties

Label 1.50.a
Level 1
Weight 50
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{50}(\Gamma_0(1))\).

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

Trace form

\( 3q - 24225168q^{2} - 326954692404q^{3} + 31502767984896q^{4} + 63884035717079250q^{5} + 8906136249246768576q^{6} + 509391477498711192q^{7} + 12774393005516465664000q^{8} + 341625690512280455369319q^{9} + O(q^{10}) \) \( 3q - 24225168q^{2} - 326954692404q^{3} + 31502767984896q^{4} + 63884035717079250q^{5} + 8906136249246768576q^{6} + 509391477498711192q^{7} + 12774393005516465664000q^{8} + 341625690512280455369319q^{9} - 1782704508900495946524000q^{10} - 20839986192711815613370524q^{11} - 101756595014280089890126848q^{12} - 187524857197340618282341014q^{13} - 9454369415029168241295938688q^{14} - 204230679539215002884128467000q^{15} - 958699697068386571735634214912q^{16} - 3305035636247475259184114894538q^{17} - 20227965712272756371678796043344q^{18} - 45882942802931464757155696518660q^{19} + 19614616637279540996591887296000q^{20} + 617794442803555252870956383262816q^{21} + 1789915191438262082177360193562944q^{22} + 4533392548023086653463590407813576q^{23} - 2513947703938508593953436558049280q^{24} - 13921151905580403838435182832171875q^{25} - 85826222237480824536590109710199264q^{26} - 316454625615073819697154158592097800q^{27} - 59263339345763305527729286468663296q^{28} + 1118089919316361970349176934175494810q^{29} + 5019197819712535273166303836165776000q^{30} + 934226816157578575389156342339525216q^{31} + 731728685735702515419064521056059392q^{32} - 24487068877327908023713382739550873968q^{33} - 44386547077569962679458968510685418528q^{34} - 118496887985378726635932152589238374000q^{35} + 37986811000754375090144064848875953408q^{36} + 242135148519373130928922851818564983602q^{37} + 1156029602765443663092741329710471492800q^{38} + 1291017145684876237790670030697385259048q^{39} + 521466322563622592824981612412298240000q^{40} - 6211739675976490815696297371880432723714q^{41} - 14577965575239226149151901656398676210176q^{42} - 7795173962887472637660666233812713782844q^{43} + 2342252938597505040255206661879982328832q^{44} + 76758284157261249391010855229819118430250q^{45} + 26787138361451470082912377308943265940096q^{46} + 72757717838871152990888073123214445891472q^{47} + 156517630976899733143320027855300206002176q^{48} - 280157966384090888246597869444400847657429q^{49} - 537697283161263679965664107392030595750000q^{50} - 1538734957331600122304206020917586572488104q^{51} - 123529799747407597799198092229890019578368q^{52} - 155850890670432045962015052011776669693854q^{53} + 8252737541190722285688590807826219494394240q^{54} + 4640026560861316230546160030865773419591000q^{55} + 6722101775263505338291028050795018900439040q^{56} - 7909742361043655643598112535376781525014800q^{57} - 16549417550295130349814323361508282251506400q^{58} - 62596927936822013761436024025606581543386380q^{59} - 8860237536153508564855019737671281942784000q^{60} - 24037263967789435219196348055077866809754374q^{61} + 188825800588885099392385526941761279238024704q^{62} - 79613772299337773967480392108297858821089864q^{63} + 514652981263254699809134733483815976080244736q^{64} - 244975900670806240216695171564505814930314500q^{65} + 520146826433507127417478540282210913545819392q^{66} - 1019040852324305100821779153549119164107459188q^{67} + 147929481309005416179289053660840704965292544q^{68} - 4867866269191569039747093560336346164459611872q^{69} + 2807457123724968989995337407321309803024672000q^{70} - 3208129335291950593367863544185773469328334504q^{71} + 10389376832262916146131931928642229401854259200q^{72} + 5668108076325412165814435883536677392865317726q^{73} + 10012493148932275797653251806650724700277983392q^{74} - 12104818074938487392123628092507024931595687500q^{75} + 753841821393723585433963960120373529551508480q^{76} - 32043339793247959180079389483660528277811891936q^{77} - 27771485354519690073790188371684714833523265408q^{78} - 7871365395380306420112251724245018715125364240q^{79} - 30434832804539714147732706266001913558351872000q^{80} + 67270927730095597004530465543730351634592280043q^{81} + 118639125705754411670909153459409940471487282784q^{82} + 94164681901361229355849854653903786480577085116q^{83} + 7700494417700399139591084235352419616317120512q^{84} + 297863286656785614551226799130947646752077538500q^{85} - 347916436657924446515600473690440815675104929984q^{86} - 175500245979701689466461020645367234129141215000q^{87} - 1057522149818049090984983413770258669302102016000q^{88} + 367807201561165354579457777200332871192114777230q^{89} - 2007318616781959003323900738484966136471887212000q^{90} + 1613372397893666548508325640004129525440352221776q^{91} + 467187868629171186038576611613065932844001728512q^{92} + 5954871901405488350696717407304686000200433533312q^{93} + 904097685092930858948295844890436017323554129152q^{94} + 1475636199748052355599968141572624301097796705000q^{95} - 2698554985048479483842995984936051823306493394944q^{96} - 10431108583780665799950508389192126734793760399578q^{97} - 2841886978251444216347516645227034756825518204176q^{98} - 11387023179914295087805432025124514927226421923052q^{99} + O(q^{100}) \)

Decomposition of \(S_{50}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.50.a.a \(3\) \(15.207\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-24225168\) \(-326954692404\) \(63\!\cdots\!50\) \(50\!\cdots\!92\) \(+\) \(q+(-8075056+\beta _{1})q^{2}+(-108984897468+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 24225168 T + 1122102928453632 T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(63\!\cdots\!84\)\( T^{4} + \)\(76\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} \)
$3$ \( 1 + 326954692404 T + \)\(24\!\cdots\!73\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!59\)\( T^{4} + \)\(18\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!87\)\( T^{6} \)
$5$ \( 1 - 63884035717079250 T + \)\(35\!\cdots\!75\)\( T^{2} - \)\(22\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!75\)\( T^{4} - \)\(20\!\cdots\!50\)\( T^{5} + \)\(56\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 509391477498711192 T + \)\(52\!\cdots\!57\)\( T^{2} - \)\(44\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!99\)\( T^{4} - \)\(33\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + \)\(20\!\cdots\!24\)\( T + \)\(19\!\cdots\!65\)\( T^{2} + \)\(41\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!15\)\( T^{4} + \)\(23\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 + \)\(18\!\cdots\!14\)\( T + \)\(76\!\cdots\!03\)\( T^{2} + \)\(72\!\cdots\!20\)\( T^{3} + \)\(29\!\cdots\!19\)\( T^{4} + \)\(27\!\cdots\!06\)\( T^{5} + \)\(56\!\cdots\!17\)\( T^{6} \)
$17$ \( 1 + \)\(33\!\cdots\!38\)\( T + \)\(65\!\cdots\!07\)\( T^{2} + \)\(96\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!79\)\( T^{4} + \)\(12\!\cdots\!42\)\( T^{5} + \)\(75\!\cdots\!73\)\( T^{6} \)
$19$ \( 1 + \)\(45\!\cdots\!60\)\( T + \)\(17\!\cdots\!37\)\( T^{2} + \)\(41\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!23\)\( T^{4} + \)\(95\!\cdots\!60\)\( T^{5} + \)\(94\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 - \)\(45\!\cdots\!76\)\( T + \)\(10\!\cdots\!33\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(56\!\cdots\!79\)\( T^{4} - \)\(12\!\cdots\!44\)\( T^{5} + \)\(14\!\cdots\!47\)\( T^{6} \)
$29$ \( 1 - \)\(11\!\cdots\!10\)\( T + \)\(17\!\cdots\!07\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(80\!\cdots\!83\)\( T^{4} - \)\(23\!\cdots\!10\)\( T^{5} + \)\(93\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 - \)\(93\!\cdots\!16\)\( T + \)\(96\!\cdots\!65\)\( T^{2} + \)\(30\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!15\)\( T^{4} - \)\(13\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 - \)\(24\!\cdots\!02\)\( T + \)\(16\!\cdots\!07\)\( T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!39\)\( T^{4} - \)\(11\!\cdots\!58\)\( T^{5} + \)\(33\!\cdots\!33\)\( T^{6} \)
$41$ \( 1 + \)\(62\!\cdots\!14\)\( T + \)\(41\!\cdots\!15\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!15\)\( T^{4} + \)\(70\!\cdots\!94\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 + \)\(77\!\cdots\!44\)\( T + \)\(27\!\cdots\!93\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!99\)\( T^{4} + \)\(93\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - \)\(72\!\cdots\!72\)\( T + \)\(24\!\cdots\!57\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!19\)\( T^{4} - \)\(53\!\cdots\!08\)\( T^{5} + \)\(62\!\cdots\!63\)\( T^{6} \)
$53$ \( 1 + \)\(15\!\cdots\!54\)\( T + \)\(82\!\cdots\!23\)\( T^{2} + \)\(50\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!59\)\( T^{4} + \)\(14\!\cdots\!06\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \)
$59$ \( 1 + \)\(62\!\cdots\!80\)\( T + \)\(28\!\cdots\!17\)\( T^{2} + \)\(75\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!63\)\( T^{4} + \)\(21\!\cdots\!80\)\( T^{5} + \)\(20\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 + \)\(24\!\cdots\!74\)\( T + \)\(61\!\cdots\!15\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} + \)\(22\!\cdots\!94\)\( T^{5} + \)\(27\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 + \)\(10\!\cdots\!88\)\( T + \)\(61\!\cdots\!57\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!79\)\( T^{4} + \)\(91\!\cdots\!92\)\( T^{5} + \)\(27\!\cdots\!23\)\( T^{6} \)
$71$ \( 1 + \)\(32\!\cdots\!04\)\( T + \)\(17\!\cdots\!65\)\( T^{2} + \)\(32\!\cdots\!80\)\( T^{3} + \)\(90\!\cdots\!15\)\( T^{4} + \)\(85\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 - \)\(56\!\cdots\!26\)\( T + \)\(30\!\cdots\!83\)\( T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(61\!\cdots\!79\)\( T^{4} - \)\(22\!\cdots\!94\)\( T^{5} + \)\(80\!\cdots\!97\)\( T^{6} \)
$79$ \( 1 + \)\(78\!\cdots\!40\)\( T + \)\(49\!\cdots\!57\)\( T^{2} + \)\(50\!\cdots\!20\)\( T^{3} + \)\(47\!\cdots\!83\)\( T^{4} + \)\(73\!\cdots\!40\)\( T^{5} + \)\(89\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 - \)\(94\!\cdots\!16\)\( T + \)\(14\!\cdots\!13\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(15\!\cdots\!39\)\( T^{4} - \)\(11\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!27\)\( T^{6} \)
$89$ \( 1 - \)\(36\!\cdots\!30\)\( T + \)\(37\!\cdots\!27\)\( T^{2} - \)\(88\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!43\)\( T^{4} - \)\(40\!\cdots\!30\)\( T^{5} + \)\(36\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 + \)\(10\!\cdots\!78\)\( T + \)\(73\!\cdots\!07\)\( T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!19\)\( T^{4} + \)\(52\!\cdots\!42\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \)
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