Properties

Label 1.50.a
Level 1
Weight 50
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 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: \(\Q\)
Newforms: \( 1 \)
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

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

Decomposition of \(S_{50}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

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 \(-24225168\) \(-326954692404\) \(63\!\cdots\!50\) \(50\!\cdots\!92\) \(+\) \(q+(-8075056+\beta _{1})q^{2}+(-108984897468+\cdots)q^{3}+\cdots\)