Properties

Label 1.58.a
Level 1
Weight 58
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 58 \)
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 \(M_{58}(\Gamma_0(1))\).

Total New Old
Modular forms 5 5 0
Cusp forms 4 4 0
Eisenstein series 1 1 0

Trace form

\(4q \) \(\mathstrut -\mathstrut 217744560q^{2} \) \(\mathstrut +\mathstrut 37475862172560q^{3} \) \(\mathstrut +\mathstrut 293124896311376128q^{4} \) \(\mathstrut -\mathstrut 106961276207842698600q^{5} \) \(\mathstrut -\mathstrut 33990084162012424888512q^{6} \) \(\mathstrut +\mathstrut 952854504056592488640800q^{7} \) \(\mathstrut -\mathstrut 88551069216810449485639680q^{8} \) \(\mathstrut +\mathstrut 807299095982681497210590132q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 217744560q^{2} \) \(\mathstrut +\mathstrut 37475862172560q^{3} \) \(\mathstrut +\mathstrut 293124896311376128q^{4} \) \(\mathstrut -\mathstrut 106961276207842698600q^{5} \) \(\mathstrut -\mathstrut 33990084162012424888512q^{6} \) \(\mathstrut +\mathstrut 952854504056592488640800q^{7} \) \(\mathstrut -\mathstrut 88551069216810449485639680q^{8} \) \(\mathstrut +\mathstrut 807299095982681497210590132q^{9} \) \(\mathstrut +\mathstrut 44476431289760320477662520800q^{10} \) \(\mathstrut -\mathstrut 462030036788542994175991820112q^{11} \) \(\mathstrut +\mathstrut 10003750272364901761239114286080q^{12} \) \(\mathstrut -\mathstrut 35670303061909387812013418679880q^{13} \) \(\mathstrut +\mathstrut 70403094050531232484583448299136q^{14} \) \(\mathstrut -\mathstrut 3873003953749638547444662603837600q^{15} \) \(\mathstrut +\mathstrut 13566910890781728442321186411577344q^{16} \) \(\mathstrut +\mathstrut 13820011728830480443629140931368520q^{17} \) \(\mathstrut -\mathstrut 1287737321088043352155941276317529840q^{18} \) \(\mathstrut +\mathstrut 336853252899071533902360882896311760q^{19} \) \(\mathstrut -\mathstrut 50382729988973225454329459567578483200q^{20} \) \(\mathstrut -\mathstrut 114908173756075652427942608537020743552q^{21} \) \(\mathstrut -\mathstrut 661915503282569637498974918979285510720q^{22} \) \(\mathstrut -\mathstrut 2975542065258688382050024994957196661920q^{23} \) \(\mathstrut -\mathstrut 10578976944220511671340845604322576875520q^{24} \) \(\mathstrut -\mathstrut 9469406228848045007865812554313674602500q^{25} \) \(\mathstrut -\mathstrut 17469964426099436483582477475571041235872q^{26} \) \(\mathstrut +\mathstrut 76349739563972016366032809192591896661920q^{27} \) \(\mathstrut +\mathstrut 649254606795882967703588744692964385720320q^{28} \) \(\mathstrut +\mathstrut 692903173502123074211699189163367889819640q^{29} \) \(\mathstrut +\mathstrut 5634330080537746852583656641742556105212800q^{30} \) \(\mathstrut -\mathstrut 899836592298317074905406089703577976621952q^{31} \) \(\mathstrut -\mathstrut 14233950964359390830124363997913062975733760q^{32} \) \(\mathstrut -\mathstrut 29363967329346027307019371741967172628226880q^{33} \) \(\mathstrut -\mathstrut 169043128317257524614754408952689830505725024q^{34} \) \(\mathstrut -\mathstrut 229326244292419606493088342114240299318011200q^{35} \) \(\mathstrut +\mathstrut 284785472553153735165042638775100342213281024q^{36} \) \(\mathstrut +\mathstrut 1088031360501840746049603861287321623324833560q^{37} \) \(\mathstrut +\mathstrut 1556247388283245464794907962939214603341474880q^{38} \) \(\mathstrut +\mathstrut 4179255936739867118004084422081931511953729504q^{39} \) \(\mathstrut +\mathstrut 11751904708902009656918381872378671214758912000q^{40} \) \(\mathstrut -\mathstrut 11103291557355181141038304173539140276092022872q^{41} \) \(\mathstrut -\mathstrut 59279988331696939901744072326763120527717946880q^{42} \) \(\mathstrut -\mathstrut 61754953860901881142563948460092952248410491600q^{43} \) \(\mathstrut -\mathstrut 133524603192894473629581294520903680469781443584q^{44} \) \(\mathstrut -\mathstrut 107702873160032711230811245899648546573795049800q^{45} \) \(\mathstrut +\mathstrut 784819010796366200631618785819800311186796983168q^{46} \) \(\mathstrut +\mathstrut 363083109323271524824455302894174707948361522880q^{47} \) \(\mathstrut +\mathstrut 2742506186706372359330741580910160874632313569280q^{48} \) \(\mathstrut +\mathstrut 1216129915477179767594559246649850886387208070628q^{49} \) \(\mathstrut -\mathstrut 5243147625423300757899019878966089303259662130000q^{50} \) \(\mathstrut -\mathstrut 764600229637750496670397948984385624631693707232q^{51} \) \(\mathstrut -\mathstrut 22843788670121162835092909423272591563858567692800q^{52} \) \(\mathstrut -\mathstrut 35290483593822076697173728695179253314838955459240q^{53} \) \(\mathstrut +\mathstrut 12366575839681539833915263850508773584456529324160q^{54} \) \(\mathstrut +\mathstrut 49196225093152182399971681538771907122934246240800q^{55} \) \(\mathstrut +\mathstrut 63731767886683182211705479053660664279663635496960q^{56} \) \(\mathstrut +\mathstrut 207676196582248616809244588606912583584671767915840q^{57} \) \(\mathstrut +\mathstrut 617562539749243989517600779192095966343296224852320q^{58} \) \(\mathstrut +\mathstrut 67768373912933539906296460571320281490527711258480q^{59} \) \(\mathstrut -\mathstrut 1815042162748644358648991332911151647292658713651200q^{60} \) \(\mathstrut -\mathstrut 1547288835446782082478245361946256543823484047403912q^{61} \) \(\mathstrut -\mathstrut 749692944011811307313821057793297922348443552785920q^{62} \) \(\mathstrut -\mathstrut 1569461042763431092010126958426126899190556123563360q^{63} \) \(\mathstrut -\mathstrut 1990114977445631941316528741938069721703684937613312q^{64} \) \(\mathstrut +\mathstrut 8063572248542729948506878297975365372089154570456400q^{65} \) \(\mathstrut +\mathstrut 16489629555376763758820085001622755676656730822766336q^{66} \) \(\mathstrut +\mathstrut 15272577061969964687199674310464241542844062558207120q^{67} \) \(\mathstrut +\mathstrut 41305128999321265264790818598430329317542515380584960q^{68} \) \(\mathstrut -\mathstrut 65786095162933072363901793014646952768970242331074176q^{69} \) \(\mathstrut -\mathstrut 24325103535157938973881571830399397699429009026566400q^{70} \) \(\mathstrut -\mathstrut 73888360588713834794031605853188887812717485611902432q^{71} \) \(\mathstrut -\mathstrut 210328265328797773276338939967226276847978373484605440q^{72} \) \(\mathstrut -\mathstrut 130297268842197178419561798573986372195905283718671320q^{73} \) \(\mathstrut +\mathstrut 7685145516671187561069432006215096921673721366642656q^{74} \) \(\mathstrut +\mathstrut 403928833669880609006163131916358629442839611491110000q^{75} \) \(\mathstrut +\mathstrut 1456086045339107958455546112995861351598648681192657920q^{76} \) \(\mathstrut -\mathstrut 106427745818663343042165176190596036157925407112924800q^{77} \) \(\mathstrut +\mathstrut 2074950811168633790188899303505400649698048509219286400q^{78} \) \(\mathstrut -\mathstrut 828943058463042070981602117491995742498435075805703360q^{79} \) \(\mathstrut -\mathstrut 2284181918093911007938097065798931209886961598847385600q^{80} \) \(\mathstrut -\mathstrut 7963307412965608591332950484302680892196600516966334236q^{81} \) \(\mathstrut -\mathstrut 1883428276461442027198836795157559789817480540746193120q^{82} \) \(\mathstrut -\mathstrut 9840480688986571601216593400698100805429733592804704560q^{83} \) \(\mathstrut +\mathstrut 13541340719968270381664710206581288963145836751957467136q^{84} \) \(\mathstrut -\mathstrut 14840134943015755845177119052592106190913842871494565200q^{85} \) \(\mathstrut +\mathstrut 67988974963281087138264344329950027971018743502811738048q^{86} \) \(\mathstrut -\mathstrut 27365055454269193487844075068414523895320708309490878240q^{87} \) \(\mathstrut +\mathstrut 61847074514255305493001192533786291945351930637515735040q^{88} \) \(\mathstrut -\mathstrut 10663966381455218428814337779016832517355842990125366680q^{89} \) \(\mathstrut +\mathstrut 144098850070698626396745850922715348604587043628376674400q^{90} \) \(\mathstrut -\mathstrut 254422712153651549623482540524255137118230327668042523712q^{91} \) \(\mathstrut -\mathstrut 274534438781355859366149237613545893970204939554260346880q^{92} \) \(\mathstrut -\mathstrut 414128946656565192633171191785208529876343680904079162880q^{93} \) \(\mathstrut +\mathstrut 632898423704876935427548603685166753697821378730383050496q^{94} \) \(\mathstrut -\mathstrut 532156934024946939894578863048125180396162968766201396000q^{95} \) \(\mathstrut +\mathstrut 663745987198780197281626102963422448954448917470231134208q^{96} \) \(\mathstrut -\mathstrut 152729877520747450623636953478949994021736793474425664120q^{97} \) \(\mathstrut +\mathstrut 2296925218269808931776551039888426925574868525366690486480q^{98} \) \(\mathstrut +\mathstrut 123613071571796047703935039848745969833181068960384152304q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{58}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.58.a.a \(4\) \(20.577\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-217744560\) \(37\!\cdots\!60\) \(-1\!\cdots\!00\) \(95\!\cdots\!00\) \(+\) \(q+(-54436140+\beta _{1})q^{2}+(9368965543140+\cdots)q^{3}+\cdots\)