Properties

Label 1.58.a
Level 1
Weight 58
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 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: \(\Q\)
Newform subspaces: \( 1 \)
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

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

Decomposition of \(S_{58}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 217744560 T + 165374274700820480 T^{2} + \)\(51\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!68\)\( T^{4} + \)\(74\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!20\)\( T^{6} + \)\(65\!\cdots\!80\)\( T^{7} + \)\(43\!\cdots\!56\)\( T^{8} \)
$3$ \( 1 - 37475862172560 T + \)\(34\!\cdots\!60\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(82\!\cdots\!38\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(84\!\cdots\!40\)\( T^{6} - \)\(14\!\cdots\!20\)\( T^{7} + \)\(60\!\cdots\!61\)\( T^{8} \)
$5$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(24\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!50\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!00\)\( T^{7} + \)\(23\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - \)\(95\!\cdots\!00\)\( T + \)\(28\!\cdots\!00\)\( T^{2} - \)\(61\!\cdots\!00\)\( T^{3} + \)\(29\!\cdots\!98\)\( T^{4} - \)\(90\!\cdots\!00\)\( T^{5} + \)\(61\!\cdots\!00\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(48\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 + \)\(46\!\cdots\!12\)\( T + \)\(37\!\cdots\!88\)\( T^{2} + \)\(52\!\cdots\!64\)\( T^{3} + \)\(56\!\cdots\!70\)\( T^{4} + \)\(11\!\cdots\!44\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} + \)\(55\!\cdots\!32\)\( T^{7} + \)\(27\!\cdots\!81\)\( T^{8} \)
$13$ \( 1 + \)\(35\!\cdots\!80\)\( T + \)\(84\!\cdots\!20\)\( T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!78\)\( T^{4} + \)\(73\!\cdots\!20\)\( T^{5} + \)\(82\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + \)\(95\!\cdots\!21\)\( T^{8} \)
$17$ \( 1 - \)\(13\!\cdots\!20\)\( T + \)\(30\!\cdots\!40\)\( T^{2} - \)\(12\!\cdots\!40\)\( T^{3} + \)\(55\!\cdots\!58\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{5} + \)\(57\!\cdots\!60\)\( T^{6} - \)\(35\!\cdots\!60\)\( T^{7} + \)\(34\!\cdots\!41\)\( T^{8} \)
$19$ \( 1 - \)\(33\!\cdots\!60\)\( T + \)\(12\!\cdots\!56\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!26\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{5} + \)\(77\!\cdots\!76\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} \)
$23$ \( 1 + \)\(29\!\cdots\!20\)\( T + \)\(47\!\cdots\!80\)\( T^{2} + \)\(49\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!18\)\( T^{4} + \)\(20\!\cdots\!80\)\( T^{5} + \)\(81\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!40\)\( T^{7} + \)\(29\!\cdots\!81\)\( T^{8} \)
$29$ \( 1 - \)\(69\!\cdots\!40\)\( T + \)\(46\!\cdots\!36\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(55\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!16\)\( T^{6} - \)\(81\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!61\)\( T^{8} \)
$31$ \( 1 + \)\(89\!\cdots\!52\)\( T + \)\(23\!\cdots\!08\)\( T^{2} - \)\(59\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!70\)\( T^{4} - \)\(61\!\cdots\!56\)\( T^{5} + \)\(23\!\cdots\!68\)\( T^{6} + \)\(94\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!41\)\( T^{8} \)
$37$ \( 1 - \)\(10\!\cdots\!60\)\( T + \)\(12\!\cdots\!20\)\( T^{2} - \)\(80\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!78\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} + \)\(77\!\cdots\!80\)\( T^{6} - \)\(15\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!21\)\( T^{8} \)
$41$ \( 1 + \)\(11\!\cdots\!72\)\( T + \)\(34\!\cdots\!68\)\( T^{2} + \)\(27\!\cdots\!24\)\( T^{3} + \)\(44\!\cdots\!70\)\( T^{4} + \)\(23\!\cdots\!44\)\( T^{5} + \)\(25\!\cdots\!48\)\( T^{6} + \)\(67\!\cdots\!52\)\( T^{7} + \)\(51\!\cdots\!21\)\( T^{8} \)
$43$ \( 1 + \)\(61\!\cdots\!00\)\( T + \)\(15\!\cdots\!00\)\( T^{2} - \)\(88\!\cdots\!00\)\( T^{3} - \)\(51\!\cdots\!02\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!00\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(36\!\cdots\!80\)\( T + \)\(38\!\cdots\!60\)\( T^{2} + \)\(33\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!38\)\( T^{4} + \)\(67\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} - \)\(30\!\cdots\!40\)\( T^{7} + \)\(17\!\cdots\!61\)\( T^{8} \)
$53$ \( 1 + \)\(35\!\cdots\!40\)\( T + \)\(91\!\cdots\!60\)\( T^{2} + \)\(16\!\cdots\!20\)\( T^{3} + \)\(24\!\cdots\!38\)\( T^{4} + \)\(30\!\cdots\!60\)\( T^{5} + \)\(33\!\cdots\!40\)\( T^{6} + \)\(25\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!61\)\( T^{8} \)
$59$ \( 1 - \)\(67\!\cdots\!80\)\( T + \)\(32\!\cdots\!76\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(40\!\cdots\!66\)\( T^{4} - \)\(15\!\cdots\!40\)\( T^{5} + \)\(24\!\cdots\!36\)\( T^{6} - \)\(44\!\cdots\!20\)\( T^{7} + \)\(56\!\cdots\!21\)\( T^{8} \)
$61$ \( 1 + \)\(15\!\cdots\!12\)\( T + \)\(31\!\cdots\!88\)\( T^{2} + \)\(28\!\cdots\!64\)\( T^{3} + \)\(29\!\cdots\!70\)\( T^{4} + \)\(16\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} + \)\(30\!\cdots\!32\)\( T^{7} + \)\(11\!\cdots\!81\)\( T^{8} \)
$67$ \( 1 - \)\(15\!\cdots\!20\)\( T + \)\(36\!\cdots\!40\)\( T^{2} - \)\(45\!\cdots\!40\)\( T^{3} + \)\(64\!\cdots\!58\)\( T^{4} - \)\(55\!\cdots\!80\)\( T^{5} + \)\(53\!\cdots\!60\)\( T^{6} - \)\(27\!\cdots\!60\)\( T^{7} + \)\(22\!\cdots\!41\)\( T^{8} \)
$71$ \( 1 + \)\(73\!\cdots\!32\)\( T + \)\(12\!\cdots\!48\)\( T^{2} + \)\(61\!\cdots\!84\)\( T^{3} + \)\(59\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!44\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} + \)\(27\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$73$ \( 1 + \)\(13\!\cdots\!20\)\( T + \)\(59\!\cdots\!80\)\( T^{2} + \)\(56\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!18\)\( T^{4} + \)\(91\!\cdots\!80\)\( T^{5} + \)\(15\!\cdots\!20\)\( T^{6} + \)\(55\!\cdots\!40\)\( T^{7} + \)\(68\!\cdots\!81\)\( T^{8} \)
$79$ \( 1 + \)\(82\!\cdots\!60\)\( T + \)\(56\!\cdots\!36\)\( T^{2} + \)\(34\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} + \)\(50\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!16\)\( T^{6} + \)\(25\!\cdots\!40\)\( T^{7} + \)\(45\!\cdots\!61\)\( T^{8} \)
$83$ \( 1 + \)\(98\!\cdots\!60\)\( T + \)\(12\!\cdots\!40\)\( T^{2} + \)\(72\!\cdots\!80\)\( T^{3} + \)\(48\!\cdots\!58\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(74\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!20\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} \)
$89$ \( 1 + \)\(10\!\cdots\!80\)\( T + \)\(35\!\cdots\!16\)\( T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(60\!\cdots\!46\)\( T^{4} + \)\(22\!\cdots\!40\)\( T^{5} + \)\(60\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!81\)\( T^{8} \)
$97$ \( 1 + \)\(15\!\cdots\!20\)\( T + \)\(10\!\cdots\!60\)\( T^{2} + \)\(68\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!38\)\( T^{4} + \)\(11\!\cdots\!80\)\( T^{5} + \)\(31\!\cdots\!40\)\( T^{6} + \)\(83\!\cdots\!60\)\( T^{7} + \)\(96\!\cdots\!61\)\( T^{8} \)
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