Properties

Label 1.42.a
Level 1
Weight 42
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 1
Sturm bound 3
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 42 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(3\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{42}(\Gamma_0(1))\).

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

Trace form

\( 3q - 344688q^{2} - 10820953044q^{3} + 6271704903936q^{4} - 212302350281550q^{5} + 4970194114982976q^{6} + 57878416258239192q^{7} - 3555831711183237120q^{8} + 13277004110931878919q^{9} + O(q^{10}) \) \( 3q - 344688q^{2} - 10820953044q^{3} + 6271704903936q^{4} - 212302350281550q^{5} + 4970194114982976q^{6} + 57878416258239192q^{7} - 3555831711183237120q^{8} + 13277004110931878919q^{9} - 911610678410074173600q^{10} - 3064080929502798535164q^{11} - 71233189317945674062848q^{12} - 98515763053518962936694q^{13} - 663773359066872932212608q^{14} + 2322231962154213782968200q^{15} + 13217913642093764057038848q^{16} + 35582277130790298867301302q^{17} - 51949630971262524069430704q^{18} - 233456308095243376865478180q^{19} - 1234817679506740618000857600q^{20} - 784315067894306487816760224q^{21} + 10093688647003206881596695744q^{22} + 2812708670812770779627026056q^{23} + 36949405163940557660187279360q^{24} - 12630087813294771903109831875q^{25} - 445502868787427067716672131104q^{26} + 40981593702376529051426180280q^{27} + 526606706694181610802628184064q^{28} - 12736322664295346653179017670q^{29} + 6115742110948196616210957398400q^{30} - 55977552834365071134409017504q^{31} - 14842037794660482009412112744448q^{32} - 13829226714693448959900097020528q^{33} - 42547131667082319398961759011808q^{34} + 35330944415319146254058128592400q^{35} + 209583308279267044115700704156928q^{36} + 4945769903781582545269841455122q^{37} + 384288369224763756133358504473920q^{38} - 546470556552953849463965686898712q^{39} - 1326138456985596139721995183104000q^{40} - 3123151500646359746292805811918274q^{41} + 5023322329604694017661024056308224q^{42} + 1491980858362212236675453498061156q^{43} + 6827579920847638909574240093432832q^{44} - 3706804708974790460103758635956150q^{45} + 34089155258950859800971794327519616q^{46} - 63038106682044077762675687404413168q^{47} - 48906179099557375982971001333612544q^{48} - 97136113131667199274890334287198229q^{49} + 286202165340727366859498513545830000q^{50} - 157319866621422817701623275797858024q^{51} + 364999309722184195149034320029732352q^{52} + 79896749344137562310547533743021506q^{53} + 279506543662409583024224141017979520q^{54} - 1107430856215374719020572615529578600q^{55} - 1507500925934438536531619079138017280q^{56} - 120534128028213908789981458568129040q^{57} + 1060511497310363735425511566792954080q^{58} + 192512048683375799719343226678403860q^{59} + 5905677886285900358494072637173094400q^{60} + 8740556115036354715092732710435455386q^{61} - 12005332293025961804297174175449366016q^{62} + 4195060549596204507618981258341049336q^{63} - 28669651485797320669854651690478731264q^{64} + 23272939413910136331858458170260680700q^{65} - 70374114793184440562716940658640274688q^{66} + 11266872945514675454702655007279407852q^{67} + 95413181576279049358373233907724317184q^{68} + 110718129232411678126981084073658946848q^{69} + 59324379124057015483688060122385068800q^{70} - 140970915469820223119231433051520161384q^{71} - 178095780793081630529589944776993320960q^{72} + 45825493210252135256288871746499029406q^{73} - 255935787825178849200244145003411599008q^{74} - 247029389196459510298088488221820147500q^{75} + 26100578647791289060370838422069130240q^{76} - 429017620126390531124795910645506217696q^{77} + 3378623821652934155171000156043557135232q^{78} - 520184835554750143118446358634240850320q^{79} + 715918562852781559130521454428746547200q^{80} - 3191652631710460747876906046824191542997q^{81} - 579816777788457983576303882284492608096q^{82} - 619777491338381742927488037142663372644q^{83} - 4198672691855986319203859063793898610688q^{84} - 106299433069331216483654652405294143100q^{85} - 4751113477095326918262488958359640807744q^{86} + 13958176049210326842816445061163645008040q^{87} + 18329036816788093472347699011602935234560q^{88} - 142389067128374093799211866144944441010q^{89} - 8753545150081132110036071066039970208800q^{90} + 25080596935320871627133801126628392473296q^{91} - 67617289414955565381239604024555789723648q^{92} + 3477058704647981071307340824432458334592q^{93} - 53008068396962431152763933809330449227008q^{94} - 33170437179755379504247502944935502503000q^{95} + 95174464657114272630879818889875681181696q^{96} + 116011691364103339137369463947331771078182q^{97} - 38494567059701712876195969460820421029616q^{98} + 45849669342635847189814810549498192693428q^{99} + O(q^{100}) \)

Decomposition of \(S_{42}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.42.a.a \(3\) \(10.647\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-344688\) \(-10820953044\) \(-2\!\cdots\!50\) \(57\!\cdots\!92\) \(+\) \(q+(-114896+\beta _{1})q^{2}+(-3606984348+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 344688 T + 222087340032 T^{2} + 995518329378570240 T^{3} + \)\(48\!\cdots\!64\)\( T^{4} + \)\(16\!\cdots\!52\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} \)
$3$ \( 1 + 10820953044 T + \)\(10\!\cdots\!13\)\( T^{2} + \)\(58\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!39\)\( T^{4} + \)\(14\!\cdots\!96\)\( T^{5} + \)\(48\!\cdots\!27\)\( T^{6} \)
$5$ \( 1 + 212302350281550 T + \)\(97\!\cdots\!75\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(44\!\cdots\!75\)\( T^{4} + \)\(43\!\cdots\!50\)\( T^{5} + \)\(94\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 57878416258239192 T + \)\(11\!\cdots\!57\)\( T^{2} - \)\(54\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!99\)\( T^{4} - \)\(11\!\cdots\!08\)\( T^{5} + \)\(88\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 + \)\(30\!\cdots\!64\)\( T + \)\(97\!\cdots\!65\)\( T^{2} + \)\(17\!\cdots\!80\)\( T^{3} + \)\(48\!\cdots\!15\)\( T^{4} + \)\(75\!\cdots\!44\)\( T^{5} + \)\(12\!\cdots\!31\)\( T^{6} \)
$13$ \( 1 + \)\(98\!\cdots\!94\)\( T + \)\(56\!\cdots\!83\)\( T^{2} + \)\(99\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!79\)\( T^{4} + \)\(21\!\cdots\!86\)\( T^{5} + \)\(10\!\cdots\!97\)\( T^{6} \)
$17$ \( 1 - \)\(35\!\cdots\!02\)\( T + \)\(12\!\cdots\!07\)\( T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!19\)\( T^{4} - \)\(28\!\cdots\!78\)\( T^{5} + \)\(22\!\cdots\!13\)\( T^{6} \)
$19$ \( 1 + \)\(23\!\cdots\!80\)\( T + \)\(86\!\cdots\!57\)\( T^{2} + \)\(11\!\cdots\!40\)\( T^{3} + \)\(23\!\cdots\!83\)\( T^{4} + \)\(16\!\cdots\!80\)\( T^{5} + \)\(19\!\cdots\!59\)\( T^{6} \)
$23$ \( 1 - \)\(28\!\cdots\!56\)\( T + \)\(85\!\cdots\!53\)\( T^{2} - \)\(72\!\cdots\!80\)\( T^{3} + \)\(57\!\cdots\!19\)\( T^{4} - \)\(12\!\cdots\!24\)\( T^{5} + \)\(31\!\cdots\!67\)\( T^{6} \)
$29$ \( 1 + \)\(12\!\cdots\!70\)\( T + \)\(15\!\cdots\!87\)\( T^{2} - \)\(39\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!23\)\( T^{4} + \)\(10\!\cdots\!70\)\( T^{5} + \)\(74\!\cdots\!89\)\( T^{6} \)
$31$ \( 1 + \)\(55\!\cdots\!04\)\( T + \)\(36\!\cdots\!65\)\( T^{2} + \)\(46\!\cdots\!80\)\( T^{3} + \)\(50\!\cdots\!15\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(27\!\cdots\!91\)\( T^{6} \)
$37$ \( 1 - \)\(49\!\cdots\!22\)\( T + \)\(47\!\cdots\!07\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!59\)\( T^{4} - \)\(19\!\cdots\!18\)\( T^{5} + \)\(77\!\cdots\!53\)\( T^{6} \)
$41$ \( 1 + \)\(31\!\cdots\!74\)\( T + \)\(70\!\cdots\!15\)\( T^{2} + \)\(92\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!15\)\( T^{4} + \)\(55\!\cdots\!94\)\( T^{5} + \)\(23\!\cdots\!21\)\( T^{6} \)
$43$ \( 1 - \)\(14\!\cdots\!56\)\( T + \)\(27\!\cdots\!93\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(26\!\cdots\!99\)\( T^{4} - \)\(13\!\cdots\!44\)\( T^{5} + \)\(82\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 + \)\(63\!\cdots\!68\)\( T + \)\(22\!\cdots\!57\)\( T^{2} + \)\(51\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!79\)\( T^{4} + \)\(81\!\cdots\!12\)\( T^{5} + \)\(46\!\cdots\!23\)\( T^{6} \)
$53$ \( 1 - \)\(79\!\cdots\!06\)\( T + \)\(59\!\cdots\!63\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(29\!\cdots\!39\)\( T^{4} - \)\(19\!\cdots\!54\)\( T^{5} + \)\(12\!\cdots\!77\)\( T^{6} \)
$59$ \( 1 - \)\(19\!\cdots\!60\)\( T + \)\(77\!\cdots\!77\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!43\)\( T^{4} - \)\(31\!\cdots\!60\)\( T^{5} + \)\(65\!\cdots\!79\)\( T^{6} \)
$61$ \( 1 - \)\(87\!\cdots\!86\)\( T + \)\(62\!\cdots\!15\)\( T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(98\!\cdots\!15\)\( T^{4} - \)\(21\!\cdots\!06\)\( T^{5} + \)\(39\!\cdots\!81\)\( T^{6} \)
$67$ \( 1 - \)\(11\!\cdots\!52\)\( T + \)\(78\!\cdots\!57\)\( T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(57\!\cdots\!19\)\( T^{4} - \)\(61\!\cdots\!28\)\( T^{5} + \)\(40\!\cdots\!63\)\( T^{6} \)
$71$ \( 1 + \)\(14\!\cdots\!84\)\( T + \)\(22\!\cdots\!65\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} + \)\(89\!\cdots\!44\)\( T^{5} + \)\(50\!\cdots\!11\)\( T^{6} \)
$73$ \( 1 - \)\(45\!\cdots\!06\)\( T + \)\(18\!\cdots\!03\)\( T^{2} - \)\(58\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!19\)\( T^{4} - \)\(28\!\cdots\!74\)\( T^{5} + \)\(15\!\cdots\!17\)\( T^{6} \)
$79$ \( 1 + \)\(52\!\cdots\!20\)\( T + \)\(83\!\cdots\!37\)\( T^{2} + \)\(82\!\cdots\!60\)\( T^{3} + \)\(52\!\cdots\!23\)\( T^{4} + \)\(20\!\cdots\!20\)\( T^{5} + \)\(25\!\cdots\!39\)\( T^{6} \)
$83$ \( 1 + \)\(61\!\cdots\!44\)\( T + \)\(61\!\cdots\!73\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(29\!\cdots\!59\)\( T^{4} + \)\(14\!\cdots\!16\)\( T^{5} + \)\(11\!\cdots\!87\)\( T^{6} \)
$89$ \( 1 + \)\(14\!\cdots\!10\)\( T + \)\(20\!\cdots\!67\)\( T^{2} + \)\(53\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!63\)\( T^{4} + \)\(10\!\cdots\!10\)\( T^{5} + \)\(59\!\cdots\!69\)\( T^{6} \)
$97$ \( 1 - \)\(11\!\cdots\!82\)\( T + \)\(11\!\cdots\!07\)\( T^{2} - \)\(67\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!79\)\( T^{4} - \)\(95\!\cdots\!38\)\( T^{5} + \)\(23\!\cdots\!73\)\( T^{6} \)
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