Properties

Label 1.40.a
Level 1
Weight 40
Character orbit a
Rep. character \(\chi_{1}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 1
Sturm bound 3
Trace bound 0

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 40 \)
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 \(M_{40}(\Gamma_0(1))\).

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

Trace form

\( 3q + 548856q^{2} + 1109442852q^{3} + 272078981184q^{4} + 17426916500490q^{5} + 1529727847867296q^{6} - 17996297718635544q^{7} + 255044243806133760q^{8} + 8305879902078677391q^{9} + O(q^{10}) \) \( 3q + 548856q^{2} + 1109442852q^{3} + 272078981184q^{4} + 17426916500490q^{5} + 1529727847867296q^{6} - 17996297718635544q^{7} + 255044243806133760q^{8} + 8305879902078677391q^{9} + 90373081165367671440q^{10} + 667814968265751536076q^{11} + 3676571280854873162496q^{12} + 3156080701447967254962q^{13} - 63063724565889303463872q^{14} - 171209278264452417200520q^{15} - 351730940560369057001472q^{16} + 890025791008793741893206q^{17} + 8758537968130529168653272q^{18} + 9227612185007697573518580q^{19} - 14662452947108747596260480q^{20} - 190208658572592931471251744q^{21} - 122745877544387429034313248q^{22} + 305308591470919570740188472q^{23} + 1672960268636194560046295040q^{24} + 1866127065486173095027030725q^{25} + 283778605804595582169202896q^{26} - 11508090459893795046427628760q^{27} - 41794856371466593385827533312q^{28} - 2492682219258648277296444030q^{29} + 105977898643957401776506418880q^{30} + 306107671747822704409554746016q^{31} - 56572224161556180973554008064q^{32} - 157376353761155567959844693616q^{33} - 1683392869529765869103799245712q^{34} - 1197259217905835166323668756560q^{35} + 188684323073267404408057366848q^{36} + 7599073412303579174888591086506q^{37} + 2479871855304922949496616465440q^{38} + 22252083518345402015145920059992q^{39} - 32669995448979982796588503833600q^{40} + 2391924206237769506232479990286q^{41} - 195689966839404680756969540713728q^{42} + 162865923671547339657067136304492q^{43} - 45577647299113910363374898556672q^{44} + 588908524142943445153326555470130q^{45} - 24296541145531030477977021520704q^{46} + 202683231875405841765643043683056q^{47} - 1747281225742357435119054532067328q^{48} + 368237681722499062385273201641179q^{49} - 2624175085265540082696197615369400q^{50} + 785651103815434341253216161914376q^{51} + 3782687188757116950236456253499776q^{52} + 5869079988764517679853946632443002q^{53} + 7930814851979058696152103557167680q^{54} - 4993109836111426485910134681304920q^{55} - 6272116703248607238900430116679680q^{56} - 46129102671186186415908500969957520q^{57} + 2579097256656295554169870267982160q^{58} - 56082681385621280317109288669637060q^{59} + 127333179651658235430567238147607040q^{60} + 39577675682126428331898689976833826q^{61} + 262011550289381451529348830657714432q^{62} - 174786720262353444072185876495505528q^{63} - 169364906963561439583712861665099776q^{64} - 264857762104053085095614223637519620q^{65} - 287121760972609093376568794235255168q^{66} - 140659929656119892877736400727430044q^{67} - 130643133997460537100958647836349312q^{68} + 1584175048695702801789842685209503392q^{69} - 480338235822643784458804278358623360q^{70} + 3849053210874068146700295241416217896q^{71} - 1699150852984576616740529956276922880q^{72} - 180754136492273067100959984036797778q^{73} - 5070707041697857186532012041702753392q^{74} - 4308561205008052508725896857031567300q^{75} - 7362591710990284189025184136271151360q^{76} + 5690389092595065468343565382905921952q^{77} + 9330105385756793042468361274821534144q^{78} + 13807772201216229289895673856694559120q^{79} + 26839954555034162361105063302056304640q^{80} - 18798023146176620962298765658838887237q^{81} + 5068915865662526415092477483885831472q^{82} - 20899400147905548803677768348186808268q^{83} - 72152575898640813291791263837177976832q^{84} - 105991789747453754784809897833488498060q^{85} + 170012097580285110854272781787916483296q^{86} - 73247996942640959139716393654519138280q^{87} + 69412125637007165807333386173055825920q^{88} + 205050039910109857136561853222076224510q^{89} + 93221656611667223268126392859978755280q^{90} - 165525104998331842571199169888366982544q^{91} + 262356454404517646077951321169615264256q^{92} - 167325068951627887866919523900021629056q^{93} - 1127972668535151884588260809571286639232q^{94} + 724803433705663618674774120451764528600q^{95} - 1451094524490643350592064137029388468224q^{96} + 173490888317728793437812209497231327206q^{97} + 2246791677555180938475077451719860538808q^{98} + 1057018158118568141770036034713090936572q^{99} + O(q^{100}) \)

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\) into newform subspaces

Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1.40.a.a \(3\) \(9.634\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(548856\) \(1109442852\) \(17\!\cdots\!90\) \(-1\!\cdots\!44\) \(+\) \(q+(182952-\beta _{1})q^{2}+(369814284+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 548856 T + 839215684608 T^{2} - 389931393425080320 T^{3} + \)\(46\!\cdots\!04\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(16\!\cdots\!72\)\( T^{6} \)
$3$ \( 1 - 1109442852 T + 2541324499416072657 T^{2} + \)\(29\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} - \)\(18\!\cdots\!28\)\( T^{5} + \)\(66\!\cdots\!63\)\( T^{6} \)
$5$ \( 1 - 17426916500490 T + \)\(19\!\cdots\!75\)\( T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!75\)\( T^{4} - \)\(57\!\cdots\!50\)\( T^{5} + \)\(60\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 + 17996297718635544 T + \)\(13\!\cdots\!93\)\( T^{2} + \)\(45\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!99\)\( T^{4} + \)\(14\!\cdots\!56\)\( T^{5} + \)\(75\!\cdots\!07\)\( T^{6} \)
$11$ \( 1 - \)\(66\!\cdots\!76\)\( T + \)\(25\!\cdots\!65\)\( T^{2} - \)\(61\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!56\)\( T^{5} + \)\(69\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 - \)\(31\!\cdots\!62\)\( T + \)\(74\!\cdots\!07\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!39\)\( T^{4} - \)\(24\!\cdots\!98\)\( T^{5} + \)\(21\!\cdots\!33\)\( T^{6} \)
$17$ \( 1 - \)\(89\!\cdots\!06\)\( T + \)\(21\!\cdots\!63\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!39\)\( T^{4} - \)\(84\!\cdots\!54\)\( T^{5} + \)\(91\!\cdots\!77\)\( T^{6} \)
$19$ \( 1 - \)\(92\!\cdots\!80\)\( T + \)\(18\!\cdots\!37\)\( T^{2} - \)\(14\!\cdots\!40\)\( T^{3} + \)\(13\!\cdots\!23\)\( T^{4} - \)\(51\!\cdots\!80\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 - \)\(30\!\cdots\!72\)\( T + \)\(35\!\cdots\!57\)\( T^{2} - \)\(67\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!59\)\( T^{4} - \)\(50\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!03\)\( T^{6} \)
$29$ \( 1 + \)\(24\!\cdots\!30\)\( T + \)\(30\!\cdots\!07\)\( T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + \)\(33\!\cdots\!83\)\( T^{4} + \)\(29\!\cdots\!30\)\( T^{5} + \)\(12\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 - \)\(30\!\cdots\!16\)\( T + \)\(58\!\cdots\!65\)\( T^{2} - \)\(75\!\cdots\!20\)\( T^{3} + \)\(85\!\cdots\!15\)\( T^{4} - \)\(64\!\cdots\!56\)\( T^{5} + \)\(30\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 - \)\(75\!\cdots\!06\)\( T + \)\(50\!\cdots\!03\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(73\!\cdots\!19\)\( T^{4} - \)\(15\!\cdots\!74\)\( T^{5} + \)\(30\!\cdots\!17\)\( T^{6} \)
$41$ \( 1 - \)\(23\!\cdots\!86\)\( T + \)\(19\!\cdots\!15\)\( T^{2} - \)\(49\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(14\!\cdots\!06\)\( T^{5} + \)\(49\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 - \)\(16\!\cdots\!92\)\( T + \)\(18\!\cdots\!57\)\( T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(94\!\cdots\!99\)\( T^{4} - \)\(41\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!43\)\( T^{6} \)
$47$ \( 1 - \)\(20\!\cdots\!56\)\( T + \)\(11\!\cdots\!73\)\( T^{2} + \)\(52\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!59\)\( T^{4} - \)\(53\!\cdots\!84\)\( T^{5} + \)\(43\!\cdots\!87\)\( T^{6} \)
$53$ \( 1 - \)\(58\!\cdots\!02\)\( T + \)\(58\!\cdots\!07\)\( T^{2} - \)\(20\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} - \)\(18\!\cdots\!78\)\( T^{5} + \)\(54\!\cdots\!13\)\( T^{6} \)
$59$ \( 1 + \)\(56\!\cdots\!60\)\( T + \)\(36\!\cdots\!17\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!63\)\( T^{4} + \)\(75\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 - \)\(39\!\cdots\!26\)\( T + \)\(57\!\cdots\!15\)\( T^{2} + \)\(46\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!15\)\( T^{4} - \)\(71\!\cdots\!06\)\( T^{5} + \)\(76\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 + \)\(14\!\cdots\!44\)\( T + \)\(49\!\cdots\!13\)\( T^{2} + \)\(46\!\cdots\!40\)\( T^{3} + \)\(82\!\cdots\!39\)\( T^{4} + \)\(38\!\cdots\!96\)\( T^{5} + \)\(44\!\cdots\!27\)\( T^{6} \)
$71$ \( 1 - \)\(38\!\cdots\!96\)\( T + \)\(86\!\cdots\!65\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(96\!\cdots\!56\)\( T^{5} + \)\(39\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 + \)\(18\!\cdots\!78\)\( T + \)\(12\!\cdots\!07\)\( T^{2} + \)\(12\!\cdots\!60\)\( T^{3} + \)\(60\!\cdots\!59\)\( T^{4} + \)\(39\!\cdots\!82\)\( T^{5} + \)\(10\!\cdots\!53\)\( T^{6} \)
$79$ \( 1 - \)\(13\!\cdots\!20\)\( T + \)\(22\!\cdots\!57\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!83\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 + \)\(20\!\cdots\!68\)\( T + \)\(13\!\cdots\!57\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(95\!\cdots\!79\)\( T^{4} + \)\(10\!\cdots\!12\)\( T^{5} + \)\(34\!\cdots\!23\)\( T^{6} \)
$89$ \( 1 - \)\(20\!\cdots\!10\)\( T + \)\(19\!\cdots\!27\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!43\)\( T^{4} - \)\(23\!\cdots\!10\)\( T^{5} + \)\(11\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 - \)\(17\!\cdots\!06\)\( T + \)\(84\!\cdots\!23\)\( T^{2} - \)\(99\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!59\)\( T^{4} - \)\(16\!\cdots\!34\)\( T^{5} + \)\(28\!\cdots\!37\)\( T^{6} \)
show more
show less