Properties

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

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Defining parameters

Level: \( N \) = \( 1 \)
Weight: \( k \) = \( 40 \)
Character orbit: \([\chi]\) = 1.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
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

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

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\) into irreducible Hecke orbits

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