Properties

Label 2.40.a
Level 2
Weight 40
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 2
Sturm bound 10
Trace bound 1

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 40 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{40}(\Gamma_0(2))\).

Total New Old
Modular forms 11 3 8
Cusp forms 9 3 6
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators.

\(2\)Dim.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 524288q^{2} \) \(\mathstrut -\mathstrut 448040028q^{3} \) \(\mathstrut +\mathstrut 824633720832q^{4} \) \(\mathstrut +\mathstrut 37396559183370q^{5} \) \(\mathstrut -\mathstrut 536281903792128q^{6} \) \(\mathstrut +\mathstrut 23548057401734376q^{7} \) \(\mathstrut -\mathstrut 144115188075855872q^{8} \) \(\mathstrut -\mathstrut 3830161076234277489q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 524288q^{2} \) \(\mathstrut -\mathstrut 448040028q^{3} \) \(\mathstrut +\mathstrut 824633720832q^{4} \) \(\mathstrut +\mathstrut 37396559183370q^{5} \) \(\mathstrut -\mathstrut 536281903792128q^{6} \) \(\mathstrut +\mathstrut 23548057401734376q^{7} \) \(\mathstrut -\mathstrut 144115188075855872q^{8} \) \(\mathstrut -\mathstrut 3830161076234277489q^{9} \) \(\mathstrut -\mathstrut 36620949074671042560q^{10} \) \(\mathstrut -\mathstrut 600358621179545885364q^{11} \) \(\mathstrut -\mathstrut 123156305123771154432q^{12} \) \(\mathstrut -\mathstrut 2434407943266402155598q^{13} \) \(\mathstrut +\mathstrut 4483749851976880357376q^{14} \) \(\mathstrut +\mathstrut 189447730968003517835640q^{15} \) \(\mathstrut +\mathstrut 226673591177742970257408q^{16} \) \(\mathstrut -\mathstrut 858208600547978241205674q^{17} \) \(\mathstrut -\mathstrut 1674130977592736752336896q^{18} \) \(\mathstrut +\mathstrut 31307516146017089016180q^{19} \) \(\mathstrut +\mathstrut 10279487915232167492321280q^{20} \) \(\mathstrut +\mathstrut 194451182132890125508828896q^{21} \) \(\mathstrut +\mathstrut 139198005730086287975645184q^{22} \) \(\mathstrut +\mathstrut 67157995958368736121296952q^{23} \) \(\mathstrut -\mathstrut 147412047246323721103736832q^{24} \) \(\mathstrut -\mathstrut 33110237710861025700572475q^{25} \) \(\mathstrut -\mathstrut 111659804005730278328762368q^{26} \) \(\mathstrut +\mathstrut 6578752508828661656159145000q^{27} \) \(\mathstrut +\mathstrut 6472840731185912230333906944q^{28} \) \(\mathstrut -\mathstrut 22265174672280926225295581310q^{29} \) \(\mathstrut -\mathstrut 86811803756312046924013240320q^{30} \) \(\mathstrut -\mathstrut 170959699274012182229038580064q^{31} \) \(\mathstrut -\mathstrut 39614081257132168796771975168q^{32} \) \(\mathstrut +\mathstrut 73625213910828578250297519504q^{33} \) \(\mathstrut -\mathstrut 70649380147397247577598459904q^{34} \) \(\mathstrut +\mathstrut 4438918404604032826461134067120q^{35} \) \(\mathstrut -\mathstrut 1052826659893656617564415983616q^{36} \) \(\mathstrut +\mathstrut 388911530082874143587136660906q^{37} \) \(\mathstrut -\mathstrut 12074810088442968904516784619520q^{38} \) \(\mathstrut +\mathstrut 9261643492039898279771447062872q^{39} \) \(\mathstrut -\mathstrut 10066289831948389744219143536640q^{40} \) \(\mathstrut +\mathstrut 68007643161987496150617761700366q^{41} \) \(\mathstrut -\mathstrut 114325973922970028993814447783936q^{42} \) \(\mathstrut +\mathstrut 68986009747378307846115681074412q^{43} \) \(\mathstrut -\mathstrut 165025321205619361393263683567616q^{44} \) \(\mathstrut +\mathstrut 97247126205573429132018612970290q^{45} \) \(\mathstrut -\mathstrut 733815434366087339292749042024448q^{46} \) \(\mathstrut +\mathstrut 915029532778958975491327534903536q^{47} \) \(\mathstrut -\mathstrut 33852947379378837790310749175808q^{48} \) \(\mathstrut +\mathstrut 2992370397058750394122773078750939q^{49} \) \(\mathstrut -\mathstrut 1613910927258641455329291154227200q^{50} \) \(\mathstrut +\mathstrut 945694449303859044594971821681416q^{51} \) \(\mathstrut -\mathstrut 669164960092916523128148052672512q^{52} \) \(\mathstrut -\mathstrut 8066347428099057831850868635343238q^{53} \) \(\mathstrut +\mathstrut 2384237163585732062941288951971840q^{54} \) \(\mathstrut -\mathstrut 8612160318480694923940207312451160q^{55} \) \(\mathstrut +\mathstrut 1232483774571874693314221592018944q^{56} \) \(\mathstrut -\mathstrut 9815465873655629297698789636095120q^{57} \) \(\mathstrut +\mathstrut 57829900979995733731911663621242880q^{58} \) \(\mathstrut -\mathstrut 33000750699110212174217716993073220q^{59} \) \(\mathstrut +\mathstrut 52074995763774818017089696206684160q^{60} \) \(\mathstrut -\mathstrut 149390285509459531187976782625971934q^{61} \) \(\mathstrut +\mathstrut 106232131988115040214806981901287424q^{62} \) \(\mathstrut +\mathstrut 1415461776129972173843762224864392q^{63} \) \(\mathstrut +\mathstrut 62307562302417931542365955950641152q^{64} \) \(\mathstrut +\mathstrut 176906949049275277480139011025791740q^{65} \) \(\mathstrut +\mathstrut 90518311945162978250837339683160064q^{66} \) \(\mathstrut +\mathstrut 26706815503945465585848770020873956q^{67} \) \(\mathstrut -\mathstrut 235902583839967630393470044136800256q^{68} \) \(\mathstrut -\mathstrut 167296225322118915115104888895533408q^{69} \) \(\mathstrut -\mathstrut 2600353600398434685866319608870338560q^{70} \) \(\mathstrut -\mathstrut 605293691077180832680380926385877464q^{71} \) \(\mathstrut -\mathstrut 460181619070804042139150073225805824q^{72} \) \(\mathstrut +\mathstrut 6037896371873457163195986157241568942q^{73} \) \(\mathstrut -\mathstrut 2916112813051584433665890834814337024q^{74} \) \(\mathstrut +\mathstrut 10468531606307457575517562794643194300q^{75} \) \(\mathstrut +\mathstrut 8605744509832662910823290550353920q^{76} \) \(\mathstrut -\mathstrut 3973784746707127600006074129626076768q^{77} \) \(\mathstrut -\mathstrut 3834959291444571989676916960785334272q^{78} \) \(\mathstrut -\mathstrut 3439639541961971841028249340726617200q^{79} \) \(\mathstrut +\mathstrut 2825604122595160296109710638390968320q^{80} \) \(\mathstrut -\mathstrut 20725384932653884806482027979285980997q^{81} \) \(\mathstrut +\mathstrut 18070629096812686567702762098219024384q^{82} \) \(\mathstrut -\mathstrut 57743921443562879954303540675175318348q^{83} \) \(\mathstrut +\mathstrut 53450333947475367361392349925106253824q^{84} \) \(\mathstrut +\mathstrut 12225312483754863430342815668998741620q^{85} \) \(\mathstrut +\mathstrut 46457578983056741804482202317956841472q^{86} \) \(\mathstrut -\mathstrut 65729394324412606646364515839930431720q^{87} \) \(\mathstrut +\mathstrut 38262456465865037447259783025913757696q^{88} \) \(\mathstrut -\mathstrut 166824435684903248525220257650708922370q^{89} \) \(\mathstrut +\mathstrut 8763159135563897597487703355141652480q^{90} \) \(\mathstrut +\mathstrut 198376560035269385711901123618380939376q^{91} \) \(\mathstrut +\mathstrut 18460249363590009545588755068446834688q^{92} \) \(\mathstrut +\mathstrut 406181486152934056717149508202299522944q^{93} \) \(\mathstrut -\mathstrut 225196019506450182093781181208056561664q^{94} \) \(\mathstrut +\mathstrut 58812152031758749530411766655084227800q^{95} \) \(\mathstrut -\mathstrut 40520315005399503255652781877161361408q^{96} \) \(\mathstrut +\mathstrut 615472286923954084663194347601246497766q^{97} \) \(\mathstrut -\mathstrut 2252471551654792460823213172884364591104q^{98} \) \(\mathstrut +\mathstrut 660551890977024558384150004591375044732q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.40.a.a \(1\) \(19.268\) \(\Q\) None \(524288\) \(-735458292\) \(-1\!\cdots\!50\) \(16\!\cdots\!64\) \(-\) \(q+2^{19}q^{2}-735458292q^{3}+2^{38}q^{4}+\cdots\)
2.40.a.b \(2\) \(19.268\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(-1048576\) \(287418264\) \(53\!\cdots\!20\) \(74\!\cdots\!12\) \(+\) \(q-2^{19}q^{2}+(143709132-\beta )q^{3}+2^{38}q^{4}+\cdots\)

Decomposition of \(S_{40}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{40}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{40}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)