Properties

Label 2.40.a
Level 2
Weight 40
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 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: \(\Q\)
Newform subspaces: \( 2 \)
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 and the Fricke involution.

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

Trace form

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

Decomposition of \(S_{40}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 524288 T \))(\( ( 1 + 524288 T )^{2} \))
$3$ (\( 1 + 735458292 T + 4052555153018976267 T^{2} \))(\( 1 - 287418264 T + 4253112193502792358 T^{2} - \)\(11\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))
$5$ (\( 1 + 16226178983250 T + \)\(18\!\cdots\!25\)\( T^{2} \))(\( 1 - 53622738166620 T + \)\(24\!\cdots\!50\)\( T^{2} - \)\(97\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!25\)\( T^{4} \))
$7$ (\( 1 - 16050065775887864 T + \)\(90\!\cdots\!43\)\( T^{2} \))(\( 1 - 7497991625846512 T - \)\(88\!\cdots\!78\)\( T^{2} - \)\(68\!\cdots\!16\)\( T^{3} + \)\(82\!\cdots\!49\)\( T^{4} \))
$11$ (\( 1 + \)\(16\!\cdots\!48\)\( T + \)\(41\!\cdots\!91\)\( T^{2} \))(\( 1 + \)\(43\!\cdots\!16\)\( T + \)\(12\!\cdots\!46\)\( T^{2} + \)\(17\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \))
$13$ (\( 1 + \)\(13\!\cdots\!42\)\( T + \)\(27\!\cdots\!77\)\( T^{2} \))(\( 1 + \)\(11\!\cdots\!56\)\( T + \)\(51\!\cdots\!38\)\( T^{2} + \)\(30\!\cdots\!12\)\( T^{3} + \)\(77\!\cdots\!29\)\( T^{4} \))
$17$ (\( 1 + \)\(49\!\cdots\!66\)\( T + \)\(97\!\cdots\!53\)\( T^{2} \))(\( 1 + \)\(36\!\cdots\!08\)\( T + \)\(19\!\cdots\!22\)\( T^{2} + \)\(35\!\cdots\!24\)\( T^{3} + \)\(94\!\cdots\!09\)\( T^{4} \))
$19$ (\( 1 + \)\(11\!\cdots\!80\)\( T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 - \)\(11\!\cdots\!60\)\( T + \)\(15\!\cdots\!58\)\( T^{2} - \)\(85\!\cdots\!40\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))
$23$ (\( 1 + \)\(66\!\cdots\!72\)\( T + \)\(12\!\cdots\!87\)\( T^{2} \))(\( 1 - \)\(73\!\cdots\!24\)\( T + \)\(35\!\cdots\!18\)\( T^{2} - \)\(93\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} \))
$29$ (\( 1 - \)\(44\!\cdots\!50\)\( T + \)\(10\!\cdots\!69\)\( T^{2} \))(\( 1 + \)\(66\!\cdots\!60\)\( T + \)\(32\!\cdots\!38\)\( T^{2} + \)\(71\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \))
$31$ (\( 1 - \)\(15\!\cdots\!92\)\( T + \)\(14\!\cdots\!71\)\( T^{2} \))(\( 1 + \)\(18\!\cdots\!56\)\( T + \)\(25\!\cdots\!26\)\( T^{2} + \)\(27\!\cdots\!76\)\( T^{3} + \)\(21\!\cdots\!41\)\( T^{4} \))
$37$ (\( 1 + \)\(25\!\cdots\!46\)\( T + \)\(14\!\cdots\!73\)\( T^{2} \))(\( 1 - \)\(29\!\cdots\!52\)\( T + \)\(15\!\cdots\!22\)\( T^{2} - \)\(42\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!29\)\( T^{4} \))
$41$ (\( 1 - \)\(51\!\cdots\!42\)\( T + \)\(79\!\cdots\!61\)\( T^{2} \))(\( 1 - \)\(16\!\cdots\!24\)\( T + \)\(89\!\cdots\!66\)\( T^{2} - \)\(13\!\cdots\!64\)\( T^{3} + \)\(62\!\cdots\!21\)\( T^{4} \))
$43$ (\( 1 - \)\(78\!\cdots\!28\)\( T + \)\(50\!\cdots\!07\)\( T^{2} \))(\( 1 + \)\(98\!\cdots\!16\)\( T + \)\(83\!\cdots\!78\)\( T^{2} + \)\(49\!\cdots\!12\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 - \)\(24\!\cdots\!04\)\( T + \)\(16\!\cdots\!83\)\( T^{2} \))(\( 1 - \)\(67\!\cdots\!32\)\( T + \)\(42\!\cdots\!22\)\( T^{2} - \)\(10\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!89\)\( T^{4} \))
$53$ (\( 1 - \)\(69\!\cdots\!98\)\( T + \)\(17\!\cdots\!17\)\( T^{2} \))(\( 1 + \)\(87\!\cdots\!36\)\( T + \)\(49\!\cdots\!58\)\( T^{2} + \)\(15\!\cdots\!12\)\( T^{3} + \)\(31\!\cdots\!89\)\( T^{4} \))
$59$ (\( 1 + \)\(20\!\cdots\!00\)\( T + \)\(11\!\cdots\!39\)\( T^{2} \))(\( 1 + \)\(12\!\cdots\!20\)\( T + \)\(23\!\cdots\!78\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!21\)\( T^{4} \))
$61$ (\( 1 + \)\(12\!\cdots\!18\)\( T + \)\(42\!\cdots\!41\)\( T^{2} \))(\( 1 + \)\(20\!\cdots\!16\)\( T + \)\(85\!\cdots\!46\)\( T^{2} + \)\(88\!\cdots\!56\)\( T^{3} + \)\(18\!\cdots\!81\)\( T^{4} \))
$67$ (\( 1 - \)\(45\!\cdots\!64\)\( T + \)\(16\!\cdots\!03\)\( T^{2} \))(\( 1 + \)\(43\!\cdots\!08\)\( T + \)\(24\!\cdots\!22\)\( T^{2} + \)\(70\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!09\)\( T^{4} \))
$71$ (\( 1 + \)\(99\!\cdots\!28\)\( T + \)\(15\!\cdots\!31\)\( T^{2} \))(\( 1 + \)\(50\!\cdots\!36\)\( T + \)\(29\!\cdots\!86\)\( T^{2} + \)\(79\!\cdots\!16\)\( T^{3} + \)\(25\!\cdots\!61\)\( T^{4} \))
$73$ (\( 1 - \)\(81\!\cdots\!18\)\( T + \)\(46\!\cdots\!37\)\( T^{2} \))(\( 1 - \)\(52\!\cdots\!24\)\( T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(24\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!69\)\( T^{4} \))
$79$ (\( 1 + \)\(85\!\cdots\!40\)\( T + \)\(10\!\cdots\!19\)\( T^{2} \))(\( 1 - \)\(51\!\cdots\!40\)\( T - \)\(73\!\cdots\!62\)\( T^{2} - \)\(52\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!61\)\( T^{4} \))
$83$ (\( 1 - \)\(71\!\cdots\!48\)\( T + \)\(69\!\cdots\!47\)\( T^{2} \))(\( 1 + \)\(64\!\cdots\!96\)\( T + \)\(24\!\cdots\!98\)\( T^{2} + \)\(45\!\cdots\!12\)\( T^{3} + \)\(48\!\cdots\!09\)\( T^{4} \))
$89$ (\( 1 + \)\(13\!\cdots\!90\)\( T + \)\(10\!\cdots\!09\)\( T^{2} \))(\( 1 + \)\(35\!\cdots\!80\)\( T + \)\(69\!\cdots\!18\)\( T^{2} + \)\(38\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} \))
$97$ (\( 1 - \)\(76\!\cdots\!34\)\( T + \)\(30\!\cdots\!33\)\( T^{2} \))(\( 1 + \)\(14\!\cdots\!68\)\( T + \)\(46\!\cdots\!22\)\( T^{2} + \)\(44\!\cdots\!44\)\( T^{3} + \)\(92\!\cdots\!89\)\( T^{4} \))
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