Properties

Label 2.34.a
Level 2
Weight 34
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 8
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(8\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{34}(\Gamma_0(2))\).

Total New Old
Modular forms 9 3 6
Cusp forms 7 3 4
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.
\(+\)\(1\)
\(-\)\(2\)

Trace form

\( 3q + 65536q^{2} - 124649076q^{3} + 12884901888q^{4} + 533466655890q^{5} + 9264303439872q^{6} + 99371784824088q^{7} + 281474976710656q^{8} + 18852405815679399q^{9} + O(q^{10}) \) \( 3q + 65536q^{2} - 124649076q^{3} + 12884901888q^{4} + 533466655890q^{5} + 9264303439872q^{6} + 99371784824088q^{7} + 281474976710656q^{8} + 18852405815679399q^{9} - 35660209141186560q^{10} - 243987735048109404q^{11} - 535363704896618496q^{12} + 6127452524374998954q^{13} + 10883328044408766464q^{14} - 540011606028256440q^{15} + 55340232221128654848q^{16} + 163351956857193229878q^{17} - 354578147647536955392q^{18} + 2316286200640058928060q^{19} + 2291221840554035773440q^{20} - 10966882762375461396384q^{21} - 4733220146206274224128q^{22} - 85544934143271027955896q^{23} + 39789880294470542426112q^{24} + 225370380364752475573725q^{25} + 251615281586057289531392q^{26} - 743752090411160780985480q^{27} + 426798565964607073026048q^{28} - 1885570868915532180931110q^{29} + 9357659564213453452738560q^{30} - 6214164899357651844971424q^{31} + 1208925819614629174706176q^{32} - 28646531485170550011011952q^{33} + 28939341541930727759806464q^{34} - 82084709605255936704167280q^{35} + 80970466429263222725935104q^{36} + 19582065655883864004622578q^{37} + 141223277382323304263843840q^{38} + 110743367844616772680078248q^{39} - 153159432029916521834741760q^{40} + 474990906888239155557354366q^{41} - 1300079482701292075135008768q^{42} - 854701774198330191704131836q^{43} - 1047919342656742876810051584q^{44} + 7113005491199426978344868970q^{45} - 3755487355665331416256217088q^{46} - 3359713735806216538266732912q^{47} - 2299369603996371501308706816q^{48} + 1027325841837036133332384171q^{49} - 8022129800291472803432038400q^{50} + 34605359459142380076105389016q^{51} + 26317208199983263347464208384q^{52} - 25719867322108276836703217886q^{53} + 65835402527390881086630789120q^{54} - 203329095921753105019451578920q^{55} + 46743538022375287618581561344q^{56} - 246920391711219760108249543440q^{57} + 90426765572487936244912619520q^{58} + 486574758853958227127412969780q^{59} - 2319332187351797861701386240q^{60} + 8583862334335294799056153146q^{61} - 158990793629680529653626830848q^{62} - 91891357726791003768354017736q^{63} + 237684487542793012780631851008q^{64} + 1570508277569118621577450564860q^{65} - 3374590807726680306939142864896q^{66} + 422874132479815900981844675148q^{67} + 701591312439247864678620069888q^{68} + 455702719474062148458385981728q^{69} - 3024467071475429121489009377280q^{70} + 5198977231352455543531112459736q^{71} - 1522901548022430558360050860032q^{72} - 8030146104418329689734403612706q^{73} + 13924466895002049051747468836864q^{74} - 23292509902333321603477068569100q^{75} + 9948373479925147363530516725760q^{76} + 27691751685039233175366988812576q^{77} + 27202320153878090082468855545856q^{78} - 25638830707110254946114730329360q^{79} + 9840722873060510167738865418240q^{80} - 26510678888959286274847485768597q^{81} - 52988841827294806721870070939648q^{82} - 53261726130515111552883263441796q^{83} - 47102402803468745970379772657664q^{84} - 16465362595617182300874127067580q^{85} + 51199962022156216766968491671552q^{86} + 401367777050797863744672651928680q^{87} - 20329025732724286262637534117888q^{88} + 99483015972296495211736491705870q^{89} - 390580869706777575871691707514880q^{90} + 75577323879611270472141157040976q^{91} - 367412694483822843534875050377216q^{92} + 194910148274720557866176228573568q^{93} + 596297622405425787577101230014464q^{94} - 943613522324582126565259935432600q^{95} + 170896234576505829355531536433152q^{96} + 294906994984199222096704110296358q^{97} + 934885914868789332728527017541632q^{98} - 1902504845231598113297079280905132q^{99} + O(q^{100}) \)

Decomposition of \(S_{34}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.34.a.a \(1\) \(13.797\) \(\Q\) None \(-65536\) \(-133005564\) \(538799132550\) \(-3\!\cdots\!68\) \(+\) \(q-2^{16}q^{2}-133005564q^{3}+2^{32}q^{4}+\cdots\)
2.34.a.b \(2\) \(13.797\) \(\mathbb{Q}[x]/(x^{2} - \cdots)\) None \(131072\) \(8356488\) \(-5332476660\) \(13\!\cdots\!56\) \(-\) \(q+2^{16}q^{2}+(4178244-\beta )q^{3}+2^{32}q^{4}+\cdots\)

Decomposition of \(S_{34}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{34}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{34}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 65536 T \))(\( ( 1 - 65536 T )^{2} \))
$3$ (\( 1 + 133005564 T + 5559060566555523 T^{2} \))(\( 1 - 8356488 T + 2233482848764182 T^{2} - \)\(46\!\cdots\!24\)\( T^{3} + \)\(30\!\cdots\!29\)\( T^{4} \))
$5$ (\( 1 - 538799132550 T + \)\(11\!\cdots\!25\)\( T^{2} \))(\( 1 + 5332476660 T + \)\(90\!\cdots\!50\)\( T^{2} + \)\(62\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!25\)\( T^{4} \))
$7$ (\( 1 + 33347311051768 T + \)\(77\!\cdots\!07\)\( T^{2} \))(\( 1 - 132719095875856 T + \)\(12\!\cdots\!98\)\( T^{2} - \)\(10\!\cdots\!92\)\( T^{3} + \)\(59\!\cdots\!49\)\( T^{4} \))
$11$ (\( 1 + 85882263625386228 T + \)\(23\!\cdots\!31\)\( T^{2} \))(\( 1 + 158105471422723176 T + \)\(90\!\cdots\!06\)\( T^{2} + \)\(36\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!61\)\( T^{4} \))
$13$ (\( 1 - 1144054008875905166 T + \)\(57\!\cdots\!53\)\( T^{2} \))(\( 1 - 4983398515499093788 T + \)\(16\!\cdots\!42\)\( T^{2} - \)\(28\!\cdots\!64\)\( T^{3} + \)\(33\!\cdots\!09\)\( T^{4} \))
$17$ (\( 1 + \)\(13\!\cdots\!98\)\( T + \)\(40\!\cdots\!37\)\( T^{2} \))(\( 1 - \)\(30\!\cdots\!76\)\( T + \)\(97\!\cdots\!18\)\( T^{2} - \)\(12\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} \))
$19$ (\( 1 - 80695000174130231060 T + \)\(15\!\cdots\!59\)\( T^{2} \))(\( 1 - \)\(22\!\cdots\!00\)\( T + \)\(27\!\cdots\!18\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!81\)\( T^{4} \))
$23$ (\( 1 + \)\(14\!\cdots\!44\)\( T + \)\(86\!\cdots\!83\)\( T^{2} \))(\( 1 + \)\(71\!\cdots\!52\)\( T + \)\(29\!\cdots\!42\)\( T^{2} + \)\(61\!\cdots\!16\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \))
$29$ (\( 1 + \)\(16\!\cdots\!90\)\( T + \)\(18\!\cdots\!89\)\( T^{2} \))(\( 1 + \)\(25\!\cdots\!20\)\( T + \)\(26\!\cdots\!78\)\( T^{2} + \)\(45\!\cdots\!80\)\( T^{3} + \)\(32\!\cdots\!21\)\( T^{4} \))
$31$ (\( 1 + \)\(18\!\cdots\!28\)\( T + \)\(16\!\cdots\!91\)\( T^{2} \))(\( 1 + \)\(43\!\cdots\!96\)\( T + \)\(37\!\cdots\!86\)\( T^{2} + \)\(70\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \))
$37$ (\( 1 + \)\(96\!\cdots\!98\)\( T + \)\(56\!\cdots\!97\)\( T^{2} \))(\( 1 - \)\(11\!\cdots\!76\)\( T + \)\(10\!\cdots\!38\)\( T^{2} - \)\(65\!\cdots\!72\)\( T^{3} + \)\(31\!\cdots\!09\)\( T^{4} \))
$41$ (\( 1 - \)\(64\!\cdots\!42\)\( T + \)\(16\!\cdots\!21\)\( T^{2} \))(\( 1 + \)\(16\!\cdots\!76\)\( T + \)\(11\!\cdots\!86\)\( T^{2} + \)\(27\!\cdots\!96\)\( T^{3} + \)\(27\!\cdots\!41\)\( T^{4} \))
$43$ (\( 1 + \)\(81\!\cdots\!84\)\( T + \)\(80\!\cdots\!43\)\( T^{2} \))(\( 1 + \)\(36\!\cdots\!52\)\( T + \)\(15\!\cdots\!62\)\( T^{2} + \)\(29\!\cdots\!36\)\( T^{3} + \)\(64\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + \)\(62\!\cdots\!68\)\( T + \)\(15\!\cdots\!27\)\( T^{2} \))(\( 1 - \)\(28\!\cdots\!56\)\( T + \)\(29\!\cdots\!38\)\( T^{2} - \)\(43\!\cdots\!12\)\( T^{3} + \)\(22\!\cdots\!29\)\( T^{4} \))
$53$ (\( 1 + \)\(21\!\cdots\!94\)\( T + \)\(79\!\cdots\!73\)\( T^{2} \))(\( 1 + \)\(43\!\cdots\!92\)\( T + \)\(15\!\cdots\!62\)\( T^{2} + \)\(35\!\cdots\!16\)\( T^{3} + \)\(63\!\cdots\!29\)\( T^{4} \))
$59$ (\( 1 - \)\(29\!\cdots\!20\)\( T + \)\(27\!\cdots\!79\)\( T^{2} \))(\( 1 - \)\(18\!\cdots\!60\)\( T + \)\(42\!\cdots\!58\)\( T^{2} - \)\(51\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!41\)\( T^{4} \))
$61$ (\( 1 + \)\(45\!\cdots\!58\)\( T + \)\(82\!\cdots\!81\)\( T^{2} \))(\( 1 - \)\(46\!\cdots\!04\)\( T + \)\(21\!\cdots\!66\)\( T^{2} - \)\(38\!\cdots\!24\)\( T^{3} + \)\(67\!\cdots\!61\)\( T^{4} \))
$67$ (\( 1 - \)\(11\!\cdots\!12\)\( T + \)\(18\!\cdots\!87\)\( T^{2} \))(\( 1 + \)\(74\!\cdots\!64\)\( T + \)\(37\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!68\)\( T^{3} + \)\(33\!\cdots\!69\)\( T^{4} \))
$71$ (\( 1 - \)\(25\!\cdots\!72\)\( T + \)\(12\!\cdots\!11\)\( T^{2} \))(\( 1 - \)\(26\!\cdots\!64\)\( T + \)\(17\!\cdots\!46\)\( T^{2} - \)\(32\!\cdots\!04\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} \))
$73$ (\( 1 + \)\(28\!\cdots\!74\)\( T + \)\(30\!\cdots\!33\)\( T^{2} \))(\( 1 + \)\(52\!\cdots\!32\)\( T + \)\(68\!\cdots\!22\)\( T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(95\!\cdots\!89\)\( T^{4} \))
$79$ (\( 1 - \)\(92\!\cdots\!20\)\( T + \)\(41\!\cdots\!39\)\( T^{2} \))(\( 1 + \)\(26\!\cdots\!80\)\( T + \)\(98\!\cdots\!78\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!21\)\( T^{4} \))
$83$ (\( 1 + \)\(16\!\cdots\!04\)\( T + \)\(21\!\cdots\!63\)\( T^{2} \))(\( 1 + \)\(37\!\cdots\!92\)\( T + \)\(35\!\cdots\!42\)\( T^{2} + \)\(79\!\cdots\!96\)\( T^{3} + \)\(45\!\cdots\!69\)\( T^{4} \))
$89$ (\( 1 + \)\(20\!\cdots\!10\)\( T + \)\(21\!\cdots\!69\)\( T^{2} \))(\( 1 - \)\(30\!\cdots\!80\)\( T + \)\(63\!\cdots\!38\)\( T^{2} - \)\(64\!\cdots\!20\)\( T^{3} + \)\(45\!\cdots\!61\)\( T^{4} \))
$97$ (\( 1 - \)\(22\!\cdots\!42\)\( T + \)\(36\!\cdots\!77\)\( T^{2} \))(\( 1 - \)\(68\!\cdots\!16\)\( T + \)\(68\!\cdots\!18\)\( T^{2} - \)\(24\!\cdots\!32\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \))
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