Properties

Label 2.34.a
Level 2
Weight 34
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 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: \(\Q\)
Newforms: \( 2 \)
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.

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

Trace form

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

Decomposition of \(S_{34}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.34.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

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