Properties

Label 2.8.a
Level 2
Weight 8
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newforms 1
Sturm bound 2
Trace bound 0

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newforms: \( 1 \)
Sturm bound: \(2\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(2))\).

Total New Old
Modular forms 3 1 2
Cusp forms 1 1 0
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.
\(+\)\(1\)

Trace form

\(q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 210q^{5} \) \(\mathstrut -\mathstrut 96q^{6} \) \(\mathstrut +\mathstrut 1016q^{7} \) \(\mathstrut -\mathstrut 512q^{8} \) \(\mathstrut -\mathstrut 2043q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut +\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 210q^{5} \) \(\mathstrut -\mathstrut 96q^{6} \) \(\mathstrut +\mathstrut 1016q^{7} \) \(\mathstrut -\mathstrut 512q^{8} \) \(\mathstrut -\mathstrut 2043q^{9} \) \(\mathstrut +\mathstrut 1680q^{10} \) \(\mathstrut +\mathstrut 1092q^{11} \) \(\mathstrut +\mathstrut 768q^{12} \) \(\mathstrut +\mathstrut 1382q^{13} \) \(\mathstrut -\mathstrut 8128q^{14} \) \(\mathstrut -\mathstrut 2520q^{15} \) \(\mathstrut +\mathstrut 4096q^{16} \) \(\mathstrut +\mathstrut 14706q^{17} \) \(\mathstrut +\mathstrut 16344q^{18} \) \(\mathstrut -\mathstrut 39940q^{19} \) \(\mathstrut -\mathstrut 13440q^{20} \) \(\mathstrut +\mathstrut 12192q^{21} \) \(\mathstrut -\mathstrut 8736q^{22} \) \(\mathstrut +\mathstrut 68712q^{23} \) \(\mathstrut -\mathstrut 6144q^{24} \) \(\mathstrut -\mathstrut 34025q^{25} \) \(\mathstrut -\mathstrut 11056q^{26} \) \(\mathstrut -\mathstrut 50760q^{27} \) \(\mathstrut +\mathstrut 65024q^{28} \) \(\mathstrut -\mathstrut 102570q^{29} \) \(\mathstrut +\mathstrut 20160q^{30} \) \(\mathstrut +\mathstrut 227552q^{31} \) \(\mathstrut -\mathstrut 32768q^{32} \) \(\mathstrut +\mathstrut 13104q^{33} \) \(\mathstrut -\mathstrut 117648q^{34} \) \(\mathstrut -\mathstrut 213360q^{35} \) \(\mathstrut -\mathstrut 130752q^{36} \) \(\mathstrut +\mathstrut 160526q^{37} \) \(\mathstrut +\mathstrut 319520q^{38} \) \(\mathstrut +\mathstrut 16584q^{39} \) \(\mathstrut +\mathstrut 107520q^{40} \) \(\mathstrut +\mathstrut 10842q^{41} \) \(\mathstrut -\mathstrut 97536q^{42} \) \(\mathstrut -\mathstrut 630748q^{43} \) \(\mathstrut +\mathstrut 69888q^{44} \) \(\mathstrut +\mathstrut 429030q^{45} \) \(\mathstrut -\mathstrut 549696q^{46} \) \(\mathstrut +\mathstrut 472656q^{47} \) \(\mathstrut +\mathstrut 49152q^{48} \) \(\mathstrut +\mathstrut 208713q^{49} \) \(\mathstrut +\mathstrut 272200q^{50} \) \(\mathstrut +\mathstrut 176472q^{51} \) \(\mathstrut +\mathstrut 88448q^{52} \) \(\mathstrut -\mathstrut 1494018q^{53} \) \(\mathstrut +\mathstrut 406080q^{54} \) \(\mathstrut -\mathstrut 229320q^{55} \) \(\mathstrut -\mathstrut 520192q^{56} \) \(\mathstrut -\mathstrut 479280q^{57} \) \(\mathstrut +\mathstrut 820560q^{58} \) \(\mathstrut +\mathstrut 2640660q^{59} \) \(\mathstrut -\mathstrut 161280q^{60} \) \(\mathstrut +\mathstrut 827702q^{61} \) \(\mathstrut -\mathstrut 1820416q^{62} \) \(\mathstrut -\mathstrut 2075688q^{63} \) \(\mathstrut +\mathstrut 262144q^{64} \) \(\mathstrut -\mathstrut 290220q^{65} \) \(\mathstrut -\mathstrut 104832q^{66} \) \(\mathstrut -\mathstrut 126004q^{67} \) \(\mathstrut +\mathstrut 941184q^{68} \) \(\mathstrut +\mathstrut 824544q^{69} \) \(\mathstrut +\mathstrut 1706880q^{70} \) \(\mathstrut -\mathstrut 1414728q^{71} \) \(\mathstrut +\mathstrut 1046016q^{72} \) \(\mathstrut +\mathstrut 980282q^{73} \) \(\mathstrut -\mathstrut 1284208q^{74} \) \(\mathstrut -\mathstrut 408300q^{75} \) \(\mathstrut -\mathstrut 2556160q^{76} \) \(\mathstrut +\mathstrut 1109472q^{77} \) \(\mathstrut -\mathstrut 132672q^{78} \) \(\mathstrut -\mathstrut 3566800q^{79} \) \(\mathstrut -\mathstrut 860160q^{80} \) \(\mathstrut +\mathstrut 3858921q^{81} \) \(\mathstrut -\mathstrut 86736q^{82} \) \(\mathstrut +\mathstrut 5672892q^{83} \) \(\mathstrut +\mathstrut 780288q^{84} \) \(\mathstrut -\mathstrut 3088260q^{85} \) \(\mathstrut +\mathstrut 5045984q^{86} \) \(\mathstrut -\mathstrut 1230840q^{87} \) \(\mathstrut -\mathstrut 559104q^{88} \) \(\mathstrut -\mathstrut 11951190q^{89} \) \(\mathstrut -\mathstrut 3432240q^{90} \) \(\mathstrut +\mathstrut 1404112q^{91} \) \(\mathstrut +\mathstrut 4397568q^{92} \) \(\mathstrut +\mathstrut 2730624q^{93} \) \(\mathstrut -\mathstrut 3781248q^{94} \) \(\mathstrut +\mathstrut 8387400q^{95} \) \(\mathstrut -\mathstrut 393216q^{96} \) \(\mathstrut +\mathstrut 8682146q^{97} \) \(\mathstrut -\mathstrut 1669704q^{98} \) \(\mathstrut -\mathstrut 2230956q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.8.a.a \(1\) \(0.625\) \(\Q\) None \(-8\) \(12\) \(-210\) \(1016\) \(+\) \(q-8q^{2}+12q^{3}+2^{6}q^{4}-210q^{5}+\cdots\)