Properties

Label 8.5
Level 8
Weight 5
Dimension 3
Nonzero newspaces 1
Newforms 2
Sturm bound 20
Trace bound 0

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 1 \)
Newforms: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(8))\).

Total New Old
Modular forms 11 5 6
Cusp forms 5 3 2
Eisenstein series 6 2 4

Trace form

\(3q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 68q^{6} \) \(\mathstrut +\mathstrut 152q^{8} \) \(\mathstrut +\mathstrut 25q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 68q^{6} \) \(\mathstrut +\mathstrut 152q^{8} \) \(\mathstrut +\mathstrut 25q^{9} \) \(\mathstrut +\mathstrut 240q^{10} \) \(\mathstrut -\mathstrut 98q^{11} \) \(\mathstrut -\mathstrut 392q^{12} \) \(\mathstrut -\mathstrut 480q^{14} \) \(\mathstrut +\mathstrut 528q^{16} \) \(\mathstrut -\mathstrut 122q^{17} \) \(\mathstrut +\mathstrut 550q^{18} \) \(\mathstrut +\mathstrut 702q^{19} \) \(\mathstrut -\mathstrut 480q^{20} \) \(\mathstrut -\mathstrut 132q^{22} \) \(\mathstrut -\mathstrut 368q^{24} \) \(\mathstrut -\mathstrut 45q^{25} \) \(\mathstrut -\mathstrut 240q^{26} \) \(\mathstrut -\mathstrut 1988q^{27} \) \(\mathstrut +\mathstrut 960q^{28} \) \(\mathstrut +\mathstrut 1440q^{30} \) \(\mathstrut -\mathstrut 928q^{32} \) \(\mathstrut +\mathstrut 332q^{33} \) \(\mathstrut -\mathstrut 2748q^{34} \) \(\mathstrut +\mathstrut 3840q^{35} \) \(\mathstrut +\mathstrut 3100q^{36} \) \(\mathstrut +\mathstrut 1468q^{38} \) \(\mathstrut -\mathstrut 2880q^{40} \) \(\mathstrut +\mathstrut 742q^{41} \) \(\mathstrut -\mathstrut 2880q^{42} \) \(\mathstrut -\mathstrut 7266q^{43} \) \(\mathstrut -\mathstrut 8q^{44} \) \(\mathstrut +\mathstrut 2400q^{46} \) \(\mathstrut -\mathstrut 1952q^{48} \) \(\mathstrut -\mathstrut 477q^{49} \) \(\mathstrut +\mathstrut 3170q^{50} \) \(\mathstrut +\mathstrut 10748q^{51} \) \(\mathstrut +\mathstrut 480q^{52} \) \(\mathstrut -\mathstrut 392q^{54} \) \(\mathstrut +\mathstrut 5760q^{56} \) \(\mathstrut -\mathstrut 4468q^{57} \) \(\mathstrut +\mathstrut 2640q^{58} \) \(\mathstrut -\mathstrut 10274q^{59} \) \(\mathstrut -\mathstrut 2880q^{60} \) \(\mathstrut -\mathstrut 9600q^{62} \) \(\mathstrut +\mathstrut 3648q^{64} \) \(\mathstrut +\mathstrut 1920q^{65} \) \(\mathstrut +\mathstrut 2888q^{66} \) \(\mathstrut +\mathstrut 10878q^{67} \) \(\mathstrut -\mathstrut 15512q^{68} \) \(\mathstrut -\mathstrut 3840q^{70} \) \(\mathstrut +\mathstrut 3400q^{72} \) \(\mathstrut +\mathstrut 10278q^{73} \) \(\mathstrut +\mathstrut 13680q^{74} \) \(\mathstrut -\mathstrut 12770q^{75} \) \(\mathstrut +\mathstrut 3192q^{76} \) \(\mathstrut -\mathstrut 1440q^{78} \) \(\mathstrut +\mathstrut 13440q^{80} \) \(\mathstrut -\mathstrut 4433q^{81} \) \(\mathstrut -\mathstrut 6972q^{82} \) \(\mathstrut +\mathstrut 6718q^{83} \) \(\mathstrut +\mathstrut 5760q^{84} \) \(\mathstrut -\mathstrut 10244q^{86} \) \(\mathstrut -\mathstrut 5232q^{88} \) \(\mathstrut -\mathstrut 14618q^{89} \) \(\mathstrut -\mathstrut 10800q^{90} \) \(\mathstrut -\mathstrut 3840q^{91} \) \(\mathstrut -\mathstrut 4800q^{92} \) \(\mathstrut +\mathstrut 16320q^{94} \) \(\mathstrut -\mathstrut 26048q^{96} \) \(\mathstrut +\mathstrut 7494q^{97} \) \(\mathstrut +\mathstrut 12482q^{98} \) \(\mathstrut -\mathstrut 2950q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(8))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
8.5.c \(\chi_{8}(7, \cdot)\) None 0 1
8.5.d \(\chi_{8}(3, \cdot)\) 8.5.d.a 1 1
8.5.d.b 2

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)