Properties

Label 8.7
Level 8
Weight 7
Dimension 5
Nonzero newspaces 1
Newforms 2
Sturm bound 28
Trace bound 0

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 1 \)
Newforms: \( 2 \)
Sturm bound: \(28\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(8))\).

Total New Old
Modular forms 15 7 8
Cusp forms 9 5 4
Eisenstein series 6 2 4

Trace form

\( 5q - 6q^{2} - 2q^{3} + 20q^{4} + 28q^{6} - 264q^{8} + 727q^{9} + O(q^{10}) \) \( 5q - 6q^{2} - 2q^{3} + 20q^{4} + 28q^{6} - 264q^{8} + 727q^{9} - 1920q^{10} - 1362q^{11} + 4312q^{12} + 5760q^{14} - 10480q^{16} + 2442q^{17} - 21506q^{18} - 3938q^{19} + 31680q^{20} + 43132q^{22} - 61808q^{24} - 8275q^{25} - 59520q^{26} + 32860q^{27} + 59520q^{28} + 90240q^{30} - 81696q^{32} - 23500q^{33} - 68108q^{34} - 49920q^{35} + 75868q^{36} + 46428q^{38} - 13440q^{40} + 16698q^{41} + 38400q^{42} + 122542q^{43} - 112488q^{44} - 213120q^{46} + 326368q^{48} + 119765q^{49} + 232650q^{50} - 465412q^{51} - 254400q^{52} - 495368q^{54} + 349440q^{56} + 12500q^{57} + 516480q^{58} + 846990q^{59} - 716160q^{60} - 407040q^{62} + 726080q^{64} - 205440q^{65} + 717608q^{66} - 1386818q^{67} - 324312q^{68} - 360960q^{70} + 95656q^{72} - 149222q^{73} - 32640q^{74} + 2483950q^{75} - 88232q^{76} + 324480q^{78} - 1032960q^{80} - 186839q^{81} - 672428q^{82} - 2786082q^{83} + 602880q^{84} + 1588860q^{86} - 753392q^{88} + 403962q^{89} - 1296000q^{90} + 3398400q^{91} + 2743680q^{92} + 971520q^{94} - 1760192q^{96} + 895978q^{97} - 3332454q^{98} - 5702086q^{99} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(8))\)

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
8.7.c \(\chi_{8}(7, \cdot)\) None 0 1
8.7.d \(\chi_{8}(3, \cdot)\) 8.7.d.a 1 1
8.7.d.b 4

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(8))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)