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

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

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

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