Properties

Label 8.8
Level 8
Weight 8
Dimension 8
Nonzero newspaces 2
Newforms 3
Sturm bound 32
Trace bound 1

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Defining parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 2 \)
Newforms: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(1\)

Dimensions

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

Total New Old
Modular forms 17 10 7
Cusp forms 11 8 3
Eisenstein series 6 2 4

Trace form

\( 8q + 6q^{2} - 40q^{3} + 116q^{4} + 348q^{5} + 268q^{6} - 2368q^{7} + 1512q^{8} + 1700q^{9} + O(q^{10}) \) \( 8q + 6q^{2} - 40q^{3} + 116q^{4} + 348q^{5} + 268q^{6} - 2368q^{7} + 1512q^{8} + 1700q^{9} - 1656q^{10} - 5688q^{11} - 4088q^{12} - 4660q^{13} + 12048q^{14} + 43680q^{15} + 35344q^{16} - 25968q^{17} - 89062q^{18} - 1352q^{19} - 114768q^{20} - 15552q^{21} + 152860q^{22} + 81984q^{23} + 282512q^{24} - 3940q^{25} - 316968q^{26} - 332560q^{27} - 480800q^{28} + 222636q^{29} + 821648q^{30} + 121600q^{31} + 817056q^{32} + 126680q^{33} - 1009108q^{34} - 488928q^{35} - 1253556q^{36} - 471780q^{37} + 974124q^{38} + 846336q^{39} + 954464q^{40} + 343248q^{41} - 1093088q^{42} + 20360q^{43} - 1096344q^{44} - 507188q^{45} + 929840q^{46} + 1168512q^{47} + 853920q^{48} - 452024q^{49} - 148626q^{50} + 220720q^{51} + 823952q^{52} + 2109180q^{53} - 1077064q^{54} - 4423808q^{55} - 2468928q^{56} - 2477256q^{57} + 3130744q^{58} - 1863576q^{59} + 5715168q^{60} + 1002860q^{61} - 7055808q^{62} + 3863776q^{63} - 4792768q^{64} + 4931016q^{65} + 7926264q^{66} + 7625560q^{67} + 6608040q^{68} - 6793792q^{69} - 7406912q^{70} - 12856128q^{71} - 11363944q^{72} - 5261616q^{73} + 7744200q^{74} + 10695784q^{75} + 9241288q^{76} + 5023680q^{77} - 9471184q^{78} + 14929792q^{79} - 12600384q^{80} + 3381600q^{81} + 10715932q^{82} - 6949320q^{83} + 4220608q^{84} - 6081800q^{85} - 5639076q^{86} - 40431936q^{87} + 1541200q^{88} + 7301712q^{89} - 121864q^{90} - 2573664q^{91} + 669600q^{92} + 14460160q^{93} + 15503712q^{94} + 45845088q^{95} + 21402176q^{96} + 3777552q^{97} - 14983242q^{98} - 11495192q^{99} + O(q^{100}) \)

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

We only show spaces with even 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
8.8.a \(\chi_{8}(1, \cdot)\) 8.8.a.a 1 1
8.8.a.b 1
8.8.b \(\chi_{8}(5, \cdot)\) 8.8.b.a 6 1

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

\( S_{8}^{\mathrm{old}}(\Gamma_1(8)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)