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Properties

Label 78.8
Level 78
Weight 8
Dimension 292
Nonzero newspaces 6
Newform subspaces 18
Sturm bound 2688
Trace bound 3

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Defining parameters

Level: \( N \) = \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 18 \)
Sturm bound: \(2688\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(78))\).

Total New Old
Modular forms 1224 292 932
Cusp forms 1128 292 836
Eisenstein series 96 0 96

Trace form

\( 292 q - 16 q^{2} - 54 q^{3} - 128 q^{4} + 228 q^{5} - 432 q^{6} - 2904 q^{7} + 2048 q^{8} - 4374 q^{9} + O(q^{10}) \) \( 292 q - 16 q^{2} - 54 q^{3} - 128 q^{4} + 228 q^{5} - 432 q^{6} - 2904 q^{7} + 2048 q^{8} - 4374 q^{9} - 18096 q^{10} + 1152 q^{11} + 3456 q^{12} + 63658 q^{13} + 3520 q^{14} - 76140 q^{15} - 40960 q^{16} - 129558 q^{17} + 23328 q^{18} + 436080 q^{19} + 25728 q^{20} - 352284 q^{21} - 117312 q^{22} - 52968 q^{23} - 27648 q^{24} + 599008 q^{25} + 30416 q^{26} + 114822 q^{27} + 201728 q^{28} - 616470 q^{29} - 758496 q^{30} - 631768 q^{31} - 65536 q^{32} + 1795908 q^{33} + 105696 q^{34} + 1029672 q^{35} - 534144 q^{36} - 1618990 q^{37} - 397760 q^{38} - 3105078 q^{39} + 116736 q^{40} + 1940082 q^{41} + 2322144 q^{42} - 71592 q^{43} - 938496 q^{44} + 4457166 q^{45} - 663168 q^{46} - 384696 q^{47} - 221184 q^{48} - 2809418 q^{49} - 3401152 q^{50} + 3540348 q^{51} + 1501696 q^{52} + 10939788 q^{53} + 3568176 q^{54} - 3326664 q^{55} - 1839104 q^{56} - 14320608 q^{57} - 10259952 q^{58} - 16591368 q^{59} - 971520 q^{60} + 7053770 q^{61} + 10339264 q^{62} + 19098336 q^{63} + 1048576 q^{64} + 25098210 q^{65} + 5864640 q^{66} + 3382944 q^{67} + 566400 q^{68} - 5277432 q^{69} - 25083456 q^{70} - 30125280 q^{71} - 10466304 q^{72} - 49654596 q^{73} - 21992240 q^{74} + 1177146 q^{75} + 27909120 q^{76} + 111483696 q^{77} + 36286992 q^{78} + 22210064 q^{79} + 1646592 q^{80} - 31010286 q^{81} - 69617136 q^{82} - 45845472 q^{83} - 6697728 q^{84} - 50104362 q^{85} + 4039744 q^{86} + 48294372 q^{87} - 7507968 q^{88} + 74105700 q^{89} + 18825696 q^{90} + 67259304 q^{91} + 22671360 q^{92} - 43765008 q^{93} + 18766656 q^{94} - 14983920 q^{95} - 1769472 q^{96} - 41578276 q^{97} - 74652432 q^{98} - 53194932 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(78))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
78.8.a \(\chi_{78}(1, \cdot)\) 78.8.a.a 1 1
78.8.a.b 1
78.8.a.c 2
78.8.a.d 2
78.8.a.e 2
78.8.a.f 2
78.8.a.g 2
78.8.a.h 2
78.8.b \(\chi_{78}(25, \cdot)\) 78.8.b.a 6 1
78.8.b.b 8
78.8.e \(\chi_{78}(55, \cdot)\) 78.8.e.a 8 2
78.8.e.b 8
78.8.e.c 10
78.8.e.d 10
78.8.g \(\chi_{78}(5, \cdot)\) 78.8.g.a 68 2
78.8.i \(\chi_{78}(43, \cdot)\) 78.8.i.a 16 2
78.8.i.b 16
78.8.k \(\chi_{78}(11, \cdot)\) 78.8.k.a 128 4

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(78))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(78)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 2}\)