# Properties

 Label 78.4 Level 78 Weight 4 Dimension 124 Nonzero newspaces 6 Newform subspaces 16 Sturm bound 1344 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$16$$ Sturm bound: $$1344$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(78))$$.

Total New Old
Modular forms 552 124 428
Cusp forms 456 124 332
Eisenstein series 96 0 96

## Trace form

 $$124 q + 4 q^{2} + 6 q^{3} - 8 q^{4} - 12 q^{5} - 12 q^{6} - 112 q^{7} - 32 q^{8} - 18 q^{9} + O(q^{10})$$ $$124 q + 4 q^{2} + 6 q^{3} - 8 q^{4} - 12 q^{5} - 12 q^{6} - 112 q^{7} - 32 q^{8} - 18 q^{9} + 204 q^{10} + 216 q^{11} + 72 q^{12} + 538 q^{13} + 176 q^{14} + 180 q^{15} - 32 q^{16} - 222 q^{17} - 72 q^{18} - 1408 q^{19} - 312 q^{20} - 216 q^{21} + 48 q^{22} + 120 q^{23} - 48 q^{24} + 928 q^{25} + 76 q^{26} + 882 q^{27} + 128 q^{28} + 330 q^{29} + 744 q^{30} + 56 q^{31} + 64 q^{32} - 960 q^{33} - 504 q^{34} - 1368 q^{35} - 840 q^{36} - 1150 q^{37} + 80 q^{38} - 2130 q^{39} + 96 q^{40} - 438 q^{41} - 744 q^{42} - 568 q^{43} - 96 q^{44} + 186 q^{45} + 672 q^{46} + 936 q^{47} + 96 q^{48} + 534 q^{49} - 1232 q^{50} + 468 q^{51} - 704 q^{52} - 2484 q^{53} - 1764 q^{54} - 144 q^{55} - 448 q^{56} + 684 q^{57} + 1308 q^{58} + 3648 q^{59} + 1200 q^{60} + 2402 q^{61} + 3824 q^{62} + 8364 q^{63} + 256 q^{64} + 5130 q^{65} + 2832 q^{66} + 2288 q^{67} + 1320 q^{68} + 4524 q^{69} + 2064 q^{70} - 528 q^{71} + 336 q^{72} - 2740 q^{73} - 4756 q^{74} - 11754 q^{75} - 5632 q^{76} - 12864 q^{77} - 6060 q^{78} - 10000 q^{79} - 1248 q^{80} - 8274 q^{81} + 444 q^{82} + 3432 q^{83} + 960 q^{84} + 8718 q^{85} + 3152 q^{86} + 6420 q^{87} + 192 q^{88} + 4260 q^{89} + 2376 q^{90} + 9968 q^{91} + 2880 q^{92} + 9540 q^{93} + 8976 q^{94} + 6480 q^{95} - 192 q^{96} + 12404 q^{97} + 1284 q^{98} + 6624 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(78))$$

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
78.4.a $$\chi_{78}(1, \cdot)$$ 78.4.a.a 1 1
78.4.a.b 1
78.4.a.c 1
78.4.a.d 1
78.4.a.e 1
78.4.a.f 1
78.4.b $$\chi_{78}(25, \cdot)$$ 78.4.b.a 2 1
78.4.b.b 4
78.4.e $$\chi_{78}(55, \cdot)$$ 78.4.e.a 2 2
78.4.e.b 4
78.4.e.c 4
78.4.e.d 6
78.4.g $$\chi_{78}(5, \cdot)$$ 78.4.g.a 28 2
78.4.i $$\chi_{78}(43, \cdot)$$ 78.4.i.a 4 2
78.4.i.b 8
78.4.k $$\chi_{78}(11, \cdot)$$ 78.4.k.a 56 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(78))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(78)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(78))$$$$^{\oplus 1}$$