## Defining parameters

 Level: $$N$$ = $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$8$$ Sturm bound: $$672$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(78))$$.

Total New Old
Modular forms 216 43 173
Cusp forms 121 43 78
Eisenstein series 95 0 95

## Trace form

 $$43q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} - 5q^{8} - 3q^{9} + O(q^{10})$$ $$43q + q^{2} + q^{3} + q^{4} + 6q^{5} + q^{6} - 5q^{8} - 3q^{9} - 24q^{10} - 12q^{11} - 3q^{12} - 35q^{13} - 16q^{14} - 18q^{15} - 7q^{16} - 12q^{17} - 5q^{18} - 36q^{19} - 20q^{21} + 12q^{22} + q^{24} + q^{25} + 13q^{26} - 35q^{27} + 8q^{28} + 30q^{30} + 8q^{31} + q^{32} + 24q^{33} + 18q^{34} + 24q^{35} + 25q^{36} + 32q^{37} + 20q^{38} + 57q^{39} + 6q^{40} + 36q^{41} + 44q^{42} + 12q^{43} + 12q^{44} + 60q^{45} + 24q^{46} + 24q^{47} + q^{48} + 49q^{49} + 25q^{50} - 6q^{51} + 7q^{52} - 18q^{53} - 35q^{54} - 48q^{55} - 16q^{56} - 44q^{57} - 24q^{58} - 36q^{59} - 18q^{60} - 16q^{61} - 40q^{62} - 56q^{63} - 5q^{64} - 24q^{65} - 36q^{66} - 36q^{67} + 12q^{68} - 24q^{69} - 24q^{70} + 24q^{71} - 23q^{72} - 6q^{73} - 40q^{74} + 3q^{75} - 36q^{76} - 48q^{77} - 47q^{78} - 64q^{79} + 21q^{81} - 60q^{82} + 12q^{83} - 20q^{84} - 42q^{85} - 4q^{86} + 6q^{87} + 12q^{88} - 30q^{89} - 18q^{90} - 24q^{92} - 8q^{93} - 72q^{94} + 24q^{95} + q^{96} - 46q^{97} + 9q^{98} + 48q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(78))$$

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 the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
78.2.a $$\chi_{78}(1, \cdot)$$ 78.2.a.a 1 1
78.2.b $$\chi_{78}(25, \cdot)$$ 78.2.b.a 2 1
78.2.e $$\chi_{78}(55, \cdot)$$ 78.2.e.a 2 2
78.2.e.b 2
78.2.g $$\chi_{78}(5, \cdot)$$ 78.2.g.a 12 2
78.2.i $$\chi_{78}(43, \cdot)$$ 78.2.i.a 4 2
78.2.i.b 4
78.2.k $$\chi_{78}(11, \cdot)$$ 78.2.k.a 16 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(78))$$ into lower level spaces

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