## 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

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

## 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

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