## Defining parameters

 Level: $$N$$ = $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$6$$ Sturm bound: $$720$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(76))$$.

Total New Old
Modular forms 225 123 102
Cusp forms 136 87 49
Eisenstein series 89 36 53

## Trace form

 $$87q - 9q^{2} - 9q^{4} - 18q^{5} - 9q^{6} - 9q^{8} - 18q^{9} + O(q^{10})$$ $$87q - 9q^{2} - 9q^{4} - 18q^{5} - 9q^{6} - 9q^{8} - 18q^{9} - 9q^{10} - 9q^{12} - 30q^{13} - 9q^{14} - 18q^{15} - 9q^{16} - 27q^{17} - 21q^{19} - 18q^{20} - 39q^{21} - 9q^{22} - 9q^{23} - 9q^{24} - 36q^{25} - 9q^{26} - 3q^{27} + 18q^{28} + 54q^{30} + 18q^{31} + 36q^{32} + 36q^{33} + 36q^{34} + 36q^{35} + 81q^{36} + 18q^{37} + 54q^{38} + 54q^{39} + 54q^{40} + 81q^{42} + 33q^{43} + 36q^{44} + 9q^{45} + 36q^{46} - 9q^{47} + 54q^{48} - 21q^{49} + 18q^{50} - 36q^{51} - 9q^{52} - 54q^{53} - 36q^{54} - 36q^{55} - 63q^{57} - 18q^{58} - 45q^{59} - 12q^{61} - 54q^{62} - 12q^{63} - 81q^{64} - 9q^{65} - 81q^{66} + 15q^{67} - 72q^{68} + 99q^{69} - 117q^{70} - 9q^{71} - 144q^{72} + 60q^{73} - 63q^{74} + 42q^{75} - 99q^{76} + 81q^{77} - 81q^{78} + 24q^{79} - 81q^{80} + 27q^{81} - 144q^{82} - 9q^{83} - 117q^{84} + 90q^{85} - 72q^{86} - 36q^{87} - 81q^{88} + 9q^{89} - 36q^{90} - 12q^{91} - 54q^{92} - 48q^{93} + 9q^{95} + 54q^{96} - 9q^{97} + 90q^{98} - 9q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(76))$$

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
76.2.a $$\chi_{76}(1, \cdot)$$ 76.2.a.a 1 1
76.2.d $$\chi_{76}(75, \cdot)$$ 76.2.d.a 8 1
76.2.e $$\chi_{76}(45, \cdot)$$ 76.2.e.a 2 2
76.2.f $$\chi_{76}(27, \cdot)$$ 76.2.f.a 16 2
76.2.i $$\chi_{76}(5, \cdot)$$ 76.2.i.a 12 6
76.2.k $$\chi_{76}(3, \cdot)$$ 76.2.k.a 48 6

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(76))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(76)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 2}$$