## Defining parameters

 Level: $$N$$ = $$57 = 3 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$11$$ Sturm bound: $$480$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(57))$$.

Total New Old
Modular forms 156 107 49
Cusp forms 85 71 14
Eisenstein series 71 36 35

## Trace form

 $$71q - 3q^{2} - 10q^{3} - 25q^{4} - 6q^{5} - 12q^{6} - 26q^{7} - 15q^{8} - 10q^{9} + O(q^{10})$$ $$71q - 3q^{2} - 10q^{3} - 25q^{4} - 6q^{5} - 12q^{6} - 26q^{7} - 15q^{8} - 10q^{9} - 36q^{10} - 12q^{11} - 4q^{12} - 8q^{13} + 12q^{14} + 3q^{15} + 23q^{16} - 3q^{18} + 5q^{19} + 30q^{20} + 4q^{21} - 6q^{23} + 12q^{24} - 13q^{25} - 6q^{26} + 2q^{27} + 4q^{28} + 6q^{29} + 36q^{30} - 14q^{31} + 27q^{32} + 33q^{33} + 18q^{34} + 24q^{35} + 56q^{36} - 2q^{37} + 69q^{38} + 22q^{39} + 54q^{40} - 6q^{41} + 57q^{42} + 10q^{43} + 6q^{44} + 12q^{45} - 12q^{47} - 40q^{48} - 39q^{49} - 39q^{50} - 63q^{51} - 116q^{52} - 54q^{53} - 66q^{54} - 90q^{55} - 120q^{56} - 73q^{57} - 126q^{58} - 60q^{59} - 114q^{60} - 20q^{61} - 6q^{62} - 41q^{63} - q^{64} + 24q^{65} - 45q^{66} + 58q^{67} + 12q^{69} + 54q^{70} + 39q^{72} + 70q^{73} + 30q^{74} + 53q^{75} + 29q^{76} + 102q^{77} + 93q^{78} + 106q^{79} + 66q^{80} + 26q^{81} + 198q^{82} + 42q^{83} + 151q^{84} + 90q^{85} + 66q^{86} + 105q^{87} + 126q^{88} + 72q^{89} + 36q^{90} + 38q^{91} + 66q^{92} + 61q^{93} + 36q^{94} - 60q^{95} - 9q^{96} - 62q^{97} - 27q^{98} - 12q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$

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
57.2.a $$\chi_{57}(1, \cdot)$$ 57.2.a.a 1 1
57.2.a.b 1
57.2.a.c 1
57.2.d $$\chi_{57}(56, \cdot)$$ 57.2.d.a 4 1
57.2.e $$\chi_{57}(7, \cdot)$$ 57.2.e.a 2 2
57.2.e.b 6
57.2.f $$\chi_{57}(8, \cdot)$$ 57.2.f.a 8 2
57.2.i $$\chi_{57}(4, \cdot)$$ 57.2.i.a 6 6
57.2.i.b 12
57.2.j $$\chi_{57}(2, \cdot)$$ 57.2.j.a 6 6
57.2.j.b 24

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(57))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(57)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 2}$$