## Defining parameters

 Level: $$N$$ = $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$5$$ Newform subspaces: $$6$$ Sturm bound: $$384$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(60))$$.

Total New Old
Modular forms 136 42 94
Cusp forms 57 34 23
Eisenstein series 79 8 71

## Trace form

 $$34q + 2q^{3} - 8q^{4} + 2q^{5} - 8q^{6} - 4q^{7} - 12q^{8} - 10q^{9} + O(q^{10})$$ $$34q + 2q^{3} - 8q^{4} + 2q^{5} - 8q^{6} - 4q^{7} - 12q^{8} - 10q^{9} - 20q^{10} - 8q^{11} - 4q^{12} - 24q^{13} - 14q^{15} - 20q^{17} + 16q^{18} + 20q^{20} - 12q^{21} + 8q^{22} + 24q^{24} - 14q^{25} + 16q^{26} + 14q^{27} + 24q^{28} + 12q^{29} + 36q^{30} + 24q^{31} + 20q^{32} + 12q^{33} + 24q^{34} + 16q^{35} + 16q^{38} + 16q^{40} - 4q^{41} + 8q^{42} - 12q^{43} - 2q^{45} - 24q^{46} - 20q^{48} - 18q^{49} - 16q^{50} - 12q^{51} - 40q^{52} + 4q^{53} - 56q^{54} - 8q^{55} - 64q^{56} + 20q^{57} - 56q^{58} - 8q^{59} - 64q^{60} - 36q^{61} - 56q^{62} + 8q^{63} - 56q^{64} + 20q^{65} - 64q^{66} - 4q^{67} - 16q^{68} + 32q^{69} - 8q^{70} + 12q^{72} + 48q^{73} + 2q^{75} + 32q^{76} + 48q^{77} + 16q^{78} + 24q^{79} + 4q^{80} + 26q^{81} + 56q^{82} + 56q^{84} + 56q^{85} + 64q^{86} + 20q^{87} + 104q^{88} + 20q^{89} + 60q^{90} + 24q^{91} + 56q^{92} + 24q^{93} + 104q^{94} + 64q^{96} - 32q^{97} + 24q^{98} + 8q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$

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
60.2.a $$\chi_{60}(1, \cdot)$$ None 0 1
60.2.d $$\chi_{60}(49, \cdot)$$ 60.2.d.a 2 1
60.2.e $$\chi_{60}(11, \cdot)$$ 60.2.e.a 8 1
60.2.h $$\chi_{60}(59, \cdot)$$ 60.2.h.a 4 1
60.2.h.b 4
60.2.i $$\chi_{60}(17, \cdot)$$ 60.2.i.a 4 2
60.2.j $$\chi_{60}(7, \cdot)$$ 60.2.j.a 12 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(60))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(60)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$