## Defining parameters

 Level: $$N$$ = $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$576$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(60))$$.

Total New Old
Modular forms 232 86 146
Cusp forms 152 70 82
Eisenstein series 80 16 64

## Trace form

 $$70q + 4q^{2} + 4q^{3} + 8q^{4} + 16q^{5} - 4q^{6} + 24q^{7} - 20q^{8} + 14q^{9} + O(q^{10})$$ $$70q + 4q^{2} + 4q^{3} + 8q^{4} + 16q^{5} - 4q^{6} + 24q^{7} - 20q^{8} + 14q^{9} - 24q^{10} - 16q^{11} - 20q^{12} + 24q^{13} - 72q^{14} - 14q^{15} - 80q^{16} - 40q^{17} - 36q^{18} - 92q^{19} + 12q^{20} - 152q^{21} - 8q^{22} - 40q^{23} - 122q^{25} + 120q^{26} - 44q^{27} - 56q^{28} + 24q^{29} + 8q^{30} + 180q^{31} - 76q^{32} + 80q^{33} - 120q^{34} + 56q^{35} + 136q^{36} - 40q^{37} - 40q^{38} + 128q^{39} + 248q^{40} + 344q^{42} + 136q^{43} + 376q^{44} + 16q^{45} + 552q^{46} + 244q^{48} + 6q^{49} + 92q^{50} - 244q^{51} + 176q^{52} + 152q^{53} + 36q^{54} - 208q^{55} - 128q^{56} + 56q^{57} - 24q^{58} - 128q^{60} - 228q^{61} - 56q^{62} + 56q^{63} - 376q^{64} - 32q^{65} - 528q^{66} + 24q^{67} - 184q^{68} + 28q^{69} - 536q^{70} - 80q^{71} - 540q^{72} - 368q^{73} - 336q^{74} - 28q^{75} - 592q^{76} - 488q^{77} - 648q^{78} - 356q^{79} - 388q^{80} - 258q^{81} - 680q^{82} - 240q^{83} - 384q^{84} + 196q^{85} - 336q^{86} - 120q^{87} - 440q^{88} + 712q^{89} + 432q^{90} + 656q^{91} + 144q^{92} + 688q^{93} + 280q^{94} + 400q^{95} + 944q^{96} + 1048q^{97} + 660q^{98} + 560q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(60))$$

We only show spaces with odd 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
60.3.b $$\chi_{60}(29, \cdot)$$ 60.3.b.a 4 1
60.3.c $$\chi_{60}(31, \cdot)$$ 60.3.c.a 8 1
60.3.f $$\chi_{60}(19, \cdot)$$ 60.3.f.a 4 1
60.3.f.b 8
60.3.g $$\chi_{60}(41, \cdot)$$ 60.3.g.a 2 1
60.3.k $$\chi_{60}(13, \cdot)$$ 60.3.k.a 4 2
60.3.l $$\chi_{60}(23, \cdot)$$ 60.3.l.a 40 2

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(60))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(60)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$