## 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

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

## 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

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