## Defining parameters

 Level: $$N$$ = $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$9$$ Newform subspaces: $$18$$ Sturm bound: $$1536$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(120))$$.

Total New Old
Modular forms 480 160 320
Cusp forms 289 136 153
Eisenstein series 191 24 167

## Trace form

 $$136 q + 4 q^{2} + 2 q^{3} + 4 q^{5} - 12 q^{6} + 8 q^{7} - 8 q^{8} + O(q^{10})$$ $$136 q + 4 q^{2} + 2 q^{3} + 4 q^{5} - 12 q^{6} + 8 q^{7} - 8 q^{8} - 12 q^{10} - 32 q^{12} + 8 q^{13} - 32 q^{14} - 18 q^{15} - 48 q^{16} - 20 q^{18} - 40 q^{19} - 32 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 8 q^{24} - 8 q^{25} - 16 q^{26} - 46 q^{27} - 24 q^{29} - 4 q^{30} - 72 q^{31} + 24 q^{32} - 40 q^{33} + 16 q^{34} - 48 q^{35} - 8 q^{36} - 40 q^{37} + 40 q^{38} - 20 q^{39} + 64 q^{40} + 32 q^{42} - 24 q^{43} + 56 q^{44} + 12 q^{45} + 64 q^{46} + 16 q^{47} + 72 q^{48} - 24 q^{49} + 52 q^{50} + 60 q^{51} - 8 q^{52} + 16 q^{53} + 36 q^{54} - 16 q^{55} + 16 q^{56} + 16 q^{58} + 24 q^{59} + 72 q^{60} + 48 q^{61} + 8 q^{62} + 64 q^{63} + 48 q^{64} - 16 q^{65} + 64 q^{66} + 24 q^{67} + 56 q^{68} + 72 q^{70} + 16 q^{71} + 72 q^{72} - 80 q^{73} + 56 q^{74} + 42 q^{75} + 32 q^{76} + 72 q^{78} + 64 q^{79} + 80 q^{80} - 48 q^{81} + 128 q^{82} + 80 q^{83} + 72 q^{84} - 64 q^{85} - 16 q^{86} + 60 q^{87} - 48 q^{88} - 40 q^{89} + 36 q^{90} - 64 q^{91} - 8 q^{92} - 32 q^{93} - 128 q^{94} + 24 q^{95} - 112 q^{96} - 24 q^{97} - 100 q^{98} + 88 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$

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
120.2.a $$\chi_{120}(1, \cdot)$$ 120.2.a.a 1 1
120.2.a.b 1
120.2.b $$\chi_{120}(11, \cdot)$$ 120.2.b.a 8 1
120.2.b.b 8
120.2.d $$\chi_{120}(109, \cdot)$$ 120.2.d.a 6 1
120.2.d.b 6
120.2.f $$\chi_{120}(49, \cdot)$$ 120.2.f.a 2 1
120.2.h $$\chi_{120}(71, \cdot)$$ None 0 1
120.2.k $$\chi_{120}(61, \cdot)$$ 120.2.k.a 2 1
120.2.k.b 6
120.2.m $$\chi_{120}(59, \cdot)$$ 120.2.m.a 4 1
120.2.m.b 16
120.2.o $$\chi_{120}(119, \cdot)$$ None 0 1
120.2.r $$\chi_{120}(17, \cdot)$$ 120.2.r.a 4 2
120.2.r.b 4
120.2.r.c 4
120.2.s $$\chi_{120}(7, \cdot)$$ None 0 2
120.2.v $$\chi_{120}(43, \cdot)$$ 120.2.v.a 24 2
120.2.w $$\chi_{120}(53, \cdot)$$ 120.2.w.a 4 2
120.2.w.b 4
120.2.w.c 32

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(120))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(120)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 2}$$