## Defining parameters

 Level: $$N$$ = $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ Sturm bound: $$1728$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(180))$$.

Total New Old
Modular forms 170 36 134
Cusp forms 10 10 0
Eisenstein series 160 26 134

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 10 0 0 0

## Trace form

 $$10q - 4q^{4} - 2q^{5} + 4q^{6} - 2q^{9} + O(q^{10})$$ $$10q - 4q^{4} - 2q^{5} + 4q^{6} - 2q^{9} - 2q^{10} - 4q^{13} + 2q^{14} - 4q^{16} - 2q^{20} - 4q^{21} - 2q^{24} - 4q^{25} + 2q^{29} - 2q^{30} + 4q^{34} + 4q^{36} - 4q^{37} + 6q^{40} + 2q^{41} + 4q^{45} - 4q^{46} - 2q^{49} + 4q^{52} - 2q^{54} + 2q^{56} + 4q^{58} - 2q^{61} + 2q^{64} + 2q^{69} + 2q^{70} + 4q^{73} + 4q^{80} - 2q^{81} - 4q^{82} + 2q^{84} + 4q^{85} - 4q^{86} - 4q^{89} + 2q^{94} - 2q^{96} + 4q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(180))$$

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
180.1.b $$\chi_{180}(89, \cdot)$$ None 0 1
180.1.c $$\chi_{180}(91, \cdot)$$ None 0 1
180.1.f $$\chi_{180}(19, \cdot)$$ 180.1.f.a 2 1
180.1.g $$\chi_{180}(161, \cdot)$$ None 0 1
180.1.l $$\chi_{180}(37, \cdot)$$ None 0 2
180.1.m $$\chi_{180}(107, \cdot)$$ 180.1.m.a 4 2
180.1.o $$\chi_{180}(41, \cdot)$$ None 0 2
180.1.p $$\chi_{180}(79, \cdot)$$ 180.1.p.a 2 2
180.1.p.b 2
180.1.s $$\chi_{180}(31, \cdot)$$ None 0 2
180.1.t $$\chi_{180}(29, \cdot)$$ None 0 2
180.1.u $$\chi_{180}(13, \cdot)$$ None 0 4
180.1.v $$\chi_{180}(23, \cdot)$$ None 0 4