## Defining parameters

 Level: $$N$$ = $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$21$$ Sturm bound: $$3456$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(180))$$.

Total New Old
Modular forms 1024 393 631
Cusp forms 705 337 368
Eisenstein series 319 56 263

## Trace form

 $$337q + 6q^{4} - 4q^{5} - 10q^{6} + 8q^{7} + 12q^{9} + O(q^{10})$$ $$337q + 6q^{4} - 4q^{5} - 10q^{6} + 8q^{7} + 12q^{9} - 6q^{10} + 22q^{11} - 20q^{12} + 22q^{13} - 36q^{14} + 3q^{15} - 30q^{16} - 8q^{17} - 52q^{18} + 12q^{19} - 58q^{20} - 26q^{21} - 42q^{22} - 24q^{23} - 42q^{24} - 14q^{25} - 68q^{26} - 24q^{27} - 56q^{28} - 72q^{29} - 30q^{30} - 18q^{31} - 10q^{32} - 78q^{33} - 42q^{34} - 56q^{35} + 18q^{36} - 32q^{37} + 10q^{38} - 50q^{39} - 42q^{40} - 100q^{41} + 16q^{42} - 12q^{43} - 111q^{45} - 40q^{46} - 60q^{47} - 34q^{48} - 81q^{49} + 26q^{50} - 48q^{51} + 8q^{52} - 104q^{53} - 34q^{54} - 26q^{55} + 68q^{56} - 100q^{57} + 20q^{58} - 38q^{59} + 46q^{60} - 96q^{61} + 100q^{62} + 2q^{63} + 60q^{64} - 37q^{65} + 88q^{66} + 20q^{67} + 134q^{68} + 34q^{69} + 42q^{70} + 84q^{71} + 138q^{72} + 40q^{73} + 132q^{74} + 69q^{75} - 26q^{76} + 30q^{77} + 152q^{78} + 30q^{79} + 112q^{80} + 96q^{81} - 76q^{82} + 84q^{83} + 100q^{84} - 54q^{85} + 22q^{86} + 130q^{87} - 42q^{88} + 98q^{89} + 138q^{90} + 48q^{91} + 56q^{92} - 2q^{93} - 48q^{94} + 96q^{95} + 104q^{96} - 34q^{97} + 126q^{98} + 94q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(180))$$

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
180.2.a $$\chi_{180}(1, \cdot)$$ 180.2.a.a 1 1
180.2.d $$\chi_{180}(109, \cdot)$$ 180.2.d.a 2 1
180.2.e $$\chi_{180}(71, \cdot)$$ 180.2.e.a 8 1
180.2.h $$\chi_{180}(179, \cdot)$$ 180.2.h.a 4 1
180.2.h.b 8
180.2.i $$\chi_{180}(61, \cdot)$$ 180.2.i.a 2 2
180.2.i.b 6
180.2.j $$\chi_{180}(17, \cdot)$$ 180.2.j.a 4 2
180.2.k $$\chi_{180}(127, \cdot)$$ 180.2.k.a 2 2
180.2.k.b 2
180.2.k.c 2
180.2.k.d 8
180.2.k.e 12
180.2.n $$\chi_{180}(59, \cdot)$$ 180.2.n.a 4 2
180.2.n.b 4
180.2.n.c 8
180.2.n.d 48
180.2.q $$\chi_{180}(11, \cdot)$$ 180.2.q.a 48 2
180.2.r $$\chi_{180}(49, \cdot)$$ 180.2.r.a 12 2
180.2.w $$\chi_{180}(77, \cdot)$$ 180.2.w.a 24 4
180.2.x $$\chi_{180}(7, \cdot)$$ 180.2.x.a 128 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(180))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(180)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$