## 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

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

## 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

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