# Properties

 Label 180.3 Level 180 Weight 3 Dimension 706 Nonzero newspaces 12 Newform subspaces 28 Sturm bound 5184 Trace bound 7

## Defining parameters

 Level: $$N$$ = $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$28$$ Sturm bound: $$5184$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(180))$$.

Total New Old
Modular forms 1888 758 1130
Cusp forms 1568 706 862
Eisenstein series 320 52 268

## Trace form

 $$706q - 8q^{2} - 6q^{3} - 14q^{4} - 29q^{5} + 2q^{6} - 12q^{7} + 64q^{8} - 6q^{9} + O(q^{10})$$ $$706q - 8q^{2} - 6q^{3} - 14q^{4} - 29q^{5} + 2q^{6} - 12q^{7} + 64q^{8} - 6q^{9} + 14q^{10} + 44q^{11} + 64q^{12} + 4q^{13} + 60q^{14} + 57q^{15} + 10q^{16} + 146q^{17} - 16q^{18} + 76q^{19} + 36q^{20} - 50q^{21} + 110q^{22} + 20q^{23} - 126q^{24} + 25q^{25} - 252q^{26} + 48q^{27} - 16q^{28} + 114q^{29} - 204q^{30} - 78q^{31} - 418q^{32} + 132q^{33} - 234q^{34} - 82q^{35} - 150q^{36} - 54q^{37} - 10q^{38} - 74q^{39} - 224q^{40} - 60q^{41} + 520q^{42} + 46q^{43} + 340q^{44} + 141q^{45} + 8q^{46} + 216q^{47} + 314q^{48} + 318q^{49} - 334q^{50} - 42q^{51} - 376q^{52} - 274q^{53} - 382q^{54} + 242q^{55} - 740q^{56} - 742q^{57} - 632q^{58} - 432q^{59} - 524q^{60} - 642q^{61} - 836q^{62} - 598q^{63} - 116q^{64} - 1019q^{65} - 644q^{66} - 726q^{67} - 10q^{68} - 806q^{69} + 846q^{70} - 428q^{71} + 378q^{72} - 706q^{73} + 924q^{74} - 471q^{75} + 886q^{76} + 358q^{77} + 356q^{78} + 394q^{79} + 392q^{80} + 390q^{81} + 904q^{82} + 300q^{83} + 580q^{84} + 596q^{85} + 558q^{86} + 22q^{87} + 86q^{88} + 568q^{89} - 36q^{90} + 416q^{91} - 492q^{92} + 1354q^{93} - 1084q^{94} - 128q^{95} - 808q^{96} + 38q^{97} - 1626q^{98} + 298q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(180))$$

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
180.3.b $$\chi_{180}(89, \cdot)$$ 180.3.b.a 4 1
180.3.c $$\chi_{180}(91, \cdot)$$ 180.3.c.a 4 1
180.3.c.b 8
180.3.c.c 8
180.3.f $$\chi_{180}(19, \cdot)$$ 180.3.f.a 1 1
180.3.f.b 1
180.3.f.c 2
180.3.f.d 4
180.3.f.e 4
180.3.f.f 4
180.3.f.g 4
180.3.f.h 8
180.3.g $$\chi_{180}(161, \cdot)$$ 180.3.g.a 4 1
180.3.l $$\chi_{180}(37, \cdot)$$ 180.3.l.a 2 2
180.3.l.b 4
180.3.l.c 4
180.3.m $$\chi_{180}(107, \cdot)$$ 180.3.m.a 4 2
180.3.m.b 4
180.3.m.c 40
180.3.o $$\chi_{180}(41, \cdot)$$ 180.3.o.a 4 2
180.3.o.b 12
180.3.p $$\chi_{180}(79, \cdot)$$ 180.3.p.a 4 2
180.3.p.b 4
180.3.p.c 128
180.3.s $$\chi_{180}(31, \cdot)$$ 180.3.s.a 96 2
180.3.t $$\chi_{180}(29, \cdot)$$ 180.3.t.a 24 2
180.3.u $$\chi_{180}(13, \cdot)$$ 180.3.u.a 48 4
180.3.v $$\chi_{180}(23, \cdot)$$ 180.3.v.a 272 4

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(180))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(180)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$