# Properties

 Label 360.2 Level 360 Weight 2 Dimension 1357 Nonzero newspaces 18 Newform subspaces 57 Sturm bound 13824 Trace bound 10

## Defining parameters

 Level: $$N$$ = $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$57$$ Sturm bound: $$13824$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(360))$$.

Total New Old
Modular forms 3840 1465 2375
Cusp forms 3073 1357 1716
Eisenstein series 767 108 659

## Trace form

 $$1357 q - 8 q^{2} - 10 q^{3} - 8 q^{4} - 5 q^{5} - 16 q^{6} - 12 q^{7} + 16 q^{8} - 14 q^{9} + O(q^{10})$$ $$1357 q - 8 q^{2} - 10 q^{3} - 8 q^{4} - 5 q^{5} - 16 q^{6} - 12 q^{7} + 16 q^{8} - 14 q^{9} - 12 q^{10} + 6 q^{11} + 12 q^{12} + 2 q^{13} + 28 q^{14} + 8 q^{15} + 12 q^{16} + 18 q^{17} - 16 q^{18} + 24 q^{19} - 2 q^{20} + 32 q^{21} - 20 q^{22} + 60 q^{23} - 64 q^{24} + q^{25} - 64 q^{26} + 8 q^{27} - 56 q^{28} + 30 q^{29} - 62 q^{30} + 48 q^{31} - 108 q^{32} + 6 q^{33} - 16 q^{34} + 12 q^{35} - 132 q^{36} + 26 q^{37} - 140 q^{38} - 72 q^{39} - 62 q^{40} - 24 q^{41} - 140 q^{42} - 38 q^{43} - 172 q^{44} - 2 q^{45} - 160 q^{46} - 116 q^{47} - 112 q^{48} + 47 q^{49} - 170 q^{50} - 98 q^{51} - 96 q^{52} - 14 q^{53} - 96 q^{54} - 84 q^{55} - 128 q^{56} - 22 q^{57} - 136 q^{58} - 150 q^{59} - 70 q^{60} - 6 q^{61} - 64 q^{62} - 152 q^{63} - 152 q^{64} - 64 q^{65} + 64 q^{66} - 74 q^{67} - 28 q^{68} - 68 q^{69} - 122 q^{70} - 152 q^{71} + 108 q^{72} - 30 q^{73} + 80 q^{74} - 122 q^{75} + 20 q^{76} - 72 q^{77} + 100 q^{78} - 40 q^{79} - 16 q^{80} - 150 q^{81} - 108 q^{82} - 172 q^{83} + 80 q^{84} + 50 q^{85} + 92 q^{86} - 156 q^{87} - 36 q^{88} - 94 q^{89} - 2 q^{90} - 96 q^{91} - 68 q^{92} - 76 q^{93} - 24 q^{94} - 204 q^{95} - 128 q^{96} - 76 q^{97} - 76 q^{98} - 228 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(360))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
360.2.a $$\chi_{360}(1, \cdot)$$ 360.2.a.a 1 1
360.2.a.b 1
360.2.a.c 1
360.2.a.d 1
360.2.a.e 1
360.2.b $$\chi_{360}(251, \cdot)$$ 360.2.b.a 2 1
360.2.b.b 2
360.2.b.c 6
360.2.b.d 6
360.2.d $$\chi_{360}(109, \cdot)$$ 360.2.d.a 4 1
360.2.d.b 4
360.2.d.c 4
360.2.d.d 4
360.2.d.e 6
360.2.d.f 6
360.2.f $$\chi_{360}(289, \cdot)$$ 360.2.f.a 2 1
360.2.f.b 2
360.2.f.c 2
360.2.f.d 2
360.2.h $$\chi_{360}(71, \cdot)$$ None 0 1
360.2.k $$\chi_{360}(181, \cdot)$$ 360.2.k.a 2 1
360.2.k.b 2
360.2.k.c 2
360.2.k.d 4
360.2.k.e 4
360.2.k.f 6
360.2.m $$\chi_{360}(179, \cdot)$$ 360.2.m.a 4 1
360.2.m.b 4
360.2.m.c 16
360.2.o $$\chi_{360}(359, \cdot)$$ None 0 1
360.2.q $$\chi_{360}(121, \cdot)$$ 360.2.q.a 2 2
360.2.q.b 4
360.2.q.c 4
360.2.q.d 6
360.2.q.e 8
360.2.s $$\chi_{360}(17, \cdot)$$ 360.2.s.a 4 2
360.2.s.b 8
360.2.t $$\chi_{360}(127, \cdot)$$ None 0 2
360.2.w $$\chi_{360}(163, \cdot)$$ 360.2.w.a 4 2
360.2.w.b 4
360.2.w.c 8
360.2.w.d 16
360.2.w.e 24
360.2.x $$\chi_{360}(53, \cdot)$$ 360.2.x.a 48 2
360.2.bb $$\chi_{360}(119, \cdot)$$ None 0 2
360.2.bd $$\chi_{360}(59, \cdot)$$ 360.2.bd.a 8 2
360.2.bd.b 128
360.2.bf $$\chi_{360}(61, \cdot)$$ 360.2.bf.a 4 2
360.2.bf.b 92
360.2.bg $$\chi_{360}(191, \cdot)$$ None 0 2
360.2.bi $$\chi_{360}(49, \cdot)$$ 360.2.bi.a 4 2
360.2.bi.b 32
360.2.bk $$\chi_{360}(229, \cdot)$$ 360.2.bk.a 136 2
360.2.bm $$\chi_{360}(11, \cdot)$$ 360.2.bm.a 48 2
360.2.bm.b 48
360.2.bo $$\chi_{360}(43, \cdot)$$ 360.2.bo.a 272 4
360.2.br $$\chi_{360}(77, \cdot)$$ 360.2.br.a 4 4
360.2.br.b 4
360.2.br.c 4
360.2.br.d 4
360.2.br.e 256
360.2.bs $$\chi_{360}(113, \cdot)$$ 360.2.bs.a 72 4
360.2.bv $$\chi_{360}(7, \cdot)$$ None 0 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(360))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(360)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 24}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 1}$$