# Properties

 Label 360.4 Level 360 Weight 4 Dimension 4177 Nonzero newspaces 18 Sturm bound 27648 Trace bound 10

## Defining parameters

 Level: $$N$$ = $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$18$$ Sturm bound: $$27648$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(360))$$.

Total New Old
Modular forms 10752 4285 6467
Cusp forms 9984 4177 5807
Eisenstein series 768 108 660

## Trace form

 $$4177 q - 4 q^{2} - 10 q^{3} + 16 q^{4} + 11 q^{5} + 8 q^{6} + 84 q^{7} - 64 q^{8} - 74 q^{9} + O(q^{10})$$ $$4177 q - 4 q^{2} - 10 q^{3} + 16 q^{4} + 11 q^{5} + 8 q^{6} + 84 q^{7} - 64 q^{8} - 74 q^{9} - 116 q^{10} - 210 q^{11} - 228 q^{12} - 74 q^{13} - 172 q^{14} - 40 q^{15} - 404 q^{16} + 174 q^{17} + 224 q^{18} + 112 q^{19} + 62 q^{20} - 112 q^{21} + 972 q^{22} - 276 q^{23} + 512 q^{24} - 1091 q^{25} + 280 q^{26} - 1144 q^{27} + 648 q^{28} - 234 q^{29} - 134 q^{30} - 80 q^{31} - 684 q^{32} + 498 q^{33} - 2488 q^{34} + 12 q^{35} + 1500 q^{36} + 382 q^{37} + 140 q^{38} + 1512 q^{39} + 738 q^{40} + 1908 q^{41} + 4 q^{42} + 2194 q^{43} + 1828 q^{44} - 446 q^{45} + 2272 q^{46} - 100 q^{47} - 2512 q^{48} - 2213 q^{49} - 1534 q^{50} - 1346 q^{51} + 1600 q^{52} - 2506 q^{53} + 1752 q^{54} + 1060 q^{55} + 2624 q^{56} - 562 q^{57} - 1400 q^{58} - 3006 q^{59} + 6266 q^{60} + 338 q^{61} + 928 q^{62} + 1000 q^{63} - 152 q^{64} + 4048 q^{65} - 3128 q^{66} + 990 q^{67} + 196 q^{68} + 6676 q^{69} - 126 q^{70} + 7544 q^{71} - 9684 q^{72} - 1810 q^{73} - 7544 q^{74} - 3914 q^{75} + 4 q^{76} - 1416 q^{77} - 9572 q^{78} - 4280 q^{79} - 6512 q^{80} - 2850 q^{81} - 11644 q^{82} - 8084 q^{83} - 832 q^{84} - 4226 q^{85} + 4852 q^{86} - 7692 q^{87} - 6980 q^{88} - 2342 q^{89} - 1886 q^{90} - 864 q^{91} - 1828 q^{92} - 100 q^{93} + 2008 q^{94} + 468 q^{95} + 8032 q^{96} + 628 q^{97} + 16168 q^{98} + 8988 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(360))$$

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
360.4.a $$\chi_{360}(1, \cdot)$$ 360.4.a.a 1 1
360.4.a.b 1
360.4.a.c 1
360.4.a.d 1
360.4.a.e 1
360.4.a.f 1
360.4.a.g 1
360.4.a.h 1
360.4.a.i 1
360.4.a.j 1
360.4.a.k 1
360.4.a.l 1
360.4.a.m 1
360.4.a.n 1
360.4.a.o 1
360.4.b $$\chi_{360}(251, \cdot)$$ 360.4.b.a 24 1
360.4.b.b 24
360.4.d $$\chi_{360}(109, \cdot)$$ 360.4.d.a 4 1
360.4.d.b 4
360.4.d.c 4
360.4.d.d 16
360.4.d.e 18
360.4.d.f 18
360.4.d.g 24
360.4.f $$\chi_{360}(289, \cdot)$$ 360.4.f.a 2 1
360.4.f.b 2
360.4.f.c 2
360.4.f.d 4
360.4.f.e 4
360.4.f.f 8
360.4.h $$\chi_{360}(71, \cdot)$$ None 0 1
360.4.k $$\chi_{360}(181, \cdot)$$ 360.4.k.a 2 1
360.4.k.b 8
360.4.k.c 12
360.4.k.d 14
360.4.k.e 24
360.4.m $$\chi_{360}(179, \cdot)$$ 360.4.m.a 4 1
360.4.m.b 4
360.4.m.c 64
360.4.o $$\chi_{360}(359, \cdot)$$ None 0 1
360.4.q $$\chi_{360}(121, \cdot)$$ 360.4.q.a 2 2
360.4.q.b 16
360.4.q.c 16
360.4.q.d 18
360.4.q.e 20
360.4.s $$\chi_{360}(17, \cdot)$$ 360.4.s.a 4 2
360.4.s.b 16
360.4.s.c 16
360.4.t $$\chi_{360}(127, \cdot)$$ None 0 2
360.4.w $$\chi_{360}(163, \cdot)$$ n/a 176 2
360.4.x $$\chi_{360}(53, \cdot)$$ n/a 144 2
360.4.bb $$\chi_{360}(119, \cdot)$$ None 0 2
360.4.bd $$\chi_{360}(59, \cdot)$$ n/a 424 2
360.4.bf $$\chi_{360}(61, \cdot)$$ n/a 288 2
360.4.bg $$\chi_{360}(191, \cdot)$$ None 0 2
360.4.bi $$\chi_{360}(49, \cdot)$$ n/a 108 2
360.4.bk $$\chi_{360}(229, \cdot)$$ n/a 424 2
360.4.bm $$\chi_{360}(11, \cdot)$$ n/a 288 2
360.4.bo $$\chi_{360}(43, \cdot)$$ n/a 848 4
360.4.br $$\chi_{360}(77, \cdot)$$ n/a 848 4
360.4.bs $$\chi_{360}(113, \cdot)$$ n/a 216 4
360.4.bv $$\chi_{360}(7, \cdot)$$ None 0 4

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(360))$$ into lower level spaces

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