# Properties

 Label 480.4 Level 480 Weight 4 Dimension 7068 Nonzero newspaces 20 Sturm bound 49152 Trace bound 17

## Defining parameters

 Level: $$N$$ = $$480 = 2^{5} \cdot 3 \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$20$$ Sturm bound: $$49152$$ Trace bound: $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(480))$$.

Total New Old
Modular forms 18944 7188 11756
Cusp forms 17920 7068 10852
Eisenstein series 1024 120 904

## Trace form

 $$7068 q - 8 q^{3} - 16 q^{4} + 4 q^{5} - 24 q^{6} + 48 q^{7} + 28 q^{9} + O(q^{10})$$ $$7068 q - 8 q^{3} - 16 q^{4} + 4 q^{5} - 24 q^{6} + 48 q^{7} + 28 q^{9} + 216 q^{10} - 104 q^{12} - 488 q^{13} - 832 q^{14} - 64 q^{15} - 1248 q^{16} - 416 q^{17} - 152 q^{18} - 56 q^{19} + 160 q^{20} - 56 q^{21} + 768 q^{22} + 1968 q^{23} - 96 q^{24} + 52 q^{25} + 80 q^{26} - 800 q^{27} + 1504 q^{28} + 568 q^{29} + 1116 q^{30} - 3240 q^{31} + 2480 q^{32} - 8 q^{33} + 2112 q^{34} - 1824 q^{35} - 1184 q^{36} - 1336 q^{37} - 880 q^{38} + 568 q^{39} - 1664 q^{40} - 240 q^{41} - 1808 q^{42} + 2352 q^{43} - 2000 q^{44} + 424 q^{45} - 48 q^{46} + 4880 q^{48} + 1844 q^{49} - 2856 q^{50} + 216 q^{51} - 6640 q^{52} + 5720 q^{53} + 1432 q^{55} + 784 q^{56} + 1312 q^{57} + 4736 q^{58} + 5504 q^{59} - 1384 q^{60} - 5608 q^{61} + 5856 q^{62} + 128 q^{63} + 4352 q^{64} - 6360 q^{65} - 1208 q^{66} + 1184 q^{67} - 1808 q^{68} - 7720 q^{69} - 384 q^{70} - 2064 q^{71} + 2752 q^{72} - 4904 q^{73} + 3360 q^{74} - 620 q^{75} - 3376 q^{76} + 13888 q^{77} + 9704 q^{78} - 5128 q^{79} + 13944 q^{80} + 1740 q^{81} + 11664 q^{82} + 5360 q^{83} + 13104 q^{84} + 11000 q^{85} + 4736 q^{86} + 5840 q^{87} + 1216 q^{88} - 1408 q^{89} - 10272 q^{90} + 2912 q^{91} - 6784 q^{92} + 128 q^{93} + 3904 q^{94} - 2624 q^{95} - 14976 q^{96} + 840 q^{97} - 4672 q^{98} + 13016 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(480))$$

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
480.4.a $$\chi_{480}(1, \cdot)$$ 480.4.a.a 1 1
480.4.a.b 1
480.4.a.c 1
480.4.a.d 1
480.4.a.e 1
480.4.a.f 1
480.4.a.g 1
480.4.a.h 1
480.4.a.i 1
480.4.a.j 1
480.4.a.k 1
480.4.a.l 1
480.4.a.m 2
480.4.a.n 2
480.4.a.o 2
480.4.a.p 2
480.4.a.q 2
480.4.a.r 2
480.4.b $$\chi_{480}(431, \cdot)$$ 480.4.b.a 24 1
480.4.b.b 24
480.4.d $$\chi_{480}(49, \cdot)$$ 480.4.d.a 18 1
480.4.d.b 18
480.4.f $$\chi_{480}(289, \cdot)$$ 480.4.f.a 2 1
480.4.f.b 2
480.4.f.c 4
480.4.f.d 6
480.4.f.e 6
480.4.f.f 8
480.4.f.g 8
480.4.h $$\chi_{480}(191, \cdot)$$ 480.4.h.a 24 1
480.4.h.b 24
480.4.k $$\chi_{480}(241, \cdot)$$ 480.4.k.a 2 1
480.4.k.b 8
480.4.k.c 14
480.4.m $$\chi_{480}(239, \cdot)$$ 480.4.m.a 4 1
480.4.m.b 64
480.4.o $$\chi_{480}(479, \cdot)$$ 480.4.o.a 72 1
480.4.s $$\chi_{480}(121, \cdot)$$ None 0 2
480.4.t $$\chi_{480}(119, \cdot)$$ None 0 2
480.4.v $$\chi_{480}(257, \cdot)$$ n/a 144 2
480.4.w $$\chi_{480}(127, \cdot)$$ 480.4.w.a 16 2
480.4.w.b 16
480.4.w.c 20
480.4.w.d 20
480.4.y $$\chi_{480}(7, \cdot)$$ None 0 2
480.4.bb $$\chi_{480}(233, \cdot)$$ None 0 2
480.4.bc $$\chi_{480}(103, \cdot)$$ None 0 2
480.4.bf $$\chi_{480}(137, \cdot)$$ None 0 2
480.4.bh $$\chi_{480}(367, \cdot)$$ 480.4.bh.a 72 2
480.4.bi $$\chi_{480}(17, \cdot)$$ n/a 136 2
480.4.bk $$\chi_{480}(71, \cdot)$$ None 0 2
480.4.bl $$\chi_{480}(169, \cdot)$$ None 0 2
480.4.bo $$\chi_{480}(43, \cdot)$$ n/a 576 4
480.4.br $$\chi_{480}(173, \cdot)$$ n/a 1136 4
480.4.bs $$\chi_{480}(59, \cdot)$$ n/a 1136 4
480.4.bv $$\chi_{480}(61, \cdot)$$ n/a 384 4
480.4.bx $$\chi_{480}(11, \cdot)$$ n/a 768 4
480.4.by $$\chi_{480}(109, \cdot)$$ n/a 576 4
480.4.cb $$\chi_{480}(53, \cdot)$$ n/a 1136 4
480.4.cc $$\chi_{480}(163, \cdot)$$ n/a 576 4

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(480))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(480)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 24}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 20}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(96))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(480))$$$$^{\oplus 1}$$