## Defining parameters

 Level: $$N$$ = $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$28$$ Sturm bound: $$110592$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(720))$$.

Total New Old
Modular forms 42368 17050 25318
Cusp forms 40576 16808 23768
Eisenstein series 1792 242 1550

## Trace form

 $$16808q - 12q^{2} - 12q^{3} - 32q^{4} - 23q^{5} - 48q^{6} - 36q^{7} + 72q^{8} + 36q^{9} + O(q^{10})$$ $$16808q - 12q^{2} - 12q^{3} - 32q^{4} - 23q^{5} - 48q^{6} - 36q^{7} + 72q^{8} + 36q^{9} + 12q^{10} + 114q^{11} - 16q^{12} - 8q^{13} - 264q^{14} + 3q^{15} + 456q^{16} + 282q^{17} + 264q^{18} - 68q^{19} - 440q^{20} - 658q^{21} - 856q^{22} - 572q^{23} - 1488q^{24} - 519q^{25} - 1688q^{26} + 744q^{27} - 376q^{28} + 834q^{29} + 476q^{30} + 6q^{31} + 2648q^{32} + 54q^{33} + 1320q^{34} + 312q^{35} - 1576q^{36} + 510q^{37} - 4440q^{38} + 438q^{39} - 2380q^{40} - 962q^{41} + 1344q^{42} - 3404q^{43} + 3344q^{44} + 733q^{45} + 3704q^{46} - 3948q^{47} + 4872q^{48} + 1070q^{49} + 6172q^{50} - 3332q^{51} + 7656q^{52} + 1142q^{53} + 424q^{54} - 606q^{55} - 5808q^{56} - 5696q^{57} - 4544q^{58} + 2294q^{59} - 7956q^{60} - 474q^{61} - 9480q^{62} + 1686q^{63} - 7088q^{64} - 415q^{65} + 9792q^{66} - 1300q^{67} + 3888q^{68} + 7174q^{69} - 4284q^{70} + 1092q^{71} + 15888q^{72} - 5690q^{73} + 12480q^{74} + 5225q^{75} + 1808q^{76} + 10922q^{77} + 888q^{78} - 5606q^{79} - 392q^{80} + 844q^{81} - 1032q^{82} - 5752q^{83} - 19120q^{84} + 1904q^{85} - 15280q^{86} + 366q^{87} + 10216q^{88} - 1840q^{89} - 1176q^{90} + 16168q^{91} + 17896q^{92} - 1654q^{93} + 16872q^{94} + 12160q^{95} + 12872q^{96} - 1260q^{97} + 14380q^{98} + 210q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(720))$$

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
720.4.a $$\chi_{720}(1, \cdot)$$ 720.4.a.a 1 1
720.4.a.b 1
720.4.a.c 1
720.4.a.d 1
720.4.a.e 1
720.4.a.f 1
720.4.a.g 1
720.4.a.h 1
720.4.a.i 1
720.4.a.j 1
720.4.a.k 1
720.4.a.l 1
720.4.a.m 1
720.4.a.n 1
720.4.a.o 1
720.4.a.p 1
720.4.a.q 1
720.4.a.r 1
720.4.a.s 1
720.4.a.t 1
720.4.a.u 1
720.4.a.v 1
720.4.a.w 1
720.4.a.x 1
720.4.a.y 1
720.4.a.z 1
720.4.a.ba 1
720.4.a.bb 1
720.4.a.bc 1
720.4.a.bd 1
720.4.b $$\chi_{720}(71, \cdot)$$ None 0 1
720.4.d $$\chi_{720}(649, \cdot)$$ None 0 1
720.4.f $$\chi_{720}(289, \cdot)$$ 720.4.f.a 2 1
720.4.f.b 2
720.4.f.c 2
720.4.f.d 2
720.4.f.e 2
720.4.f.f 2
720.4.f.g 2
720.4.f.h 2
720.4.f.i 4
720.4.f.j 4
720.4.f.k 4
720.4.f.l 4
720.4.f.m 4
720.4.f.n 8
720.4.h $$\chi_{720}(431, \cdot)$$ 720.4.h.a 8 1
720.4.h.b 16
720.4.k $$\chi_{720}(361, \cdot)$$ None 0 1
720.4.m $$\chi_{720}(359, \cdot)$$ None 0 1
720.4.o $$\chi_{720}(719, \cdot)$$ 720.4.o.a 4 1
720.4.o.b 8
720.4.o.c 24
720.4.q $$\chi_{720}(241, \cdot)$$ n/a 144 2
720.4.t $$\chi_{720}(181, \cdot)$$ n/a 240 2
720.4.u $$\chi_{720}(179, \cdot)$$ n/a 288 2
720.4.w $$\chi_{720}(17, \cdot)$$ 720.4.w.a 4 2
720.4.w.b 4
720.4.w.c 8
720.4.w.d 12
720.4.w.e 12
720.4.w.f 16
720.4.w.g 16
720.4.x $$\chi_{720}(127, \cdot)$$ 720.4.x.a 2 2
720.4.x.b 2
720.4.x.c 2
720.4.x.d 4
720.4.x.e 8
720.4.x.f 12
720.4.x.g 12
720.4.x.h 24
720.4.x.i 24
720.4.z $$\chi_{720}(163, \cdot)$$ n/a 356 2
720.4.bc $$\chi_{720}(197, \cdot)$$ n/a 288 2
720.4.bd $$\chi_{720}(307, \cdot)$$ n/a 356 2
720.4.bg $$\chi_{720}(53, \cdot)$$ n/a 288 2
720.4.bi $$\chi_{720}(343, \cdot)$$ None 0 2
720.4.bj $$\chi_{720}(233, \cdot)$$ None 0 2
720.4.bl $$\chi_{720}(251, \cdot)$$ n/a 192 2
720.4.bm $$\chi_{720}(109, \cdot)$$ n/a 356 2
720.4.br $$\chi_{720}(239, \cdot)$$ n/a 216 2
720.4.bt $$\chi_{720}(119, \cdot)$$ None 0 2
720.4.bv $$\chi_{720}(121, \cdot)$$ None 0 2
720.4.bw $$\chi_{720}(191, \cdot)$$ n/a 144 2
720.4.by $$\chi_{720}(49, \cdot)$$ n/a 212 2
720.4.ca $$\chi_{720}(169, \cdot)$$ None 0 2
720.4.cc $$\chi_{720}(311, \cdot)$$ None 0 2
720.4.ce $$\chi_{720}(229, \cdot)$$ n/a 1712 4
720.4.cf $$\chi_{720}(11, \cdot)$$ n/a 1152 4
720.4.ci $$\chi_{720}(7, \cdot)$$ None 0 4
720.4.cl $$\chi_{720}(137, \cdot)$$ None 0 4
720.4.cm $$\chi_{720}(77, \cdot)$$ n/a 1712 4
720.4.cp $$\chi_{720}(43, \cdot)$$ n/a 1712 4
720.4.cq $$\chi_{720}(173, \cdot)$$ n/a 1712 4
720.4.ct $$\chi_{720}(187, \cdot)$$ n/a 1712 4
720.4.cu $$\chi_{720}(113, \cdot)$$ n/a 424 4
720.4.cx $$\chi_{720}(223, \cdot)$$ n/a 432 4
720.4.da $$\chi_{720}(59, \cdot)$$ n/a 1712 4
720.4.db $$\chi_{720}(61, \cdot)$$ n/a 1152 4

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(720))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(720)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 15}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(240))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 2}$$