# Properties

 Label 720.2 Level 720 Weight 2 Dimension 5522 Nonzero newspaces 28 Newform subspaces 120 Sturm bound 55296 Trace bound 9

## Defining parameters

 Level: $$N$$ = $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$28$$ Newform subspaces: $$120$$ Sturm bound: $$55296$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(720))$$.

Total New Old
Modular forms 14720 5764 8956
Cusp forms 12929 5522 7407
Eisenstein series 1791 242 1549

## Trace form

 $$5522 q - 12 q^{2} - 12 q^{3} - 16 q^{4} - 25 q^{5} - 48 q^{6} - 20 q^{7} - 24 q^{8} - 12 q^{9} + O(q^{10})$$ $$5522 q - 12 q^{2} - 12 q^{3} - 16 q^{4} - 25 q^{5} - 48 q^{6} - 20 q^{7} - 24 q^{8} - 12 q^{9} - 56 q^{10} - 54 q^{11} - 16 q^{12} - 36 q^{13} + 24 q^{14} - 21 q^{15} + 8 q^{16} - 18 q^{17} + 24 q^{18} - 44 q^{19} + 32 q^{20} - 34 q^{21} + 80 q^{22} + 20 q^{23} + 48 q^{24} + 19 q^{25} + 80 q^{26} + 24 q^{27} + 72 q^{28} + 54 q^{29} - 4 q^{30} + 6 q^{31} + 88 q^{32} + 6 q^{33} + 72 q^{34} + 72 q^{35} - 40 q^{36} - 22 q^{37} - 48 q^{38} + 102 q^{39} + 4 q^{40} + 50 q^{41} - 96 q^{42} + 124 q^{43} - 128 q^{44} + 13 q^{45} - 168 q^{46} + 180 q^{47} - 120 q^{48} - 16 q^{49} - 52 q^{50} + 76 q^{51} - 104 q^{52} + 82 q^{53} - 104 q^{54} + 106 q^{55} - 240 q^{56} + 16 q^{57} - 88 q^{58} + 142 q^{59} - 132 q^{60} + 50 q^{61} - 216 q^{62} + 54 q^{63} + 80 q^{64} - 35 q^{65} - 288 q^{66} + 100 q^{67} - 240 q^{68} - 122 q^{69} - 52 q^{70} - 12 q^{71} - 240 q^{72} + 66 q^{73} - 216 q^{74} - 31 q^{75} - 32 q^{76} - 38 q^{77} - 264 q^{78} - 6 q^{79} - 232 q^{80} - 260 q^{81} - 120 q^{82} - 128 q^{83} - 304 q^{84} + 12 q^{85} - 392 q^{86} - 162 q^{87} - 248 q^{88} - 68 q^{89} - 168 q^{90} - 216 q^{91} - 376 q^{92} + 74 q^{93} - 264 q^{94} - 112 q^{95} - 184 q^{96} - 32 q^{97} - 388 q^{98} - 78 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(720))$$

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
720.2.a $$\chi_{720}(1, \cdot)$$ 720.2.a.a 1 1
720.2.a.b 1
720.2.a.c 1
720.2.a.d 1
720.2.a.e 1
720.2.a.f 1
720.2.a.g 1
720.2.a.h 1
720.2.a.i 1
720.2.a.j 1
720.2.b $$\chi_{720}(71, \cdot)$$ None 0 1
720.2.d $$\chi_{720}(649, \cdot)$$ None 0 1
720.2.f $$\chi_{720}(289, \cdot)$$ 720.2.f.a 2 1
720.2.f.b 2
720.2.f.c 2
720.2.f.d 2
720.2.f.e 2
720.2.f.f 2
720.2.f.g 2
720.2.h $$\chi_{720}(431, \cdot)$$ 720.2.h.a 8 1
720.2.k $$\chi_{720}(361, \cdot)$$ None 0 1
720.2.m $$\chi_{720}(359, \cdot)$$ None 0 1
720.2.o $$\chi_{720}(719, \cdot)$$ 720.2.o.a 4 1
720.2.o.b 8
720.2.q $$\chi_{720}(241, \cdot)$$ 720.2.q.a 2 2
720.2.q.b 2
720.2.q.c 2
720.2.q.d 2
720.2.q.e 2
720.2.q.f 4
720.2.q.g 4
720.2.q.h 4
720.2.q.i 6
720.2.q.j 6
720.2.q.k 6
720.2.q.l 8
720.2.t $$\chi_{720}(181, \cdot)$$ 720.2.t.a 4 2
720.2.t.b 8
720.2.t.c 16
720.2.t.d 20
720.2.t.e 32
720.2.u $$\chi_{720}(179, \cdot)$$ 720.2.u.a 96 2
720.2.w $$\chi_{720}(17, \cdot)$$ 720.2.w.a 4 2
720.2.w.b 4
720.2.w.c 4
720.2.w.d 4
720.2.w.e 8
720.2.x $$\chi_{720}(127, \cdot)$$ 720.2.x.a 2 2
720.2.x.b 2
720.2.x.c 2
720.2.x.d 4
720.2.x.e 4
720.2.x.f 8
720.2.x.g 8
720.2.z $$\chi_{720}(163, \cdot)$$ 720.2.z.a 2 2
720.2.z.b 2
720.2.z.c 2
720.2.z.d 2
720.2.z.e 6
720.2.z.f 16
720.2.z.g 18
720.2.z.h 20
720.2.z.i 48
720.2.bc $$\chi_{720}(197, \cdot)$$ 720.2.bc.a 96 2
720.2.bd $$\chi_{720}(307, \cdot)$$ 720.2.bd.a 2 2
720.2.bd.b 2
720.2.bd.c 2
720.2.bd.d 2
720.2.bd.e 6
720.2.bd.f 16
720.2.bd.g 18
720.2.bd.h 20
720.2.bd.i 48
720.2.bg $$\chi_{720}(53, \cdot)$$ 720.2.bg.a 96 2
720.2.bi $$\chi_{720}(343, \cdot)$$ None 0 2
720.2.bj $$\chi_{720}(233, \cdot)$$ None 0 2
720.2.bl $$\chi_{720}(251, \cdot)$$ 720.2.bl.a 8 2
720.2.bl.b 24
720.2.bl.c 32
720.2.bm $$\chi_{720}(109, \cdot)$$ 720.2.bm.a 2 2
720.2.bm.b 2
720.2.bm.c 4
720.2.bm.d 4
720.2.bm.e 8
720.2.bm.f 16
720.2.bm.g 32
720.2.bm.h 48
720.2.br $$\chi_{720}(239, \cdot)$$ 720.2.br.a 8 2
720.2.br.b 16
720.2.br.c 24
720.2.br.d 24
720.2.bt $$\chi_{720}(119, \cdot)$$ None 0 2
720.2.bv $$\chi_{720}(121, \cdot)$$ None 0 2
720.2.bw $$\chi_{720}(191, \cdot)$$ 720.2.bw.a 16 2
720.2.bw.b 16
720.2.bw.c 16
720.2.by $$\chi_{720}(49, \cdot)$$ 720.2.by.a 4 2
720.2.by.b 4
720.2.by.c 8
720.2.by.d 8
720.2.by.e 12
720.2.by.f 32
720.2.ca $$\chi_{720}(169, \cdot)$$ None 0 2
720.2.cc $$\chi_{720}(311, \cdot)$$ None 0 2
720.2.ce $$\chi_{720}(229, \cdot)$$ 720.2.ce.a 560 4
720.2.cf $$\chi_{720}(11, \cdot)$$ 720.2.cf.a 384 4
720.2.ci $$\chi_{720}(7, \cdot)$$ None 0 4
720.2.cl $$\chi_{720}(137, \cdot)$$ None 0 4
720.2.cm $$\chi_{720}(77, \cdot)$$ 720.2.cm.a 560 4
720.2.cp $$\chi_{720}(43, \cdot)$$ 720.2.cp.a 4 4
720.2.cp.b 4
720.2.cp.c 552
720.2.cq $$\chi_{720}(173, \cdot)$$ 720.2.cq.a 560 4
720.2.ct $$\chi_{720}(187, \cdot)$$ 720.2.ct.a 4 4
720.2.ct.b 4
720.2.ct.c 552
720.2.cu $$\chi_{720}(113, \cdot)$$ 720.2.cu.a 8 4
720.2.cu.b 16
720.2.cu.c 16
720.2.cu.d 24
720.2.cu.e 72
720.2.cx $$\chi_{720}(223, \cdot)$$ 720.2.cx.a 8 4
720.2.cx.b 8
720.2.cx.c 40
720.2.cx.d 40
720.2.cx.e 48
720.2.da $$\chi_{720}(59, \cdot)$$ 720.2.da.a 4 4
720.2.da.b 4
720.2.da.c 4
720.2.da.d 4
720.2.da.e 544
720.2.db $$\chi_{720}(61, \cdot)$$ 720.2.db.a 384 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(720))$$ into lower level spaces

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