# Properties

 Label 900.2 Level 900 Weight 2 Dimension 9010 Nonzero newspaces 24 Newform subspaces 96 Sturm bound 86400 Trace bound 16

## Defining parameters

 Level: $$N$$ = $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Newform subspaces: $$96$$ Sturm bound: $$86400$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(900))$$.

Total New Old
Modular forms 22720 9378 13342
Cusp forms 20481 9010 11471
Eisenstein series 2239 368 1871

## Trace form

 $$9010q - 21q^{2} - 27q^{4} - 47q^{5} - 43q^{6} - 7q^{7} - 30q^{8} - 68q^{9} + O(q^{10})$$ $$9010q - 21q^{2} - 27q^{4} - 47q^{5} - 43q^{6} - 7q^{7} - 30q^{8} - 68q^{9} - 80q^{10} - 35q^{11} - 18q^{12} - 83q^{13} + 6q^{14} - 12q^{15} - 15q^{16} - 90q^{17} + 10q^{18} - 44q^{19} + 10q^{20} - 83q^{21} + 45q^{22} + 19q^{23} + 35q^{24} - 57q^{25} + 52q^{26} + 48q^{27} + 58q^{28} + 55q^{29} + 8q^{30} + 43q^{31} + 89q^{32} + 31q^{33} + 93q^{34} + 52q^{35} - 9q^{36} - 81q^{37} + 61q^{38} + 109q^{39} + 60q^{40} + 49q^{41} - 18q^{42} + 37q^{43} + 40q^{44} + 16q^{45} - 58q^{46} + 133q^{47} - 35q^{48} + 110q^{49} - 2q^{50} + 96q^{51} - 52q^{52} + 229q^{53} - 89q^{54} + 48q^{55} - 184q^{56} + 102q^{57} - 96q^{58} + 137q^{59} - 92q^{60} + 149q^{61} - 180q^{62} + 105q^{63} - 210q^{64} + 159q^{65} - 176q^{66} + 107q^{67} - 253q^{68} + 109q^{69} - 66q^{70} + 24q^{71} - 253q^{72} + 34q^{73} - 234q^{74} + 88q^{75} - 55q^{76} + 201q^{77} - 308q^{78} + 105q^{79} - 142q^{80} - 220q^{81} - 20q^{82} + 35q^{83} - 330q^{84} + 179q^{85} - 221q^{86} - 133q^{87} - 3q^{88} - 7q^{89} - 184q^{90} + 22q^{91} - 314q^{92} - 191q^{93} - 40q^{94} - 68q^{95} - 316q^{96} + 159q^{97} - 482q^{98} - 215q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(900))$$

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
900.2.a $$\chi_{900}(1, \cdot)$$ 900.2.a.a 1 1
900.2.a.b 1
900.2.a.c 1
900.2.a.d 1
900.2.a.e 1
900.2.a.f 1
900.2.a.g 1
900.2.a.h 1
900.2.d $$\chi_{900}(649, \cdot)$$ 900.2.d.a 2 1
900.2.d.b 2
900.2.d.c 2
900.2.d.d 2
900.2.e $$\chi_{900}(251, \cdot)$$ 900.2.e.a 2 1
900.2.e.b 2
900.2.e.c 2
900.2.e.d 8
900.2.e.e 8
900.2.e.f 8
900.2.e.g 8
900.2.h $$\chi_{900}(899, \cdot)$$ 900.2.h.a 4 1
900.2.h.b 8
900.2.h.c 8
900.2.h.d 16
900.2.i $$\chi_{900}(301, \cdot)$$ 900.2.i.a 2 2
900.2.i.b 2
900.2.i.c 6
900.2.i.d 8
900.2.i.e 8
900.2.i.f 12
900.2.j $$\chi_{900}(557, \cdot)$$ 900.2.j.a 4 2
900.2.j.b 8
900.2.k $$\chi_{900}(307, \cdot)$$ 900.2.k.a 2 2
900.2.k.b 2
900.2.k.c 2
900.2.k.d 2
900.2.k.e 2
900.2.k.f 8
900.2.k.g 8
900.2.k.h 8
900.2.k.i 8
900.2.k.j 8
900.2.k.k 8
900.2.k.l 8
900.2.k.m 8
900.2.k.n 12
900.2.n $$\chi_{900}(181, \cdot)$$ 900.2.n.a 8 4
900.2.n.b 8
900.2.n.c 12
900.2.n.d 24
900.2.o $$\chi_{900}(299, \cdot)$$ 900.2.o.a 16 2
900.2.o.b 48
900.2.o.c 48
900.2.o.d 96
900.2.r $$\chi_{900}(551, \cdot)$$ 900.2.r.a 8 2
900.2.r.b 8
900.2.r.c 8
900.2.r.d 48
900.2.r.e 48
900.2.r.f 48
900.2.r.g 48
900.2.s $$\chi_{900}(49, \cdot)$$ 900.2.s.a 4 2
900.2.s.b 4
900.2.s.c 12
900.2.s.d 16
900.2.v $$\chi_{900}(71, \cdot)$$ 900.2.v.a 16 4
900.2.v.b 224
900.2.w $$\chi_{900}(109, \cdot)$$ 900.2.w.a 8 4
900.2.w.b 16
900.2.w.c 24
900.2.z $$\chi_{900}(179, \cdot)$$ 900.2.z.a 16 4
900.2.z.b 224
900.2.be $$\chi_{900}(257, \cdot)$$ 900.2.be.a 4 4
900.2.be.b 4
900.2.be.c 4
900.2.be.d 4
900.2.be.e 24
900.2.be.f 32
900.2.bf $$\chi_{900}(7, \cdot)$$ 900.2.bf.a 8 4
900.2.bf.b 8
900.2.bf.c 16
900.2.bf.d 64
900.2.bf.e 128
900.2.bf.f 192
900.2.bg $$\chi_{900}(61, \cdot)$$ 900.2.bg.a 240 8
900.2.bj $$\chi_{900}(127, \cdot)$$ 900.2.bj.a 8 8
900.2.bj.b 8
900.2.bj.c 8
900.2.bj.d 96
900.2.bj.e 224
900.2.bj.f 240
900.2.bk $$\chi_{900}(17, \cdot)$$ 900.2.bk.a 80 8
900.2.bn $$\chi_{900}(59, \cdot)$$ 900.2.bn.a 1408 8
900.2.bq $$\chi_{900}(169, \cdot)$$ 900.2.bq.a 240 8
900.2.br $$\chi_{900}(11, \cdot)$$ 900.2.br.a 1408 8
900.2.bs $$\chi_{900}(67, \cdot)$$ 900.2.bs.a 2816 16
900.2.bt $$\chi_{900}(77, \cdot)$$ 900.2.bt.a 480 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(900))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(900)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(150))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(450))$$$$^{\oplus 2}$$