## Defining parameters

 Level: $$N$$ = $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$11$$ Newforms: $$24$$ Sturm bound: $$172800$$ Trace bound: $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1800))$$.

Total New Old
Modular forms 3098 477 2621
Cusp forms 410 108 302
Eisenstein series 2688 369 2319

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 84 0 8 16

## Trace form

 $$108q + q^{2} + 5q^{3} - 3q^{4} + q^{6} + 8q^{7} - 2q^{8} - 3q^{9} + O(q^{10})$$ $$108q + q^{2} + 5q^{3} - 3q^{4} + q^{6} + 8q^{7} - 2q^{8} - 3q^{9} + 9q^{11} - 2q^{14} - 3q^{16} - 2q^{17} - 2q^{18} + 2q^{19} - 9q^{22} + q^{24} - 2q^{25} - 8q^{26} + 2q^{27} - 12q^{28} + 16q^{30} + 16q^{31} + q^{32} + 3q^{33} - q^{34} + 4q^{35} - q^{36} - q^{38} - 8q^{40} + 7q^{41} + 2q^{42} - 9q^{43} - 6q^{44} - 16q^{46} - q^{48} + 3q^{49} - 17q^{51} + q^{54} + 4q^{55} + 2q^{56} - 17q^{57} + 4q^{58} - q^{59} + 8q^{61} - 4q^{62} + 4q^{65} - 22q^{66} + 5q^{67} + 7q^{68} + 4q^{70} + q^{72} - 6q^{73} + 2q^{75} - q^{76} - 12q^{78} - 3q^{81} + 2q^{82} + 6q^{83} - 33q^{86} - 9q^{88} - 8q^{89} + 4q^{90} - 2q^{96} - 13q^{97} - 2q^{98} - 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1800))$$

We only show spaces with odd 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
1800.1.c $$\chi_{1800}(449, \cdot)$$ 1800.1.c.a 4 1
1800.1.e $$\chi_{1800}(1351, \cdot)$$ None 0 1
1800.1.g $$\chi_{1800}(451, \cdot)$$ 1800.1.g.a 1 1
1800.1.g.b 1
1800.1.g.c 2
1800.1.i $$\chi_{1800}(1349, \cdot)$$ None 0 1
1800.1.j $$\chi_{1800}(199, \cdot)$$ None 0 1
1800.1.l $$\chi_{1800}(1601, \cdot)$$ 1800.1.l.a 2 1
1800.1.l.b 2
1800.1.n $$\chi_{1800}(701, \cdot)$$ None 0 1
1800.1.p $$\chi_{1800}(1099, \cdot)$$ 1800.1.p.a 2 1
1800.1.r $$\chi_{1800}(107, \cdot)$$ 1800.1.r.a 4 2
1800.1.r.b 4
1800.1.u $$\chi_{1800}(757, \cdot)$$ 1800.1.u.a 4 2
1800.1.u.b 4
1800.1.v $$\chi_{1800}(793, \cdot)$$ None 0 2
1800.1.y $$\chi_{1800}(143, \cdot)$$ None 0 2
1800.1.ba $$\chi_{1800}(499, \cdot)$$ 1800.1.ba.a 4 2
1800.1.ba.b 4
1800.1.ba.c 4
1800.1.bb $$\chi_{1800}(101, \cdot)$$ None 0 2
1800.1.bd $$\chi_{1800}(401, \cdot)$$ None 0 2
1800.1.bf $$\chi_{1800}(799, \cdot)$$ None 0 2
1800.1.bi $$\chi_{1800}(149, \cdot)$$ None 0 2
1800.1.bk $$\chi_{1800}(1051, \cdot)$$ 1800.1.bk.a 2 2
1800.1.bk.b 2
1800.1.bk.c 2
1800.1.bk.d 2
1800.1.bk.e 2
1800.1.bm $$\chi_{1800}(151, \cdot)$$ None 0 2
1800.1.bo $$\chi_{1800}(1049, \cdot)$$ None 0 2
1800.1.bp $$\chi_{1800}(269, \cdot)$$ None 0 4
1800.1.br $$\chi_{1800}(91, \cdot)$$ None 0 4
1800.1.bt $$\chi_{1800}(271, \cdot)$$ None 0 4
1800.1.bv $$\chi_{1800}(89, \cdot)$$ None 0 4
1800.1.bx $$\chi_{1800}(19, \cdot)$$ None 0 4
1800.1.bz $$\chi_{1800}(341, \cdot)$$ None 0 4
1800.1.cb $$\chi_{1800}(161, \cdot)$$ None 0 4
1800.1.cd $$\chi_{1800}(559, \cdot)$$ None 0 4
1800.1.cf $$\chi_{1800}(193, \cdot)$$ None 0 4
1800.1.cg $$\chi_{1800}(407, \cdot)$$ None 0 4
1800.1.cj $$\chi_{1800}(443, \cdot)$$ 1800.1.cj.a 8 4
1800.1.cj.b 8
1800.1.cj.c 8
1800.1.ck $$\chi_{1800}(157, \cdot)$$ None 0 4
1800.1.cn $$\chi_{1800}(287, \cdot)$$ None 0 8
1800.1.cq $$\chi_{1800}(73, \cdot)$$ None 0 8
1800.1.cr $$\chi_{1800}(37, \cdot)$$ 1800.1.cr.a 16 8
1800.1.cu $$\chi_{1800}(323, \cdot)$$ None 0 8
1800.1.cw $$\chi_{1800}(79, \cdot)$$ None 0 8
1800.1.cy $$\chi_{1800}(41, \cdot)$$ None 0 8
1800.1.da $$\chi_{1800}(221, \cdot)$$ None 0 8
1800.1.db $$\chi_{1800}(139, \cdot)$$ None 0 8
1800.1.dc $$\chi_{1800}(209, \cdot)$$ None 0 8
1800.1.de $$\chi_{1800}(31, \cdot)$$ None 0 8
1800.1.dg $$\chi_{1800}(211, \cdot)$$ 1800.1.dg.a 16 8
1800.1.di $$\chi_{1800}(29, \cdot)$$ None 0 8
1800.1.dl $$\chi_{1800}(13, \cdot)$$ None 0 16
1800.1.dm $$\chi_{1800}(83, \cdot)$$ None 0 16
1800.1.dp $$\chi_{1800}(23, \cdot)$$ None 0 16
1800.1.dq $$\chi_{1800}(97, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1800))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1800)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(360))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(600))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(900))$$$$^{\oplus 2}$$