## Defining parameters

 Level: $$N$$ = $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$7$$ Newform subspaces: $$9$$ Sturm bound: $$216000$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1900))$$.

Total New Old
Modular forms 2641 739 1902
Cusp forms 121 43 78
Eisenstein series 2520 696 1824

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 39 0 4 0

## Trace form

 $$43 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + O(q^{10})$$ $$43 q + 2 q^{3} + 2 q^{5} + 3 q^{7} + 5 q^{11} + 2 q^{13} - 5 q^{17} - 6 q^{19} - 6 q^{23} + 2 q^{25} - 16 q^{26} + 3 q^{29} - 4 q^{31} + 2 q^{33} - 8 q^{35} - 16 q^{36} + 3 q^{43} + 5 q^{47} + 6 q^{49} - 4 q^{51} - 2 q^{53} + 4 q^{57} + 3 q^{59} - 4 q^{61} + 3 q^{63} + 16 q^{66} - 2 q^{67} + q^{71} + 3 q^{73} - 3 q^{77} + 3 q^{79} - 18 q^{81} + 14 q^{83} - 4 q^{85} - 4 q^{87} - 3 q^{89} - 2 q^{93} + 2 q^{95} + 16 q^{96} + 6 q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1900))$$

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
1900.1.b $$\chi_{1900}(951, \cdot)$$ None 0 1
1900.1.e $$\chi_{1900}(1101, \cdot)$$ 1900.1.e.a 1 1
1900.1.e.b 2
1900.1.g $$\chi_{1900}(949, \cdot)$$ 1900.1.g.a 2 1
1900.1.h $$\chi_{1900}(799, \cdot)$$ None 0 1
1900.1.j $$\chi_{1900}(607, \cdot)$$ 1900.1.j.a 8 2
1900.1.j.b 8
1900.1.m $$\chi_{1900}(457, \cdot)$$ None 0 2
1900.1.p $$\chi_{1900}(449, \cdot)$$ None 0 2
1900.1.q $$\chi_{1900}(999, \cdot)$$ None 0 2
1900.1.r $$\chi_{1900}(1151, \cdot)$$ None 0 2
1900.1.u $$\chi_{1900}(601, \cdot)$$ 1900.1.u.a 2 2
1900.1.w $$\chi_{1900}(189, \cdot)$$ 1900.1.w.a 8 4
1900.1.y $$\chi_{1900}(39, \cdot)$$ None 0 4
1900.1.ba $$\chi_{1900}(191, \cdot)$$ None 0 4
1900.1.bb $$\chi_{1900}(341, \cdot)$$ 1900.1.bb.a 8 4
1900.1.be $$\chi_{1900}(107, \cdot)$$ None 0 4
1900.1.bf $$\chi_{1900}(657, \cdot)$$ 1900.1.bf.a 4 4
1900.1.bi $$\chi_{1900}(401, \cdot)$$ None 0 6
1900.1.bj $$\chi_{1900}(99, \cdot)$$ None 0 6
1900.1.bl $$\chi_{1900}(249, \cdot)$$ None 0 6
1900.1.bo $$\chi_{1900}(251, \cdot)$$ None 0 6
1900.1.bp $$\chi_{1900}(77, \cdot)$$ None 0 8
1900.1.bs $$\chi_{1900}(227, \cdot)$$ None 0 8
1900.1.bu $$\chi_{1900}(11, \cdot)$$ None 0 8
1900.1.bv $$\chi_{1900}(141, \cdot)$$ None 0 8
1900.1.bx $$\chi_{1900}(69, \cdot)$$ None 0 8
1900.1.bz $$\chi_{1900}(159, \cdot)$$ None 0 8
1900.1.ca $$\chi_{1900}(93, \cdot)$$ None 0 12
1900.1.cc $$\chi_{1900}(143, \cdot)$$ None 0 12
1900.1.cg $$\chi_{1900}(197, \cdot)$$ None 0 16
1900.1.ch $$\chi_{1900}(27, \cdot)$$ None 0 16
1900.1.ck $$\chi_{1900}(119, \cdot)$$ None 0 24
1900.1.cl $$\chi_{1900}(21, \cdot)$$ None 0 24
1900.1.cm $$\chi_{1900}(111, \cdot)$$ None 0 24
1900.1.cp $$\chi_{1900}(29, \cdot)$$ None 0 24
1900.1.cr $$\chi_{1900}(3, \cdot)$$ None 0 48
1900.1.ct $$\chi_{1900}(17, \cdot)$$ None 0 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1900))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1900)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(95))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(380))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(475))$$$$^{\oplus 3}$$