## Defining parameters

 Level: $$N$$ = $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$7$$ Newform subspaces: $$14$$ Sturm bound: $$153600$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1600))$$.

Total New Old
Modular forms 2198 510 1688
Cusp forms 182 59 123
Eisenstein series 2016 451 1565

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 35 0 0 24

## Trace form

 $$59 q + 2 q^{5} + 9 q^{9} + O(q^{10})$$ $$59 q + 2 q^{5} + 9 q^{9} - 8 q^{13} - 10 q^{17} + 4 q^{21} + 2 q^{25} + 12 q^{29} + 12 q^{33} + 4 q^{37} - 6 q^{41} + 4 q^{45} - 3 q^{49} + 4 q^{53} + 12 q^{57} + 4 q^{61} - 4 q^{65} - 2 q^{69} - 8 q^{73} - 7 q^{81} - 6 q^{89} + 8 q^{93} - 10 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1600))$$

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
1600.1.b $$\chi_{1600}(1151, \cdot)$$ 1600.1.b.a 1 1
1600.1.e $$\chi_{1600}(799, \cdot)$$ None 0 1
1600.1.g $$\chi_{1600}(351, \cdot)$$ None 0 1
1600.1.h $$\chi_{1600}(1599, \cdot)$$ None 0 1
1600.1.i $$\chi_{1600}(593, \cdot)$$ None 0 2
1600.1.k $$\chi_{1600}(399, \cdot)$$ None 0 2
1600.1.m $$\chi_{1600}(993, \cdot)$$ 1600.1.m.a 2 2
1600.1.m.b 2
1600.1.m.c 4
1600.1.m.d 4
1600.1.p $$\chi_{1600}(193, \cdot)$$ 1600.1.p.a 2 2
1600.1.p.b 2
1600.1.p.c 2
1600.1.r $$\chi_{1600}(751, \cdot)$$ None 0 2
1600.1.t $$\chi_{1600}(657, \cdot)$$ None 0 2
1600.1.w $$\chi_{1600}(57, \cdot)$$ None 0 4
1600.1.x $$\chi_{1600}(151, \cdot)$$ None 0 4
1600.1.z $$\chi_{1600}(199, \cdot)$$ None 0 4
1600.1.bc $$\chi_{1600}(457, \cdot)$$ None 0 4
1600.1.bd $$\chi_{1600}(31, \cdot)$$ 1600.1.bd.a 8 4
1600.1.bd.b 8
1600.1.bf $$\chi_{1600}(319, \cdot)$$ 1600.1.bf.a 4 4
1600.1.bh $$\chi_{1600}(191, \cdot)$$ 1600.1.bh.a 4 4
1600.1.bh.b 8
1600.1.bi $$\chi_{1600}(159, \cdot)$$ None 0 4
1600.1.bk $$\chi_{1600}(157, \cdot)$$ None 0 8
1600.1.bo $$\chi_{1600}(51, \cdot)$$ None 0 8
1600.1.bp $$\chi_{1600}(99, \cdot)$$ None 0 8
1600.1.bq $$\chi_{1600}(93, \cdot)$$ None 0 8
1600.1.bs $$\chi_{1600}(17, \cdot)$$ None 0 8
1600.1.bv $$\chi_{1600}(79, \cdot)$$ None 0 8
1600.1.bw $$\chi_{1600}(513, \cdot)$$ 1600.1.bw.a 8 8
1600.1.bz $$\chi_{1600}(33, \cdot)$$ None 0 8
1600.1.ca $$\chi_{1600}(111, \cdot)$$ None 0 8
1600.1.cd $$\chi_{1600}(177, \cdot)$$ None 0 8
1600.1.ce $$\chi_{1600}(73, \cdot)$$ None 0 16
1600.1.ch $$\chi_{1600}(39, \cdot)$$ None 0 16
1600.1.cj $$\chi_{1600}(71, \cdot)$$ None 0 16
1600.1.ck $$\chi_{1600}(137, \cdot)$$ None 0 16
1600.1.cn $$\chi_{1600}(13, \cdot)$$ None 0 32
1600.1.co $$\chi_{1600}(19, \cdot)$$ None 0 32
1600.1.cp $$\chi_{1600}(11, \cdot)$$ None 0 32
1600.1.ct $$\chi_{1600}(53, \cdot)$$ None 0 32

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1600))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1600)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(320))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(800))$$$$^{\oplus 2}$$