## Defining parameters

 Level: $$N$$ = $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$18$$ Sturm bound: $$614400$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(3200))$$.

Total New Old
Modular forms 4864 1053 3811
Cusp forms 384 69 315
Eisenstein series 4480 984 3496

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 69 0 0 0

## Trace form

 $$69 q + 3 q^{9} + O(q^{10})$$ $$69 q + 3 q^{9} - 2 q^{17} - 14 q^{41} - 3 q^{49} + 8 q^{65} + 14 q^{73} - 11 q^{81} + 6 q^{89} + 14 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(3200))$$

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
3200.1.b $$\chi_{3200}(1151, \cdot)$$ None 0 1
3200.1.e $$\chi_{3200}(1599, \cdot)$$ 3200.1.e.a 2 1
3200.1.e.b 4
3200.1.g $$\chi_{3200}(2751, \cdot)$$ 3200.1.g.a 1 1
3200.1.g.b 2
3200.1.g.c 2
3200.1.g.d 2
3200.1.g.e 4
3200.1.h $$\chi_{3200}(3199, \cdot)$$ None 0 1
3200.1.i $$\chi_{3200}(2593, \cdot)$$ None 0 2
3200.1.k $$\chi_{3200}(799, \cdot)$$ None 0 2
3200.1.m $$\chi_{3200}(193, \cdot)$$ 3200.1.m.a 2 2
3200.1.m.b 2
3200.1.m.c 4
3200.1.m.d 4
3200.1.m.e 8
3200.1.p $$\chi_{3200}(257, \cdot)$$ None 0 2
3200.1.r $$\chi_{3200}(351, \cdot)$$ None 0 2
3200.1.t $$\chi_{3200}(993, \cdot)$$ None 0 2
3200.1.w $$\chi_{3200}(593, \cdot)$$ None 0 4
3200.1.x $$\chi_{3200}(751, \cdot)$$ None 0 4
3200.1.z $$\chi_{3200}(399, \cdot)$$ None 0 4
3200.1.bc $$\chi_{3200}(657, \cdot)$$ None 0 4
3200.1.bd $$\chi_{3200}(191, \cdot)$$ 3200.1.bd.a 4 4
3200.1.bd.b 4
3200.1.bf $$\chi_{3200}(639, \cdot)$$ None 0 4
3200.1.bh $$\chi_{3200}(511, \cdot)$$ None 0 4
3200.1.bi $$\chi_{3200}(319, \cdot)$$ 3200.1.bi.a 4 4
3200.1.bi.b 4
3200.1.bk $$\chi_{3200}(457, \cdot)$$ None 0 8
3200.1.bo $$\chi_{3200}(151, \cdot)$$ None 0 8
3200.1.bp $$\chi_{3200}(199, \cdot)$$ None 0 8
3200.1.bq $$\chi_{3200}(57, \cdot)$$ None 0 8
3200.1.bs $$\chi_{3200}(353, \cdot)$$ None 0 8
3200.1.bv $$\chi_{3200}(159, \cdot)$$ None 0 8
3200.1.bw $$\chi_{3200}(513, \cdot)$$ None 0 8
3200.1.bz $$\chi_{3200}(577, \cdot)$$ 3200.1.bz.a 8 8
3200.1.bz.b 8
3200.1.ca $$\chi_{3200}(31, \cdot)$$ None 0 8
3200.1.cd $$\chi_{3200}(33, \cdot)$$ None 0 8
3200.1.ce $$\chi_{3200}(93, \cdot)$$ None 0 16
3200.1.ch $$\chi_{3200}(99, \cdot)$$ None 0 16
3200.1.cj $$\chi_{3200}(51, \cdot)$$ None 0 16
3200.1.ck $$\chi_{3200}(157, \cdot)$$ None 0 16
3200.1.cm $$\chi_{3200}(177, \cdot)$$ None 0 16
3200.1.cp $$\chi_{3200}(79, \cdot)$$ None 0 16
3200.1.cr $$\chi_{3200}(111, \cdot)$$ None 0 16
3200.1.cs $$\chi_{3200}(17, \cdot)$$ None 0 16
3200.1.cv $$\chi_{3200}(73, \cdot)$$ None 0 32
3200.1.cw $$\chi_{3200}(39, \cdot)$$ None 0 32
3200.1.cx $$\chi_{3200}(71, \cdot)$$ None 0 32
3200.1.db $$\chi_{3200}(137, \cdot)$$ None 0 32
3200.1.dd $$\chi_{3200}(53, \cdot)$$ None 0 64
3200.1.de $$\chi_{3200}(11, \cdot)$$ None 0 64
3200.1.dg $$\chi_{3200}(19, \cdot)$$ None 0 64
3200.1.dj $$\chi_{3200}(13, \cdot)$$ None 0 64

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(3200))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(3200)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(128))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(320))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(640))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(800))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1600))$$$$^{\oplus 2}$$