# Properties

 Label 800.1 Level 800 Weight 1 Dimension 26 Nonzero newspaces 5 Newform subspaces 8 Sturm bound 38400 Trace bound 19

## Defining parameters

 Level: $$N$$ = $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$5$$ Newform subspaces: $$8$$ Sturm bound: $$38400$$ Trace bound: $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(800))$$.

Total New Old
Modular forms 968 231 737
Cusp forms 72 26 46
Eisenstein series 896 205 691

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 18 0 0 8

## Trace form

 $$26q - 2q^{5} + O(q^{10})$$ $$26q - 2q^{5} + 4q^{11} + 10q^{13} + 4q^{17} - 4q^{21} - 6q^{29} - 2q^{37} - 4q^{41} - 2q^{45} - 4q^{49} - 4q^{51} - 2q^{53} - 4q^{57} - 2q^{61} - 6q^{65} + 2q^{69} - 6q^{73} + 2q^{81} - 8q^{85} - 10q^{89} - 8q^{93} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(800))$$

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
800.1.b $$\chi_{800}(351, \cdot)$$ None 0 1
800.1.e $$\chi_{800}(399, \cdot)$$ 800.1.e.a 2 1
800.1.g $$\chi_{800}(751, \cdot)$$ 800.1.g.a 1 1
800.1.g.b 1
800.1.h $$\chi_{800}(799, \cdot)$$ None 0 1
800.1.i $$\chi_{800}(57, \cdot)$$ None 0 2
800.1.k $$\chi_{800}(199, \cdot)$$ None 0 2
800.1.m $$\chi_{800}(593, \cdot)$$ None 0 2
800.1.p $$\chi_{800}(193, \cdot)$$ 800.1.p.a 2 2
800.1.p.b 2
800.1.p.c 2
800.1.r $$\chi_{800}(151, \cdot)$$ None 0 2
800.1.t $$\chi_{800}(457, \cdot)$$ None 0 2
800.1.w $$\chi_{800}(93, \cdot)$$ None 0 4
800.1.x $$\chi_{800}(51, \cdot)$$ None 0 4
800.1.z $$\chi_{800}(99, \cdot)$$ None 0 4
800.1.bc $$\chi_{800}(157, \cdot)$$ None 0 4
800.1.bd $$\chi_{800}(111, \cdot)$$ None 0 4
800.1.bf $$\chi_{800}(159, \cdot)$$ None 0 4
800.1.bh $$\chi_{800}(31, \cdot)$$ 800.1.bh.a 8 4
800.1.bi $$\chi_{800}(79, \cdot)$$ None 0 4
800.1.bk $$\chi_{800}(137, \cdot)$$ None 0 8
800.1.bn $$\chi_{800}(39, \cdot)$$ None 0 8
800.1.bo $$\chi_{800}(33, \cdot)$$ 800.1.bo.a 8 8
800.1.br $$\chi_{800}(17, \cdot)$$ None 0 8
800.1.bs $$\chi_{800}(71, \cdot)$$ None 0 8
800.1.bv $$\chi_{800}(73, \cdot)$$ None 0 8
800.1.bw $$\chi_{800}(13, \cdot)$$ None 0 16
800.1.bz $$\chi_{800}(19, \cdot)$$ None 0 16
800.1.cb $$\chi_{800}(11, \cdot)$$ None 0 16
800.1.cc $$\chi_{800}(53, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(800))$$ into lower level spaces

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