# Properties

 Label 400.1 Level 400 Weight 1 Dimension 5 Nonzero newspaces 2 Newforms 2 Sturm bound 9600 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newforms: $$2$$ Sturm bound: $$9600$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(400))$$.

Total New Old
Modular forms 419 97 322
Cusp forms 27 5 22
Eisenstein series 392 92 300

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 5 0 0 0

## Trace form

 $$5q$$ $$\mathstrut +\mathstrut q^{5}$$ $$\mathstrut +\mathstrut 2q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$5q$$ $$\mathstrut +\mathstrut q^{5}$$ $$\mathstrut +\mathstrut 2q^{9}$$ $$\mathstrut -\mathstrut q^{25}$$ $$\mathstrut -\mathstrut 4q^{29}$$ $$\mathstrut -\mathstrut 5q^{37}$$ $$\mathstrut -\mathstrut q^{45}$$ $$\mathstrut -\mathstrut 3q^{49}$$ $$\mathstrut -\mathstrut 5q^{53}$$ $$\mathstrut -\mathstrut 5q^{65}$$ $$\mathstrut +\mathstrut 5q^{85}$$ $$\mathstrut +\mathstrut q^{89}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$

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
400.1.b $$\chi_{400}(351, \cdot)$$ 400.1.b.a 1 1
400.1.e $$\chi_{400}(199, \cdot)$$ None 0 1
400.1.g $$\chi_{400}(151, \cdot)$$ None 0 1
400.1.h $$\chi_{400}(399, \cdot)$$ None 0 1
400.1.i $$\chi_{400}(93, \cdot)$$ None 0 2
400.1.k $$\chi_{400}(99, \cdot)$$ None 0 2
400.1.m $$\chi_{400}(57, \cdot)$$ None 0 2
400.1.p $$\chi_{400}(193, \cdot)$$ None 0 2
400.1.r $$\chi_{400}(51, \cdot)$$ None 0 2
400.1.t $$\chi_{400}(157, \cdot)$$ None 0 2
400.1.v $$\chi_{400}(71, \cdot)$$ None 0 4
400.1.x $$\chi_{400}(79, \cdot)$$ 400.1.x.a 4 4
400.1.z $$\chi_{400}(31, \cdot)$$ None 0 4
400.1.ba $$\chi_{400}(39, \cdot)$$ None 0 4
400.1.bc $$\chi_{400}(53, \cdot)$$ None 0 8
400.1.bf $$\chi_{400}(19, \cdot)$$ None 0 8
400.1.bg $$\chi_{400}(17, \cdot)$$ None 0 8
400.1.bj $$\chi_{400}(73, \cdot)$$ None 0 8
400.1.bk $$\chi_{400}(11, \cdot)$$ None 0 8
400.1.bn $$\chi_{400}(13, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(400))$$ into lower level spaces

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