## Defining parameters

 Level: $$N$$ = $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$2$$ Sturm bound: $$11664$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(405))$$.

Total New Old
Modular forms 440 172 268
Cusp forms 8 4 4
Eisenstein series 432 168 264

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

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

## Trace form

 $$4 q + O(q^{10})$$ $$4 q - 4 q^{10} + 2 q^{16} - 4 q^{19} - 2 q^{25} + 2 q^{31} - 2 q^{34} - 2 q^{40} + 4 q^{46} - 2 q^{49} + 2 q^{61} + 4 q^{64} + 2 q^{79} + 2 q^{85} + 4 q^{94} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(405))$$

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
405.1.c $$\chi_{405}(161, \cdot)$$ None 0 1
405.1.d $$\chi_{405}(404, \cdot)$$ None 0 1
405.1.g $$\chi_{405}(82, \cdot)$$ None 0 2
405.1.h $$\chi_{405}(134, \cdot)$$ 405.1.h.a 2 2
405.1.h.b 2
405.1.i $$\chi_{405}(26, \cdot)$$ None 0 2
405.1.l $$\chi_{405}(28, \cdot)$$ None 0 4
405.1.n $$\chi_{405}(44, \cdot)$$ None 0 6
405.1.o $$\chi_{405}(71, \cdot)$$ None 0 6
405.1.s $$\chi_{405}(37, \cdot)$$ None 0 12
405.1.u $$\chi_{405}(11, \cdot)$$ None 0 18
405.1.v $$\chi_{405}(14, \cdot)$$ None 0 18
405.1.w $$\chi_{405}(7, \cdot)$$ None 0 36

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(405))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(405)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 2}$$