## Defining parameters

 Level: $$N$$ = $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newforms: $$1$$ Sturm bound: $$1008$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(117))$$.

Total New Old
Modular forms 100 52 48
Cusp forms 4 2 2
Eisenstein series 96 50 46

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 2 0 0 0

## Trace form

 $$2q - 2q^{7} + O(q^{10})$$ $$2q - 2q^{7} - 2q^{16} - 2q^{19} + 2q^{28} + 2q^{31} + 2q^{37} + 2q^{52} - 2q^{67} - 2q^{73} - 2q^{76} - 2q^{91} + 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(117))$$

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
117.1.c $$\chi_{117}(53, \cdot)$$ None 0 1
117.1.d $$\chi_{117}(116, \cdot)$$ None 0 1
117.1.j $$\chi_{117}(73, \cdot)$$ 117.1.j.a 2 2
117.1.k $$\chi_{117}(29, \cdot)$$ None 0 2
117.1.m $$\chi_{117}(23, \cdot)$$ None 0 2
117.1.n $$\chi_{117}(38, \cdot)$$ None 0 2
117.1.o $$\chi_{117}(17, \cdot)$$ None 0 2
117.1.p $$\chi_{117}(35, \cdot)$$ None 0 2
117.1.s $$\chi_{117}(14, \cdot)$$ None 0 2
117.1.u $$\chi_{117}(68, \cdot)$$ None 0 2
117.1.v $$\chi_{117}(95, \cdot)$$ None 0 2
117.1.w $$\chi_{117}(58, \cdot)$$ None 0 4
117.1.y $$\chi_{117}(31, \cdot)$$ None 0 4
117.1.bb $$\chi_{117}(7, \cdot)$$ None 0 4
117.1.bd $$\chi_{117}(19, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(117))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(117)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$