## Defining parameters

 Level: $$N$$ = $$39 = 3 \cdot 13$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newforms: $$1$$ Sturm bound: $$112$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(39))$$.

Total New Old
Modular forms 25 11 14
Cusp forms 1 1 0
Eisenstein series 24 10 14

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 1 0 0 0

## Trace form

 $$q - q^{3} - q^{4} + q^{9} + O(q^{10})$$ $$q - q^{3} - q^{4} + q^{9} + q^{12} - q^{13} + q^{16} - q^{25} - q^{27} - q^{36} + q^{39} + 2q^{43} - q^{48} + q^{49} + q^{52} - 2q^{61} - q^{64} + q^{75} - 2q^{79} + q^{81} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(39))$$

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
39.1.c $$\chi_{39}(14, \cdot)$$ None 0 1
39.1.d $$\chi_{39}(38, \cdot)$$ 39.1.d.a 1 1
39.1.g $$\chi_{39}(31, \cdot)$$ None 0 2
39.1.h $$\chi_{39}(17, \cdot)$$ None 0 2
39.1.i $$\chi_{39}(29, \cdot)$$ None 0 2
39.1.l $$\chi_{39}(7, \cdot)$$ None 0 4