## Defining parameters

 Level: $$N$$ = $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$288$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(68))$$.

Total New Old
Modular forms 43 17 26
Cusp forms 3 3 0
Eisenstein series 40 14 26

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 3 0 0 0

## Trace form

 $$3 q - q^{2} - q^{4} - 2 q^{5} - q^{8} - q^{9} + O(q^{10})$$ $$3 q - q^{2} - q^{4} - 2 q^{5} - q^{8} - q^{9} + 2 q^{10} - 2 q^{13} + 3 q^{16} - q^{17} - q^{18} + 2 q^{20} + q^{25} + 2 q^{26} + 2 q^{29} - q^{32} - q^{34} - q^{36} - 2 q^{37} - 2 q^{40} + 2 q^{41} + 2 q^{45} - q^{49} - 3 q^{50} - 2 q^{52} + 2 q^{53} - 2 q^{58} - 2 q^{61} - q^{64} + 3 q^{68} + 3 q^{72} + 2 q^{73} + 2 q^{74} - 2 q^{80} - q^{81} + 2 q^{82} + 2 q^{85} - 2 q^{89} + 2 q^{90} - 2 q^{97} + 3 q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(68))$$

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
68.1.c $$\chi_{68}(35, \cdot)$$ None 0 1
68.1.d $$\chi_{68}(67, \cdot)$$ 68.1.d.a 1 1
68.1.f $$\chi_{68}(47, \cdot)$$ 68.1.f.a 2 2
68.1.g $$\chi_{68}(15, \cdot)$$ None 0 4
68.1.j $$\chi_{68}(5, \cdot)$$ None 0 8