## Defining parameters

 Level: $$N$$ = $$88 = 2^{3} \cdot 11$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$480$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(88))$$.

Total New Old
Modular forms 66 22 44
Cusp forms 6 4 2
Eisenstein series 60 18 42

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

 $$4q - q^{2} - 2q^{3} - q^{4} + 3q^{6} - q^{8} - 3q^{9} + O(q^{10})$$ $$4q - q^{2} - 2q^{3} - q^{4} + 3q^{6} - q^{8} - 3q^{9} - q^{11} - 2q^{12} - q^{16} - 2q^{17} + 2q^{18} + 3q^{19} - q^{22} + 3q^{24} - q^{25} + q^{27} + 4q^{32} + 3q^{33} - 2q^{34} + 2q^{36} - 2q^{38} - 2q^{41} - 2q^{43} + 4q^{44} - 2q^{48} - q^{49} - q^{50} + q^{51} - 4q^{54} + q^{57} + 3q^{59} - q^{64} - 2q^{66} - 2q^{67} - 2q^{68} - 3q^{72} - 2q^{73} + 3q^{75} - 2q^{76} + 3q^{82} + 3q^{83} + 3q^{86} - q^{88} - 2q^{89} - 2q^{96} + 3q^{97} + 4q^{98} - 3q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(88))$$

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
88.1.b $$\chi_{88}(21, \cdot)$$ None 0 1
88.1.d $$\chi_{88}(23, \cdot)$$ None 0 1
88.1.f $$\chi_{88}(67, \cdot)$$ None 0 1
88.1.h $$\chi_{88}(65, \cdot)$$ None 0 1
88.1.j $$\chi_{88}(17, \cdot)$$ None 0 4
88.1.l $$\chi_{88}(3, \cdot)$$ 88.1.l.a 4 4
88.1.n $$\chi_{88}(15, \cdot)$$ None 0 4
88.1.p $$\chi_{88}(13, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(88))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(88)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(44))$$$$^{\oplus 2}$$