## Defining parameters

 Level: $$N$$ = $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$768$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(112))$$.

Total New Old
Modular forms 88 25 63
Cusp forms 4 2 2
Eisenstein series 84 23 61

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

 $$2 q - 2 q^{4} + O(q^{10})$$ $$2 q - 2 q^{4} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 2 q^{18} + 2 q^{22} - 2 q^{29} - 2 q^{37} + 2 q^{43} + 2 q^{44} - 2 q^{49} - 2 q^{50} + 2 q^{53} + 2 q^{56} - 2 q^{58} + 2 q^{63} - 2 q^{64} + 2 q^{67} - 2 q^{72} + 2 q^{74} + 2 q^{77} - 2 q^{81} - 2 q^{86} - 2 q^{88} - 2 q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(112))$$

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
112.1.c $$\chi_{112}(97, \cdot)$$ None 0 1
112.1.d $$\chi_{112}(15, \cdot)$$ None 0 1
112.1.g $$\chi_{112}(71, \cdot)$$ None 0 1
112.1.h $$\chi_{112}(41, \cdot)$$ None 0 1
112.1.k $$\chi_{112}(43, \cdot)$$ None 0 2
112.1.l $$\chi_{112}(13, \cdot)$$ 112.1.l.a 2 2
112.1.n $$\chi_{112}(73, \cdot)$$ None 0 2
112.1.o $$\chi_{112}(23, \cdot)$$ None 0 2
112.1.r $$\chi_{112}(79, \cdot)$$ None 0 2
112.1.s $$\chi_{112}(17, \cdot)$$ None 0 2
112.1.u $$\chi_{112}(11, \cdot)$$ None 0 4
112.1.x $$\chi_{112}(5, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(112))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(112)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 2}$$