## Defining parameters

 Level: $$N$$ = $$111 = 3 \cdot 37$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ Sturm bound: $$912$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(111))$$.

Total New Old
Modular forms 79 41 38
Cusp forms 7 7 0
Eisenstein series 72 34 38

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 7 0 0 0

## Trace form

 $$7q - 3q^{3} + q^{4} - 2q^{7} + q^{9} + O(q^{10})$$ $$7q - 3q^{3} + q^{4} - 2q^{7} + q^{9} - 4q^{10} - 3q^{12} - 2q^{13} - 3q^{16} - 2q^{19} - 2q^{21} + q^{25} + 3q^{27} + 4q^{28} + 4q^{30} - 2q^{31} + 4q^{34} + q^{36} + 3q^{37} + 4q^{39} - 2q^{43} - 4q^{46} + 7q^{48} + q^{49} - 2q^{52} - 2q^{57} + 4q^{58} - 2q^{61} - 2q^{63} - 3q^{64} - 2q^{67} - 2q^{73} - 3q^{75} - 2q^{76} - 2q^{79} + q^{81} - 2q^{84} - 4q^{85} - 4q^{90} + 2q^{91} - 2q^{93} - 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(111))$$

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
111.1.b $$\chi_{111}(38, \cdot)$$ None 0 1
111.1.d $$\chi_{111}(110, \cdot)$$ 111.1.d.a 1 1
111.1.d.b 2
111.1.f $$\chi_{111}(31, \cdot)$$ None 0 2
111.1.h $$\chi_{111}(11, \cdot)$$ 111.1.h.a 2 2
111.1.i $$\chi_{111}(26, \cdot)$$ 111.1.i.a 2 2
111.1.l $$\chi_{111}(82, \cdot)$$ None 0 4
111.1.n $$\chi_{111}(41, \cdot)$$ None 0 6
111.1.p $$\chi_{111}(44, \cdot)$$ None 0 6
111.1.r $$\chi_{111}(13, \cdot)$$ None 0 12