## Defining parameters

 Level: $$N$$ = $$309 = 3 \cdot 103$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$7072$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(309))$$.

Total New Old
Modular forms 228 120 108
Cusp forms 24 20 4
Eisenstein series 204 100 104

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 16 4 0 0

## Trace form

 $$20q - 5q^{3} - q^{4} + 3q^{9} + O(q^{10})$$ $$20q - 5q^{3} - q^{4} + 3q^{9} + 4q^{10} - q^{12} - 2q^{13} + q^{16} - 4q^{19} - 4q^{21} - 4q^{22} - q^{25} - 5q^{27} - 2q^{28} - 4q^{30} - 2q^{31} - 4q^{34} - q^{36} - 2q^{37} - 2q^{39} - 2q^{40} - 4q^{43} + 4q^{46} - 3q^{48} - 3q^{49} - 2q^{52} - 2q^{55} - 2q^{58} - 2q^{61} - 5q^{64} + 4q^{66} - 4q^{67} + 2q^{70} - 2q^{73} - q^{75} - 2q^{76} + 6q^{79} + 3q^{81} + 2q^{82} + 15q^{84} - 2q^{85} + 2q^{88} + 4q^{90} + 13q^{91} - 2q^{93} - 4q^{94} + 17q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(309))$$

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
309.1.b $$\chi_{309}(104, \cdot)$$ None 0 1
309.1.d $$\chi_{309}(205, \cdot)$$ None 0 1
309.1.f $$\chi_{309}(160, \cdot)$$ None 0 2
309.1.h $$\chi_{309}(56, \cdot)$$ 309.1.h.a 4 2
309.1.j $$\chi_{309}(10, \cdot)$$ None 0 16
309.1.l $$\chi_{309}(8, \cdot)$$ 309.1.l.a 16 16
309.1.n $$\chi_{309}(2, \cdot)$$ None 0 32
309.1.p $$\chi_{309}(40, \cdot)$$ None 0 32

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(309))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(309)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(103))$$$$^{\oplus 2}$$