## Defining parameters

 Level: $$N$$ = $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$2$$ Sturm bound: $$1536$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(168))$$.

Total New Old
Modular forms 154 24 130
Cusp forms 10 4 6
Eisenstein series 144 20 124

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

 $$4 q - 2 q^{4} + 4 q^{6} - 2 q^{7} - 2 q^{9} + O(q^{10})$$ $$4 q - 2 q^{4} + 4 q^{6} - 2 q^{7} - 2 q^{9} + 2 q^{10} - 4 q^{15} - 2 q^{16} - 4 q^{22} - 2 q^{24} - 2 q^{28} + 2 q^{31} + 2 q^{33} + 4 q^{36} + 2 q^{40} - 2 q^{42} - 2 q^{49} - 2 q^{54} + 4 q^{55} + 2 q^{58} + 2 q^{60} + 4 q^{63} + 4 q^{64} + 2 q^{70} - 4 q^{73} + 2 q^{79} - 2 q^{81} + 2 q^{87} + 2 q^{88} - 4 q^{90} - 2 q^{96} - 4 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(168))$$

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
168.1.d $$\chi_{168}(113, \cdot)$$ None 0 1
168.1.e $$\chi_{168}(83, \cdot)$$ None 0 1
168.1.f $$\chi_{168}(97, \cdot)$$ None 0 1
168.1.g $$\chi_{168}(43, \cdot)$$ None 0 1
168.1.l $$\chi_{168}(13, \cdot)$$ None 0 1
168.1.m $$\chi_{168}(127, \cdot)$$ None 0 1
168.1.n $$\chi_{168}(29, \cdot)$$ None 0 1
168.1.o $$\chi_{168}(167, \cdot)$$ None 0 1
168.1.r $$\chi_{168}(47, \cdot)$$ None 0 2
168.1.s $$\chi_{168}(53, \cdot)$$ 168.1.s.a 2 2
168.1.s.b 2
168.1.w $$\chi_{168}(79, \cdot)$$ None 0 2
168.1.x $$\chi_{168}(61, \cdot)$$ None 0 2
168.1.y $$\chi_{168}(67, \cdot)$$ None 0 2
168.1.z $$\chi_{168}(73, \cdot)$$ None 0 2
168.1.be $$\chi_{168}(59, \cdot)$$ None 0 2
168.1.bf $$\chi_{168}(65, \cdot)$$ None 0 2

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(168))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(168)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 2}$$