## Defining parameters

 Level: $$N$$ = $$1340 = 2^{2} \cdot 5 \cdot 67$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$6$$ Sturm bound: $$107712$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1340))$$.

Total New Old
Modular forms 1410 456 954
Cusp forms 90 64 26
Eisenstein series 1320 392 928

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 64 0 0 0

## Trace form

 $$64q - 2q^{4} - 2q^{5} - 4q^{6} - 6q^{9} + O(q^{10})$$ $$64q - 2q^{4} - 2q^{5} - 4q^{6} - 6q^{9} - 4q^{14} - 2q^{16} - 2q^{20} - 8q^{21} - 4q^{24} - 2q^{25} - 4q^{29} - 4q^{30} - 6q^{36} - 4q^{41} - 6q^{45} - 4q^{46} - 6q^{49} - 8q^{54} - 4q^{56} - 4q^{61} - 2q^{64} - 8q^{69} - 4q^{70} - 2q^{80} - 10q^{81} - 8q^{84} - 4q^{86} - 4q^{89} - 4q^{94} - 4q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1340))$$

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
1340.1.b $$\chi_{1340}(401, \cdot)$$ None 0 1
1340.1.c $$\chi_{1340}(671, \cdot)$$ None 0 1
1340.1.f $$\chi_{1340}(939, \cdot)$$ None 0 1
1340.1.g $$\chi_{1340}(669, \cdot)$$ None 0 1
1340.1.l $$\chi_{1340}(537, \cdot)$$ None 0 2
1340.1.m $$\chi_{1340}(267, \cdot)$$ None 0 2
1340.1.p $$\chi_{1340}(171, \cdot)$$ None 0 2
1340.1.q $$\chi_{1340}(641, \cdot)$$ None 0 2
1340.1.s $$\chi_{1340}(909, \cdot)$$ None 0 2
1340.1.t $$\chi_{1340}(439, \cdot)$$ 1340.1.t.a 2 2
1340.1.t.b 2
1340.1.v $$\chi_{1340}(507, \cdot)$$ None 0 4
1340.1.w $$\chi_{1340}(37, \cdot)$$ None 0 4
1340.1.ba $$\chi_{1340}(109, \cdot)$$ None 0 10
1340.1.bb $$\chi_{1340}(59, \cdot)$$ 1340.1.bb.a 10 10
1340.1.bb.b 10
1340.1.be $$\chi_{1340}(91, \cdot)$$ None 0 10
1340.1.bf $$\chi_{1340}(161, \cdot)$$ None 0 10
1340.1.bh $$\chi_{1340}(3, \cdot)$$ None 0 20
1340.1.bi $$\chi_{1340}(193, \cdot)$$ None 0 20
1340.1.bl $$\chi_{1340}(19, \cdot)$$ 1340.1.bl.a 20 20
1340.1.bl.b 20
1340.1.bm $$\chi_{1340}(69, \cdot)$$ None 0 20
1340.1.bo $$\chi_{1340}(41, \cdot)$$ None 0 20
1340.1.bp $$\chi_{1340}(71, \cdot)$$ None 0 20
1340.1.bu $$\chi_{1340}(17, \cdot)$$ None 0 40
1340.1.bv $$\chi_{1340}(7, \cdot)$$ None 0 40

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1340))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1340)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(268))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(335))$$$$^{\oplus 3}$$