## Defining parameters

 Level: $$N$$ = $$1352 = 2^{3} \cdot 13^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$4$$ Newform subspaces: $$10$$ Sturm bound: $$113568$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1352))$$.

Total New Old
Modular forms 1518 455 1063
Cusp forms 150 46 104
Eisenstein series 1368 409 959

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 46 0 0 0

## Trace form

 $$46 q + 2 q^{3} - 2 q^{4} + O(q^{10})$$ $$46 q + 2 q^{3} - 2 q^{4} + 2 q^{10} + 2 q^{12} - 10 q^{14} - 2 q^{16} + 2 q^{17} - 14 q^{27} - 2 q^{30} - 2 q^{35} - 10 q^{40} - 2 q^{42} + 2 q^{43} + 2 q^{48} - 2 q^{51} + 2 q^{56} - 4 q^{62} - 2 q^{64} - 24 q^{66} + 2 q^{68} + 2 q^{74} + 2 q^{81} + 2 q^{94} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1352))$$

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
1352.1.c $$\chi_{1352}(1351, \cdot)$$ None 0 1
1352.1.d $$\chi_{1352}(1015, \cdot)$$ None 0 1
1352.1.g $$\chi_{1352}(339, \cdot)$$ 1352.1.g.a 2 1
1352.1.g.b 3
1352.1.g.c 3
1352.1.h $$\chi_{1352}(675, \cdot)$$ 1352.1.h.a 6 1
1352.1.j $$\chi_{1352}(437, \cdot)$$ None 0 2
1352.1.l $$\chi_{1352}(577, \cdot)$$ None 0 2
1352.1.n $$\chi_{1352}(315, \cdot)$$ 1352.1.n.a 4 2
1352.1.n.b 6
1352.1.n.c 6
1352.1.p $$\chi_{1352}(147, \cdot)$$ 1352.1.p.a 2 2
1352.1.p.b 2
1352.1.p.c 12
1352.1.q $$\chi_{1352}(23, \cdot)$$ None 0 2
1352.1.t $$\chi_{1352}(191, \cdot)$$ None 0 2
1352.1.v $$\chi_{1352}(89, \cdot)$$ None 0 4
1352.1.x $$\chi_{1352}(357, \cdot)$$ None 0 4
1352.1.z $$\chi_{1352}(51, \cdot)$$ None 0 12
1352.1.ba $$\chi_{1352}(27, \cdot)$$ None 0 12
1352.1.bd $$\chi_{1352}(79, \cdot)$$ None 0 12
1352.1.be $$\chi_{1352}(103, \cdot)$$ None 0 12
1352.1.bh $$\chi_{1352}(57, \cdot)$$ None 0 24
1352.1.bj $$\chi_{1352}(5, \cdot)$$ None 0 24
1352.1.bl $$\chi_{1352}(55, \cdot)$$ None 0 24
1352.1.bo $$\chi_{1352}(95, \cdot)$$ None 0 24
1352.1.bp $$\chi_{1352}(43, \cdot)$$ None 0 24
1352.1.br $$\chi_{1352}(3, \cdot)$$ None 0 24
1352.1.bt $$\chi_{1352}(37, \cdot)$$ None 0 48
1352.1.bv $$\chi_{1352}(33, \cdot)$$ None 0 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1352))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1352)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(104))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(676))$$$$^{\oplus 2}$$