## Defining parameters

 Level: $$N$$ = $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$8$$ Newform subspaces: $$15$$ Sturm bound: $$301056$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2352))$$.

Total New Old
Modular forms 3822 507 3315
Cusp forms 462 70 392
Eisenstein series 3360 437 2923

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 70 0 0 0

## Trace form

 $$70q - q^{3} + q^{9} + O(q^{10})$$ $$70q - q^{3} + q^{9} + 2q^{13} + 4q^{19} + 3q^{21} + q^{25} + 5q^{27} + 4q^{31} + 8q^{37} + 4q^{39} + 4q^{43} - 4q^{57} + 2q^{61} + 4q^{67} + 8q^{73} - q^{75} + 4q^{79} + q^{81} + 3q^{91} - 16q^{93} - 10q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2352))$$

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
2352.1.d $$\chi_{2352}(785, \cdot)$$ 2352.1.d.a 1 1
2352.1.d.b 1
2352.1.e $$\chi_{2352}(1175, \cdot)$$ None 0 1
2352.1.f $$\chi_{2352}(97, \cdot)$$ None 0 1
2352.1.g $$\chi_{2352}(295, \cdot)$$ None 0 1
2352.1.l $$\chi_{2352}(1273, \cdot)$$ None 0 1
2352.1.m $$\chi_{2352}(1471, \cdot)$$ None 0 1
2352.1.n $$\chi_{2352}(1961, \cdot)$$ None 0 1
2352.1.o $$\chi_{2352}(2351, \cdot)$$ 2352.1.o.a 2 1
2352.1.o.b 2
2352.1.r $$\chi_{2352}(685, \cdot)$$ None 0 2
2352.1.t $$\chi_{2352}(197, \cdot)$$ None 0 2
2352.1.v $$\chi_{2352}(587, \cdot)$$ None 0 2
2352.1.x $$\chi_{2352}(883, \cdot)$$ None 0 2
2352.1.z $$\chi_{2352}(815, \cdot)$$ 2352.1.z.a 2 2
2352.1.z.b 2
2352.1.z.c 2
2352.1.z.d 2
2352.1.ba $$\chi_{2352}(569, \cdot)$$ None 0 2
2352.1.be $$\chi_{2352}(79, \cdot)$$ None 0 2
2352.1.bf $$\chi_{2352}(313, \cdot)$$ None 0 2
2352.1.bg $$\chi_{2352}(1255, \cdot)$$ None 0 2
2352.1.bh $$\chi_{2352}(913, \cdot)$$ None 0 2
2352.1.bm $$\chi_{2352}(215, \cdot)$$ None 0 2
2352.1.bn $$\chi_{2352}(1745, \cdot)$$ 2352.1.bn.a 2 2
2352.1.bq $$\chi_{2352}(67, \cdot)$$ None 0 4
2352.1.bs $$\chi_{2352}(227, \cdot)$$ None 0 4
2352.1.bu $$\chi_{2352}(557, \cdot)$$ None 0 4
2352.1.bw $$\chi_{2352}(325, \cdot)$$ None 0 4
2352.1.by $$\chi_{2352}(335, \cdot)$$ 2352.1.by.a 6 6
2352.1.by.b 6
2352.1.bz $$\chi_{2352}(281, \cdot)$$ None 0 6
2352.1.ca $$\chi_{2352}(127, \cdot)$$ None 0 6
2352.1.cb $$\chi_{2352}(265, \cdot)$$ None 0 6
2352.1.cg $$\chi_{2352}(631, \cdot)$$ None 0 6
2352.1.ch $$\chi_{2352}(433, \cdot)$$ None 0 6
2352.1.ci $$\chi_{2352}(167, \cdot)$$ None 0 6
2352.1.cj $$\chi_{2352}(113, \cdot)$$ 2352.1.cj.a 6 6
2352.1.co $$\chi_{2352}(43, \cdot)$$ None 0 12
2352.1.cq $$\chi_{2352}(83, \cdot)$$ None 0 12
2352.1.cs $$\chi_{2352}(29, \cdot)$$ None 0 12
2352.1.cu $$\chi_{2352}(13, \cdot)$$ None 0 12
2352.1.cv $$\chi_{2352}(65, \cdot)$$ 2352.1.cv.a 12 12
2352.1.cw $$\chi_{2352}(311, \cdot)$$ None 0 12
2352.1.db $$\chi_{2352}(145, \cdot)$$ None 0 12
2352.1.dc $$\chi_{2352}(151, \cdot)$$ None 0 12
2352.1.dd $$\chi_{2352}(73, \cdot)$$ None 0 12
2352.1.de $$\chi_{2352}(319, \cdot)$$ None 0 12
2352.1.di $$\chi_{2352}(137, \cdot)$$ None 0 12
2352.1.dj $$\chi_{2352}(47, \cdot)$$ 2352.1.dj.a 12 12
2352.1.dj.b 12
2352.1.dk $$\chi_{2352}(61, \cdot)$$ None 0 24
2352.1.dm $$\chi_{2352}(53, \cdot)$$ None 0 24
2352.1.do $$\chi_{2352}(59, \cdot)$$ None 0 24
2352.1.dq $$\chi_{2352}(163, \cdot)$$ None 0 24

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2352))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2352)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(392))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(588))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(784))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1176))$$$$^{\oplus 2}$$