## Defining parameters

 Level: $$N$$ = $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$14$$ Sturm bound: $$75264$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1176))$$.

Total New Old
Modular forms 1630 274 1356
Cusp forms 190 80 110
Eisenstein series 1440 194 1246

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 80 0 0 0

## Trace form

 $$80 q + 2 q^{4} - 10 q^{6} + 2 q^{9} + O(q^{10})$$ $$80 q + 2 q^{4} - 10 q^{6} + 2 q^{9} - 8 q^{10} - 20 q^{15} + 2 q^{16} - 8 q^{22} + 2 q^{24} - 6 q^{25} - 8 q^{31} - 8 q^{33} + 2 q^{36} + 34 q^{40} + 2 q^{54} - 16 q^{55} - 12 q^{57} - 8 q^{58} + 34 q^{60} - 6 q^{63} - 34 q^{64} - 6 q^{70} + 4 q^{73} + 12 q^{78} - 8 q^{79} + 2 q^{81} - 8 q^{87} - 8 q^{88} + 4 q^{90} + 2 q^{96} + 4 q^{97} - 24 q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1176))$$

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
1176.1.d $$\chi_{1176}(785, \cdot)$$ None 0 1
1176.1.e $$\chi_{1176}(587, \cdot)$$ 1176.1.e.a 8 1
1176.1.f $$\chi_{1176}(97, \cdot)$$ None 0 1
1176.1.g $$\chi_{1176}(883, \cdot)$$ None 0 1
1176.1.l $$\chi_{1176}(685, \cdot)$$ None 0 1
1176.1.m $$\chi_{1176}(295, \cdot)$$ None 0 1
1176.1.n $$\chi_{1176}(197, \cdot)$$ 1176.1.n.a 1 1
1176.1.n.b 1
1176.1.n.c 1
1176.1.n.d 1
1176.1.n.e 4
1176.1.o $$\chi_{1176}(1175, \cdot)$$ None 0 1
1176.1.r $$\chi_{1176}(215, \cdot)$$ None 0 2
1176.1.s $$\chi_{1176}(557, \cdot)$$ 1176.1.s.a 2 2
1176.1.s.b 2
1176.1.s.c 8
1176.1.w $$\chi_{1176}(79, \cdot)$$ None 0 2
1176.1.x $$\chi_{1176}(325, \cdot)$$ None 0 2
1176.1.y $$\chi_{1176}(67, \cdot)$$ None 0 2
1176.1.z $$\chi_{1176}(313, \cdot)$$ None 0 2
1176.1.be $$\chi_{1176}(227, \cdot)$$ 1176.1.be.a 16 2
1176.1.bf $$\chi_{1176}(569, \cdot)$$ None 0 2
1176.1.bi $$\chi_{1176}(167, \cdot)$$ None 0 6
1176.1.bj $$\chi_{1176}(29, \cdot)$$ 1176.1.bj.a 6 6
1176.1.bj.b 6
1176.1.bk $$\chi_{1176}(127, \cdot)$$ None 0 6
1176.1.bl $$\chi_{1176}(13, \cdot)$$ None 0 6
1176.1.bq $$\chi_{1176}(43, \cdot)$$ None 0 6
1176.1.br $$\chi_{1176}(265, \cdot)$$ None 0 6
1176.1.bs $$\chi_{1176}(83, \cdot)$$ None 0 6
1176.1.bt $$\chi_{1176}(113, \cdot)$$ None 0 6
1176.1.bx $$\chi_{1176}(65, \cdot)$$ None 0 12
1176.1.by $$\chi_{1176}(59, \cdot)$$ None 0 12
1176.1.cd $$\chi_{1176}(73, \cdot)$$ None 0 12
1176.1.ce $$\chi_{1176}(163, \cdot)$$ None 0 12
1176.1.cf $$\chi_{1176}(61, \cdot)$$ None 0 12
1176.1.cg $$\chi_{1176}(151, \cdot)$$ None 0 12
1176.1.ck $$\chi_{1176}(53, \cdot)$$ 1176.1.ck.a 12 12
1176.1.ck.b 12
1176.1.cl $$\chi_{1176}(47, \cdot)$$ None 0 12

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1176))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1176)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(392))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(588))$$$$^{\oplus 2}$$