## Defining parameters

 Level: $$N$$ = $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$4$$ Newform subspaces: $$8$$ Sturm bound: $$31104$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(756))$$.

Total New Old
Modular forms 950 172 778
Cusp forms 50 12 38
Eisenstein series 900 160 740

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 12 0 0 0

## Trace form

 $$12q + 4q^{4} - q^{7} + O(q^{10})$$ $$12q + 4q^{4} - q^{7} + 4q^{13} + 4q^{16} + 4q^{19} - 4q^{22} - 4q^{25} - 2q^{31} - 2q^{43} - 4q^{46} + 3q^{49} - 2q^{61} + 4q^{64} - 2q^{67} - 4q^{70} - 2q^{73} - 8q^{79} - 14q^{85} - 4q^{88} - 5q^{91} - 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(756))$$

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
756.1.c $$\chi_{756}(701, \cdot)$$ None 0 1
756.1.d $$\chi_{756}(433, \cdot)$$ 756.1.d.a 2 1
756.1.d.b 2
756.1.g $$\chi_{756}(379, \cdot)$$ None 0 1
756.1.h $$\chi_{756}(755, \cdot)$$ 756.1.h.a 1 1
756.1.h.b 1
756.1.h.c 1
756.1.h.d 1
756.1.m $$\chi_{756}(557, \cdot)$$ None 0 2
756.1.p $$\chi_{756}(397, \cdot)$$ None 0 2
756.1.q $$\chi_{756}(215, \cdot)$$ None 0 2
756.1.r $$\chi_{756}(143, \cdot)$$ None 0 2
756.1.s $$\chi_{756}(251, \cdot)$$ None 0 2
756.1.u $$\chi_{756}(235, \cdot)$$ None 0 2
756.1.v $$\chi_{756}(127, \cdot)$$ None 0 2
756.1.y $$\chi_{756}(163, \cdot)$$ None 0 2
756.1.z $$\chi_{756}(325, \cdot)$$ 756.1.z.a 2 2
756.1.bc $$\chi_{756}(181, \cdot)$$ None 0 2
756.1.bd $$\chi_{756}(73, \cdot)$$ None 0 2
756.1.bg $$\chi_{756}(197, \cdot)$$ None 0 2
756.1.bh $$\chi_{756}(233, \cdot)$$ None 0 2
756.1.bk $$\chi_{756}(53, \cdot)$$ 756.1.bk.a 2 2
756.1.bl $$\chi_{756}(667, \cdot)$$ None 0 2
756.1.bn $$\chi_{756}(395, \cdot)$$ None 0 2
756.1.br $$\chi_{756}(229, \cdot)$$ None 0 6
756.1.bu $$\chi_{756}(137, \cdot)$$ None 0 6
756.1.bv $$\chi_{756}(83, \cdot)$$ None 0 6
756.1.bw $$\chi_{756}(47, \cdot)$$ None 0 6
756.1.by $$\chi_{756}(43, \cdot)$$ None 0 6
756.1.bz $$\chi_{756}(67, \cdot)$$ None 0 6
756.1.cb $$\chi_{756}(29, \cdot)$$ None 0 6
756.1.ce $$\chi_{756}(65, \cdot)$$ None 0 6
756.1.cg $$\chi_{756}(13, \cdot)$$ None 0 6
756.1.ch $$\chi_{756}(61, \cdot)$$ None 0 6
756.1.cj $$\chi_{756}(151, \cdot)$$ None 0 6
756.1.cl $$\chi_{756}(131, \cdot)$$ None 0 6

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(756))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(756)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(189))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(252))$$$$^{\oplus 2}$$