## Defining parameters

 Level: $$N$$ = $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$5$$ Newform subspaces: $$7$$ Sturm bound: $$92160$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1216))$$.

Total New Old
Modular forms 1400 402 998
Cusp forms 104 28 76
Eisenstein series 1296 374 922

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 24 4 0 0

## Trace form

 $$28 q - 2 q^{9} + O(q^{10})$$ $$28 q - 2 q^{9} + 2 q^{13} - 8 q^{17} - 8 q^{25} + 2 q^{29} - 2 q^{41} + 2 q^{45} + 12 q^{49} - 2 q^{53} - 2 q^{57} - 4 q^{65} + 4 q^{69} - 14 q^{73} - 2 q^{77} - 10 q^{81} - 4 q^{85} - 2 q^{89} + 4 q^{93} - 2 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1216))$$

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
1216.1.d $$\chi_{1216}(191, \cdot)$$ None 0 1
1216.1.e $$\chi_{1216}(1025, \cdot)$$ 1216.1.e.a 1 1
1216.1.e.b 1
1216.1.f $$\chi_{1216}(799, \cdot)$$ None 0 1
1216.1.g $$\chi_{1216}(417, \cdot)$$ 1216.1.g.a 2 1
1216.1.g.b 4
1216.1.j $$\chi_{1216}(113, \cdot)$$ None 0 2
1216.1.l $$\chi_{1216}(495, \cdot)$$ None 0 2
1216.1.o $$\chi_{1216}(159, \cdot)$$ None 0 2
1216.1.p $$\chi_{1216}(673, \cdot)$$ 1216.1.p.a 4 2
1216.1.q $$\chi_{1216}(767, \cdot)$$ 1216.1.q.a 4 2
1216.1.r $$\chi_{1216}(65, \cdot)$$ None 0 2
1216.1.w $$\chi_{1216}(265, \cdot)$$ None 0 4
1216.1.x $$\chi_{1216}(39, \cdot)$$ None 0 4
1216.1.ba $$\chi_{1216}(145, \cdot)$$ None 0 4
1216.1.bc $$\chi_{1216}(239, \cdot)$$ None 0 4
1216.1.bf $$\chi_{1216}(115, \cdot)$$ None 0 8
1216.1.bg $$\chi_{1216}(37, \cdot)$$ None 0 8
1216.1.bh $$\chi_{1216}(129, \cdot)$$ None 0 6
1216.1.bi $$\chi_{1216}(33, \cdot)$$ 1216.1.bi.a 12 6
1216.1.bk $$\chi_{1216}(351, \cdot)$$ None 0 6
1216.1.bn $$\chi_{1216}(63, \cdot)$$ None 0 6
1216.1.bo $$\chi_{1216}(7, \cdot)$$ None 0 8
1216.1.bp $$\chi_{1216}(217, \cdot)$$ None 0 8
1216.1.bt $$\chi_{1216}(47, \cdot)$$ None 0 12
1216.1.bv $$\chi_{1216}(241, \cdot)$$ None 0 12
1216.1.by $$\chi_{1216}(69, \cdot)$$ None 0 16
1216.1.bz $$\chi_{1216}(11, \cdot)$$ None 0 16
1216.1.cc $$\chi_{1216}(23, \cdot)$$ None 0 24
1216.1.cd $$\chi_{1216}(41, \cdot)$$ None 0 24
1216.1.ce $$\chi_{1216}(35, \cdot)$$ None 0 48
1216.1.cf $$\chi_{1216}(13, \cdot)$$ None 0 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1216))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1216)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(304))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(608))$$$$^{\oplus 2}$$