# Properties

 Label 1216.1 Level 1216 Weight 1 Dimension 28 Nonzero newspaces 5 Newform subspaces 7 Sturm bound 92160 Trace bound 5

## 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 available 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

$$S_{1}^{\mathrm{old}}(\Gamma_1(1216)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 14}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 10}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 7}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(304))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(608))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1216))$$$$^{\oplus 1}$$