## Defining parameters

 Level: $$N$$ = $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$41472$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(864))$$.

Total New Old
Modular forms 1054 174 880
Cusp forms 94 14 80
Eisenstein series 960 160 800

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 10 4 0 0

## Trace form

 $$14q - 2q^{5} + 2q^{7} + O(q^{10})$$ $$14q - 2q^{5} + 2q^{7} + 4q^{11} - 2q^{13} + 2q^{17} + 2q^{19} - q^{25} + 3q^{27} + 2q^{29} + 2q^{31} - 3q^{33} - 2q^{41} + 2q^{43} - q^{49} - 6q^{51} - 2q^{55} - 3q^{57} - 5q^{59} - 2q^{61} - 2q^{65} + 2q^{67} - 4q^{73} + 2q^{77} - 4q^{79} - 2q^{83} - q^{89} - 6q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(864))$$

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
864.1.b $$\chi_{864}(271, \cdot)$$ None 0 1
864.1.e $$\chi_{864}(161, \cdot)$$ None 0 1
864.1.g $$\chi_{864}(703, \cdot)$$ None 0 1
864.1.h $$\chi_{864}(593, \cdot)$$ 864.1.h.a 1 1
864.1.h.b 1
864.1.j $$\chi_{864}(377, \cdot)$$ None 0 2
864.1.m $$\chi_{864}(55, \cdot)$$ None 0 2
864.1.n $$\chi_{864}(17, \cdot)$$ None 0 2
864.1.o $$\chi_{864}(127, \cdot)$$ 864.1.o.a 4 2
864.1.q $$\chi_{864}(449, \cdot)$$ None 0 2
864.1.t $$\chi_{864}(559, \cdot)$$ 864.1.t.a 2 2
864.1.u $$\chi_{864}(163, \cdot)$$ None 0 4
864.1.x $$\chi_{864}(53, \cdot)$$ None 0 4
864.1.ba $$\chi_{864}(199, \cdot)$$ None 0 4
864.1.bb $$\chi_{864}(89, \cdot)$$ None 0 4
864.1.bd $$\chi_{864}(79, \cdot)$$ 864.1.bd.a 6 6
864.1.be $$\chi_{864}(31, \cdot)$$ None 0 6
864.1.bg $$\chi_{864}(65, \cdot)$$ None 0 6
864.1.bj $$\chi_{864}(113, \cdot)$$ None 0 6
864.1.bl $$\chi_{864}(19, \cdot)$$ None 0 8
864.1.bm $$\chi_{864}(125, \cdot)$$ None 0 8
864.1.bp $$\chi_{864}(7, \cdot)$$ None 0 12
864.1.bq $$\chi_{864}(41, \cdot)$$ None 0 12
864.1.bs $$\chi_{864}(5, \cdot)$$ None 0 24
864.1.bv $$\chi_{864}(43, \cdot)$$ None 0 24

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(864))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(864)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(432))$$$$^{\oplus 2}$$