## Defining parameters

 Level: $$N$$ = $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$8$$ Newform subspaces: $$10$$ Sturm bound: $$165888$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1728))$$.

Total New Old
Modular forms 2380 392 1988
Cusp forms 220 40 180
Eisenstein series 2160 352 1808

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 32 4 4 0

## Trace form

 $$40q + 2q^{5} + O(q^{10})$$ $$40q + 2q^{5} - 2q^{13} - 6q^{25} - 2q^{29} - 4q^{31} + 6q^{33} + 4q^{37} + 14q^{41} + 4q^{43} + 6q^{49} - 6q^{57} + 6q^{61} - 2q^{65} - 4q^{67} - 2q^{77} - 4q^{85} - 18q^{89} + 4q^{91} + 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1728))$$

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
1728.1.b $$\chi_{1728}(1567, \cdot)$$ 1728.1.b.a 4 1
1728.1.b.b 4
1728.1.e $$\chi_{1728}(1025, \cdot)$$ 1728.1.e.a 1 1
1728.1.e.b 1
1728.1.g $$\chi_{1728}(703, \cdot)$$ 1728.1.g.a 2 1
1728.1.h $$\chi_{1728}(161, \cdot)$$ 1728.1.h.a 4 1
1728.1.j $$\chi_{1728}(593, \cdot)$$ 1728.1.j.a 4 2
1728.1.m $$\chi_{1728}(271, \cdot)$$ None 0 2
1728.1.n $$\chi_{1728}(737, \cdot)$$ 1728.1.n.a 4 2
1728.1.o $$\chi_{1728}(127, \cdot)$$ 1728.1.o.a 4 2
1728.1.q $$\chi_{1728}(449, \cdot)$$ None 0 2
1728.1.t $$\chi_{1728}(415, \cdot)$$ None 0 2
1728.1.u $$\chi_{1728}(55, \cdot)$$ None 0 4
1728.1.x $$\chi_{1728}(377, \cdot)$$ None 0 4
1728.1.ba $$\chi_{1728}(559, \cdot)$$ None 0 4
1728.1.bb $$\chi_{1728}(17, \cdot)$$ None 0 4
1728.1.bd $$\chi_{1728}(53, \cdot)$$ None 0 8
1728.1.bg $$\chi_{1728}(163, \cdot)$$ None 0 8
1728.1.bh $$\chi_{1728}(31, \cdot)$$ None 0 6
1728.1.bi $$\chi_{1728}(319, \cdot)$$ None 0 6
1728.1.bk $$\chi_{1728}(65, \cdot)$$ None 0 6
1728.1.bn $$\chi_{1728}(353, \cdot)$$ 1728.1.bn.a 12 6
1728.1.bp $$\chi_{1728}(199, \cdot)$$ None 0 8
1728.1.bq $$\chi_{1728}(89, \cdot)$$ None 0 8
1728.1.bt $$\chi_{1728}(79, \cdot)$$ None 0 12
1728.1.bu $$\chi_{1728}(113, \cdot)$$ None 0 12
1728.1.bw $$\chi_{1728}(19, \cdot)$$ None 0 16
1728.1.bz $$\chi_{1728}(125, \cdot)$$ None 0 16
1728.1.ca $$\chi_{1728}(41, \cdot)$$ None 0 24
1728.1.cd $$\chi_{1728}(7, \cdot)$$ None 0 24
1728.1.cf $$\chi_{1728}(5, \cdot)$$ None 0 48
1728.1.cg $$\chi_{1728}(43, \cdot)$$ None 0 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1728))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1728)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(192))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(432))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(576))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(864))$$$$^{\oplus 2}$$