## Defining parameters

 Level: $$N$$ = $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$127008$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1323))$$.

Total New Old
Modular forms 1890 808 1082
Cusp forms 90 32 58
Eisenstein series 1800 776 1024

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

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

## Trace form

 $$32 q + O(q^{10})$$ $$32 q + 4 q^{13} + 4 q^{16} + 4 q^{19} + 4 q^{25} + 2 q^{28} + 4 q^{31} - 4 q^{49} - 8 q^{52} - 8 q^{61} - 4 q^{67} + 4 q^{73} - 8 q^{76} - 4 q^{79} + 2 q^{91} + 4 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1323))$$

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
1323.1.b $$\chi_{1323}(1079, \cdot)$$ 1323.1.b.a 1 1
1323.1.b.b 1
1323.1.d $$\chi_{1323}(244, \cdot)$$ 1323.1.d.a 2 1
1323.1.j $$\chi_{1323}(116, \cdot)$$ None 0 2
1323.1.k $$\chi_{1323}(19, \cdot)$$ None 0 2
1323.1.l $$\chi_{1323}(685, \cdot)$$ None 0 2
1323.1.m $$\chi_{1323}(325, \cdot)$$ 1323.1.m.a 2 2
1323.1.n $$\chi_{1323}(557, \cdot)$$ None 0 2
1323.1.q $$\chi_{1323}(863, \cdot)$$ 1323.1.q.a 2 2
1323.1.r $$\chi_{1323}(197, \cdot)$$ None 0 2
1323.1.t $$\chi_{1323}(901, \cdot)$$ None 0 2
1323.1.y $$\chi_{1323}(55, \cdot)$$ None 0 6
1323.1.ba $$\chi_{1323}(134, \cdot)$$ None 0 6
1323.1.bb $$\chi_{1323}(166, \cdot)$$ None 0 6
1323.1.bc $$\chi_{1323}(97, \cdot)$$ None 0 6
1323.1.bd $$\chi_{1323}(31, \cdot)$$ None 0 6
1323.1.bf $$\chi_{1323}(50, \cdot)$$ None 0 6
1323.1.bg $$\chi_{1323}(128, \cdot)$$ None 0 6
1323.1.bj $$\chi_{1323}(263, \cdot)$$ None 0 6
1323.1.bo $$\chi_{1323}(73, \cdot)$$ None 0 12
1323.1.bq $$\chi_{1323}(8, \cdot)$$ None 0 12
1323.1.br $$\chi_{1323}(53, \cdot)$$ 1323.1.br.a 12 12
1323.1.bu $$\chi_{1323}(170, \cdot)$$ None 0 12
1323.1.bv $$\chi_{1323}(82, \cdot)$$ 1323.1.bv.a 12 12
1323.1.bw $$\chi_{1323}(118, \cdot)$$ None 0 12
1323.1.bx $$\chi_{1323}(10, \cdot)$$ None 0 12
1323.1.by $$\chi_{1323}(44, \cdot)$$ None 0 12
1323.1.cd $$\chi_{1323}(11, \cdot)$$ None 0 36
1323.1.cg $$\chi_{1323}(2, \cdot)$$ None 0 36
1323.1.ch $$\chi_{1323}(29, \cdot)$$ None 0 36
1323.1.cj $$\chi_{1323}(61, \cdot)$$ None 0 36
1323.1.ck $$\chi_{1323}(13, \cdot)$$ None 0 36
1323.1.cl $$\chi_{1323}(40, \cdot)$$ None 0 36

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1323))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1323)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(189))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(441))$$$$^{\oplus 2}$$