## Defining parameters

 Level: $$N$$ = $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$87120$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(1089))$$.

Total New Old
Modular forms 1342 680 662
Cusp forms 62 48 14
Eisenstein series 1280 632 648

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 28 0 20 0

## Trace form

 $$48q + q^{3} + q^{4} - q^{5} + q^{9} + O(q^{10})$$ $$48q + q^{3} + q^{4} - q^{5} + q^{9} - 9q^{12} - q^{15} + q^{16} - q^{20} - 8q^{23} - 2q^{27} - q^{31} - 2q^{36} + 2q^{37} - 28q^{45} - q^{47} - 2q^{48} + q^{49} + 2q^{53} - q^{59} + 2q^{60} - 2q^{64} + 9q^{67} + 2q^{69} + 2q^{71} + 2q^{80} + q^{81} + 16q^{89} + 2q^{92} - q^{93} - q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(1089))$$

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
1089.1.b $$\chi_{1089}(485, \cdot)$$ None 0 1
1089.1.c $$\chi_{1089}(604, \cdot)$$ 1089.1.c.a 2 1
1089.1.h $$\chi_{1089}(241, \cdot)$$ 1089.1.h.a 4 2
1089.1.i $$\chi_{1089}(122, \cdot)$$ 1089.1.i.a 2 2
1089.1.k $$\chi_{1089}(118, \cdot)$$ 1089.1.k.a 8 4
1089.1.l $$\chi_{1089}(251, \cdot)$$ None 0 4
1089.1.p $$\chi_{1089}(10, \cdot)$$ None 0 10
1089.1.q $$\chi_{1089}(89, \cdot)$$ None 0 10
1089.1.r $$\chi_{1089}(245, \cdot)$$ 1089.1.r.a 8 8
1089.1.s $$\chi_{1089}(40, \cdot)$$ 1089.1.s.a 8 8
1089.1.s.b 16
1089.1.w $$\chi_{1089}(23, \cdot)$$ None 0 20
1089.1.x $$\chi_{1089}(43, \cdot)$$ None 0 20
1089.1.z $$\chi_{1089}(26, \cdot)$$ None 0 40
1089.1.ba $$\chi_{1089}(19, \cdot)$$ None 0 40
1089.1.be $$\chi_{1089}(7, \cdot)$$ None 0 80
1089.1.bf $$\chi_{1089}(5, \cdot)$$ None 0 80

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(1089))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(1089)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(99))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(363))$$$$^{\oplus 2}$$