## Defining parameters

 Level: $$N$$ = $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ Sturm bound: $$2592$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(216))$$.

Total New Old
Modular forms 196 42 154
Cusp forms 16 10 6
Eisenstein series 180 32 148

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

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

## Trace form

 $$10q + q^{2} + q^{4} - 2q^{7} - 5q^{8} + O(q^{10})$$ $$10q + q^{2} + q^{4} - 2q^{7} - 5q^{8} - 2q^{10} - 4q^{11} - 3q^{12} + q^{16} + 2q^{17} + 6q^{18} - 2q^{19} - 4q^{22} - q^{25} - 3q^{27} - 2q^{28} - 2q^{31} + q^{32} - 3q^{33} - 2q^{34} + 5q^{38} - 2q^{40} - 4q^{41} - 2q^{43} + 2q^{44} - q^{49} + q^{50} + 6q^{51} + 2q^{55} - 3q^{57} + 4q^{58} + 5q^{59} + q^{64} - 2q^{67} + 5q^{68} + 2q^{70} - 4q^{73} + 7q^{76} + 4q^{79} - 2q^{82} + 2q^{83} - 4q^{86} + 5q^{88} - q^{89} + 6q^{96} - 4q^{97} - 5q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(216))$$

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
216.1.b $$\chi_{216}(163, \cdot)$$ None 0 1
216.1.e $$\chi_{216}(161, \cdot)$$ None 0 1
216.1.g $$\chi_{216}(55, \cdot)$$ None 0 1
216.1.h $$\chi_{216}(53, \cdot)$$ 216.1.h.a 1 1
216.1.h.b 1
216.1.j $$\chi_{216}(125, \cdot)$$ None 0 2
216.1.k $$\chi_{216}(127, \cdot)$$ None 0 2
216.1.m $$\chi_{216}(17, \cdot)$$ None 0 2
216.1.p $$\chi_{216}(19, \cdot)$$ 216.1.p.a 2 2
216.1.r $$\chi_{216}(43, \cdot)$$ 216.1.r.a 6 6
216.1.s $$\chi_{216}(7, \cdot)$$ None 0 6
216.1.u $$\chi_{216}(41, \cdot)$$ None 0 6
216.1.x $$\chi_{216}(5, \cdot)$$ None 0 6

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(216))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(216)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 2}$$