## Defining parameters

 Level: $$N$$ = $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$4$$ Newform subspaces: $$6$$ Sturm bound: $$23328$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(648))$$.

Total New Old
Modular forms 724 142 582
Cusp forms 76 30 46
Eisenstein series 648 112 536

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 30 0 0 0

## Trace form

 $$30q + 2q^{7} + 3q^{8} + O(q^{10})$$ $$30q + 2q^{7} + 3q^{8} - 4q^{10} + 3q^{11} - 9q^{18} - 2q^{19} - 3q^{22} + 2q^{25} - 4q^{28} + 2q^{31} - 5q^{34} - 15q^{38} + 2q^{40} + 3q^{41} - 5q^{43} + 2q^{49} - 9q^{51} + 4q^{55} - 4q^{58} - 15q^{59} + 3q^{64} - 5q^{67} + 12q^{68} - 2q^{70} - 6q^{73} - 5q^{76} - 4q^{79} - 2q^{82} + 3q^{86} - 3q^{88} - 12q^{89} + 18q^{96} - 3q^{97} + 3q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(648))$$

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
648.1.b $$\chi_{648}(163, \cdot)$$ 648.1.b.a 1 1
648.1.b.b 1
648.1.e $$\chi_{648}(161, \cdot)$$ None 0 1
648.1.g $$\chi_{648}(487, \cdot)$$ None 0 1
648.1.h $$\chi_{648}(485, \cdot)$$ None 0 1
648.1.j $$\chi_{648}(53, \cdot)$$ 648.1.j.a 2 2
648.1.j.b 2
648.1.k $$\chi_{648}(55, \cdot)$$ None 0 2
648.1.m $$\chi_{648}(377, \cdot)$$ None 0 2
648.1.p $$\chi_{648}(379, \cdot)$$ None 0 2
648.1.r $$\chi_{648}(19, \cdot)$$ 648.1.r.a 6 6
648.1.s $$\chi_{648}(127, \cdot)$$ None 0 6
648.1.u $$\chi_{648}(17, \cdot)$$ None 0 6
648.1.x $$\chi_{648}(125, \cdot)$$ None 0 6
648.1.z $$\chi_{648}(5, \cdot)$$ None 0 18
648.1.ba $$\chi_{648}(7, \cdot)$$ None 0 18
648.1.bc $$\chi_{648}(41, \cdot)$$ None 0 18
648.1.bf $$\chi_{648}(43, \cdot)$$ 648.1.bf.a 18 18

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(648))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(648)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 2}$$