## Defining parameters

 Level: $$N$$ = $$9 = 3^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$18$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(9))$$.

Total New Old
Modular forms 10 6 4
Cusp forms 2 2 0
Eisenstein series 8 4 4

## Trace form

 $$2q - 3q^{2} - 3q^{3} - q^{4} + 6q^{5} + 9q^{6} - 2q^{7} - 9q^{9} + O(q^{10})$$ $$2q - 3q^{2} - 3q^{3} - q^{4} + 6q^{5} + 9q^{6} - 2q^{7} - 9q^{9} - 12q^{10} - 3q^{11} + 6q^{12} + 4q^{13} + 6q^{14} + 11q^{16} + 22q^{19} - 6q^{20} - 6q^{21} + 3q^{22} - 48q^{23} - 45q^{24} - 13q^{25} + 54q^{27} + 4q^{28} + 78q^{29} + 18q^{30} - 32q^{31} + 27q^{32} + 9q^{33} - 27q^{34} - 9q^{36} - 68q^{37} - 33q^{38} - 24q^{39} + 30q^{40} - 21q^{41} + 61q^{43} - 54q^{45} + 96q^{46} - 84q^{47} + 33q^{48} + 45q^{49} + 39q^{50} + 81q^{51} + 4q^{52} - 81q^{54} - 12q^{55} - 30q^{56} - 33q^{57} - 78q^{58} + 87q^{59} + 18q^{60} - 56q^{61} + 36q^{63} - 142q^{64} + 24q^{65} - 18q^{66} + 31q^{67} - 27q^{68} + 12q^{70} + 135q^{72} + 130q^{73} + 102q^{74} - 39q^{75} - 11q^{76} + 6q^{77} + 36q^{78} - 38q^{79} - 81q^{81} + 42q^{82} - 84q^{83} - 6q^{84} - 54q^{85} - 183q^{86} - 234q^{87} + 15q^{88} + 54q^{90} - 16q^{91} + 48q^{92} + 192q^{93} + 84q^{94} + 66q^{95} + 115q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$

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
9.3.b $$\chi_{9}(8, \cdot)$$ None 0 1
9.3.d $$\chi_{9}(2, \cdot)$$ 9.3.d.a 2 2