## Defining parameters

 Level: $$N$$ = $$9 = 3^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newforms: $$2$$ Sturm bound: $$24$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(9))$$.

Total New Old
Modular forms 13 10 3
Cusp forms 5 5 0
Eisenstein series 8 5 3

## Trace form

 $$5q - 3q^{2} - 3q^{3} - 13q^{4} - 15q^{5} + 9q^{6} + 13q^{7} + 66q^{8} + 45q^{9} + O(q^{10})$$ $$5q - 3q^{2} - 3q^{3} - 13q^{4} - 15q^{5} + 9q^{6} + 13q^{7} + 66q^{8} + 45q^{9} + 12q^{10} - 66q^{11} - 156q^{12} - 59q^{13} - 60q^{14} + 27q^{15} + 71q^{16} + 198q^{17} + 216q^{18} - 98q^{19} + 12q^{20} + 21q^{21} + 33q^{22} - 33q^{23} - 99q^{24} - 4q^{25} - 528q^{26} - 432q^{27} + 172q^{28} + 51q^{29} + 288q^{30} + 265q^{31} + 423q^{32} + 198q^{33} - 297q^{34} + 6q^{35} - 225q^{36} + 10q^{37} + 561q^{38} + 759q^{39} - 264q^{40} - 132q^{41} - 486q^{42} - 608q^{43} - 462q^{44} - 675q^{45} - 528q^{46} - 399q^{47} - 21q^{48} + 570q^{49} + 429q^{50} + 297q^{51} + 1330q^{52} + 108q^{53} + 1215q^{54} + 1254q^{55} - 66q^{56} - 1221q^{57} + 60q^{58} - 798q^{59} - 36q^{60} - 257q^{61} + 228q^{62} + 603q^{63} - 1966q^{64} - 165q^{65} - 990q^{66} - 1868q^{67} - 693q^{68} + 891q^{69} - 318q^{70} + 2736q^{71} + 891q^{72} + 280q^{73} - 816q^{74} - 363q^{75} + 1081q^{76} + 165q^{77} - 990q^{78} + 1687q^{79} + 192q^{80} - 567q^{81} + 3630q^{82} - 813q^{83} + 642q^{84} - 594q^{85} - 33q^{86} - 153q^{87} - 1221q^{88} - 792q^{89} - 756q^{90} - 2962q^{91} + 858q^{92} - 213q^{93} - 2100q^{94} + 132q^{95} + 1080q^{96} - 2066q^{97} - 846q^{98} + 297q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$

We only show spaces with even 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
9.4.a $$\chi_{9}(1, \cdot)$$ 9.4.a.a 1 1
9.4.c $$\chi_{9}(4, \cdot)$$ 9.4.c.a 4 2