## Defining parameters

 Level: $$N$$ = $$9 = 3^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ 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

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

## 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