## Defining parameters

 Level: $$N$$ = $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$8$$ Newform subspaces: $$13$$ Sturm bound: $$2160$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(99))$$.

Total New Old
Modular forms 800 603 197
Cusp forms 640 521 119
Eisenstein series 160 82 78

## Trace form

 $$521 q - 9 q^{2} - 14 q^{3} - 13 q^{4} - 27 q^{5} - 38 q^{6} + 4 q^{7} + 5 q^{8} - 2 q^{9} + O(q^{10})$$ $$521 q - 9 q^{2} - 14 q^{3} - 13 q^{4} - 27 q^{5} - 38 q^{6} + 4 q^{7} + 5 q^{8} - 2 q^{9} - 16 q^{10} - 17 q^{11} - 52 q^{12} - 38 q^{13} - 32 q^{14} - 20 q^{15} + 3 q^{16} - 10 q^{17} - 20 q^{18} - 59 q^{19} - 38 q^{20} - 8 q^{21} - 48 q^{22} - 89 q^{23} - 170 q^{24} - 279 q^{25} - 500 q^{26} - 278 q^{27} - 588 q^{28} - 386 q^{29} - 176 q^{30} - 81 q^{31} - 84 q^{32} + 21 q^{33} + 324 q^{34} + 370 q^{35} + 278 q^{36} + 361 q^{37} + 716 q^{38} + 238 q^{39} + 830 q^{40} + 492 q^{41} + 340 q^{42} + 308 q^{43} + 690 q^{44} + 198 q^{45} - 202 q^{46} + 178 q^{47} + 74 q^{48} - 160 q^{49} - 633 q^{50} - 322 q^{51} - 948 q^{52} - 920 q^{53} - 738 q^{54} - 893 q^{55} - 1660 q^{56} - 594 q^{57} - 934 q^{58} - 894 q^{59} - 756 q^{60} - 208 q^{61} - 710 q^{62} - 252 q^{63} - 161 q^{64} - 78 q^{65} + 128 q^{66} + 43 q^{67} + 644 q^{68} + 300 q^{69} + 1126 q^{70} + 965 q^{71} + 760 q^{72} + 640 q^{73} + 2036 q^{74} + 758 q^{75} + 1772 q^{76} + 1704 q^{77} + 1188 q^{78} + 1256 q^{79} + 2930 q^{80} + 1462 q^{81} + 301 q^{82} + 1193 q^{83} + 2722 q^{84} - 162 q^{85} + 1291 q^{86} + 1548 q^{87} - 770 q^{88} + 455 q^{89} + 1472 q^{90} - 928 q^{91} + 1114 q^{92} + 96 q^{93} - 378 q^{94} - 2 q^{95} + 530 q^{96} - 350 q^{97} - 180 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(99))$$

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
99.3.b $$\chi_{99}(89, \cdot)$$ 99.3.b.a 8 1
99.3.c $$\chi_{99}(10, \cdot)$$ 99.3.c.a 1 1
99.3.c.b 4
99.3.c.c 4
99.3.h $$\chi_{99}(43, \cdot)$$ 99.3.h.a 4 2
99.3.h.b 40
99.3.i $$\chi_{99}(23, \cdot)$$ 99.3.i.a 40 2
99.3.k $$\chi_{99}(19, \cdot)$$ 99.3.k.a 4 4
99.3.k.b 16
99.3.k.c 16
99.3.l $$\chi_{99}(26, \cdot)$$ 99.3.l.a 32 4
99.3.n $$\chi_{99}(5, \cdot)$$ 99.3.n.a 176 8
99.3.o $$\chi_{99}(7, \cdot)$$ 99.3.o.a 176 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(99))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(99)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$