## Defining parameters

 Level: $$N$$ = $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$6$$ Newform subspaces: $$12$$ Sturm bound: $$1944$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(108))$$.

Total New Old
Modular forms 723 315 408
Cusp forms 573 283 290
Eisenstein series 150 32 118

## Trace form

 $$283 q - 5 q^{2} - 9 q^{4} - 28 q^{5} - 6 q^{6} - 4 q^{7} + 19 q^{8} - 6 q^{9} + O(q^{10})$$ $$283 q - 5 q^{2} - 9 q^{4} - 28 q^{5} - 6 q^{6} - 4 q^{7} + 19 q^{8} - 6 q^{9} + 15 q^{10} + 72 q^{11} + 39 q^{12} + 50 q^{13} + 63 q^{14} + 45 q^{15} + 3 q^{16} + 26 q^{17} - 27 q^{18} - 19 q^{19} - 121 q^{20} + 30 q^{21} - 117 q^{22} - 117 q^{23} - 138 q^{24} - 35 q^{25} - 334 q^{26} - 198 q^{28} + 131 q^{29} - 153 q^{30} + 62 q^{31} - 215 q^{32} - 105 q^{33} - 45 q^{34} - 243 q^{35} + 24 q^{36} - 55 q^{37} + 225 q^{38} - 123 q^{39} + 243 q^{40} - 430 q^{41} - 126 q^{42} - 22 q^{43} + 171 q^{44} - 465 q^{45} + 273 q^{46} - 162 q^{47} - 219 q^{48} - 111 q^{49} + 72 q^{50} - 99 q^{51} + 87 q^{52} - 160 q^{53} + 78 q^{54} - 72 q^{55} - 171 q^{56} + 93 q^{57} - 429 q^{58} + 126 q^{59} + 210 q^{60} - 214 q^{61} - 270 q^{62} + 381 q^{63} - 327 q^{64} + 274 q^{65} + 393 q^{66} + 239 q^{67} + 44 q^{68} + 525 q^{69} - 141 q^{70} + 324 q^{71} + 228 q^{72} + 290 q^{73} + 851 q^{74} + 597 q^{75} + 615 q^{76} + 1152 q^{77} + 750 q^{78} + 350 q^{79} + 1646 q^{80} + 174 q^{81} + 786 q^{82} + 54 q^{83} + 762 q^{84} + 1053 q^{86} - 441 q^{87} + 219 q^{88} - 289 q^{89} + 894 q^{90} - 146 q^{91} + 297 q^{92} - 1719 q^{93} - 609 q^{94} - 900 q^{95} + 474 q^{96} - 325 q^{97} - 512 q^{98} - 945 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(108))$$

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
108.3.c $$\chi_{108}(53, \cdot)$$ 108.3.c.a 1 1
108.3.c.b 2
108.3.d $$\chi_{108}(55, \cdot)$$ 108.3.d.a 2 1
108.3.d.b 2
108.3.d.c 4
108.3.d.d 8
108.3.f $$\chi_{108}(19, \cdot)$$ 108.3.f.a 2 2
108.3.f.b 2
108.3.f.c 16
108.3.g $$\chi_{108}(17, \cdot)$$ 108.3.g.a 4 2
108.3.j $$\chi_{108}(7, \cdot)$$ 108.3.j.a 204 6
108.3.k $$\chi_{108}(5, \cdot)$$ 108.3.k.a 36 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(108))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(108)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$