## Defining parameters

 Level: $$N$$ = $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$9$$ Newform subspaces: $$20$$ Sturm bound: $$7776$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(216))$$.

Total New Old
Modular forms 2772 1184 1588
Cusp forms 2412 1120 1292
Eisenstein series 360 64 296

## Trace form

 $$1120q - 8q^{2} - 12q^{3} - 14q^{4} - 12q^{6} - 26q^{7} - 26q^{8} - 24q^{9} + O(q^{10})$$ $$1120q - 8q^{2} - 12q^{3} - 14q^{4} - 12q^{6} - 26q^{7} - 26q^{8} - 24q^{9} - 22q^{10} - 46q^{11} - 12q^{12} - 28q^{13} + 30q^{14} - 36q^{15} + 54q^{16} - 16q^{17} - 12q^{18} - 18q^{19} + 66q^{20} + 48q^{21} + 14q^{22} + 138q^{23} + 42q^{24} + 54q^{25} + 222q^{26} + 162q^{27} + 132q^{28} + 216q^{29} + 228q^{30} + 150q^{31} + 202q^{32} - 6q^{33} - 14q^{34} + 102q^{35} - 72q^{36} - 84q^{37} - 190q^{38} - 156q^{39} - 122q^{40} - 100q^{41} - 432q^{42} - 194q^{43} - 694q^{44} + 108q^{45} - 418q^{46} - 330q^{47} - 246q^{48} - 118q^{49} - 524q^{50} + 126q^{51} - 298q^{52} - 12q^{54} - 88q^{55} - 762q^{56} - 42q^{57} - 630q^{58} + 440q^{59} - 672q^{60} + 380q^{61} - 990q^{62} + 108q^{63} - 146q^{64} + 300q^{65} - 462q^{66} + 414q^{67} - 346q^{68} - 120q^{69} - 82q^{70} - 18q^{71} + 72q^{72} - 128q^{73} + 330q^{74} - 600q^{75} + 366q^{76} - 1008q^{77} + 426q^{78} - 738q^{79} + 1518q^{80} - 576q^{81} + 600q^{82} - 1330q^{83} + 846q^{84} - 392q^{85} + 1826q^{86} - 648q^{87} + 1046q^{88} - 406q^{89} - 954q^{90} + 150q^{91} - 810q^{92} - 360q^{93} + 726q^{94} - 594q^{95} - 1728q^{96} + 400q^{97} - 1058q^{98} - 576q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(216))$$

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
216.3.b $$\chi_{216}(163, \cdot)$$ 216.3.b.a 16 1
216.3.b.b 16
216.3.e $$\chi_{216}(161, \cdot)$$ 216.3.e.a 2 1
216.3.e.b 2
216.3.e.c 4
216.3.g $$\chi_{216}(55, \cdot)$$ None 0 1
216.3.h $$\chi_{216}(53, \cdot)$$ 216.3.h.a 2 1
216.3.h.b 2
216.3.h.c 6
216.3.h.d 6
216.3.h.e 8
216.3.h.f 8
216.3.j $$\chi_{216}(125, \cdot)$$ 216.3.j.a 44 2
216.3.k $$\chi_{216}(127, \cdot)$$ None 0 2
216.3.m $$\chi_{216}(17, \cdot)$$ 216.3.m.a 4 2
216.3.m.b 8
216.3.p $$\chi_{216}(19, \cdot)$$ 216.3.p.a 4 2
216.3.p.b 40
216.3.r $$\chi_{216}(43, \cdot)$$ 216.3.r.a 12 6
216.3.r.b 408
216.3.s $$\chi_{216}(7, \cdot)$$ None 0 6
216.3.u $$\chi_{216}(41, \cdot)$$ 216.3.u.a 108 6
216.3.x $$\chi_{216}(5, \cdot)$$ 216.3.x.a 420 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(216))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(216)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 2}$$