## Defining parameters

 Level: $$N$$ = $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$9$$ Newform subspaces: $$20$$ Sturm bound: $$5184$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(216))$$.

Total New Old
Modular forms 1476 608 868
Cusp forms 1117 544 573
Eisenstein series 359 64 295

## Trace form

 $$544 q - 8 q^{2} - 12 q^{3} - 14 q^{4} - 2 q^{5} - 12 q^{6} - 14 q^{7} - 2 q^{8} - 24 q^{9} + O(q^{10})$$ $$544 q - 8 q^{2} - 12 q^{3} - 14 q^{4} - 2 q^{5} - 12 q^{6} - 14 q^{7} - 2 q^{8} - 24 q^{9} - 6 q^{10} + 4 q^{11} - 12 q^{12} + 10 q^{13} - 2 q^{14} - 10 q^{16} + 10 q^{17} - 12 q^{18} + 2 q^{19} - 14 q^{20} + 12 q^{21} - 34 q^{22} + 18 q^{23} - 30 q^{24} - 17 q^{25} - 50 q^{26} - 27 q^{27} - 76 q^{28} - 30 q^{29} - 60 q^{30} - 34 q^{31} - 118 q^{32} - 33 q^{33} - 62 q^{34} - 90 q^{35} - 72 q^{36} - 30 q^{37} - 118 q^{38} - 48 q^{39} - 74 q^{40} - 38 q^{41} - 72 q^{42} - 50 q^{43} - 102 q^{44} - 54 q^{45} - 50 q^{46} - 84 q^{47} - 30 q^{48} + 5 q^{49} - 68 q^{50} - 72 q^{51} - 26 q^{52} - 20 q^{53} - 12 q^{54} - 58 q^{55} + 70 q^{56} - 69 q^{57} + 26 q^{58} - 95 q^{59} + 48 q^{60} - 14 q^{61} + 122 q^{62} - 108 q^{63} + 46 q^{64} - 82 q^{65} + 78 q^{66} - 70 q^{67} + 150 q^{68} - 48 q^{69} + 110 q^{70} - 130 q^{71} + 72 q^{72} - 42 q^{73} + 154 q^{74} - 78 q^{75} + 110 q^{76} - 24 q^{77} + 102 q^{78} - 58 q^{79} + 174 q^{80} - 36 q^{81} + 8 q^{82} - 70 q^{83} + 54 q^{84} - 4 q^{85} + 106 q^{86} - 18 q^{87} - 10 q^{88} + 25 q^{89} + 126 q^{90} - 36 q^{91} + 54 q^{92} + 72 q^{93} - 42 q^{94} + 40 q^{95} + 144 q^{96} - 42 q^{97} + 114 q^{98} + 126 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(216))$$

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
216.2.a $$\chi_{216}(1, \cdot)$$ 216.2.a.a 1 1
216.2.a.b 1
216.2.a.c 1
216.2.a.d 1
216.2.c $$\chi_{216}(215, \cdot)$$ None 0 1
216.2.d $$\chi_{216}(109, \cdot)$$ 216.2.d.a 4 1
216.2.d.b 4
216.2.d.c 8
216.2.f $$\chi_{216}(107, \cdot)$$ 216.2.f.a 8 1
216.2.f.b 8
216.2.i $$\chi_{216}(73, \cdot)$$ 216.2.i.a 2 2
216.2.i.b 4
216.2.l $$\chi_{216}(35, \cdot)$$ 216.2.l.a 4 2
216.2.l.b 16
216.2.n $$\chi_{216}(37, \cdot)$$ 216.2.n.a 4 2
216.2.n.b 16
216.2.o $$\chi_{216}(71, \cdot)$$ None 0 2
216.2.q $$\chi_{216}(25, \cdot)$$ 216.2.q.a 24 6
216.2.q.b 30
216.2.t $$\chi_{216}(13, \cdot)$$ 216.2.t.a 204 6
216.2.v $$\chi_{216}(11, \cdot)$$ 216.2.v.a 12 6
216.2.v.b 192
216.2.w $$\chi_{216}(23, \cdot)$$ None 0 6

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(216))$$ into lower level spaces

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