# Properties

 Label 216.4 Level 216 Weight 4 Dimension 1696 Nonzero newspaces 9 Newform subspaces 24 Sturm bound 10368 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$9$$ Newform subspaces: $$24$$ Sturm bound: $$10368$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(216))$$.

Total New Old
Modular forms 4068 1760 2308
Cusp forms 3708 1696 2012
Eisenstein series 360 64 296

## Trace form

 $$1696q - 8q^{2} - 12q^{3} - 14q^{4} + 10q^{5} - 12q^{6} + 10q^{7} - 2q^{8} - 24q^{9} + O(q^{10})$$ $$1696q - 8q^{2} - 12q^{3} - 14q^{4} + 10q^{5} - 12q^{6} + 10q^{7} - 2q^{8} - 24q^{9} - 86q^{10} + 28q^{11} - 12q^{12} + 6q^{13} - 74q^{14} - 144q^{15} - 74q^{16} - 242q^{17} - 12q^{18} - 38q^{19} + 274q^{20} - 60q^{21} + 206q^{22} - 78q^{23} - 174q^{24} + 269q^{25} - 1370q^{26} - 459q^{27} - 668q^{28} + 6q^{29} + 228q^{30} + 550q^{31} + 1802q^{32} + 237q^{33} + 1090q^{34} + 2070q^{35} + 2088q^{36} + 894q^{37} + 2306q^{38} + 816q^{39} + 502q^{40} - 698q^{41} - 432q^{42} + 454q^{43} - 2070q^{44} - 1674q^{45} - 1170q^{46} - 1068q^{47} - 2190q^{48} - 841q^{49} + 412q^{50} + 1800q^{51} + 102q^{52} + 340q^{53} - 12q^{54} - 1810q^{55} + 1606q^{56} + 201q^{57} + 1162q^{58} - 6383q^{59} + 3648q^{60} - 450q^{61} - 502q^{62} - 1404q^{63} - 1490q^{64} + 26q^{65} - 3702q^{66} + 4738q^{67} - 5898q^{68} + 96q^{69} - 4066q^{70} + 12254q^{71} - 5976q^{72} + 3962q^{73} - 6878q^{74} + 7122q^{75} - 4706q^{76} + 4728q^{77} - 1950q^{78} - 1058q^{79} - 4626q^{80} - 1116q^{81} - 2312q^{82} - 11182q^{83} + 5598q^{84} - 5532q^{85} + 11266q^{86} - 8586q^{87} + 3638q^{88} - 6509q^{89} + 21078q^{90} - 3900q^{91} + 13350q^{92} + 936q^{93} + 198q^{94} + 2248q^{95} + 3888q^{96} + 74q^{97} + 2526q^{98} - 3762q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(216))$$

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
216.4.a $$\chi_{216}(1, \cdot)$$ 216.4.a.a 1 1
216.4.a.b 1
216.4.a.c 1
216.4.a.d 1
216.4.a.e 2
216.4.a.f 2
216.4.a.g 2
216.4.a.h 2
216.4.c $$\chi_{216}(215, \cdot)$$ None 0 1
216.4.d $$\chi_{216}(109, \cdot)$$ 216.4.d.a 4 1
216.4.d.b 4
216.4.d.c 16
216.4.d.d 24
216.4.f $$\chi_{216}(107, \cdot)$$ 216.4.f.a 24 1
216.4.f.b 24
216.4.i $$\chi_{216}(73, \cdot)$$ 216.4.i.a 8 2
216.4.i.b 10
216.4.l $$\chi_{216}(35, \cdot)$$ 216.4.l.a 4 2
216.4.l.b 64
216.4.n $$\chi_{216}(37, \cdot)$$ 216.4.n.a 68 2
216.4.o $$\chi_{216}(71, \cdot)$$ None 0 2
216.4.q $$\chi_{216}(25, \cdot)$$ 216.4.q.a 78 6
216.4.q.b 84
216.4.t $$\chi_{216}(13, \cdot)$$ 216.4.t.a 636 6
216.4.v $$\chi_{216}(11, \cdot)$$ 216.4.v.a 12 6
216.4.v.b 624
216.4.w $$\chi_{216}(23, \cdot)$$ None 0 6

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(216))$$ into lower level spaces

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