## Defining parameters

 Level: $$N$$ = $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$66$$ Sturm bound: $$46656$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(648))$$.

Total New Old
Modular forms 12312 5296 7016
Cusp forms 11017 5072 5945
Eisenstein series 1295 224 1071

## Trace form

 $$5072 q - 24 q^{2} - 36 q^{3} - 40 q^{4} - 36 q^{6} - 40 q^{7} - 24 q^{8} - 72 q^{9} + O(q^{10})$$ $$5072 q - 24 q^{2} - 36 q^{3} - 40 q^{4} - 36 q^{6} - 40 q^{7} - 24 q^{8} - 72 q^{9} - 58 q^{10} - 27 q^{11} - 36 q^{12} - 6 q^{13} - 24 q^{14} - 36 q^{15} - 40 q^{16} - 60 q^{17} - 36 q^{18} - 64 q^{19} - 24 q^{20} - 32 q^{22} - 36 q^{23} - 36 q^{24} - 86 q^{25} - 6 q^{26} - 36 q^{27} - 34 q^{28} + 18 q^{29} - 36 q^{30} - 22 q^{31} + 36 q^{32} - 72 q^{33} - 4 q^{34} + 18 q^{35} - 36 q^{36} + 18 q^{37} + 36 q^{38} - 36 q^{39} - 4 q^{40} - 9 q^{41} - 36 q^{42} - q^{43} + 24 q^{44} + 54 q^{45} - 42 q^{46} + 84 q^{47} - 36 q^{48} - 38 q^{49} + 6 q^{50} + 27 q^{51} - 24 q^{52} + 90 q^{53} - 36 q^{54} + 34 q^{55} - 66 q^{56} - 18 q^{57} - 64 q^{58} + 117 q^{59} - 36 q^{60} + 60 q^{61} - 90 q^{62} + 18 q^{63} - 94 q^{64} + 78 q^{65} - 36 q^{66} + 29 q^{67} - 102 q^{68} + 18 q^{69} - 140 q^{70} + 54 q^{71} - 36 q^{72} - 104 q^{73} - 96 q^{74} - 36 q^{75} - 112 q^{76} + 12 q^{77} - 90 q^{78} - 10 q^{79} - 114 q^{80} - 72 q^{81} - 148 q^{82} + 6 q^{83} - 36 q^{84} - 84 q^{86} - 36 q^{87} - 176 q^{88} - 120 q^{89} - 162 q^{90} - 126 q^{91} - 324 q^{92} - 54 q^{93} - 174 q^{94} - 294 q^{95} - 270 q^{96} - 161 q^{97} - 594 q^{98} - 198 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(648))$$

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
648.2.a $$\chi_{648}(1, \cdot)$$ 648.2.a.a 1 1
648.2.a.b 1
648.2.a.c 1
648.2.a.d 1
648.2.a.e 2
648.2.a.f 2
648.2.a.g 2
648.2.a.h 2
648.2.c $$\chi_{648}(647, \cdot)$$ None 0 1
648.2.d $$\chi_{648}(325, \cdot)$$ 648.2.d.a 2 1
648.2.d.b 2
648.2.d.c 2
648.2.d.d 2
648.2.d.e 4
648.2.d.f 4
648.2.d.g 4
648.2.d.h 4
648.2.d.i 4
648.2.d.j 8
648.2.d.k 8
648.2.f $$\chi_{648}(323, \cdot)$$ 648.2.f.a 4 1
648.2.f.b 16
648.2.f.c 24
648.2.i $$\chi_{648}(217, \cdot)$$ 648.2.i.a 2 2
648.2.i.b 2
648.2.i.c 2
648.2.i.d 2
648.2.i.e 2
648.2.i.f 2
648.2.i.g 2
648.2.i.h 2
648.2.i.i 4
648.2.i.j 4
648.2.l $$\chi_{648}(107, \cdot)$$ 648.2.l.a 4 2
648.2.l.b 4
648.2.l.c 4
648.2.l.d 8
648.2.l.e 8
648.2.l.f 16
648.2.l.g 48
648.2.n $$\chi_{648}(109, \cdot)$$ 648.2.n.a 4 2
648.2.n.b 4
648.2.n.c 4
648.2.n.d 4
648.2.n.e 4
648.2.n.f 4
648.2.n.g 4
648.2.n.h 4
648.2.n.i 4
648.2.n.j 4
648.2.n.k 4
648.2.n.l 4
648.2.n.m 4
648.2.n.n 8
648.2.n.o 8
648.2.n.p 8
648.2.n.q 16
648.2.o $$\chi_{648}(215, \cdot)$$ None 0 2
648.2.q $$\chi_{648}(73, \cdot)$$ 648.2.q.a 24 6
648.2.q.b 30
648.2.t $$\chi_{648}(37, \cdot)$$ 648.2.t.a 204 6
648.2.v $$\chi_{648}(35, \cdot)$$ 648.2.v.a 12 6
648.2.v.b 192
648.2.w $$\chi_{648}(71, \cdot)$$ None 0 6
648.2.y $$\chi_{648}(25, \cdot)$$ 648.2.y.a 234 18
648.2.y.b 252
648.2.bb $$\chi_{648}(11, \cdot)$$ 648.2.bb.a 36 18
648.2.bb.b 1872
648.2.bd $$\chi_{648}(13, \cdot)$$ 648.2.bd.a 1908 18
648.2.be $$\chi_{648}(23, \cdot)$$ None 0 18

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(648))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(648)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 2}$$