# Properties

 Label 648.4 Level 648 Weight 4 Dimension 15440 Nonzero newspaces 12 Sturm bound 93312 Trace bound 7

## Defining parameters

 Level: $$N$$ = $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$93312$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(648))$$.

Total New Old
Modular forms 35640 15664 19976
Cusp forms 34344 15440 18904
Eisenstein series 1296 224 1072

## Trace form

 $$15440 q - 24 q^{2} - 36 q^{3} - 40 q^{4} - 36 q^{6} - 40 q^{7} - 24 q^{8} - 72 q^{9} + O(q^{10})$$ $$15440 q - 24 q^{2} - 36 q^{3} - 40 q^{4} - 36 q^{6} - 40 q^{7} - 24 q^{8} - 72 q^{9} - 58 q^{10} - 99 q^{11} - 36 q^{12} - 18 q^{13} - 24 q^{14} - 36 q^{15} - 40 q^{16} + 156 q^{17} - 36 q^{18} + 32 q^{19} - 24 q^{20} - 8 q^{22} - 180 q^{23} - 36 q^{24} - 314 q^{25} + 966 q^{26} - 36 q^{27} + 446 q^{28} + 126 q^{29} - 36 q^{30} - 310 q^{31} - 1404 q^{32} - 72 q^{33} - 1228 q^{34} - 1278 q^{35} - 36 q^{36} - 594 q^{37} - 1764 q^{38} - 36 q^{39} - 580 q^{40} + 2997 q^{41} - 36 q^{42} + 1535 q^{43} + 1752 q^{44} + 1026 q^{45} + 1710 q^{46} + 732 q^{47} - 36 q^{48} - 1466 q^{49} + 726 q^{50} - 2997 q^{51} + 216 q^{52} - 4770 q^{53} - 36 q^{54} - 3230 q^{55} - 2082 q^{56} - 2286 q^{57} - 2704 q^{58} + 621 q^{59} - 36 q^{60} + 828 q^{61} - 1278 q^{62} + 1962 q^{63} + 482 q^{64} + 6018 q^{65} - 36 q^{66} + 221 q^{67} + 4578 q^{68} + 3366 q^{69} + 4816 q^{70} - 5994 q^{71} - 36 q^{72} - 2456 q^{73} + 5484 q^{74} - 36 q^{75} + 3632 q^{76} - 3084 q^{77} - 522 q^{78} - 1138 q^{79} - 114 q^{80} - 72 q^{81} - 1516 q^{82} + 5046 q^{83} - 36 q^{84} + 3240 q^{85} - 8364 q^{86} - 36 q^{87} - 12032 q^{88} - 5304 q^{89} - 25110 q^{90} - 8478 q^{91} - 39924 q^{92} - 5562 q^{93} - 9522 q^{94} - 10734 q^{95} - 5886 q^{96} + 325 q^{97} + 9666 q^{98} + 5634 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(648))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
648.4.a $$\chi_{648}(1, \cdot)$$ 648.4.a.a 1 1
648.4.a.b 1
648.4.a.c 2
648.4.a.d 2
648.4.a.e 2
648.4.a.f 2
648.4.a.g 4
648.4.a.h 4
648.4.a.i 4
648.4.a.j 4
648.4.a.k 5
648.4.a.l 5
648.4.c $$\chi_{648}(647, \cdot)$$ None 0 1
648.4.d $$\chi_{648}(325, \cdot)$$ n/a 140 1
648.4.f $$\chi_{648}(323, \cdot)$$ n/a 140 1
648.4.i $$\chi_{648}(217, \cdot)$$ 648.4.i.a 2 2
648.4.i.b 2
648.4.i.c 2
648.4.i.d 2
648.4.i.e 2
648.4.i.f 2
648.4.i.g 2
648.4.i.h 2
648.4.i.i 2
648.4.i.j 2
648.4.i.k 2
648.4.i.l 2
648.4.i.m 4
648.4.i.n 4
648.4.i.o 4
648.4.i.p 4
648.4.i.q 4
648.4.i.r 4
648.4.i.s 4
648.4.i.t 4
648.4.i.u 8
648.4.i.v 8
648.4.l $$\chi_{648}(107, \cdot)$$ n/a 284 2
648.4.n $$\chi_{648}(109, \cdot)$$ n/a 284 2
648.4.o $$\chi_{648}(215, \cdot)$$ None 0 2
648.4.q $$\chi_{648}(73, \cdot)$$ n/a 162 6
648.4.t $$\chi_{648}(37, \cdot)$$ n/a 636 6
648.4.v $$\chi_{648}(35, \cdot)$$ n/a 636 6
648.4.w $$\chi_{648}(71, \cdot)$$ None 0 6
648.4.y $$\chi_{648}(25, \cdot)$$ n/a 1458 18
648.4.bb $$\chi_{648}(11, \cdot)$$ n/a 5796 18
648.4.bd $$\chi_{648}(13, \cdot)$$ n/a 5796 18
648.4.be $$\chi_{648}(23, \cdot)$$ None 0 18

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(648))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(648)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 20}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 15}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(216))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(648))$$$$^{\oplus 1}$$