## 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

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

## 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 the 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

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