Properties

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

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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

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