Properties

Label 1764.4
Level 1764
Weight 4
Dimension 112131
Nonzero newspaces 40
Sturm bound 677376
Trace bound 13

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Defining parameters

Level: \( N \) = \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 40 \)
Sturm bound: \(677376\)
Trace bound: \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(1764))\).

Total New Old
Modular forms 256416 113003 143413
Cusp forms 251616 112131 139485
Eisenstein series 4800 872 3928

Trace form

\( 112131q - 48q^{2} - 3q^{3} - 34q^{4} - 96q^{5} - 39q^{6} - 24q^{7} - 183q^{8} - 75q^{9} + O(q^{10}) \) \( 112131q - 48q^{2} - 3q^{3} - 34q^{4} - 96q^{5} - 39q^{6} - 24q^{7} - 183q^{8} - 75q^{9} - 269q^{10} + 183q^{11} - 66q^{12} - 18q^{13} + 102q^{14} - 420q^{15} + 386q^{16} + 114q^{17} - 180q^{18} + 302q^{19} - 261q^{20} + 216q^{21} - 810q^{22} + 978q^{23} - 57q^{24} + 1078q^{25} - 651q^{26} - 144q^{27} - 810q^{28} - 114q^{29} + 1878q^{30} + 1952q^{31} + 4842q^{32} + 369q^{33} + 2098q^{34} - 426q^{35} - 405q^{36} - 5424q^{37} - 2880q^{38} - 2556q^{39} - 6065q^{40} - 5451q^{41} - 2778q^{42} - 1519q^{43} - 5127q^{44} - 1716q^{45} - 2337q^{46} + 1014q^{47} - 1311q^{48} + 3090q^{49} + 2409q^{50} + 6921q^{51} + 2113q^{52} + 6612q^{53} + 3723q^{54} + 3078q^{55} + 522q^{56} - 3531q^{57} - 5165q^{58} - 6207q^{59} - 918q^{60} + 3744q^{61} - 63q^{62} - 5208q^{63} + 2777q^{64} + 2166q^{65} + 786q^{66} + 8351q^{67} - 1878q^{68} + 6636q^{69} - 2217q^{70} + 6960q^{71} - 1053q^{72} + 5346q^{73} + 7299q^{74} + 11409q^{75} + 7230q^{76} - 162q^{77} + 11280q^{78} - 8008q^{79} + 31428q^{80} - 18615q^{81} + 23368q^{82} - 5544q^{83} + 8892q^{84} - 18562q^{85} + 15519q^{86} - 13770q^{87} + 6987q^{88} - 36264q^{89} - 1914q^{90} - 7866q^{91} - 8238q^{92} - 3240q^{93} - 13524q^{94} - 7680q^{95} - 16224q^{96} + 19839q^{97} - 35580q^{98} + 12786q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(1764))\)

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
1764.4.a \(\chi_{1764}(1, \cdot)\) 1764.4.a.a 1 1
1764.4.a.b 1
1764.4.a.c 1
1764.4.a.d 1
1764.4.a.e 1
1764.4.a.f 1
1764.4.a.g 1
1764.4.a.h 1
1764.4.a.i 1
1764.4.a.j 1
1764.4.a.k 1
1764.4.a.l 1
1764.4.a.m 1
1764.4.a.n 2
1764.4.a.o 2
1764.4.a.p 2
1764.4.a.q 2
1764.4.a.r 2
1764.4.a.s 2
1764.4.a.t 2
1764.4.a.u 2
1764.4.a.v 2
1764.4.a.w 2
1764.4.a.x 2
1764.4.a.y 2
1764.4.a.z 2
1764.4.a.ba 4
1764.4.a.bb 4
1764.4.a.bc 4
1764.4.b \(\chi_{1764}(1567, \cdot)\) n/a 296 1
1764.4.e \(\chi_{1764}(1079, \cdot)\) n/a 246 1
1764.4.f \(\chi_{1764}(881, \cdot)\) 1764.4.f.a 16 1
1764.4.f.b 24
1764.4.i \(\chi_{1764}(373, \cdot)\) n/a 240 2
1764.4.j \(\chi_{1764}(589, \cdot)\) n/a 246 2
1764.4.k \(\chi_{1764}(361, \cdot)\) 1764.4.k.a 2 2
1764.4.k.b 2
1764.4.k.c 2
1764.4.k.d 2
1764.4.k.e 2
1764.4.k.f 2
1764.4.k.g 2
1764.4.k.h 2
1764.4.k.i 2
1764.4.k.j 2
1764.4.k.k 2
1764.4.k.l 2
1764.4.k.m 2
1764.4.k.n 2
1764.4.k.o 2
1764.4.k.p 2
1764.4.k.q 4
1764.4.k.r 4
1764.4.k.s 4
1764.4.k.t 4
1764.4.k.u 4
1764.4.k.v 4
1764.4.k.w 4
1764.4.k.x 4
1764.4.k.y 4
1764.4.k.z 4
1764.4.k.ba 4
1764.4.k.bb 8
1764.4.k.bc 8
1764.4.k.bd 8
1764.4.l \(\chi_{1764}(949, \cdot)\) n/a 240 2
1764.4.n \(\chi_{1764}(31, \cdot)\) n/a 1424 2
1764.4.o \(\chi_{1764}(851, \cdot)\) n/a 1424 2
1764.4.t \(\chi_{1764}(521, \cdot)\) 1764.4.t.a 16 2
1764.4.t.b 16
1764.4.t.c 48
1764.4.w \(\chi_{1764}(509, \cdot)\) n/a 240 2
1764.4.x \(\chi_{1764}(293, \cdot)\) n/a 240 2
1764.4.ba \(\chi_{1764}(491, \cdot)\) n/a 1456 2
1764.4.bb \(\chi_{1764}(263, \cdot)\) n/a 1424 2
1764.4.be \(\chi_{1764}(863, \cdot)\) n/a 480 2
1764.4.bf \(\chi_{1764}(19, \cdot)\) n/a 592 2
1764.4.bi \(\chi_{1764}(391, \cdot)\) n/a 1424 2
1764.4.bj \(\chi_{1764}(607, \cdot)\) n/a 1424 2
1764.4.bm \(\chi_{1764}(1685, \cdot)\) n/a 240 2
1764.4.bo \(\chi_{1764}(253, \cdot)\) n/a 420 6
1764.4.br \(\chi_{1764}(125, \cdot)\) n/a 336 6
1764.4.bs \(\chi_{1764}(71, \cdot)\) n/a 2016 6
1764.4.bv \(\chi_{1764}(55, \cdot)\) n/a 2508 6
1764.4.bw \(\chi_{1764}(193, \cdot)\) n/a 2016 12
1764.4.bx \(\chi_{1764}(37, \cdot)\) n/a 840 12
1764.4.by \(\chi_{1764}(85, \cdot)\) n/a 2016 12
1764.4.bz \(\chi_{1764}(25, \cdot)\) n/a 2016 12
1764.4.cb \(\chi_{1764}(173, \cdot)\) n/a 2016 12
1764.4.ce \(\chi_{1764}(103, \cdot)\) n/a 12048 12
1764.4.cf \(\chi_{1764}(139, \cdot)\) n/a 12048 12
1764.4.ci \(\chi_{1764}(199, \cdot)\) n/a 5016 12
1764.4.cj \(\chi_{1764}(107, \cdot)\) n/a 4032 12
1764.4.cm \(\chi_{1764}(11, \cdot)\) n/a 12048 12
1764.4.cn \(\chi_{1764}(155, \cdot)\) n/a 12048 12
1764.4.cq \(\chi_{1764}(41, \cdot)\) n/a 2016 12
1764.4.cr \(\chi_{1764}(5, \cdot)\) n/a 2016 12
1764.4.cu \(\chi_{1764}(17, \cdot)\) n/a 672 12
1764.4.cz \(\chi_{1764}(95, \cdot)\) n/a 12048 12
1764.4.da \(\chi_{1764}(187, \cdot)\) n/a 12048 12

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(1764))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(1764)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(147))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(294))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(441))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(588))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(882))\)\(^{\oplus 2}\)