Properties

Label 1728.4
Level 1728
Weight 4
Dimension 110240
Nonzero newspaces 24
Sturm bound 663552
Trace bound 52

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Defining parameters

Level: \( N \) = \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 24 \)
Sturm bound: \(663552\)
Trace bound: \(52\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(1728))\).

Total New Old
Modular forms 250992 110944 140048
Cusp forms 246672 110240 136432
Eisenstein series 4320 704 3616

Trace form

\( 110240q - 64q^{2} - 72q^{3} - 112q^{4} - 64q^{5} - 96q^{6} - 84q^{7} - 64q^{8} - 120q^{9} + O(q^{10}) \) \( 110240q - 64q^{2} - 72q^{3} - 112q^{4} - 64q^{5} - 96q^{6} - 84q^{7} - 64q^{8} - 120q^{9} - 112q^{10} - 48q^{11} - 96q^{12} - 112q^{13} - 64q^{14} - 72q^{15} - 112q^{16} - 112q^{17} - 96q^{18} - 84q^{19} - 64q^{20} - 96q^{21} - 112q^{22} - 52q^{23} - 96q^{24} - 140q^{25} - 64q^{26} - 72q^{27} - 256q^{28} - 64q^{29} - 96q^{30} - 84q^{31} - 64q^{32} - 72q^{33} - 112q^{34} - 548q^{35} - 96q^{36} - 1120q^{37} - 64q^{38} - 72q^{39} - 112q^{40} + 80q^{41} - 96q^{42} + 780q^{43} - 64q^{44} - 96q^{45} - 112q^{46} + 1836q^{47} - 96q^{48} + 1244q^{49} - 64q^{50} - 72q^{51} - 112q^{52} + 752q^{53} - 96q^{54} - 480q^{55} - 64q^{56} - 120q^{57} - 112q^{58} - 2088q^{59} - 96q^{60} - 2272q^{61} + 64q^{62} - 72q^{63} - 112q^{64} - 676q^{65} - 96q^{66} - 84q^{67} - 64q^{68} - 96q^{69} - 112q^{70} - 28q^{71} - 96q^{72} - 140q^{73} - 64q^{74} - 72q^{75} - 112q^{76} + 2680q^{77} - 96q^{78} + 2748q^{79} + 18496q^{80} - 168q^{81} + 13664q^{82} + 2392q^{83} - 96q^{84} + 848q^{85} - 1104q^{86} - 72q^{87} - 6352q^{88} - 6864q^{89} - 96q^{90} - 3684q^{91} - 25296q^{92} - 96q^{93} - 17968q^{94} - 6108q^{95} - 96q^{96} - 11988q^{97} - 24272q^{98} - 72q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(1728))\)

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
1728.4.a \(\chi_{1728}(1, \cdot)\) 1728.4.a.a 1 1
1728.4.a.b 1
1728.4.a.c 1
1728.4.a.d 1
1728.4.a.e 1
1728.4.a.f 1
1728.4.a.g 1
1728.4.a.h 1
1728.4.a.i 1
1728.4.a.j 1
1728.4.a.k 1
1728.4.a.l 1
1728.4.a.m 1
1728.4.a.n 1
1728.4.a.o 1
1728.4.a.p 1
1728.4.a.q 1
1728.4.a.r 1
1728.4.a.s 1
1728.4.a.t 1
1728.4.a.u 1
1728.4.a.v 1
1728.4.a.w 1
1728.4.a.x 1
1728.4.a.y 1
1728.4.a.z 1
1728.4.a.ba 1
1728.4.a.bb 1
1728.4.a.bc 1
1728.4.a.bd 1
1728.4.a.be 1
1728.4.a.bf 1
1728.4.a.bg 2
1728.4.a.bh 2
1728.4.a.bi 2
1728.4.a.bj 2
1728.4.a.bk 2
1728.4.a.bl 2
1728.4.a.bm 2
1728.4.a.bn 2
1728.4.a.bo 2
1728.4.a.bp 2
1728.4.a.bq 2
1728.4.a.br 2
1728.4.a.bs 2
1728.4.a.bt 2
1728.4.a.bu 3
1728.4.a.bv 3
1728.4.a.bw 3
1728.4.a.bx 3
1728.4.a.by 3
1728.4.a.bz 3
1728.4.a.ca 3
1728.4.a.cb 3
1728.4.a.cc 3
1728.4.a.cd 3
1728.4.a.ce 3
1728.4.a.cf 3
1728.4.c \(\chi_{1728}(1727, \cdot)\) 1728.4.c.a 2 1
1728.4.c.b 2
1728.4.c.c 2
1728.4.c.d 2
1728.4.c.e 4
1728.4.c.f 4
1728.4.c.g 4
1728.4.c.h 4
1728.4.c.i 12
1728.4.c.j 12
1728.4.c.k 24
1728.4.c.l 24
1728.4.d \(\chi_{1728}(865, \cdot)\) 1728.4.d.a 4 1
1728.4.d.b 4
1728.4.d.c 4
1728.4.d.d 4
1728.4.d.e 8
1728.4.d.f 8
1728.4.d.g 8
1728.4.d.h 8
1728.4.d.i 16
1728.4.d.j 16
1728.4.d.k 16
1728.4.f \(\chi_{1728}(863, \cdot)\) 1728.4.f.a 4 1
1728.4.f.b 4
1728.4.f.c 8
1728.4.f.d 8
1728.4.f.e 8
1728.4.f.f 8
1728.4.f.g 8
1728.4.f.h 16
1728.4.f.i 16
1728.4.f.j 16
1728.4.i \(\chi_{1728}(577, \cdot)\) n/a 140 2
1728.4.k \(\chi_{1728}(433, \cdot)\) n/a 192 2
1728.4.l \(\chi_{1728}(431, \cdot)\) n/a 192 2
1728.4.p \(\chi_{1728}(287, \cdot)\) n/a 144 2
1728.4.r \(\chi_{1728}(289, \cdot)\) n/a 144 2
1728.4.s \(\chi_{1728}(575, \cdot)\) n/a 140 2
1728.4.v \(\chi_{1728}(217, \cdot)\) None 0 4
1728.4.w \(\chi_{1728}(215, \cdot)\) None 0 4
1728.4.y \(\chi_{1728}(193, \cdot)\) n/a 1284 6
1728.4.z \(\chi_{1728}(143, \cdot)\) n/a 280 4
1728.4.bc \(\chi_{1728}(145, \cdot)\) n/a 280 4
1728.4.be \(\chi_{1728}(109, \cdot)\) n/a 3072 8
1728.4.bf \(\chi_{1728}(107, \cdot)\) n/a 3072 8
1728.4.bj \(\chi_{1728}(97, \cdot)\) n/a 1296 6
1728.4.bl \(\chi_{1728}(95, \cdot)\) n/a 1296 6
1728.4.bm \(\chi_{1728}(191, \cdot)\) n/a 1284 6
1728.4.bo \(\chi_{1728}(73, \cdot)\) None 0 8
1728.4.br \(\chi_{1728}(71, \cdot)\) None 0 8
1728.4.bs \(\chi_{1728}(49, \cdot)\) n/a 2568 12
1728.4.bv \(\chi_{1728}(47, \cdot)\) n/a 2568 12
1728.4.bx \(\chi_{1728}(35, \cdot)\) n/a 4576 16
1728.4.by \(\chi_{1728}(37, \cdot)\) n/a 4576 16
1728.4.cb \(\chi_{1728}(23, \cdot)\) None 0 24
1728.4.cc \(\chi_{1728}(25, \cdot)\) None 0 24
1728.4.ce \(\chi_{1728}(13, \cdot)\) n/a 41376 48
1728.4.ch \(\chi_{1728}(11, \cdot)\) n/a 41376 48

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(1728))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(1728)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 14}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(144))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(288))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(432))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(576))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(864))\)\(^{\oplus 2}\)