↑

Properties

Label 688.3
Level 688
Weight 3
Dimension 16429
Nonzero newspaces 16
Sturm bound 88704
Trace bound 2

Downloads

Learn more

Defining parameters

Level: \( N \) = \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(88704\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(688))\).

Total New Old
Modular forms 30156 16799 13357
Cusp forms 28980 16429 12551
Eisenstein series 1176 370 806

Trace form

\( 16429 q - 80 q^{2} - 59 q^{3} - 68 q^{4} - 89 q^{5} - 68 q^{6} - 55 q^{7} - 92 q^{8} - 39 q^{9} + O(q^{10}) \) \( 16429 q - 80 q^{2} - 59 q^{3} - 68 q^{4} - 89 q^{5} - 68 q^{6} - 55 q^{7} - 92 q^{8} - 39 q^{9} - 156 q^{10} - 27 q^{11} - 188 q^{12} - 121 q^{13} - 108 q^{14} - 63 q^{15} - 4 q^{16} - 121 q^{17} + 64 q^{18} - 123 q^{19} + 84 q^{20} - 65 q^{21} + 20 q^{22} - 183 q^{23} - 180 q^{24} - 43 q^{25} - 276 q^{26} - 191 q^{27} - 196 q^{28} - 153 q^{29} - 188 q^{30} - 63 q^{31} - 100 q^{32} - 181 q^{33} + 68 q^{34} + 137 q^{35} + 20 q^{36} - 57 q^{37} - 164 q^{38} + 329 q^{39} - 164 q^{40} - 57 q^{41} - 36 q^{42} + 51 q^{43} - 208 q^{44} - 129 q^{45} - 140 q^{46} - 63 q^{47} - 36 q^{48} - 195 q^{49} - 176 q^{50} - 375 q^{51} - 284 q^{52} - 441 q^{53} - 148 q^{54} - 567 q^{55} + 252 q^{56} - 21 q^{57} + 268 q^{58} - 475 q^{59} + 236 q^{60} - 121 q^{61} + 204 q^{62} - 63 q^{63} - 212 q^{64} - 93 q^{65} - 476 q^{66} + 389 q^{67} - 308 q^{68} + 127 q^{69} - 52 q^{70} + 457 q^{71} - 188 q^{72} + 199 q^{73} + 100 q^{74} + 413 q^{75} + 292 q^{76} + 319 q^{77} + 84 q^{78} - 63 q^{79} - 548 q^{80} - 523 q^{81} - 692 q^{82} - 699 q^{83} - 548 q^{84} - 20 q^{85} - 352 q^{86} - 1014 q^{87} - 68 q^{88} + 135 q^{89} + 236 q^{90} - 439 q^{91} + 252 q^{92} - 41 q^{93} - 180 q^{94} - 63 q^{95} + 76 q^{96} - 441 q^{97} - 104 q^{98} + 389 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(688))\)

We only show spaces with odd 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
688.3.b \(\chi_{688}(257, \cdot)\) 688.3.b.a 1 1
688.3.b.b 2
688.3.b.c 2
688.3.b.d 4
688.3.b.e 6
688.3.b.f 6
688.3.b.g 22
688.3.d \(\chi_{688}(431, \cdot)\) 688.3.d.a 14 1
688.3.d.b 28
688.3.f \(\chi_{688}(87, \cdot)\) None 0 1
688.3.h \(\chi_{688}(601, \cdot)\) None 0 1
688.3.l \(\chi_{688}(259, \cdot)\) n/a 336 2
688.3.m \(\chi_{688}(85, \cdot)\) n/a 348 2
688.3.n \(\chi_{688}(265, \cdot)\) None 0 2
688.3.p \(\chi_{688}(135, \cdot)\) None 0 2
688.3.r \(\chi_{688}(79, \cdot)\) 688.3.r.a 4 2
688.3.r.b 4
688.3.r.c 26
688.3.r.d 26
688.3.r.e 28
688.3.t \(\chi_{688}(209, \cdot)\) 688.3.t.a 2 2
688.3.t.b 12
688.3.t.c 12
688.3.t.d 16
688.3.t.e 44
688.3.v \(\chi_{688}(37, \cdot)\) n/a 696 4
688.3.w \(\chi_{688}(251, \cdot)\) n/a 696 4
688.3.z \(\chi_{688}(137, \cdot)\) None 0 6
688.3.ba \(\chi_{688}(183, \cdot)\) None 0 6
688.3.bc \(\chi_{688}(47, \cdot)\) n/a 264 6
688.3.be \(\chi_{688}(65, \cdot)\) n/a 258 6
688.3.bj \(\chi_{688}(45, \cdot)\) n/a 2088 12
688.3.bk \(\chi_{688}(11, \cdot)\) n/a 2088 12
688.3.bm \(\chi_{688}(33, \cdot)\) n/a 516 12
688.3.bo \(\chi_{688}(15, \cdot)\) n/a 528 12
688.3.bq \(\chi_{688}(23, \cdot)\) None 0 12
688.3.br \(\chi_{688}(73, \cdot)\) None 0 12
688.3.bs \(\chi_{688}(67, \cdot)\) n/a 4176 24
688.3.bt \(\chi_{688}(5, \cdot)\) n/a 4176 24

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

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(688))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(688)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(43))\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(86))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(172))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(344))\)\(^{\oplus 2}\)