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Properties

Label 688.5
Level 688
Weight 5
Dimension 33043
Nonzero newspaces 16
Sturm bound 147840
Trace bound 2

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Defining parameters

Level: \( N \) = \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 16 \)
Sturm bound: \(147840\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(688))\).

Total New Old
Modular forms 59724 33413 26311
Cusp forms 58548 33043 25505
Eisenstein series 1176 370 806

Trace form

\( 33043 q - 80 q^{2} - 59 q^{3} - 68 q^{4} - 173 q^{5} - 212 q^{6} - 55 q^{7} + 100 q^{8} + 423 q^{9} + O(q^{10}) \) \( 33043 q - 80 q^{2} - 59 q^{3} - 68 q^{4} - 173 q^{5} - 212 q^{6} - 55 q^{7} + 100 q^{8} + 423 q^{9} + 116 q^{10} - 251 q^{11} + 580 q^{12} - 813 q^{13} - 172 q^{14} - 63 q^{15} + 252 q^{16} + 323 q^{17} - 2864 q^{18} + 1349 q^{19} - 3884 q^{20} - 1313 q^{21} - 1884 q^{22} - 2359 q^{23} + 3660 q^{24} + 1183 q^{25} + 6748 q^{26} + 3265 q^{27} + 7484 q^{28} + 3859 q^{29} + 7396 q^{30} - 63 q^{31} - 6500 q^{32} - 7093 q^{33} - 15100 q^{34} - 2743 q^{35} - 23020 q^{36} + 1427 q^{37} - 7220 q^{38} - 5431 q^{39} + 10204 q^{40} - 669 q^{41} + 34044 q^{42} - 1757 q^{43} + 29104 q^{44} + 5067 q^{45} + 10548 q^{46} - 63 q^{47} - 13860 q^{48} - 8085 q^{49} - 40224 q^{50} + 5961 q^{51} - 40988 q^{52} - 1517 q^{53} - 21652 q^{54} + 23497 q^{55} + 13820 q^{56} + 22251 q^{57} + 40828 q^{58} + 5509 q^{59} + 59756 q^{60} + 4947 q^{61} + 22860 q^{62} - 63 q^{63} - 31700 q^{64} - 8965 q^{65} - 60380 q^{66} - 16059 q^{67} - 36148 q^{68} - 60833 q^{69} - 30676 q^{70} - 39991 q^{71} + 35332 q^{72} + 26723 q^{73} + 47476 q^{74} - 35203 q^{75} + 47908 q^{76} + 32735 q^{77} + 16020 q^{78} - 63 q^{79} - 2852 q^{80} + 9827 q^{81} - 32116 q^{82} + 34501 q^{83} - 39332 q^{84} - 10908 q^{85} + 4712 q^{86} + 98442 q^{87} - 14660 q^{88} - 32925 q^{89} + 10748 q^{90} + 56009 q^{91} + 29180 q^{92} + 535 q^{93} - 948 q^{94} - 63 q^{95} - 12212 q^{96} + 6211 q^{97} + 24408 q^{98} - 98491 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(688))\)

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
688.5.b \(\chi_{688}(257, \cdot)\) 688.5.b.a 1 1
688.5.b.b 2
688.5.b.c 12
688.5.b.d 12
688.5.b.e 16
688.5.b.f 44
688.5.d \(\chi_{688}(431, \cdot)\) 688.5.d.a 28 1
688.5.d.b 56
688.5.f \(\chi_{688}(87, \cdot)\) None 0 1
688.5.h \(\chi_{688}(601, \cdot)\) None 0 1
688.5.l \(\chi_{688}(259, \cdot)\) n/a 672 2
688.5.m \(\chi_{688}(85, \cdot)\) n/a 700 2
688.5.n \(\chi_{688}(265, \cdot)\) None 0 2
688.5.p \(\chi_{688}(135, \cdot)\) None 0 2
688.5.r \(\chi_{688}(79, \cdot)\) n/a 176 2
688.5.t \(\chi_{688}(209, \cdot)\) n/a 174 2
688.5.v \(\chi_{688}(37, \cdot)\) n/a 1400 4
688.5.w \(\chi_{688}(251, \cdot)\) n/a 1400 4
688.5.z \(\chi_{688}(137, \cdot)\) None 0 6
688.5.ba \(\chi_{688}(183, \cdot)\) None 0 6
688.5.bc \(\chi_{688}(47, \cdot)\) n/a 528 6
688.5.be \(\chi_{688}(65, \cdot)\) n/a 522 6
688.5.bj \(\chi_{688}(45, \cdot)\) n/a 4200 12
688.5.bk \(\chi_{688}(11, \cdot)\) n/a 4200 12
688.5.bm \(\chi_{688}(33, \cdot)\) n/a 1044 12
688.5.bo \(\chi_{688}(15, \cdot)\) n/a 1056 12
688.5.bq \(\chi_{688}(23, \cdot)\) None 0 12
688.5.br \(\chi_{688}(73, \cdot)\) None 0 12
688.5.bs \(\chi_{688}(67, \cdot)\) n/a 8400 24
688.5.bt \(\chi_{688}(5, \cdot)\) n/a 8400 24

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

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(688))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(688)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(43))\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(86))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(172))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(344))\)\(^{\oplus 2}\)