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Properties

Label 688.2
Level 688
Weight 2
Dimension 8123
Nonzero newspaces 16
Newform subspaces 58
Sturm bound 59136
Trace bound 2

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Defining parameters

Level: \( N \) = \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 16 \)
Newform subspaces: \( 58 \)
Sturm bound: \(59136\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(688))\).

Total New Old
Modular forms 15372 8491 6881
Cusp forms 14197 8123 6074
Eisenstein series 1175 368 807

Trace form

\( 8123 q - 80 q^{2} - 59 q^{3} - 84 q^{4} - 101 q^{5} - 92 q^{6} - 63 q^{7} - 92 q^{8} - 21 q^{9} + O(q^{10}) \) \( 8123 q - 80 q^{2} - 59 q^{3} - 84 q^{4} - 101 q^{5} - 92 q^{6} - 63 q^{7} - 92 q^{8} - 21 q^{9} - 84 q^{10} - 67 q^{11} - 76 q^{12} - 101 q^{13} - 76 q^{14} - 71 q^{15} - 68 q^{16} - 181 q^{17} - 88 q^{18} - 75 q^{19} - 92 q^{20} - 113 q^{21} - 84 q^{22} - 63 q^{23} - 84 q^{24} - 21 q^{25} - 92 q^{26} - 47 q^{27} - 100 q^{28} - 117 q^{29} - 76 q^{30} - 31 q^{31} - 100 q^{32} - 181 q^{33} - 92 q^{34} - 55 q^{35} - 76 q^{36} - 117 q^{37} - 60 q^{38} - 63 q^{39} - 68 q^{40} - 21 q^{41} - 84 q^{42} - 73 q^{43} - 160 q^{44} - 109 q^{45} - 108 q^{46} - 95 q^{47} - 100 q^{48} - 201 q^{49} - 72 q^{50} - 71 q^{51} - 76 q^{52} - 85 q^{53} - 84 q^{54} - 63 q^{55} - 68 q^{56} - 21 q^{57} - 60 q^{58} - 51 q^{59} - 84 q^{60} - 69 q^{61} - 116 q^{62} - 71 q^{63} - 84 q^{64} - 197 q^{65} - 92 q^{66} - 43 q^{67} - 84 q^{68} - 81 q^{69} - 100 q^{70} - 63 q^{71} - 92 q^{72} - 21 q^{73} - 84 q^{74} - 75 q^{75} - 108 q^{76} - 113 q^{77} - 76 q^{78} - 63 q^{79} - 100 q^{80} - 209 q^{81} - 84 q^{82} - 59 q^{83} - 68 q^{84} - 134 q^{85} - 84 q^{86} - 126 q^{87} - 100 q^{88} - 21 q^{89} - 76 q^{90} - 71 q^{91} - 36 q^{92} - 137 q^{93} - 52 q^{94} - 39 q^{95} - 52 q^{96} - 181 q^{97} - 72 q^{98} - 59 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(688))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
688.2.a \(\chi_{688}(1, \cdot)\) 688.2.a.a 1 1
688.2.a.b 1
688.2.a.c 1
688.2.a.d 2
688.2.a.e 2
688.2.a.f 2
688.2.a.g 2
688.2.a.h 2
688.2.a.i 3
688.2.a.j 5
688.2.c \(\chi_{688}(345, \cdot)\) None 0 1
688.2.e \(\chi_{688}(343, \cdot)\) None 0 1
688.2.g \(\chi_{688}(687, \cdot)\) 688.2.g.a 2 1
688.2.g.b 2
688.2.g.c 2
688.2.g.d 4
688.2.g.e 12
688.2.i \(\chi_{688}(49, \cdot)\) 688.2.i.a 2 2
688.2.i.b 2
688.2.i.c 2
688.2.i.d 2
688.2.i.e 4
688.2.i.f 4
688.2.i.g 8
688.2.i.h 8
688.2.i.i 10
688.2.j \(\chi_{688}(171, \cdot)\) 688.2.j.a 172 2
688.2.k \(\chi_{688}(173, \cdot)\) 688.2.k.a 168 2
688.2.o \(\chi_{688}(351, \cdot)\) 688.2.o.a 2 2
688.2.o.b 2
688.2.o.c 6
688.2.o.d 6
688.2.o.e 6
688.2.o.f 6
688.2.o.g 16
688.2.q \(\chi_{688}(7, \cdot)\) None 0 2
688.2.s \(\chi_{688}(393, \cdot)\) None 0 2
688.2.u \(\chi_{688}(97, \cdot)\) 688.2.u.a 6 6
688.2.u.b 6
688.2.u.c 6
688.2.u.d 12
688.2.u.e 12
688.2.u.f 18
688.2.u.g 30
688.2.u.h 36
688.2.x \(\chi_{688}(165, \cdot)\) 688.2.x.a 344 4
688.2.y \(\chi_{688}(123, \cdot)\) 688.2.y.a 344 4
688.2.bb \(\chi_{688}(223, \cdot)\) 688.2.bb.a 36 6
688.2.bb.b 96
688.2.bd \(\chi_{688}(39, \cdot)\) None 0 6
688.2.bf \(\chi_{688}(41, \cdot)\) None 0 6
688.2.bg \(\chi_{688}(17, \cdot)\) 688.2.bg.a 12 12
688.2.bg.b 24
688.2.bg.c 36
688.2.bg.d 48
688.2.bg.e 60
688.2.bg.f 72
688.2.bh \(\chi_{688}(21, \cdot)\) 688.2.bh.a 1032 12
688.2.bi \(\chi_{688}(27, \cdot)\) 688.2.bi.a 1032 12
688.2.bl \(\chi_{688}(9, \cdot)\) None 0 12
688.2.bn \(\chi_{688}(55, \cdot)\) None 0 12
688.2.bp \(\chi_{688}(63, \cdot)\) 688.2.bp.a 84 12
688.2.bp.b 84
688.2.bp.c 96
688.2.bu \(\chi_{688}(3, \cdot)\) 688.2.bu.a 2064 24
688.2.bv \(\chi_{688}(13, \cdot)\) 688.2.bv.a 2064 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(688))\) into lower level spaces

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