Properties

Label 656.2
Level 656
Weight 2
Dimension 7376
Nonzero newspaces 20
Newform subspaces 71
Sturm bound 53760
Trace bound 5

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Defining parameters

Level: \( N \) = \( 656 = 2^{4} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 20 \)
Newform subspaces: \( 71 \)
Sturm bound: \(53760\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(656))\).

Total New Old
Modular forms 14000 7726 6274
Cusp forms 12881 7376 5505
Eisenstein series 1119 350 769

Trace form

\( 7376 q - 76 q^{2} - 56 q^{3} - 80 q^{4} - 96 q^{5} - 88 q^{6} - 60 q^{7} - 88 q^{8} - 20 q^{9} + O(q^{10}) \) \( 7376 q - 76 q^{2} - 56 q^{3} - 80 q^{4} - 96 q^{5} - 88 q^{6} - 60 q^{7} - 88 q^{8} - 20 q^{9} - 80 q^{10} - 64 q^{11} - 72 q^{12} - 96 q^{13} - 72 q^{14} - 68 q^{15} - 64 q^{16} - 172 q^{17} - 84 q^{18} - 72 q^{19} - 88 q^{20} - 108 q^{21} - 80 q^{22} - 60 q^{23} - 80 q^{24} - 20 q^{25} - 88 q^{26} - 44 q^{27} - 96 q^{28} - 112 q^{29} - 72 q^{30} - 28 q^{31} - 96 q^{32} - 172 q^{33} - 88 q^{34} - 52 q^{35} - 72 q^{36} - 112 q^{37} - 56 q^{38} - 60 q^{39} - 64 q^{40} - 20 q^{41} - 160 q^{42} - 80 q^{43} - 72 q^{44} - 104 q^{45} - 104 q^{46} - 92 q^{47} - 96 q^{48} - 192 q^{49} - 68 q^{50} - 68 q^{51} - 72 q^{52} - 80 q^{53} - 80 q^{54} - 60 q^{55} - 64 q^{56} - 20 q^{57} - 56 q^{58} - 48 q^{59} - 80 q^{60} - 64 q^{61} - 112 q^{62} - 68 q^{63} - 80 q^{64} - 188 q^{65} - 88 q^{66} - 40 q^{67} - 80 q^{68} - 76 q^{69} - 96 q^{70} - 60 q^{71} - 88 q^{72} - 20 q^{73} - 80 q^{74} - 72 q^{75} - 104 q^{76} - 108 q^{77} - 72 q^{78} - 60 q^{79} - 96 q^{80} - 220 q^{81} - 80 q^{82} - 116 q^{83} - 64 q^{84} - 108 q^{85} - 80 q^{86} - 60 q^{87} - 96 q^{88} - 20 q^{89} - 72 q^{90} - 68 q^{91} - 32 q^{92} - 132 q^{93} - 48 q^{94} - 36 q^{95} - 48 q^{96} - 172 q^{97} - 68 q^{98} - 56 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(656))\)

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
656.2.a \(\chi_{656}(1, \cdot)\) 656.2.a.a 1 1
656.2.a.b 1
656.2.a.c 1
656.2.a.d 2
656.2.a.e 2
656.2.a.f 3
656.2.a.g 3
656.2.a.h 3
656.2.a.i 4
656.2.b \(\chi_{656}(329, \cdot)\) None 0 1
656.2.d \(\chi_{656}(81, \cdot)\) 656.2.d.a 2 1
656.2.d.b 2
656.2.d.c 2
656.2.d.d 4
656.2.d.e 4
656.2.d.f 6
656.2.g \(\chi_{656}(409, \cdot)\) None 0 1
656.2.i \(\chi_{656}(173, \cdot)\) 656.2.i.a 2 2
656.2.i.b 162
656.2.l \(\chi_{656}(337, \cdot)\) 656.2.l.a 2 2
656.2.l.b 2
656.2.l.c 2
656.2.l.d 2
656.2.l.e 2
656.2.l.f 6
656.2.l.g 6
656.2.l.h 8
656.2.l.i 10
656.2.n \(\chi_{656}(165, \cdot)\) 656.2.n.a 2 2
656.2.n.b 2
656.2.n.c 2
656.2.n.d 4
656.2.n.e 4
656.2.n.f 66
656.2.n.g 80
656.2.o \(\chi_{656}(245, \cdot)\) 656.2.o.a 164 2
656.2.r \(\chi_{656}(9, \cdot)\) None 0 2
656.2.t \(\chi_{656}(237, \cdot)\) 656.2.t.a 2 2
656.2.t.b 162
656.2.u \(\chi_{656}(305, \cdot)\) 656.2.u.a 4 4
656.2.u.b 4
656.2.u.c 8
656.2.u.d 8
656.2.u.e 8
656.2.u.f 12
656.2.u.g 16
656.2.u.h 20
656.2.v \(\chi_{656}(331, \cdot)\) 656.2.v.a 328 4
656.2.z \(\chi_{656}(55, \cdot)\) None 0 4
656.2.ba \(\chi_{656}(79, \cdot)\) 656.2.ba.a 4 4
656.2.ba.b 8
656.2.ba.c 24
656.2.ba.d 48
656.2.bb \(\chi_{656}(3, \cdot)\) 656.2.bb.a 328 4
656.2.be \(\chi_{656}(113, \cdot)\) 656.2.be.a 8 4
656.2.be.b 8
656.2.be.c 8
656.2.be.d 16
656.2.be.e 16
656.2.be.f 24
656.2.bg \(\chi_{656}(57, \cdot)\) None 0 4
656.2.bi \(\chi_{656}(25, \cdot)\) None 0 4
656.2.bl \(\chi_{656}(197, \cdot)\) 656.2.bl.a 656 8
656.2.bm \(\chi_{656}(121, \cdot)\) None 0 8
656.2.bp \(\chi_{656}(45, \cdot)\) 656.2.bp.a 656 8
656.2.bq \(\chi_{656}(37, \cdot)\) 656.2.bq.a 656 8
656.2.bs \(\chi_{656}(33, \cdot)\) 656.2.bs.a 8 8
656.2.bs.b 16
656.2.bs.c 24
656.2.bs.d 24
656.2.bs.e 40
656.2.bs.f 48
656.2.bu \(\chi_{656}(5, \cdot)\) 656.2.bu.a 656 8
656.2.bx \(\chi_{656}(259, \cdot)\) 656.2.bx.a 1312 16
656.2.by \(\chi_{656}(15, \cdot)\) 656.2.by.a 16 16
656.2.by.b 96
656.2.by.c 224
656.2.bz \(\chi_{656}(7, \cdot)\) None 0 16
656.2.cd \(\chi_{656}(11, \cdot)\) 656.2.cd.a 1312 16

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(656))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(656)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(41))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(82))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(164))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(328))\)\(^{\oplus 2}\)