Properties

Label 840.2
Level 840
Weight 2
Dimension 6712
Nonzero newspaces 36
Newform subspaces 110
Sturm bound 73728
Trace bound 12

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Defining parameters

Level: \( N \) = \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 36 \)
Newform subspaces: \( 110 \)
Sturm bound: \(73728\)
Trace bound: \(12\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(840))\).

Total New Old
Modular forms 19584 6952 12632
Cusp forms 17281 6712 10569
Eisenstein series 2303 240 2063

Trace form

\( 6712q - 8q^{2} - 16q^{3} - 24q^{4} - 8q^{5} - 12q^{6} - 32q^{7} + 16q^{8} - 36q^{9} + O(q^{10}) \) \( 6712q - 8q^{2} - 16q^{3} - 24q^{4} - 8q^{5} - 12q^{6} - 32q^{7} + 16q^{8} - 36q^{9} - 12q^{10} - 24q^{11} + 52q^{12} - 40q^{13} + 32q^{14} - 12q^{15} + 24q^{16} - 24q^{17} + 28q^{18} + 56q^{19} + 100q^{20} + 16q^{21} + 136q^{22} + 120q^{23} + 76q^{24} - 20q^{25} + 152q^{26} + 104q^{27} + 168q^{28} + 48q^{29} + 38q^{30} + 168q^{31} + 72q^{32} + 116q^{33} + 88q^{34} + 84q^{35} - 32q^{36} + 104q^{37} - 8q^{38} + 112q^{39} - 104q^{40} - 116q^{42} + 48q^{43} - 208q^{44} + 18q^{45} - 320q^{46} + 112q^{47} - 276q^{48} + 48q^{49} - 128q^{50} - 224q^{52} + 40q^{53} - 276q^{54} + 116q^{55} - 184q^{56} + 48q^{57} - 176q^{58} + 192q^{59} - 264q^{60} + 48q^{61} - 232q^{62} + 56q^{63} - 312q^{64} + 176q^{65} - 356q^{66} + 168q^{67} - 232q^{68} + 48q^{69} - 156q^{70} + 64q^{71} - 384q^{72} + 208q^{73} - 112q^{74} - 84q^{75} - 136q^{76} + 72q^{77} - 336q^{78} - 56q^{79} - 160q^{80} - 36q^{81} - 280q^{82} - 160q^{83} - 300q^{84} + 152q^{85} + 32q^{86} - 240q^{87} + 72q^{88} + 32q^{89} - 204q^{90} - 224q^{91} - 8q^{92} - 68q^{93} + 88q^{94} - 168q^{95} - 100q^{96} + 168q^{97} - 40q^{98} - 416q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(840))\)

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
840.2.a \(\chi_{840}(1, \cdot)\) 840.2.a.a 1 1
840.2.a.b 1
840.2.a.c 1
840.2.a.d 1
840.2.a.e 1
840.2.a.f 1
840.2.a.g 1
840.2.a.h 1
840.2.a.i 1
840.2.a.j 1
840.2.a.k 2
840.2.d \(\chi_{840}(391, \cdot)\) None 0 1
840.2.e \(\chi_{840}(491, \cdot)\) 840.2.e.a 4 1
840.2.e.b 4
840.2.e.c 44
840.2.e.d 44
840.2.f \(\chi_{840}(41, \cdot)\) 840.2.f.a 16 1
840.2.f.b 16
840.2.g \(\chi_{840}(421, \cdot)\) 840.2.g.a 8 1
840.2.g.b 12
840.2.g.c 12
840.2.g.d 16
840.2.j \(\chi_{840}(589, \cdot)\) 840.2.j.a 2 1
840.2.j.b 2
840.2.j.c 2
840.2.j.d 2
840.2.j.e 32
840.2.j.f 32
840.2.k \(\chi_{840}(209, \cdot)\) 840.2.k.a 24 1
840.2.k.b 24
840.2.p \(\chi_{840}(659, \cdot)\) 840.2.p.a 72 1
840.2.p.b 72
840.2.q \(\chi_{840}(559, \cdot)\) None 0 1
840.2.t \(\chi_{840}(169, \cdot)\) 840.2.t.a 2 1
840.2.t.b 2
840.2.t.c 4
840.2.t.d 6
840.2.t.e 6
840.2.u \(\chi_{840}(629, \cdot)\) 840.2.u.a 4 1
840.2.u.b 4
840.2.u.c 8
840.2.u.d 8
840.2.u.e 160
840.2.v \(\chi_{840}(239, \cdot)\) None 0 1
840.2.w \(\chi_{840}(139, \cdot)\) 840.2.w.a 48 1
840.2.w.b 48
840.2.z \(\chi_{840}(811, \cdot)\) 840.2.z.a 4 1
840.2.z.b 4
840.2.z.c 28
840.2.z.d 28
840.2.ba \(\chi_{840}(71, \cdot)\) None 0 1
840.2.bf \(\chi_{840}(461, \cdot)\) 840.2.bf.a 8 1
840.2.bf.b 120
840.2.bg \(\chi_{840}(121, \cdot)\) 840.2.bg.a 2 2
840.2.bg.b 2
840.2.bg.c 2
840.2.bg.d 2
840.2.bg.e 2
840.2.bg.f 2
840.2.bg.g 4
840.2.bg.h 4
840.2.bg.i 6
840.2.bg.j 6
840.2.bj \(\chi_{840}(13, \cdot)\) 840.2.bj.a 8 2
840.2.bj.b 184
840.2.bk \(\chi_{840}(113, \cdot)\) 840.2.bk.a 4 2
840.2.bk.b 4
840.2.bk.c 32
840.2.bk.d 32
840.2.bl \(\chi_{840}(127, \cdot)\) None 0 2
840.2.bm \(\chi_{840}(83, \cdot)\) 840.2.bm.a 368 2
840.2.br \(\chi_{840}(43, \cdot)\) 840.2.br.a 72 2
840.2.br.b 72
840.2.bs \(\chi_{840}(167, \cdot)\) None 0 2
840.2.bt \(\chi_{840}(97, \cdot)\) 840.2.bt.a 24 2
840.2.bt.b 24
840.2.bu \(\chi_{840}(197, \cdot)\) 840.2.bu.a 288 2
840.2.bz \(\chi_{840}(19, \cdot)\) 840.2.bz.a 96 2
840.2.bz.b 96
840.2.ca \(\chi_{840}(359, \cdot)\) None 0 2
840.2.cb \(\chi_{840}(269, \cdot)\) 840.2.cb.a 8 2
840.2.cb.b 8
840.2.cb.c 352
840.2.cc \(\chi_{840}(289, \cdot)\) 840.2.cc.a 4 2
840.2.cc.b 20
840.2.cc.c 24
840.2.cf \(\chi_{840}(101, \cdot)\) 840.2.cf.a 256 2
840.2.ck \(\chi_{840}(191, \cdot)\) None 0 2
840.2.cl \(\chi_{840}(451, \cdot)\) 840.2.cl.a 64 2
840.2.cl.b 64
840.2.co \(\chi_{840}(541, \cdot)\) 840.2.co.a 4 2
840.2.co.b 4
840.2.co.c 60
840.2.co.d 60
840.2.cp \(\chi_{840}(521, \cdot)\) 840.2.cp.a 32 2
840.2.cp.b 32
840.2.cq \(\chi_{840}(11, \cdot)\) 840.2.cq.a 128 2
840.2.cq.b 128
840.2.cr \(\chi_{840}(31, \cdot)\) None 0 2
840.2.cu \(\chi_{840}(199, \cdot)\) None 0 2
840.2.cv \(\chi_{840}(179, \cdot)\) 840.2.cv.a 368 2
840.2.da \(\chi_{840}(89, \cdot)\) 840.2.da.a 48 2
840.2.da.b 48
840.2.db \(\chi_{840}(109, \cdot)\) 840.2.db.a 4 2
840.2.db.b 4
840.2.db.c 4
840.2.db.d 4
840.2.db.e 88
840.2.db.f 88
840.2.dc \(\chi_{840}(53, \cdot)\) 840.2.dc.a 8 4
840.2.dc.b 8
840.2.dc.c 8
840.2.dc.d 8
840.2.dc.e 704
840.2.dd \(\chi_{840}(73, \cdot)\) 840.2.dd.a 48 4
840.2.dd.b 48
840.2.di \(\chi_{840}(47, \cdot)\) None 0 4
840.2.dj \(\chi_{840}(67, \cdot)\) 840.2.dj.a 384 4
840.2.dk \(\chi_{840}(227, \cdot)\) 840.2.dk.a 736 4
840.2.dl \(\chi_{840}(247, \cdot)\) None 0 4
840.2.dq \(\chi_{840}(137, \cdot)\) 840.2.dq.a 192 4
840.2.dr \(\chi_{840}(157, \cdot)\) 840.2.dr.a 384 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(840))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(840)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(210))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(420))\)\(^{\oplus 2}\)