Properties

Label 735.2
Level 735
Weight 2
Dimension 12120
Nonzero newspaces 24
Newform subspaces 118
Sturm bound 75264
Trace bound 4

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Defining parameters

Level: \( N \) = \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 24 \)
Newform subspaces: \( 118 \)
Sturm bound: \(75264\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(735))\).

Total New Old
Modular forms 19776 12704 7072
Cusp forms 17857 12120 5737
Eisenstein series 1919 584 1335

Trace form

\( 12120q - 4q^{2} - 28q^{3} - 36q^{4} + 12q^{5} - 56q^{6} - 56q^{7} + 60q^{8} - 2q^{9} + O(q^{10}) \) \( 12120q - 4q^{2} - 28q^{3} - 36q^{4} + 12q^{5} - 56q^{6} - 56q^{7} + 60q^{8} - 2q^{9} - 34q^{10} + 56q^{11} + 16q^{12} + 4q^{13} + 48q^{14} - 71q^{15} - 68q^{16} + 32q^{17} - 22q^{18} + 4q^{19} + 64q^{20} - 94q^{21} + 4q^{22} + 72q^{23} - 60q^{24} - 54q^{25} + 32q^{26} - 76q^{27} - 24q^{28} + 16q^{29} - 137q^{30} - 132q^{31} - 92q^{32} - 74q^{33} - 92q^{34} - 30q^{35} - 346q^{36} - 180q^{37} - 208q^{38} - 192q^{39} - 462q^{40} - 200q^{41} - 246q^{42} - 204q^{43} - 440q^{44} - 141q^{45} - 540q^{46} - 40q^{47} - 454q^{48} - 300q^{49} - 52q^{50} - 218q^{51} - 572q^{52} - 40q^{53} - 20q^{54} - 256q^{55} - 396q^{56} - 58q^{57} - 508q^{58} - 88q^{59} - 315q^{60} - 456q^{61} - 168q^{62} - 24q^{63} - 508q^{64} - 124q^{65} - 262q^{66} - 212q^{67} - 128q^{68} - 162q^{69} - 354q^{70} - 32q^{71} - 150q^{72} - 308q^{73} - 224q^{74} - 241q^{75} - 316q^{76} - 386q^{78} - 188q^{79} - 230q^{80} - 278q^{81} - 496q^{82} - 264q^{83} - 424q^{84} - 274q^{85} - 412q^{86} - 262q^{87} - 768q^{88} - 192q^{89} - 226q^{90} - 332q^{91} - 84q^{92} - 354q^{93} - 320q^{94} - 124q^{95} - 166q^{96} - 136q^{97} - 876q^{98} - 184q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(735))\)

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
735.2.a \(\chi_{735}(1, \cdot)\) 735.2.a.a 1 1
735.2.a.b 1
735.2.a.c 1
735.2.a.d 1
735.2.a.e 1
735.2.a.f 1
735.2.a.g 2
735.2.a.h 2
735.2.a.i 2
735.2.a.j 2
735.2.a.k 2
735.2.a.l 2
735.2.a.m 2
735.2.a.n 4
735.2.a.o 4
735.2.b \(\chi_{735}(146, \cdot)\) 735.2.b.a 2 1
735.2.b.b 2
735.2.b.c 8
735.2.b.d 8
735.2.b.e 16
735.2.b.f 16
735.2.d \(\chi_{735}(589, \cdot)\) 735.2.d.a 2 1
735.2.d.b 6
735.2.d.c 8
735.2.d.d 8
735.2.d.e 8
735.2.d.f 8
735.2.g \(\chi_{735}(734, \cdot)\) 735.2.g.a 16 1
735.2.g.b 24
735.2.g.c 32
735.2.i \(\chi_{735}(226, \cdot)\) 735.2.i.a 2 2
735.2.i.b 2
735.2.i.c 2
735.2.i.d 2
735.2.i.e 2
735.2.i.f 2
735.2.i.g 4
735.2.i.h 4
735.2.i.i 4
735.2.i.j 4
735.2.i.k 4
735.2.i.l 4
735.2.i.m 8
735.2.i.n 8
735.2.j \(\chi_{735}(197, \cdot)\) 735.2.j.a 8 2
735.2.j.b 8
735.2.j.c 16
735.2.j.d 16
735.2.j.e 24
735.2.j.f 24
735.2.j.g 24
735.2.j.h 24
735.2.m \(\chi_{735}(97, \cdot)\) 735.2.m.a 24 2
735.2.m.b 24
735.2.m.c 32
735.2.p \(\chi_{735}(374, \cdot)\) 735.2.p.a 8 2
735.2.p.b 8
735.2.p.c 8
735.2.p.d 16
735.2.p.e 16
735.2.p.f 24
735.2.p.g 64
735.2.q \(\chi_{735}(79, \cdot)\) 735.2.q.a 4 2
735.2.q.b 4
735.2.q.c 8
735.2.q.d 8
735.2.q.e 12
735.2.q.f 12
735.2.q.g 16
735.2.q.h 16
735.2.s \(\chi_{735}(521, \cdot)\) 735.2.s.a 2 2
735.2.s.b 2
735.2.s.c 2
735.2.s.d 2
735.2.s.e 2
735.2.s.f 2
735.2.s.g 4
735.2.s.h 4
735.2.s.i 4
735.2.s.j 4
735.2.s.k 8
735.2.s.l 8
735.2.s.m 32
735.2.s.n 32
735.2.u \(\chi_{735}(106, \cdot)\) 735.2.u.a 48 6
735.2.u.b 48
735.2.u.c 60
735.2.u.d 60
735.2.v \(\chi_{735}(178, \cdot)\) 735.2.v.a 32 4
735.2.v.b 32
735.2.v.c 48
735.2.v.d 48
735.2.y \(\chi_{735}(128, \cdot)\) 735.2.y.a 8 4
735.2.y.b 8
735.2.y.c 8
735.2.y.d 8
735.2.y.e 32
735.2.y.f 32
735.2.y.g 48
735.2.y.h 48
735.2.y.i 48
735.2.y.j 48
735.2.ba \(\chi_{735}(104, \cdot)\) 735.2.ba.a 648 6
735.2.bd \(\chi_{735}(64, \cdot)\) 735.2.bd.a 336 6
735.2.bf \(\chi_{735}(41, \cdot)\) 735.2.bf.a 228 6
735.2.bf.b 228
735.2.bg \(\chi_{735}(16, \cdot)\) 735.2.bg.a 108 12
735.2.bg.b 108
735.2.bg.c 120
735.2.bg.d 120
735.2.bi \(\chi_{735}(13, \cdot)\) 735.2.bi.a 672 12
735.2.bj \(\chi_{735}(8, \cdot)\) 735.2.bj.a 1296 12
735.2.bm \(\chi_{735}(26, \cdot)\) 735.2.bm.a 444 12
735.2.bm.b 444
735.2.bo \(\chi_{735}(4, \cdot)\) 735.2.bo.a 672 12
735.2.bp \(\chi_{735}(59, \cdot)\) 735.2.bp.a 1296 12
735.2.bt \(\chi_{735}(2, \cdot)\) 735.2.bt.a 2592 24
735.2.bu \(\chi_{735}(52, \cdot)\) 735.2.bu.a 1344 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(735))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(735)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(147))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(245))\)\(^{\oplus 2}\)