Properties

Label 315.2
Level 315
Weight 2
Dimension 2268
Nonzero newspaces 30
Newform subspaces 93
Sturm bound 13824
Trace bound 9

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 30 \)
Newform subspaces: \( 93 \)
Sturm bound: \(13824\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(315))\).

Total New Old
Modular forms 3840 2540 1300
Cusp forms 3073 2268 805
Eisenstein series 767 272 495

Trace form

\( 2268q - 4q^{2} - 8q^{3} + 12q^{4} - 6q^{5} - 40q^{6} + 2q^{7} - 12q^{8} - 16q^{9} + O(q^{10}) \) \( 2268q - 4q^{2} - 8q^{3} + 12q^{4} - 6q^{5} - 40q^{6} + 2q^{7} - 12q^{8} - 16q^{9} - 36q^{10} - 28q^{11} - 64q^{12} - 42q^{14} - 92q^{15} - 64q^{16} - 64q^{17} - 104q^{18} - 36q^{19} - 122q^{20} - 108q^{21} - 88q^{22} - 72q^{23} - 168q^{24} - 30q^{25} - 172q^{26} - 92q^{27} - 98q^{28} - 104q^{29} - 142q^{30} - 60q^{31} - 212q^{32} - 88q^{33} - 148q^{34} - 88q^{35} - 200q^{36} - 120q^{37} - 160q^{38} - 32q^{39} - 110q^{40} - 116q^{41} - 120q^{42} - 16q^{43} - 80q^{44} - 28q^{45} - 208q^{46} - 4q^{47} + 80q^{48} - 52q^{49} - 16q^{50} - 40q^{51} - 104q^{52} + 44q^{53} + 44q^{54} - 108q^{55} + 30q^{56} + 16q^{57} - 112q^{58} + 20q^{59} + 26q^{60} - 152q^{61} + 84q^{63} - 204q^{64} - 160q^{65} - 164q^{66} - 156q^{67} + 4q^{68} - 96q^{69} - 300q^{70} - 296q^{71} + 36q^{72} - 228q^{73} - 188q^{74} - 32q^{75} - 364q^{76} - 168q^{77} - 40q^{78} - 180q^{79} + 34q^{80} - 160q^{81} - 204q^{82} - 72q^{83} - 96q^{84} - 132q^{85} - 100q^{86} - 44q^{87} - 72q^{88} + 12q^{89} + 202q^{90} - 100q^{91} + 288q^{92} + 36q^{93} + 80q^{94} + 218q^{95} + 16q^{96} + 64q^{97} + 268q^{98} - 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(315))\)

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
315.2.a \(\chi_{315}(1, \cdot)\) 315.2.a.a 1 1
315.2.a.b 1
315.2.a.c 2
315.2.a.d 2
315.2.a.e 2
315.2.a.f 2
315.2.b \(\chi_{315}(251, \cdot)\) 315.2.b.a 4 1
315.2.b.b 4
315.2.d \(\chi_{315}(64, \cdot)\) 315.2.d.a 2 1
315.2.d.b 2
315.2.d.c 2
315.2.d.d 2
315.2.d.e 6
315.2.g \(\chi_{315}(314, \cdot)\) 315.2.g.a 16 1
315.2.i \(\chi_{315}(106, \cdot)\) 315.2.i.a 2 2
315.2.i.b 2
315.2.i.c 8
315.2.i.d 8
315.2.i.e 12
315.2.i.f 16
315.2.j \(\chi_{315}(46, \cdot)\) 315.2.j.a 2 2
315.2.j.b 2
315.2.j.c 4
315.2.j.d 4
315.2.j.e 4
315.2.j.f 6
315.2.j.g 6
315.2.k \(\chi_{315}(16, \cdot)\) 315.2.k.a 4 2
315.2.k.b 24
315.2.k.c 36
315.2.l \(\chi_{315}(121, \cdot)\) 315.2.l.a 4 2
315.2.l.b 24
315.2.l.c 36
315.2.m \(\chi_{315}(8, \cdot)\) 315.2.m.a 12 2
315.2.m.b 12
315.2.p \(\chi_{315}(118, \cdot)\) 315.2.p.a 4 2
315.2.p.b 4
315.2.p.c 4
315.2.p.d 8
315.2.p.e 16
315.2.r \(\chi_{315}(184, \cdot)\) 315.2.r.a 4 2
315.2.r.b 84
315.2.t \(\chi_{315}(101, \cdot)\) 315.2.t.a 2 2
315.2.t.b 30
315.2.t.c 32
315.2.u \(\chi_{315}(59, \cdot)\) 315.2.u.a 88 2
315.2.z \(\chi_{315}(104, \cdot)\) 315.2.z.a 8 2
315.2.z.b 80
315.2.bb \(\chi_{315}(89, \cdot)\) 315.2.bb.a 8 2
315.2.bb.b 24
315.2.be \(\chi_{315}(236, \cdot)\) 315.2.be.a 2 2
315.2.be.b 30
315.2.be.c 32
315.2.bf \(\chi_{315}(109, \cdot)\) 315.2.bf.a 4 2
315.2.bf.b 16
315.2.bf.c 16
315.2.bh \(\chi_{315}(169, \cdot)\) 315.2.bh.a 4 2
315.2.bh.b 4
315.2.bh.c 64
315.2.bj \(\chi_{315}(26, \cdot)\) 315.2.bj.a 12 2
315.2.bj.b 12
315.2.bl \(\chi_{315}(41, \cdot)\) 315.2.bl.a 2 2
315.2.bl.b 2
315.2.bl.c 2
315.2.bl.d 2
315.2.bl.e 2
315.2.bl.f 2
315.2.bl.g 2
315.2.bl.h 2
315.2.bl.i 24
315.2.bl.j 24
315.2.bo \(\chi_{315}(4, \cdot)\) 315.2.bo.a 4 2
315.2.bo.b 84
315.2.bq \(\chi_{315}(164, \cdot)\) 315.2.bq.a 88 2
315.2.bs \(\chi_{315}(52, \cdot)\) 315.2.bs.a 4 4
315.2.bs.b 4
315.2.bs.c 4
315.2.bs.d 4
315.2.bs.e 160
315.2.bv \(\chi_{315}(23, \cdot)\) 315.2.bv.a 176 4
315.2.bx \(\chi_{315}(2, \cdot)\) 315.2.bx.a 176 4
315.2.bz \(\chi_{315}(73, \cdot)\) 315.2.bz.a 4 4
315.2.bz.b 4
315.2.bz.c 32
315.2.bz.d 32
315.2.cb \(\chi_{315}(13, \cdot)\) 315.2.cb.a 176 4
315.2.cc \(\chi_{315}(92, \cdot)\) 315.2.cc.a 144 4
315.2.ce \(\chi_{315}(53, \cdot)\) 315.2.ce.a 64 4
315.2.cg \(\chi_{315}(157, \cdot)\) 315.2.cg.a 4 4
315.2.cg.b 4
315.2.cg.c 4
315.2.cg.d 4
315.2.cg.e 160

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(315))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(315)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 2}\)