Properties

Label 315.2
Level 315
Weight 2
Dimension 2268
Nonzero newspaces 30
Newforms 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: \( 30 \)
Newforms: \( 93 \)
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

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

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

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