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Properties

Label 399.2
Level 399
Weight 2
Dimension 3951
Nonzero newspaces 32
Newform subspaces 100
Sturm bound 23040
Trace bound 12

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Defining parameters

Level: \( N \) = \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 32 \)
Newform subspaces: \( 100 \)
Sturm bound: \(23040\)
Trace bound: \(12\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(399))\).

Total New Old
Modular forms 6192 4287 1905
Cusp forms 5329 3951 1378
Eisenstein series 863 336 527

Trace form

\( 3951 q + 9 q^{2} - 29 q^{3} - 55 q^{4} + 6 q^{5} - 39 q^{6} - 79 q^{7} + 9 q^{8} - 41 q^{9} + O(q^{10}) \) \( 3951 q + 9 q^{2} - 29 q^{3} - 55 q^{4} + 6 q^{5} - 39 q^{6} - 79 q^{7} + 9 q^{8} - 41 q^{9} - 66 q^{10} - 59 q^{12} - 106 q^{13} - 39 q^{14} - 108 q^{15} - 183 q^{16} - 6 q^{17} - 45 q^{18} - 131 q^{19} - 90 q^{20} - 59 q^{21} - 228 q^{22} - 12 q^{23} - 99 q^{24} - 123 q^{25} - 30 q^{26} - 65 q^{27} - 195 q^{28} - 54 q^{29} - 156 q^{30} - 124 q^{31} - 123 q^{32} - 144 q^{33} - 210 q^{34} - 66 q^{35} - 209 q^{36} - 146 q^{37} - 189 q^{38} - 166 q^{39} - 294 q^{40} + 6 q^{41} - 114 q^{42} - 272 q^{43} - 96 q^{44} - 84 q^{45} - 180 q^{46} - 36 q^{47} + 13 q^{48} - 123 q^{49} - 9 q^{50} + 66 q^{51} + 38 q^{52} + 66 q^{53} + 81 q^{54} + 21 q^{56} - 5 q^{57} - 42 q^{58} + 12 q^{59} + 120 q^{60} - 130 q^{61} - 156 q^{62} - 50 q^{63} - 355 q^{64} - 144 q^{65} - 288 q^{67} - 138 q^{68} - 102 q^{69} - 252 q^{70} - 48 q^{71} - 153 q^{72} - 262 q^{73} - 150 q^{74} - 161 q^{75} - 299 q^{76} - 162 q^{77} - 348 q^{78} - 396 q^{79} - 306 q^{80} - 89 q^{81} - 666 q^{82} - 168 q^{83} - 248 q^{84} - 528 q^{85} - 252 q^{86} - 282 q^{87} - 564 q^{88} - 174 q^{89} - 132 q^{90} - 256 q^{91} - 300 q^{92} - 160 q^{93} - 348 q^{94} + 12 q^{95} - 159 q^{96} - 94 q^{97} - 147 q^{98} - 96 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(399))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
399.2.a \(\chi_{399}(1, \cdot)\) 399.2.a.a 1 1
399.2.a.b 1
399.2.a.c 1
399.2.a.d 3
399.2.a.e 3
399.2.a.f 5
399.2.a.g 5
399.2.c \(\chi_{399}(265, \cdot)\) 399.2.c.a 2 1
399.2.c.b 2
399.2.c.c 12
399.2.c.d 12
399.2.d \(\chi_{399}(20, \cdot)\) 399.2.d.a 48 1
399.2.f \(\chi_{399}(113, \cdot)\) 399.2.f.a 4 1
399.2.f.b 16
399.2.f.c 20
399.2.i \(\chi_{399}(163, \cdot)\) 399.2.i.a 2 2
399.2.i.b 22
399.2.i.c 28
399.2.j \(\chi_{399}(58, \cdot)\) 399.2.j.a 2 2
399.2.j.b 2
399.2.j.c 4
399.2.j.d 8
399.2.j.e 8
399.2.j.f 8
399.2.j.g 16
399.2.k \(\chi_{399}(64, \cdot)\) 399.2.k.a 8 2
399.2.k.b 8
399.2.k.c 12
399.2.k.d 12
399.2.l \(\chi_{399}(121, \cdot)\) 399.2.l.a 2 2
399.2.l.b 22
399.2.l.c 28
399.2.m \(\chi_{399}(145, \cdot)\) 399.2.m.a 2 2
399.2.m.b 4
399.2.m.c 22
399.2.m.d 24
399.2.p \(\chi_{399}(26, \cdot)\) 399.2.p.a 2 2
399.2.p.b 2
399.2.p.c 96
399.2.t \(\chi_{399}(8, \cdot)\) 399.2.t.a 40 2
399.2.t.b 40
399.2.w \(\chi_{399}(170, \cdot)\) 399.2.w.a 2 2
399.2.w.b 2
399.2.w.c 8
399.2.w.d 88
399.2.x \(\chi_{399}(65, \cdot)\) 399.2.x.a 2 2
399.2.x.b 2
399.2.x.c 96
399.2.z \(\chi_{399}(83, \cdot)\) 399.2.z.a 96 2
399.2.bc \(\chi_{399}(248, \cdot)\) 399.2.bc.a 96 2
399.2.bd \(\chi_{399}(68, \cdot)\) 399.2.bd.a 2 2
399.2.bd.b 2
399.2.bd.c 96
399.2.bg \(\chi_{399}(31, \cdot)\) 399.2.bg.a 2 2
399.2.bg.b 4
399.2.bg.c 22
399.2.bg.d 24
399.2.bh \(\chi_{399}(94, \cdot)\) 399.2.bh.a 2 2
399.2.bh.b 2
399.2.bh.c 4
399.2.bh.d 4
399.2.bh.e 20
399.2.bh.f 20
399.2.bk \(\chi_{399}(160, \cdot)\) 399.2.bk.a 2 2
399.2.bk.b 2
399.2.bk.c 2
399.2.bk.d 2
399.2.bk.e 24
399.2.bk.f 24
399.2.bm \(\chi_{399}(107, \cdot)\) 399.2.bm.a 2 2
399.2.bm.b 2
399.2.bm.c 96
399.2.bo \(\chi_{399}(43, \cdot)\) 399.2.bo.a 6 6
399.2.bo.b 18
399.2.bo.c 24
399.2.bo.d 36
399.2.bo.e 36
399.2.bp \(\chi_{399}(130, \cdot)\) 399.2.bp.a 72 6
399.2.bp.b 90
399.2.bq \(\chi_{399}(4, \cdot)\) 399.2.bq.a 72 6
399.2.bq.b 90
399.2.br \(\chi_{399}(2, \cdot)\) 399.2.br.a 6 6
399.2.br.b 288
399.2.bv \(\chi_{399}(86, \cdot)\) 399.2.bv.a 6 6
399.2.bv.b 288
399.2.bw \(\chi_{399}(29, \cdot)\) 399.2.bw.a 120 6
399.2.bw.b 120
399.2.cb \(\chi_{399}(5, \cdot)\) 399.2.cb.a 6 6
399.2.cb.b 288
399.2.cc \(\chi_{399}(13, \cdot)\) 399.2.cc.a 78 6
399.2.cc.b 78
399.2.cd \(\chi_{399}(52, \cdot)\) 399.2.cd.a 78 6
399.2.cd.b 84
399.2.ci \(\chi_{399}(17, \cdot)\) 399.2.ci.a 6 6
399.2.ci.b 288
399.2.cj \(\chi_{399}(62, \cdot)\) 399.2.cj.a 6 6
399.2.cj.b 6
399.2.cj.c 288
399.2.ck \(\chi_{399}(10, \cdot)\) 399.2.ck.a 78 6
399.2.ck.b 84

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(399))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(399)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(57))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(133))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(399))\)\(^{\oplus 1}\)