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Properties

Label 351.2
Level 351
Weight 2
Dimension 3366
Nonzero newspaces 24
Newform subspaces 61
Sturm bound 18144
Trace bound 10

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Defining parameters

Level: \( N \) = \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 24 \)
Newform subspaces: \( 61 \)
Sturm bound: \(18144\)
Trace bound: \(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(351))\).

Total New Old
Modular forms 4896 3718 1178
Cusp forms 4177 3366 811
Eisenstein series 719 352 367

Trace form

\( 3366 q - 36 q^{2} - 60 q^{3} - 68 q^{4} - 42 q^{5} - 72 q^{6} - 70 q^{7} - 60 q^{8} - 72 q^{9} + O(q^{10}) \) \( 3366 q - 36 q^{2} - 60 q^{3} - 68 q^{4} - 42 q^{5} - 72 q^{6} - 70 q^{7} - 60 q^{8} - 72 q^{9} - 78 q^{10} - 54 q^{11} - 96 q^{12} - 83 q^{13} - 126 q^{14} - 90 q^{15} - 92 q^{16} - 66 q^{17} - 90 q^{18} - 64 q^{19} - 42 q^{20} - 48 q^{21} - 90 q^{22} - 24 q^{23} - 36 q^{24} - 80 q^{25} - 18 q^{26} - 126 q^{27} - 188 q^{28} - 60 q^{29} - 54 q^{30} - 110 q^{31} - 120 q^{32} - 72 q^{33} - 150 q^{34} - 132 q^{35} - 108 q^{36} - 128 q^{37} - 216 q^{38} - 105 q^{39} - 306 q^{40} - 126 q^{41} - 108 q^{42} - 134 q^{43} - 150 q^{44} - 54 q^{45} - 126 q^{46} - 54 q^{47} - 42 q^{48} - 124 q^{49} - 54 q^{50} - 36 q^{51} - 99 q^{52} - 60 q^{53} + 36 q^{54} - 168 q^{55} - 30 q^{56} - 66 q^{57} - 126 q^{58} - 24 q^{59} - 72 q^{60} - 106 q^{61} - 60 q^{62} - 90 q^{63} - 164 q^{64} - 123 q^{65} - 126 q^{66} - 112 q^{67} - 222 q^{68} - 90 q^{69} - 198 q^{70} - 198 q^{71} - 108 q^{72} - 178 q^{73} - 282 q^{74} - 150 q^{75} - 88 q^{76} - 186 q^{77} - 90 q^{78} - 166 q^{79} - 384 q^{80} - 144 q^{81} - 288 q^{82} - 246 q^{83} - 84 q^{84} - 126 q^{85} - 306 q^{86} - 90 q^{87} - 168 q^{88} - 138 q^{89} - 108 q^{90} - 133 q^{91} - 138 q^{92} + 6 q^{93} - 126 q^{94} - 30 q^{95} + 216 q^{96} - 8 q^{97} + 198 q^{98} + 90 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(351))\)

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
351.2.a \(\chi_{351}(1, \cdot)\) 351.2.a.a 2 1
351.2.a.b 2
351.2.a.c 2
351.2.a.d 2
351.2.a.e 4
351.2.a.f 4
351.2.b \(\chi_{351}(298, \cdot)\) 351.2.b.a 2 1
351.2.b.b 4
351.2.b.c 4
351.2.b.d 4
351.2.b.e 4
351.2.e \(\chi_{351}(118, \cdot)\) 351.2.e.a 2 2
351.2.e.b 10
351.2.e.c 12
351.2.f \(\chi_{351}(100, \cdot)\) 351.2.f.a 24 2
351.2.g \(\chi_{351}(55, \cdot)\) 351.2.g.a 2 2
351.2.g.b 2
351.2.g.c 2
351.2.g.d 10
351.2.g.e 10
351.2.g.f 12
351.2.h \(\chi_{351}(289, \cdot)\) 351.2.h.a 24 2
351.2.i \(\chi_{351}(161, \cdot)\) 351.2.i.a 4 2
351.2.i.b 16
351.2.i.c 16
351.2.l \(\chi_{351}(127, \cdot)\) 351.2.l.a 2 2
351.2.l.b 22
351.2.q \(\chi_{351}(82, \cdot)\) 351.2.q.a 2 2
351.2.q.b 2
351.2.q.c 2
351.2.q.d 2
351.2.q.e 2
351.2.q.f 4
351.2.q.g 8
351.2.q.h 8
351.2.q.i 8
351.2.r \(\chi_{351}(10, \cdot)\) 351.2.r.a 2 2
351.2.r.b 22
351.2.t \(\chi_{351}(64, \cdot)\) 351.2.t.a 2 2
351.2.t.b 2
351.2.t.c 20
351.2.w \(\chi_{351}(40, \cdot)\) 351.2.w.a 6 6
351.2.w.b 102
351.2.w.c 108
351.2.x \(\chi_{351}(16, \cdot)\) 351.2.x.a 240 6
351.2.y \(\chi_{351}(61, \cdot)\) 351.2.y.a 240 6
351.2.ba \(\chi_{351}(71, \cdot)\) 351.2.ba.a 48 4
351.2.bc \(\chi_{351}(8, \cdot)\) 351.2.bc.a 4 4
351.2.bc.b 44
351.2.bd \(\chi_{351}(80, \cdot)\) 351.2.bd.a 4 4
351.2.bd.b 16
351.2.bd.c 16
351.2.bd.d 20
351.2.bd.e 20
351.2.bf \(\chi_{351}(206, \cdot)\) 351.2.bf.a 48 4
351.2.bl \(\chi_{351}(25, \cdot)\) 351.2.bl.a 240 6
351.2.bn \(\chi_{351}(4, \cdot)\) 351.2.bn.a 240 6
351.2.bo \(\chi_{351}(43, \cdot)\) 351.2.bo.a 240 6
351.2.bq \(\chi_{351}(20, \cdot)\) 351.2.bq.a 480 12
351.2.bt \(\chi_{351}(5, \cdot)\) 351.2.bt.a 480 12
351.2.bv \(\chi_{351}(2, \cdot)\) 351.2.bv.a 480 12

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(351))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(351)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(117))\)\(^{\oplus 2}\)