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Properties

Label 351.4
Level 351
Weight 4
Dimension 10448
Nonzero newspaces 24
Sturm bound 36288
Trace bound 10

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Defining parameters

Level: \( N \) = \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 24 \)
Sturm bound: \(36288\)
Trace bound: \(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(351))\).

Total New Old
Modular forms 13968 10800 3168
Cusp forms 13248 10448 2800
Eisenstein series 720 352 368

Trace form

\( 10448 q - 42 q^{2} - 60 q^{3} - 106 q^{4} - 90 q^{5} - 36 q^{6} - 2 q^{7} + 234 q^{8} + 36 q^{9} + O(q^{10}) \) \( 10448 q - 42 q^{2} - 60 q^{3} - 106 q^{4} - 90 q^{5} - 36 q^{6} - 2 q^{7} + 234 q^{8} + 36 q^{9} + 6 q^{10} - 294 q^{11} - 366 q^{12} - 187 q^{13} - 318 q^{14} + 326 q^{16} + 762 q^{17} + 1206 q^{18} + 106 q^{19} + 1170 q^{20} + 204 q^{21} - 138 q^{22} - 918 q^{23} - 2412 q^{24} - 934 q^{25} - 2490 q^{26} - 2394 q^{27} - 2324 q^{28} - 2154 q^{29} - 990 q^{30} + 286 q^{31} + 1806 q^{32} + 1206 q^{33} + 798 q^{34} + 4038 q^{35} + 4104 q^{36} + 2518 q^{37} + 6846 q^{38} + 696 q^{39} + 4530 q^{40} + 4686 q^{41} + 6084 q^{42} + 70 q^{43} + 4218 q^{44} + 648 q^{45} - 2694 q^{46} - 4038 q^{47} - 4650 q^{48} - 2166 q^{49} - 12516 q^{50} - 5382 q^{51} - 2347 q^{52} - 5352 q^{53} - 10980 q^{54} - 624 q^{55} - 12762 q^{56} - 6924 q^{57} - 3378 q^{58} - 6120 q^{59} - 2700 q^{60} - 866 q^{61} - 1224 q^{62} + 5148 q^{63} - 3166 q^{64} + 1089 q^{65} + 22086 q^{66} + 4714 q^{67} + 18876 q^{68} + 12204 q^{69} + 11310 q^{70} + 14994 q^{71} + 3958 q^{73} + 2154 q^{74} - 5280 q^{75} - 1970 q^{76} - 1782 q^{77} - 6138 q^{78} - 3830 q^{79} - 3312 q^{80} - 6948 q^{81} + 4380 q^{82} - 2970 q^{83} - 15420 q^{84} - 9714 q^{85} - 6582 q^{86} - 5832 q^{87} - 9930 q^{88} + 4812 q^{89} + 24948 q^{90} - 3467 q^{91} + 22566 q^{92} + 23532 q^{93} + 3966 q^{94} + 9450 q^{95} + 46548 q^{96} + 20902 q^{97} + 57168 q^{98} + 22932 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(351))\)

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
351.4.a \(\chi_{351}(1, \cdot)\) 351.4.a.a 1 1
351.4.a.b 1
351.4.a.c 2
351.4.a.d 4
351.4.a.e 6
351.4.a.f 6
351.4.a.g 6
351.4.a.h 6
351.4.a.i 8
351.4.a.j 8
351.4.b \(\chi_{351}(298, \cdot)\) 351.4.b.a 4 1
351.4.b.b 4
351.4.b.c 8
351.4.b.d 12
351.4.b.e 14
351.4.b.f 14
351.4.e \(\chi_{351}(118, \cdot)\) 351.4.e.a 36 2
351.4.e.b 36
351.4.f \(\chi_{351}(100, \cdot)\) 351.4.f.a 2 2
351.4.f.b 78
351.4.g \(\chi_{351}(55, \cdot)\) n/a 112 2
351.4.h \(\chi_{351}(289, \cdot)\) 351.4.h.a 2 2
351.4.h.b 78
351.4.i \(\chi_{351}(161, \cdot)\) n/a 112 2
351.4.l \(\chi_{351}(127, \cdot)\) 351.4.l.a 80 2
351.4.q \(\chi_{351}(82, \cdot)\) n/a 112 2
351.4.r \(\chi_{351}(10, \cdot)\) 351.4.r.a 80 2
351.4.t \(\chi_{351}(64, \cdot)\) 351.4.t.a 80 2
351.4.w \(\chi_{351}(40, \cdot)\) n/a 648 6
351.4.x \(\chi_{351}(16, \cdot)\) n/a 744 6
351.4.y \(\chi_{351}(61, \cdot)\) n/a 744 6
351.4.ba \(\chi_{351}(71, \cdot)\) n/a 160 4
351.4.bc \(\chi_{351}(8, \cdot)\) n/a 160 4
351.4.bd \(\chi_{351}(80, \cdot)\) n/a 224 4
351.4.bf \(\chi_{351}(206, \cdot)\) n/a 160 4
351.4.bl \(\chi_{351}(25, \cdot)\) n/a 744 6
351.4.bn \(\chi_{351}(4, \cdot)\) n/a 744 6
351.4.bo \(\chi_{351}(43, \cdot)\) n/a 744 6
351.4.bq \(\chi_{351}(20, \cdot)\) n/a 1488 12
351.4.bt \(\chi_{351}(5, \cdot)\) n/a 1488 12
351.4.bv \(\chi_{351}(2, \cdot)\) n/a 1488 12

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(351))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(351)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(117))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(351))\)\(^{\oplus 1}\)