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Properties

Label 168.3
Level 168
Weight 3
Dimension 600
Nonzero newspaces 12
Newform subspaces 21
Sturm bound 4608
Trace bound 3

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Defining parameters

Level: \( N \) = \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 21 \)
Sturm bound: \(4608\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(168))\).

Total New Old
Modular forms 1680 640 1040
Cusp forms 1392 600 792
Eisenstein series 288 40 248

Trace form

\( 600 q - 4 q^{2} - 10 q^{3} + 12 q^{4} + 22 q^{6} + 4 q^{7} + 8 q^{8} - 28 q^{9} + O(q^{10}) \) \( 600 q - 4 q^{2} - 10 q^{3} + 12 q^{4} + 22 q^{6} + 4 q^{7} + 8 q^{8} - 28 q^{9} + 12 q^{10} + 40 q^{11} - 38 q^{12} - 40 q^{13} - 36 q^{14} - 12 q^{15} - 76 q^{16} + 64 q^{17} - 130 q^{18} + 84 q^{19} + 108 q^{20} + 96 q^{21} + 180 q^{22} + 144 q^{23} + 206 q^{24} + 192 q^{25} + 252 q^{26} + 92 q^{27} + 68 q^{28} + 194 q^{30} - 236 q^{31} - 124 q^{32} - 64 q^{33} - 380 q^{34} - 384 q^{35} - 336 q^{36} - 96 q^{37} - 476 q^{38} - 252 q^{39} - 808 q^{40} - 80 q^{41} - 442 q^{42} - 128 q^{43} - 656 q^{44} - 44 q^{45} - 856 q^{46} + 576 q^{47} - 282 q^{48} + 488 q^{49} - 184 q^{50} + 502 q^{51} - 16 q^{52} + 72 q^{53} + 290 q^{54} + 400 q^{55} + 252 q^{56} - 264 q^{57} + 184 q^{58} - 368 q^{59} + 264 q^{60} - 160 q^{61} + 348 q^{62} - 346 q^{63} + 252 q^{64} - 696 q^{65} + 6 q^{66} - 468 q^{67} + 556 q^{68} - 376 q^{69} + 668 q^{70} - 336 q^{71} - 100 q^{72} - 712 q^{73} + 240 q^{74} - 800 q^{75} + 132 q^{76} + 288 q^{77} - 112 q^{78} - 332 q^{79} + 192 q^{80} - 456 q^{81} + 684 q^{82} - 320 q^{83} + 14 q^{84} - 704 q^{85} - 176 q^{86} - 960 q^{87} + 84 q^{88} + 208 q^{89} + 704 q^{90} - 912 q^{91} - 420 q^{92} + 412 q^{93} - 900 q^{94} - 1248 q^{95} + 266 q^{96} + 1568 q^{97} - 1360 q^{98} + 436 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(168))\)

We only show spaces with odd 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
168.3.d \(\chi_{168}(113, \cdot)\) 168.3.d.a 12 1
168.3.e \(\chi_{168}(83, \cdot)\) 168.3.e.a 1 1
168.3.e.b 1
168.3.e.c 1
168.3.e.d 1
168.3.e.e 8
168.3.e.f 48
168.3.f \(\chi_{168}(97, \cdot)\) 168.3.f.a 8 1
168.3.g \(\chi_{168}(43, \cdot)\) 168.3.g.a 24 1
168.3.l \(\chi_{168}(13, \cdot)\) 168.3.l.a 32 1
168.3.m \(\chi_{168}(127, \cdot)\) None 0 1
168.3.n \(\chi_{168}(29, \cdot)\) 168.3.n.a 48 1
168.3.o \(\chi_{168}(167, \cdot)\) None 0 1
168.3.r \(\chi_{168}(47, \cdot)\) None 0 2
168.3.s \(\chi_{168}(53, \cdot)\) 168.3.s.a 4 2
168.3.s.b 4
168.3.s.c 112
168.3.w \(\chi_{168}(79, \cdot)\) None 0 2
168.3.x \(\chi_{168}(61, \cdot)\) 168.3.x.a 4 2
168.3.x.b 60
168.3.y \(\chi_{168}(67, \cdot)\) 168.3.y.a 64 2
168.3.z \(\chi_{168}(73, \cdot)\) 168.3.z.a 8 2
168.3.z.b 8
168.3.be \(\chi_{168}(59, \cdot)\) 168.3.be.a 120 2
168.3.bf \(\chi_{168}(65, \cdot)\) 168.3.bf.a 32 2

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(168))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(168)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(168))\)\(^{\oplus 1}\)