Properties

Label 384.3
Level 384
Weight 3
Dimension 3408
Nonzero newspaces 10
Newform subspaces 26
Sturm bound 24576
Trace bound 25

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Defining parameters

Level: \( N \) = \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 10 \)
Newform subspaces: \( 26 \)
Sturm bound: \(24576\)
Trace bound: \(25\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(384))\).

Total New Old
Modular forms 8512 3504 5008
Cusp forms 7872 3408 4464
Eisenstein series 640 96 544

Trace form

\( 3408q - 12q^{3} - 32q^{4} - 16q^{6} - 24q^{7} - 20q^{9} + O(q^{10}) \) \( 3408q - 12q^{3} - 32q^{4} - 16q^{6} - 24q^{7} - 20q^{9} - 32q^{10} - 16q^{12} - 32q^{13} - 16q^{15} - 32q^{16} - 16q^{18} - 24q^{19} - 52q^{21} - 32q^{22} - 128q^{23} - 16q^{24} - 232q^{25} - 108q^{27} - 32q^{28} - 64q^{29} - 16q^{30} - 16q^{31} + 32q^{33} - 32q^{34} + 192q^{35} - 16q^{36} + 160q^{37} + 180q^{39} - 32q^{40} + 320q^{41} - 16q^{42} + 168q^{43} + 84q^{45} - 32q^{46} - 16q^{48} + 344q^{49} + 1248q^{50} + 216q^{51} + 2080q^{52} + 640q^{53} + 272q^{54} + 488q^{55} + 1568q^{56} + 364q^{57} + 1408q^{58} + 256q^{59} + 560q^{60} + 224q^{61} + 192q^{62} + 8q^{63} - 416q^{64} - 256q^{65} - 592q^{66} - 344q^{67} - 960q^{68} - 364q^{69} - 2720q^{70} - 512q^{71} - 16q^{72} - 1320q^{73} - 2464q^{74} - 296q^{75} - 3360q^{76} - 896q^{77} - 1456q^{78} - 544q^{79} - 1632q^{80} + 424q^{81} - 32q^{82} - 16q^{84} - 432q^{85} + 436q^{87} - 32q^{88} - 16q^{90} - 24q^{91} + 80q^{93} - 32q^{94} - 16q^{96} - 64q^{97} - 268q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(384))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
384.3.b \(\chi_{384}(319, \cdot)\) 384.3.b.a 4 1
384.3.b.b 4
384.3.b.c 8
384.3.e \(\chi_{384}(257, \cdot)\) 384.3.e.a 8 1
384.3.e.b 8
384.3.e.c 8
384.3.e.d 8
384.3.g \(\chi_{384}(127, \cdot)\) 384.3.g.a 8 1
384.3.g.b 8
384.3.h \(\chi_{384}(65, \cdot)\) 384.3.h.a 2 1
384.3.h.b 2
384.3.h.c 2
384.3.h.d 2
384.3.h.e 4
384.3.h.f 4
384.3.h.g 16
384.3.i \(\chi_{384}(161, \cdot)\) 384.3.i.a 8 2
384.3.i.b 8
384.3.i.c 20
384.3.i.d 20
384.3.l \(\chi_{384}(31, \cdot)\) 384.3.l.a 16 2
384.3.l.b 16
384.3.m \(\chi_{384}(79, \cdot)\) 384.3.m.a 64 4
384.3.p \(\chi_{384}(17, \cdot)\) 384.3.p.a 120 4
384.3.q \(\chi_{384}(41, \cdot)\) None 0 8
384.3.t \(\chi_{384}(7, \cdot)\) None 0 8
384.3.u \(\chi_{384}(19, \cdot)\) 384.3.u.a 1024 16
384.3.x \(\chi_{384}(5, \cdot)\) 384.3.x.a 2016 16

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(384))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(384)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 2}\)