Properties

Label 768.3
Level 768
Weight 3
Dimension 13720
Nonzero newspaces 12
Sturm bound 98304
Trace bound 49

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Defining parameters

Level: \( N \) = \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(98304\)
Trace bound: \(49\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(768))\).

Total New Old
Modular forms 33472 13928 19544
Cusp forms 32064 13720 18344
Eisenstein series 1408 208 1200

Trace form

\( 13720q - 24q^{3} - 64q^{4} - 32q^{6} - 48q^{7} - 40q^{9} + O(q^{10}) \) \( 13720q - 24q^{3} - 64q^{4} - 32q^{6} - 48q^{7} - 40q^{9} - 64q^{10} - 32q^{12} - 64q^{13} - 24q^{15} - 64q^{16} - 32q^{18} - 48q^{19} - 32q^{21} - 64q^{22} - 32q^{24} - 80q^{25} - 24q^{27} - 64q^{28} - 32q^{30} - 64q^{31} - 56q^{33} - 64q^{34} - 32q^{36} - 64q^{37} - 24q^{39} - 64q^{40} - 32q^{42} - 48q^{43} - 32q^{45} - 64q^{46} - 32q^{48} - 488q^{49} - 408q^{51} - 64q^{52} - 640q^{53} - 32q^{54} - 1072q^{55} - 424q^{57} - 64q^{58} - 512q^{59} - 32q^{60} - 320q^{61} - 16q^{63} - 64q^{64} + 256q^{65} - 32q^{66} + 592q^{67} + 352q^{69} - 64q^{70} + 1024q^{71} - 32q^{72} + 1200q^{73} + 744q^{75} - 64q^{76} + 896q^{77} - 32q^{78} + 976q^{79} - 48q^{81} - 64q^{82} - 32q^{84} + 336q^{85} - 24q^{87} - 64q^{88} - 32q^{90} - 48q^{91} + 112q^{93} - 64q^{94} - 32q^{96} - 112q^{97} - 24q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(768))\)

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
768.3.b \(\chi_{768}(127, \cdot)\) 768.3.b.a 4 1
768.3.b.b 4
768.3.b.c 4
768.3.b.d 4
768.3.b.e 8
768.3.b.f 8
768.3.e \(\chi_{768}(257, \cdot)\) 768.3.e.a 2 1
768.3.e.b 2
768.3.e.c 2
768.3.e.d 2
768.3.e.e 2
768.3.e.f 2
768.3.e.g 4
768.3.e.h 4
768.3.e.i 4
768.3.e.j 4
768.3.e.k 4
768.3.e.l 4
768.3.e.m 4
768.3.e.n 4
768.3.e.o 8
768.3.e.p 8
768.3.g \(\chi_{768}(511, \cdot)\) 768.3.g.a 2 1
768.3.g.b 2
768.3.g.c 4
768.3.g.d 4
768.3.g.e 4
768.3.g.f 4
768.3.g.g 4
768.3.g.h 8
768.3.h \(\chi_{768}(641, \cdot)\) 768.3.h.a 2 1
768.3.h.b 2
768.3.h.c 4
768.3.h.d 4
768.3.h.e 8
768.3.h.f 8
768.3.h.g 16
768.3.h.h 16
768.3.i \(\chi_{768}(65, \cdot)\) n/a 128 2
768.3.l \(\chi_{768}(319, \cdot)\) 768.3.l.a 8 2
768.3.l.b 8
768.3.l.c 8
768.3.l.d 8
768.3.l.e 16
768.3.l.f 16
768.3.m \(\chi_{768}(31, \cdot)\) n/a 128 4
768.3.p \(\chi_{768}(161, \cdot)\) n/a 240 4
768.3.q \(\chi_{768}(17, \cdot)\) n/a 496 8
768.3.t \(\chi_{768}(79, \cdot)\) n/a 256 8
768.3.u \(\chi_{768}(7, \cdot)\) None 0 16
768.3.x \(\chi_{768}(41, \cdot)\) None 0 16
768.3.y \(\chi_{768}(5, \cdot)\) n/a 8128 32
768.3.bb \(\chi_{768}(19, \cdot)\) n/a 4096 32

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

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(768))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(768)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 7}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(384))\)\(^{\oplus 2}\)