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Properties

Label 768.5
Level 768
Weight 5
Dimension 27544
Nonzero newspaces 12
Sturm bound 163840
Trace bound 49

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Defining parameters

Level: \( N \) = \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) = \( 5 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(163840\)
Trace bound: \(49\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(\Gamma_1(768))\).

Total New Old
Modular forms 66240 27752 38488
Cusp forms 64832 27544 37288
Eisenstein series 1408 208 1200

Trace form

\( 27544 q - 24 q^{3} - 64 q^{4} - 32 q^{6} - 48 q^{7} - 40 q^{9} + O(q^{10}) \) \( 27544 q - 24 q^{3} - 64 q^{4} - 32 q^{6} - 48 q^{7} - 40 q^{9} - 64 q^{10} - 32 q^{12} - 64 q^{13} - 24 q^{15} - 64 q^{16} - 32 q^{18} - 48 q^{19} - 32 q^{21} - 64 q^{22} - 32 q^{24} - 80 q^{25} - 24 q^{27} - 64 q^{28} - 32 q^{30} - 64 q^{31} - 56 q^{33} - 64 q^{34} - 32 q^{36} - 64 q^{37} - 24 q^{39} - 64 q^{40} - 32 q^{42} - 48 q^{43} - 32 q^{45} - 64 q^{46} - 32 q^{48} - 19304 q^{49} - 16152 q^{51} - 64 q^{52} + 3840 q^{53} - 32 q^{54} + 47056 q^{55} + 29912 q^{57} - 64 q^{58} + 52224 q^{59} - 32 q^{60} + 30144 q^{61} - 16 q^{63} - 64 q^{64} - 32256 q^{65} - 32 q^{66} - 75568 q^{67} - 39200 q^{69} - 64 q^{70} - 79872 q^{71} - 32 q^{72} - 58960 q^{73} - 4632 q^{75} - 64 q^{76} + 37632 q^{77} - 32 q^{78} + 100304 q^{79} - 48 q^{81} - 64 q^{82} - 32 q^{84} + 9936 q^{85} - 24 q^{87} - 64 q^{88} - 32 q^{90} - 48 q^{91} + 1264 q^{93} - 64 q^{94} - 32 q^{96} - 112 q^{97} - 24 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{5}^{\mathrm{new}}(\Gamma_1(768))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
768.5.b \(\chi_{768}(127, \cdot)\) 768.5.b.a 4 1
768.5.b.b 4
768.5.b.c 4
768.5.b.d 4
768.5.b.e 4
768.5.b.f 4
768.5.b.g 8
768.5.b.h 16
768.5.b.i 16
768.5.e \(\chi_{768}(257, \cdot)\) n/a 124 1
768.5.g \(\chi_{768}(511, \cdot)\) 768.5.g.a 2 1
768.5.g.b 2
768.5.g.c 4
768.5.g.d 4
768.5.g.e 4
768.5.g.f 4
768.5.g.g 4
768.5.g.h 8
768.5.g.i 8
768.5.g.j 8
768.5.g.k 16
768.5.h \(\chi_{768}(641, \cdot)\) n/a 124 1
768.5.i \(\chi_{768}(65, \cdot)\) n/a 256 2
768.5.l \(\chi_{768}(319, \cdot)\) n/a 128 2
768.5.m \(\chi_{768}(31, \cdot)\) n/a 256 4
768.5.p \(\chi_{768}(161, \cdot)\) n/a 496 4
768.5.q \(\chi_{768}(17, \cdot)\) n/a 1008 8
768.5.t \(\chi_{768}(79, \cdot)\) n/a 512 8
768.5.u \(\chi_{768}(7, \cdot)\) None 0 16
768.5.x \(\chi_{768}(41, \cdot)\) None 0 16
768.5.y \(\chi_{768}(5, \cdot)\) n/a 16320 32
768.5.bb \(\chi_{768}(19, \cdot)\) n/a 8192 32

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

Decomposition of \(S_{5}^{\mathrm{old}}(\Gamma_1(768))\) into lower level spaces

\( S_{5}^{\mathrm{old}}(\Gamma_1(768)) \cong \) \(S_{5}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 14}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 7}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 10}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(\Gamma_1(384))\)\(^{\oplus 2}\)