Properties

Label 768.4.f
Level $768$
Weight $4$
Character orbit 768.f
Rep. character $\chi_{768}(383,\cdot)$
Character field $\Q$
Dimension $92$
Newform subspaces $9$
Sturm bound $512$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(512\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(5\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(768, [\chi])\).

Total New Old
Modular forms 408 100 308
Cusp forms 360 92 268
Eisenstein series 48 8 40

Trace form

\( 92 q + 4 q^{9} + O(q^{10}) \) \( 92 q + 4 q^{9} + 1908 q^{25} - 112 q^{33} - 4780 q^{49} - 104 q^{57} + 872 q^{73} - 4 q^{81} - 3176 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(768, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
768.4.f.a 768.f 24.f $4$ $45.313$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\zeta_{12}^{2}q^{3}+3\zeta_{12}^{3}q^{7}+3^{3}q^{9}+35\zeta_{12}q^{13}+\cdots\)
768.4.f.b 768.f 24.f $8$ $45.313$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{24}^{4}q^{3}+\zeta_{24}^{6}q^{5}+5\zeta_{24}^{3}q^{7}+\cdots\)
768.4.f.c 768.f 24.f $8$ $45.313$ 8.0.12960000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+\beta _{1}q^{5}+\beta _{2}q^{7}+(3-3\beta _{7})q^{9}+\cdots\)
768.4.f.d 768.f 24.f $12$ $45.313$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+\beta _{10}q^{5}+(\beta _{1}+\beta _{2})q^{7}+(2+\cdots)q^{9}+\cdots\)
768.4.f.e 768.f 24.f $12$ $45.313$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-2\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{5}q^{3}+(-1-\beta _{2})q^{5}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
768.4.f.f 768.f 24.f $12$ $45.313$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-2\) \(12\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{5}q^{3}+(1+\beta _{2})q^{5}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
768.4.f.g 768.f 24.f $12$ $45.313$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(2\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{3}+(-1-\beta _{2})q^{5}+(\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)
768.4.f.h 768.f 24.f $12$ $45.313$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(2\) \(12\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{3}+(1+\beta _{2})q^{5}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
768.4.f.i 768.f 24.f $12$ $45.313$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+\beta _{10}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(768, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)