Properties

Label 768.3.l
Level $768$
Weight $3$
Character orbit 768.l
Rep. character $\chi_{768}(319,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $64$
Newform subspaces $6$
Sturm bound $384$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 768.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(384\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(768, [\chi])\).

Total New Old
Modular forms 560 64 496
Cusp forms 464 64 400
Eisenstein series 96 0 96

Trace form

\( 64q + O(q^{10}) \) \( 64q + 448q^{49} + 384q^{65} - 576q^{81} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(768, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
768.3.l.a \(8\) \(20.926\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(-16\) \(0\) \(q+\zeta_{24}^{3}q^{3}+(-2+2\zeta_{24}^{2})q^{5}+(\zeta_{24}+\cdots)q^{7}+\cdots\)
768.3.l.b \(8\) \(20.926\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(-8\) \(0\) \(q-\zeta_{24}q^{3}+(-1-\zeta_{24}^{2}-\zeta_{24}^{5})q^{5}+\cdots\)
768.3.l.c \(8\) \(20.926\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(8\) \(0\) \(q-\zeta_{24}q^{3}+(1+\zeta_{24}^{2}+\zeta_{24}^{5})q^{5}+\cdots\)
768.3.l.d \(8\) \(20.926\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(16\) \(0\) \(q-\zeta_{24}q^{3}+(2+2\zeta_{24}^{2})q^{5}+(-\zeta_{24}+\cdots)q^{7}+\cdots\)
768.3.l.e \(16\) \(20.926\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(-8\) \(0\) \(q+\beta _{8}q^{3}+(-1+\beta _{10})q^{5}+(-2\beta _{7}+\cdots)q^{7}+\cdots\)
768.3.l.f \(16\) \(20.926\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(8\) \(0\) \(q+\beta _{7}q^{3}+(1+\beta _{1}-\beta _{6})q^{5}+(2\beta _{7}+2\beta _{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(768, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)