Properties

Label 768.2.f
Level $768$
Weight $2$
Character orbit 768.f
Rep. character $\chi_{768}(383,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $7$
Sturm bound $256$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(256\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(768, [\chi])\).

Total New Old
Modular forms 152 36 116
Cusp forms 104 28 76
Eisenstein series 48 8 40

Trace form

\( 28 q + 4 q^{9} + O(q^{10}) \) \( 28 q + 4 q^{9} + 20 q^{25} - 16 q^{33} + 20 q^{49} - 8 q^{57} + 40 q^{73} - 4 q^{81} - 40 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(768, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
768.2.f.a \(4\) \(6.133\) \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(0\) \(q+(-1-\zeta_{8}^{2})q^{3}-\zeta_{8}^{3}q^{5}+\zeta_{8}q^{7}+\cdots\)
768.2.f.b \(4\) \(6.133\) \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(-4\) \(0\) \(q+(-1+\beta _{3})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
768.2.f.c \(4\) \(6.133\) \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(4\) \(0\) \(q+(-1-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+\cdots\)
768.2.f.d \(4\) \(6.133\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{7}+3q^{9}+\zeta_{12}^{3}q^{13}+\cdots\)
768.2.f.e \(4\) \(6.133\) \(\Q(i, \sqrt{5})\) None \(0\) \(2\) \(-4\) \(0\) \(q+(1-\beta _{3})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
768.2.f.f \(4\) \(6.133\) \(\Q(i, \sqrt{5})\) None \(0\) \(2\) \(4\) \(0\) \(q+(1+\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
768.2.f.g \(4\) \(6.133\) \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(0\) \(q+(1+\zeta_{8}^{2})q^{3}-\zeta_{8}^{3}q^{5}-\zeta_{8}q^{7}+(-1+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

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