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 Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
768.2.f.a 768.f 24.f $4$ $6.133$ \(\Q(\zeta_{8})\) None 96.2.c.a \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta_{2}-1)q^{3}-\beta_{3} q^{5}+\beta_1 q^{7}+\cdots\)
768.2.f.b 768.f 24.f $4$ $6.133$ \(\Q(i, \sqrt{5})\) None 384.2.c.a \(0\) \(-2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{3})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
768.2.f.c 768.f 24.f $4$ $6.133$ \(\Q(i, \sqrt{5})\) None 384.2.c.a \(0\) \(-2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+\cdots\)
768.2.f.d 768.f 24.f $4$ $6.133$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) 48.2.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta_1 q^{3}+\beta_{2} q^{7}+3 q^{9}+\beta_{3} q^{13}+\cdots\)
768.2.f.e 768.f 24.f $4$ $6.133$ \(\Q(i, \sqrt{5})\) None 384.2.c.a \(0\) \(2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta _{3})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
768.2.f.f 768.f 24.f $4$ $6.133$ \(\Q(i, \sqrt{5})\) None 384.2.c.a \(0\) \(2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
768.2.f.g 768.f 24.f $4$ $6.133$ \(\Q(\zeta_{8})\) None 96.2.c.a \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta_{2}+1)q^{3}-\beta_{3} q^{5}-\beta_1 q^{7}+\cdots\)

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

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