Properties

Label 768.2.j
Level $768$
Weight $2$
Character orbit 768.j
Rep. character $\chi_{768}(193,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $32$
Newform subspaces $6$
Sturm bound $256$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.j (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(256\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(13\)

Dimensions

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

Total New Old
Modular forms 304 32 272
Cusp forms 208 32 176
Eisenstein series 96 0 96

Trace form

\( 32 q + O(q^{10}) \) \( 32 q - 32 q^{49} + 192 q^{65} - 32 q^{81} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
768.2.j.a 768.j 16.e $4$ $6.133$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}q^{3}+(-2+2\zeta_{8}^{2})q^{5}+(3\zeta_{8}+3\zeta_{8}^{3})q^{7}+\cdots\)
768.2.j.b 768.j 16.e $4$ $6.133$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{3}q^{3}+(\zeta_{8}+\zeta_{8}^{3})q^{7}-\zeta_{8}^{2}q^{9}+\cdots\)
768.2.j.c 768.j 16.e $4$ $6.133$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{3}q^{3}+(-\zeta_{8}-\zeta_{8}^{3})q^{7}-\zeta_{8}^{2}q^{9}+\cdots\)
768.2.j.d 768.j 16.e $4$ $6.133$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{3}+(2-2\zeta_{8}^{2})q^{5}+(3\zeta_{8}+3\zeta_{8}^{3})q^{7}+\cdots\)
768.2.j.e 768.j 16.e $8$ $6.133$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{24}q^{3}+(-1+\zeta_{24}^{2}-\zeta_{24}^{6})q^{5}+\cdots\)
768.2.j.f 768.j 16.e $8$ $6.133$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{24}q^{3}+(1-\zeta_{24}^{2}+\zeta_{24}^{6})q^{5}+\cdots\)

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

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