Space of modular forms of level 768, weight 5, and Character 768.b

• Label: 768.5.b
• Level: $768$
• Weight: $5$
• Character orbit: 768.b
• Rep. character: $\chi_{768}(127,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $9$
• Sturm bound: $640$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(640\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(768, [\chi])\).

	Total	New	Old
Modular forms	536	64	472
Cusp forms	488	64	424
Eisenstein series	48	0	48

Trace form

\( 64 q + 1728 q^{9} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(768, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.5.b.a	$768$	$5$	768.b	8.d	$2$	$4$	$4$	$79.388$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{2}q^{3}+(-11\zeta_{12}-2\zeta_{12}^{3})q^{5}+\cdots\)
768.5.b.b	$768$	$5$	768.b	8.d	$2$	$4$	$4$	$79.388$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\zeta_{12}q^{3}+(2\zeta_{12}^{2}-5\zeta_{12}^{3})q^{5}+\cdots\)
768.5.b.c	$768$	$5$	768.b	8.d	$2$	$4$	$4$	$79.388$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\zeta_{12}q^{3}+21\zeta_{12}^{3}q^{5}-22\zeta_{12}^{2}q^{7}+\cdots\)
768.5.b.d	$768$	$5$	768.b	8.d	$2$	$4$	$4$	$79.388$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{2}q^{3}-3\zeta_{12}q^{5}-6\zeta_{12}^{3}q^{7}+\cdots\)
768.5.b.e	$768$	$5$	768.b	8.d	$2$	$4$	$4$	$79.388$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\zeta_{12}q^{3}+(2\zeta_{12}^{2}-5\zeta_{12}^{3})q^{5}+\cdots\)
768.5.b.f	$768$	$5$	768.b	8.d	$2$	$4$	$4$	$79.388$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}^{2}q^{3}+(-11\zeta_{12}-2\zeta_{12}^{3})q^{5}+\cdots\)
768.5.b.g	$768$	$5$	768.b	8.d	$2$	$8$	$8$	$79.388$	8.0.592240896.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{24}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(3\beta _{3}+\beta _{6})q^{5}+(-\beta _{2}-\beta _{5}+\cdots)q^{7}+\cdots\)
768.5.b.h	$768$	$5$	768.b	8.d	$2$	$16$	$16$	$79.388$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{70}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(\beta _{9}-\beta _{10})q^{5}+(-\beta _{9}-\beta _{10}+\cdots)q^{7}+\cdots\)
768.5.b.i	$768$	$5$	768.b	8.d	$2$	$16$	$16$	$79.388$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{70}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(\beta _{9}-\beta _{10})q^{5}+(\beta _{9}+\beta _{10}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(768, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)