Space of modular forms of level 768, weight 7, and Character 768.b

• Label: 768.7.b
• Level: $768$
• Weight: $7$
• Character orbit: 768.b
• Rep. character: $\chi_{768}(127,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $10$
• Sturm bound: $896$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(896\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(768, [\chi])\).

	Total	New	Old
Modular forms	792	96	696
Cusp forms	744	96	648
Eisenstein series	48	0	48

Trace form

\( 96 q + 23328 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(768, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.7.b.a	$768$	$7$	768.b	8.d	$2$	$4$	$4$	$176.682$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\zeta_{12}^{2}q^{3}+(-5^{2}\zeta_{12}-10\zeta_{12}^{3})q^{5}+\cdots\)
768.7.b.b	$768$	$7$	768.b	8.d	$2$	$4$	$4$	$176.682$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\zeta_{12}q^{3}+3\zeta_{12}^{3}q^{5}+58\zeta_{12}^{2}q^{7}+\cdots\)
768.7.b.c	$768$	$7$	768.b	8.d	$2$	$4$	$4$	$176.682$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{2}q^{3}-45\zeta_{12}q^{5}-6\zeta_{12}^{3}q^{7}+\cdots\)
768.7.b.d	$768$	$7$	768.b	8.d	$2$	$4$	$4$	$176.682$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\zeta_{12}q^{3}+75\zeta_{12}^{3}q^{5}+94\zeta_{12}^{2}q^{7}+\cdots\)
768.7.b.e	$768$	$7$	768.b	8.d	$2$	$4$	$4$	$176.682$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\zeta_{12}^{2}q^{3}+(-5^{2}\zeta_{12}-10\zeta_{12}^{3})q^{5}+\cdots\)
768.7.b.f	$768$	$7$	768.b	8.d	$2$	$8$	$8$	$176.682$	8.0.\(\cdots\).5	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{26}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{3}+(7\beta _{1}-\beta _{4}+\beta _{7})q^{5}+(-22\beta _{1}+\cdots)q^{7}+\cdots\)
768.7.b.g	$768$	$7$	768.b	8.d	$2$	$8$	$8$	$176.682$	8.0.\(\cdots\).5	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{26}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{3}+(7\beta _{1}-\beta _{4}+\beta _{7})q^{5}+(22\beta _{1}+\cdots)q^{7}+\cdots\)
768.7.b.h	$768$	$7$	768.b	8.d	$2$	$12$	$12$	$176.682$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{64}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(-\beta _{5}-\beta _{8})q^{5}+(-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
768.7.b.i	$768$	$7$	768.b	8.d	$2$	$24$	$24$	$176.682$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
768.7.b.j	$768$	$7$	768.b	8.d	$2$	$24$	$24$	$176.682$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(768, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)