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

• Label: 768.7.g
• Level: $768$
• Weight: $7$
• Character orbit: 768.g
• Rep. character: $\chi_{768}(511,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $12$
• Sturm bound: $896$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(896\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\)

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.g.a	$768$	$7$	768.g	4.b	$2$	$2$	$2$	$176.682$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-392\)	\(0\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\zeta_{6}q^{3}-14^{2}q^{5}+52\zeta_{6}q^{7}-3^{5}q^{9}+\cdots\)
768.7.g.b	$768$	$7$	768.g	4.b	$2$	$2$	$2$	$176.682$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(392\)	\(0\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\zeta_{6}q^{3}+14^{2}q^{5}+52\zeta_{6}q^{7}-3^{5}q^{9}+\cdots\)
768.7.g.c	$768$	$7$	768.g	4.b	$2$	$4$	$4$	$176.682$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(0\)	\(-80\)	\(0\)	$2^{12}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-20q^{5}+(12\beta _{1}-\beta _{2})q^{7}+\cdots\)
768.7.g.d	$768$	$7$	768.g	4.b	$2$	$4$	$4$	$176.682$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\zeta_{12}^{2}q^{3}+5\zeta_{12}q^{5}+\zeta_{12}^{3}q^{7}+\cdots\)
768.7.g.e	$768$	$7$	768.g	4.b	$2$	$4$	$4$	$176.682$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(0\)	\(80\)	\(0\)	$2^{12}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+20q^{5}+(12\beta _{1}-\beta _{2})q^{7}+\cdots\)
768.7.g.f	$768$	$7$	768.g	4.b	$2$	$8$	$8$	$176.682$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\beta _{1}q^{3}+(-4\beta _{2}-\beta _{3})q^{5}+(-83\beta _{4}+\cdots)q^{7}+\cdots\)
768.7.g.g	$768$	$7$	768.g	4.b	$2$	$8$	$8$	$176.682$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{32}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}-3^{5}q^{9}+\cdots\)
768.7.g.h	$768$	$7$	768.g	4.b	$2$	$8$	$8$	$176.682$	8.0.\(\cdots\).10	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\beta _{1}q^{3}+(10\beta _{2}+5\beta _{3})q^{5}+(5^{2}\beta _{4}+\cdots)q^{7}+\cdots\)
768.7.g.i	$768$	$7$	768.g	4.b	$2$	$8$	$8$	$176.682$	8.0.\(\cdots\).4	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-\beta _{2}+\beta _{4})q^{5}+(13\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots\)
768.7.g.j	$768$	$7$	768.g	4.b	$2$	$12$	$12$	$176.682$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-176\)	\(0\)	$2^{60}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-15-\beta _{1})q^{5}+\beta _{4}q^{7}+\cdots\)
768.7.g.k	$768$	$7$	768.g	4.b	$2$	$12$	$12$	$176.682$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(176\)	\(0\)	$2^{60}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(15+\beta _{1})q^{5}-\beta _{4}q^{7}-3^{5}q^{9}+\cdots\)
768.7.g.l	$768$	$7$	768.g	4.b	$2$	$24$	$24$	$176.682$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

\( S_{7}^{\mathrm{old}}(768, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 14}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)