Space of modular forms of level 768, weight 3, and Character 768.g

• Label: 768.3.g
• Level: $768$
• Weight: $3$
• Character orbit: 768.g
• Rep. character: $\chi_{768}(511,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $8$
• Sturm bound: $384$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(384\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(768, [\chi])\).

	Total	New	Old
Modular forms	280	32	248
Cusp forms	232	32	200
Eisenstein series	48	0	48

Trace form

\( 32 q - 96 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(768, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.3.g.a	$768$	$3$	768.g	4.b	$2$	$2$	$2$	$20.926$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{6}q^{3}-4q^{5}+4\zeta_{6}q^{7}-3q^{9}-4\zeta_{6}q^{11}+\cdots\)
768.3.g.b	$768$	$3$	768.g	4.b	$2$	$2$	$2$	$20.926$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{6}q^{3}+4q^{5}-4\zeta_{6}q^{7}-3q^{9}-4\zeta_{6}q^{11}+\cdots\)
768.3.g.c	$768$	$3$	768.g	4.b	$2$	$4$	$4$	$20.926$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-4+\beta _{3})q^{5}-\beta _{1}q^{7}-3q^{9}+\cdots\)
768.3.g.d	$768$	$3$	768.g	4.b	$2$	$4$	$4$	$20.926$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{5}-\zeta_{12}q^{7}-3q^{9}+\cdots\)
768.3.g.e	$768$	$3$	768.g	4.b	$2$	$4$	$4$	$20.926$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}q^{3}-\zeta_{12}^{2}q^{5}+\zeta_{12}^{3}q^{7}-3q^{9}+\cdots\)
768.3.g.f	$768$	$3$	768.g	4.b	$2$	$4$	$4$	$20.926$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{5}-5\zeta_{12}q^{7}+\cdots\)
768.3.g.g	$768$	$3$	768.g	4.b	$2$	$4$	$4$	$20.926$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(16\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(4+\beta _{3})q^{5}-\beta _{1}q^{7}-3q^{9}+\cdots\)
768.3.g.h	$768$	$3$	768.g	4.b	$2$	$8$	$8$	$20.926$	8.0.22581504.2	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{1}q^{5}-\beta _{7}q^{7}-3q^{9}-\beta _{6}q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

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