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

• Label: 768.5.g
• Level: $768$
• Weight: $5$
• Character orbit: 768.g
• Rep. character: $\chi_{768}(511,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $11$
• Sturm bound: $640$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(640\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\)

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.g.a	$768$	$5$	768.g	4.b	$2$	$2$	$2$	$79.388$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-24\)	\(0\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}-12q^{5}+12\zeta_{6}q^{7}-3^{3}q^{9}+\cdots\)
768.5.g.b	$768$	$5$	768.g	4.b	$2$	$2$	$2$	$79.388$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(24\)	\(0\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}+12q^{5}-12\zeta_{6}q^{7}-3^{3}q^{9}+\cdots\)
768.5.g.c	$768$	$5$	768.g	4.b	$2$	$4$	$4$	$79.388$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(-32\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-8-\beta _{3})q^{5}+(\beta _{1}+5\beta _{2}+\cdots)q^{7}+\cdots\)
768.5.g.d	$768$	$5$	768.g	4.b	$2$	$4$	$4$	$79.388$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{3}q^{7}-3^{3}q^{9}+\cdots\)
768.5.g.e	$768$	$5$	768.g	4.b	$2$	$4$	$4$	$79.388$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\zeta_{12}^{2}q^{3}+7\zeta_{12}^{3}q^{5}+29\zeta_{12}q^{7}+\cdots\)
768.5.g.f	$768$	$5$	768.g	4.b	$2$	$4$	$4$	$79.388$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{5}-\zeta_{12}q^{7}+\cdots\)
768.5.g.g	$768$	$5$	768.g	4.b	$2$	$4$	$4$	$79.388$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$2$	$0$	\(0\)	\(0\)	\(32\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(8-\beta _{3})q^{5}+(-\beta _{1}+5\beta _{2}+\cdots)q^{7}+\cdots\)
768.5.g.h	$768$	$5$	768.g	4.b	$2$	$8$	$8$	$79.388$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-96\)	\(0\)	$2^{24}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-12-\beta _{3})q^{5}+(-\beta _{4}+\beta _{5}+\cdots)q^{7}+\cdots\)
768.5.g.i	$768$	$5$	768.g	4.b	$2$	$8$	$8$	$79.388$	8.0.\(\cdots\).10	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{20}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(2\beta _{2}-\beta _{5})q^{5}+(5\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
768.5.g.j	$768$	$5$	768.g	4.b	$2$	$8$	$8$	$79.388$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(96\)	\(0\)	$2^{24}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(12+\beta _{3})q^{5}+(\beta _{4}-\beta _{5}+\beta _{7})q^{7}+\cdots\)
768.5.g.k	$768$	$5$	768.g	4.b	$2$	$16$	$16$	$79.388$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{88}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{6}q^{7}-3^{3}q^{9}+\cdots\)

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

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