Space of modular forms of level 320, weight 6, and Character 320.d

• Label: 320.6.d
• Level: $320$
• Weight: $6$
• Character orbit: 320.d
• Rep. character: $\chi_{320}(161,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $4$
• Sturm bound: $288$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(288\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(320, [\chi])\).

	Total	New	Old
Modular forms	252	40	212
Cusp forms	228	40	188
Eisenstein series	24	0	24

Trace form

\( 40 q - 3240 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(320, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
320.6.d.a	$320$	$6$	320.d	8.b	$2$	$8$	$8$	$51.323$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-320\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+5^{2}\beta _{1}q^{5}+(-40-3\beta _{2}+\cdots)q^{7}+\cdots\)
320.6.d.b	$320$	$6$	320.d	8.b	$2$	$8$	$8$	$51.323$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(320\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}-5^{2}\beta _{1}q^{5}+(40+3\beta _{2}+\beta _{6}+\cdots)q^{7}+\cdots\)
320.6.d.c	$320$	$6$	320.d	8.b	$2$	$12$	$12$	$51.323$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-268\)	$2^{30}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3\beta _{1}-\beta _{4})q^{3}-5^{2}\beta _{1}q^{5}+(-22+\cdots)q^{7}+\cdots\)
320.6.d.d	$320$	$6$	320.d	8.b	$2$	$12$	$12$	$51.323$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(268\)	$2^{30}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3\beta _{1}-\beta _{4})q^{3}+5^{2}\beta _{1}q^{5}+(22-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(320, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(320, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(160, [\chi])\)\(^{\oplus 2}\)