Space of modular forms of level 384, weight 7, and Character 384.b

• Label: 384.7.b
• Level: $384$
• Weight: $7$
• Character orbit: 384.b
• Rep. character: $\chi_{384}(319,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $5$
• Sturm bound: $448$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(448\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(384, [\chi])\).

	Total	New	Old
Modular forms	400	48	352
Cusp forms	368	48	320
Eisenstein series	32	0	32

Trace form

\( 48 q + 11664 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(384, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.7.b.a	$384$	$7$	384.b	8.d	$2$	$4$	$4$	$88.341$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\zeta_{12}q^{3}-7^{2}\zeta_{12}^{2}q^{5}-13\zeta_{12}^{3}q^{7}+\cdots\)
384.7.b.b	$384$	$7$	384.b	8.d	$2$	$4$	$4$	$88.341$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\zeta_{12}q^{3}-5\zeta_{12}^{2}q^{5}-\zeta_{12}^{3}q^{7}+\cdots\)
384.7.b.c	$384$	$7$	384.b	8.d	$2$	$8$	$8$	$88.341$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{32}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}-\beta _{1}q^{5}+\beta _{5}q^{7}+3^{5}q^{9}+\cdots\)
384.7.b.d	$384$	$7$	384.b	8.d	$2$	$8$	$8$	$88.341$	8.0.1731891456.1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{32}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+5\beta _{3}q^{5}+(3\beta _{2}+\beta _{4})q^{7}+\cdots\)
384.7.b.e	$384$	$7$	384.b	8.d	$2$	$24$	$24$	$88.341$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

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