Space of modular forms of level 384, weight 6, and Character 384.f

• Label: 384.6.f
• Level: $384$
• Weight: $6$
• Character orbit: 384.f
• Rep. character: $\chi_{384}(191,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $6$
• Sturm bound: $384$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(384\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(5\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(384, [\chi])\).

	Total	New	Old
Modular forms	336	80	256
Cusp forms	304	80	224
Eisenstein series	32	0	32

Trace form

\( 80 q + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(384, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.6.f.a	$384$	$6$	384.f	24.f	$2$	$4$	$4$	$61.587$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+9\beta _{2}q^{3}+11\beta _{1}q^{5}-19\beta _{3}q^{7}-3^{5}q^{9}+\cdots\)
384.6.f.b	$384$	$6$	384.f	24.f	$2$	$4$	$4$	$61.587$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q+(-7\zeta_{8}-2\zeta_{8}^{3})q^{3}+(241+11\zeta_{8}^{2}+\cdots)q^{9}+\cdots\)
384.6.f.c	$384$	$6$	384.f	24.f	$2$	$16$	$16$	$61.587$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{60}\cdot 3^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-\beta _{9}q^{5}-\beta _{10}q^{7}+(-105+\cdots)q^{9}+\cdots\)
384.6.f.d	$384$	$6$	384.f	24.f	$2$	$16$	$16$	$61.587$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{60}\cdot 3^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+\beta _{12}q^{5}+\beta _{3}q^{7}+(111-\beta _{1}+\cdots)q^{9}+\cdots\)
384.6.f.e	$384$	$6$	384.f	24.f	$2$	$20$	$20$	$61.587$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{87}\cdot 3^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(\beta _{2}-\beta _{4})q^{5}+\beta _{5}q^{7}+(-2+\cdots)q^{9}+\cdots\)
384.6.f.f	$384$	$6$	384.f	24.f	$2$	$20$	$20$	$61.587$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{87}\cdot 3^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-\beta _{2}+\beta _{4})q^{5}+\beta _{5}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(384, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 5}\)