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

• Label: 384.6.d
• Level: $384$
• Weight: $6$
• Character orbit: 384.d
• Rep. character: $\chi_{384}(193,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $10$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	336	40	296
Cusp forms	304	40	264
Eisenstein series	32	0	32

Trace form

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

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.d.a	$384$	$6$	384.d	8.b	$2$	$2$	$2$	$61.587$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-424\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+96iq^{5}-212q^{7}-3^{4}q^{9}+\cdots\)
384.6.d.b	$384$	$6$	384.d	8.b	$2$	$2$	$2$	$61.587$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(-136\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+24iq^{5}-68q^{7}-3^{4}q^{9}+\cdots\)
384.6.d.c	$384$	$6$	384.d	8.b	$2$	$2$	$2$	$61.587$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-72\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+80iq^{5}-6^{2}q^{7}-3^{4}q^{9}+\cdots\)
384.6.d.d	$384$	$6$	384.d	8.b	$2$	$2$	$2$	$61.587$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(72\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+80iq^{5}+6^{2}q^{7}-3^{4}q^{9}+\cdots\)
384.6.d.e	$384$	$6$	384.d	8.b	$2$	$2$	$2$	$61.587$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(136\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+9iq^{3}+24iq^{5}+68q^{7}-3^{4}q^{9}+\cdots\)
384.6.d.f	$384$	$6$	384.d	8.b	$2$	$2$	$2$	$61.587$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(424\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-9iq^{3}+96iq^{5}+212q^{7}-3^{4}q^{9}+\cdots\)
384.6.d.g	$384$	$6$	384.d	8.b	$2$	$4$	$4$	$61.587$	\(\Q(i, \sqrt{61})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+9\beta _{1}q^{3}+(-20\beta _{1}+\beta _{2})q^{5}+(-8+\cdots)q^{7}+\cdots\)
384.6.d.h	$384$	$6$	384.d	8.b	$2$	$4$	$4$	$61.587$	\(\Q(i, \sqrt{61})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(32\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-9\beta _{1}q^{3}+(-20\beta _{1}+\beta _{2})q^{5}+(8-5\beta _{3})q^{7}+\cdots\)
384.6.d.i	$384$	$6$	384.d	8.b	$2$	$8$	$8$	$61.587$	8.0.110166016.2	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{4}q^{5}+\beta _{5}q^{7}-3^{4}q^{9}+\cdots\)
384.6.d.j	$384$	$6$	384.d	8.b	$2$	$12$	$12$	$61.587$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{57}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{3}+\beta _{8}q^{5}+\beta _{4}q^{7}-3^{4}q^{9}+\cdots\)

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

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