Space of modular forms of level 384, weight 3, and Character 384.e

• Label: 384.3.e
• Level: $384$
• Weight: $3$
• Character orbit: 384.e
• Rep. character: $\chi_{384}(257,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $4$
• Sturm bound: $192$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(192\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(384, [\chi])\).

	Total	New	Old
Modular forms	144	32	112
Cusp forms	112	32	80
Eisenstein series	32	0	32

Trace form

\( 32 q + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(384, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.3.e.a	$384$	$3$	384.e	3.b	$2$	$8$	$8$	$10.463$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(-8\)	$2^{10}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+\beta _{3}q^{5}+(-1-\beta _{2})q^{7}+(\beta _{2}+\cdots)q^{9}+\cdots\)
384.3.e.b	$384$	$3$	384.e	3.b	$2$	$8$	$8$	$10.463$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(8\)	$2^{10}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+\beta _{2}q^{5}+(1+\beta _{3})q^{7}+(\beta _{3}+\cdots)q^{9}+\cdots\)
384.3.e.c	$384$	$3$	384.e	3.b	$2$	$8$	$8$	$10.463$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(-8\)	$2^{10}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-1-\beta _{3})q^{7}+(\beta _{3}+\cdots)q^{9}+\cdots\)
384.3.e.d	$384$	$3$	384.e	3.b	$2$	$8$	$8$	$10.463$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(8\)	$2^{10}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{3}q^{5}+(1+\beta _{2})q^{7}+(\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(384, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)