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

• Label: 384.3.h
• Level: $384$
• Weight: $3$
• Character orbit: 384.h
• Rep. character: $\chi_{384}(65,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $7$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(11\)

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.h.a	$384$	$3$	384.h	24.h	$2$	$2$	$2$	$10.463$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-6}) \)	✓	$4$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-3q^{3}+\beta q^{5}+\beta q^{7}+9q^{9}-10q^{11}+\cdots\)
384.3.h.b	$384$	$3$	384.h	24.h	$2$	$2$	$2$	$10.463$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(-1+\beta )q^{3}+(-7-2\beta )q^{9}-14q^{11}+\cdots\)
384.3.h.c	$384$	$3$	384.h	24.h	$2$	$2$	$2$	$10.463$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+(1+\beta )q^{3}+(-7+2\beta )q^{9}+14q^{11}+\cdots\)
384.3.h.d	$384$	$3$	384.h	24.h	$2$	$2$	$2$	$10.463$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-6}) \)	✓	$4$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+3q^{3}+\beta q^{5}-\beta q^{7}+9q^{9}+10q^{11}+\cdots\)
384.3.h.e	$384$	$3$	384.h	24.h	$2$	$4$	$4$	$10.463$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-4q^{5}+(-2\beta _{1}-2\beta _{3})q^{7}+\cdots\)
384.3.h.f	$384$	$3$	384.h	24.h	$2$	$4$	$4$	$10.463$	\(\Q(i, \sqrt{5})\)	None		$4$	$0$	\(0\)	\(0\)	\(16\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+4q^{5}+(-2\beta _{1}-2\beta _{3})q^{7}+\cdots\)
384.3.h.g	$384$	$3$	384.h	24.h	$2$	$16$	$16$	$10.463$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{46}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{8}q^{3}+\beta _{2}q^{5}+\beta _{1}q^{7}+(-1-\beta _{5}+\cdots)q^{9}+\cdots\)

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

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