Space of modular forms of level 384, weight 5, and Character 384.h

• Label: 384.5.h
• Level: $384$
• Weight: $5$
• Character orbit: 384.h
• Rep. character: $\chi_{384}(65,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $8$
• Sturm bound: $320$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	384.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(320\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(384, [\chi])\).

	Total	New	Old
Modular forms	272	64	208
Cusp forms	240	64	176
Eisenstein series	32	0	32

Trace form

\( 64 q + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(384, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.5.h.a	$384$	$5$	384.h	24.h	$2$	$2$	$2$	$39.694$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-6}) \)	✓	$4$	$0$	\(0\)	\(-18\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q-9q^{3}+\beta q^{5}-5\beta q^{7}+3^{4}q^{9}-142q^{11}+\cdots\)
384.5.h.b	$384$	$5$	384.h	24.h	$2$	$2$	$2$	$39.694$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(-14\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-7-\beta )q^{3}+(17+14\beta )q^{9}+46q^{11}+\cdots\)
384.5.h.c	$384$	$5$	384.h	24.h	$2$	$2$	$2$	$39.694$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(14\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(7+\beta )q^{3}+(17+14\beta )q^{9}-46q^{11}+\cdots\)
384.5.h.d	$384$	$5$	384.h	24.h	$2$	$2$	$2$	$39.694$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-6}) \)	✓	$4$	$0$	\(0\)	\(18\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+9q^{3}+\beta q^{5}+5\beta q^{7}+3^{4}q^{9}+142q^{11}+\cdots\)
384.5.h.e	$384$	$5$	384.h	24.h	$2$	$4$	$4$	$39.694$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3-\beta _{1})q^{3}+\beta _{2}q^{5}+\beta _{2}q^{7}+(-63+\cdots)q^{9}+\cdots\)
384.5.h.f	$384$	$5$	384.h	24.h	$2$	$4$	$4$	$39.694$	\(\Q(\sqrt{-2}, \sqrt{-5})\)	None		$4$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{8}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3-\beta _{1})q^{3}-\beta _{2}q^{5}+\beta _{2}q^{7}+(-63+\cdots)q^{9}+\cdots\)
384.5.h.g	$384$	$5$	384.h	24.h	$2$	$16$	$16$	$39.694$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{70}\cdot 3^{4}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+\beta _{8}q^{5}+\beta _{7}q^{7}+(-7-\beta _{5}+\cdots)q^{9}+\cdots\)
384.5.h.h	$384$	$5$	384.h	24.h	$2$	$32$	$32$	$39.694$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{5}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

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