Space of modular forms of level 384, weight 7, and Character 384.h

• Label: 384.7.h
• Level: $384$
• Weight: $7$
• Character orbit: 384.h
• Rep. character: $\chi_{384}(65,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $8$
• Sturm bound: $448$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
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:			\(448\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(384, [\chi])\).

	Total	New	Old
Modular forms	400	96	304
Cusp forms	368	96	272
Eisenstein series	32	0	32

Trace form

\( 96 q + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(384, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.7.h.a	$384$	$7$	384.h	24.h	$2$	$2$	$2$	$88.341$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-6}) \)	✓	$4$	$0$	\(0\)	\(-54\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q-3^{3}q^{3}+7\beta q^{5}-17\beta q^{7}+3^{6}q^{9}+\cdots\)
384.7.h.b	$384$	$7$	384.h	24.h	$2$	$2$	$2$	$88.341$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(-46\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q+(-23+\beta )q^{3}+(329-46\beta )q^{9}-2338q^{11}+\cdots\)
384.7.h.c	$384$	$7$	384.h	24.h	$2$	$2$	$2$	$88.341$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(46\)	\(0\)	\(0\)	$2\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q+(23-\beta )q^{3}+(329-46\beta )q^{9}+2338q^{11}+\cdots\)
384.7.h.d	$384$	$7$	384.h	24.h	$2$	$2$	$2$	$88.341$	\(\Q(\sqrt{6}) \)	\(\Q(\sqrt{-6}) \)	✓	$4$	$0$	\(0\)	\(54\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{3}q^{3}+7\beta q^{5}+17\beta q^{7}+3^{6}q^{9}+\cdots\)
384.7.h.e	$384$	$7$	384.h	24.h	$2$	$8$	$8$	$88.341$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(-72\)	\(0\)	\(0\)	$2^{26}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-9+\beta _{5})q^{3}-\beta _{6}q^{5}+(\beta _{3}+\beta _{6}+\cdots)q^{7}+\cdots\)
384.7.h.f	$384$	$7$	384.h	24.h	$2$	$8$	$8$	$88.341$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(72\)	\(0\)	\(0\)	$2^{26}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(9-\beta _{4})q^{3}-\beta _{6}q^{5}+(-\beta _{3}-\beta _{6}+\cdots)q^{7}+\cdots\)
384.7.h.g	$384$	$7$	384.h	24.h	$2$	$24$	$24$	$88.341$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
384.7.h.h	$384$	$7$	384.h	24.h	$2$	$48$	$48$	$88.341$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(384, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)