Space of modular forms of level 384, weight 4, and Character 384.f

• Label: 384.4.f
• Level: $384$
• Weight: $4$
• Character orbit: 384.f
• Rep. character: $\chi_{384}(191,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $9$
• Sturm bound: $256$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(256\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(5\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(384, [\chi])\).

	Total	New	Old
Modular forms	208	48	160
Cusp forms	176	48	128
Eisenstein series	32	0	32

Trace form

\( 48 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(384, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.4.f.a	$384$	$4$	384.f	24.f	$2$	$4$	$4$	$22.657$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}-7\beta _{1}q^{5}-\beta _{3}q^{7}-3^{3}q^{9}+\cdots\)
384.4.f.b	$384$	$4$	384.f	24.f	$2$	$4$	$4$	$22.657$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\zeta_{8}^{2}q^{3}+(-23+5\zeta_{8})q^{9}+(3\zeta_{8}^{2}+\cdots)q^{11}+\cdots\)
384.4.f.c	$384$	$4$	384.f	24.f	$2$	$4$	$4$	$22.657$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\zeta_{8}+2\zeta_{8}^{2})q^{3}-2\zeta_{8}^{2}q^{5}+2\zeta_{8}^{3}q^{7}+\cdots\)
384.4.f.d	$384$	$4$	384.f	24.f	$2$	$4$	$4$	$22.657$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\zeta_{8}-2\zeta_{8}^{2})q^{3}-2\zeta_{8}^{2}q^{5}-2\zeta_{8}^{3}q^{7}+\cdots\)
384.4.f.e	$384$	$4$	384.f	24.f	$2$	$4$	$4$	$22.657$	\(\Q(i, \sqrt{26})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+2\beta _{2})q^{5}+\beta _{3}q^{7}+\cdots\)
384.4.f.f	$384$	$4$	384.f	24.f	$2$	$4$	$4$	$22.657$	\(\Q(i, \sqrt{26})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(\beta _{1}+2\beta _{2})q^{5}+\beta _{3}q^{7}+(5^{2}+\cdots)q^{9}+\cdots\)
384.4.f.g	$384$	$4$	384.f	24.f	$2$	$8$	$8$	$22.657$	8.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{3}+\beta _{4})q^{3}+(-2\beta _{3}+\beta _{6})q^{5}+(-3\beta _{2}+\cdots)q^{7}+\cdots\)
384.4.f.h	$384$	$4$	384.f	24.f	$2$	$8$	$8$	$22.657$	8.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}+(-2\beta _{3}+\beta _{6})q^{5}+(-3\beta _{2}+\cdots)q^{7}+\cdots\)
384.4.f.i	$384$	$4$	384.f	24.f	$2$	$8$	$8$	$22.657$	8.0.\(\cdots\).6	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{2})q^{3}+\beta _{3}q^{5}+\beta _{7}q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

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