Space of modular forms of level 384, weight 2, and Character 384.f

• Label: 384.2.f
• Level: $384$
• Weight: $2$
• Character orbit: 384.f
• Rep. character: $\chi_{384}(191,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $4$
• Sturm bound: $128$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 24 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(128\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(5\), \(7\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(384, [\chi])\).

	Total	New	Old
Modular forms	80	16	64
Cusp forms	48	16	32
Eisenstein series	32	0	32

Trace form

\( 16 q + 16 q^{25} + 16 q^{33} - 48 q^{49} + 16 q^{57} - 32 q^{73} - 48 q^{81}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(384, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.2.f.a	$384$	$2$	384.f	24.f	$2$	$4$	$4$	$3.066$	\(\Q(\sqrt{2}, \sqrt{-3})\)	\(\Q(\sqrt{-6}) \)		✓			384.2.f.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{3}+\beta _{1}q^{5}-\beta _{3}q^{7}-3q^{9}-2\beta _{2}q^{11}+\cdots\)
384.2.f.b	$384$	$2$	384.f	24.f	$2$	$4$	$4$	$3.066$	\(\Q(\zeta_{8})\)	None		✓			384.2.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{3}+(\beta_{2}+\beta_1)q^{5}-\beta_{3} q^{7}+\cdots\)
384.2.f.c	$384$	$2$	384.f	24.f	$2$	$4$	$4$	$3.066$	\(\Q(\zeta_{8})\)	\(\Q(\sqrt{-2}) \)		✓			384.2.f.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta_1 q^{3}+(-\beta_{2}+1)q^{9}+(\beta_{3}-2\beta_1)q^{11}+\cdots\)
384.2.f.d	$384$	$2$	384.f	24.f	$2$	$4$	$4$	$3.066$	\(\Q(\zeta_{8})\)	None		✓			384.2.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{3}+(-\beta_{2}-\beta_1)q^{5}+\beta_{3} q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(384, [\chi])\) into lower level spaces

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