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

• Label: 384.3.b
• Level: $384$
• Weight: $3$
• Character orbit: 384.b
• Rep. character: $\chi_{384}(319,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $3$
• Sturm bound: $192$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(192\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(384, [\chi])\).

	Total	New	Old
Modular forms	144	16	128
Cusp forms	112	16	96
Eisenstein series	32	0	32

Trace form

\( 16 q + 48 q^{9} - 32 q^{17} - 176 q^{25} + 160 q^{41} - 48 q^{49} + 192 q^{57} + 128 q^{65} - 608 q^{73} + 144 q^{81} + 480 q^{89} - 672 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.3.b.a	$384$	$3$	384.b	8.d	$2$	$4$	$4$	$10.463$	\(\Q(\zeta_{12})\)	None		✓			384.3.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}+\beta_{3} q^{5}+\beta_{2} q^{7}+3 q^{9}+\cdots\)
384.3.b.b	$384$	$3$	384.b	8.d	$2$	$4$	$4$	$10.463$	\(\Q(\zeta_{12})\)	None		✓			384.3.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}+\beta_{2} q^{5}-\beta_{3} q^{7}+3 q^{9}+\cdots\)
384.3.b.c	$384$	$3$	384.b	8.d	$2$	$8$	$8$	$10.463$	\(\Q(\zeta_{24})\)	None		✓	✓		384.3.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{18}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_{2} q^{3}+(\beta_{5}-\beta_{3})q^{5}+\beta_1 q^{7}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(384, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 2}\)