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

• Label: 384.4.d
• Level: $384$
• Weight: $4$
• Character orbit: 384.d
• Rep. character: $\chi_{384}(193,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $6$
• Sturm bound: $256$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(256\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	208	24	184
Cusp forms	176	24	152
Eisenstein series	32	0	32

Trace form

\( 24 q - 216 q^{9} + 208 q^{17} - 424 q^{25} + 944 q^{41} - 264 q^{49} + 672 q^{57} + 832 q^{65} - 2640 q^{73} + 1944 q^{81} + 528 q^{89} - 3440 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
384.4.d.a	$384$	$4$	384.d	8.b	$2$	$2$	$2$	$22.657$	\(\Q(\sqrt{-1}) \)	None		✓			384.4.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 i q^{3}+8 i q^{5}-12 q^{7}-9 q^{9}+\cdots\)
384.4.d.b	$384$	$4$	384.d	8.b	$2$	$2$	$2$	$22.657$	\(\Q(\sqrt{-1}) \)	None		✓			384.4.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{3}+8 i q^{5}+12 q^{7}-9 q^{9}+\cdots\)
384.4.d.c	$384$	$4$	384.d	8.b	$2$	$4$	$4$	$22.657$	\(\Q(i, \sqrt{13})\)	None		✓			384.4.d.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-32\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-3\beta _{1}q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+(-8+\cdots)q^{7}+\cdots\)
384.4.d.d	$384$	$4$	384.d	8.b	$2$	$4$	$4$	$22.657$	\(\Q(\zeta_{8})\)	None		✓			384.4.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta_1 q^{3}+\beta_{2} q^{5}-5\beta_{3} q^{7}-9 q^{9}+\cdots\)
384.4.d.e	$384$	$4$	384.d	8.b	$2$	$4$	$4$	$22.657$	\(\Q(i, \sqrt{13})\)	None		✓			384.4.d.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(32\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+(8-\beta _{3})q^{7}+\cdots\)
384.4.d.f	$384$	$4$	384.d	8.b	$2$	$8$	$8$	$22.657$	8.0.1534132224.8	None		✓	✓		384.4.d.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{6}q^{5}-\beta _{2}q^{7}-9q^{9}+(-4\beta _{1}+\cdots)q^{11}+\cdots\)

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

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