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

• Label: 384.2.d
• Level: $384$
• Weight: $2$
• Character orbit: 384.d
• Rep. character: $\chi_{384}(193,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $128$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(128\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	80	8	72
Cusp forms	48	8	40
Eisenstein series	32	0	32

Trace form

\( 8 q - 8 q^{9} - 16 q^{17} + 8 q^{25} + 16 q^{41} + 40 q^{49} - 32 q^{57} - 64 q^{65} + 16 q^{73} + 8 q^{81} + 48 q^{89} + 48 q^{97}+O(q^{100}) \)

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.d.a	$384$	$2$	384.d	8.b	$2$	$2$	$2$	$3.066$	\(\Q(\sqrt{-1}) \)	None		✓			384.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-4 q^{7}-q^{9}+4 i q^{11}+4 i q^{13}+\cdots\)
384.2.d.b	$384$	$2$	384.d	8.b	$2$	$2$	$2$	$3.066$	\(\Q(\sqrt{-1}) \)	None		✓			384.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+4 q^{7}-q^{9}+4 i q^{11}-4 i q^{13}+\cdots\)
384.2.d.c	$384$	$2$	384.d	8.b	$2$	$4$	$4$	$3.066$	\(\Q(\zeta_{8})\)	None		✓	✓		384.2.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{3}+\beta_{2} q^{5}+\beta_{3} q^{7}-q^{9}+\cdots\)

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

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