Space of modular forms of level 384, weight 2, and trivial character

• Label: 384.2.a
• Level: $384$
• Weight: $2$
• Character orbit: 384.a
• Rep. character: $\chi_{384}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $8$
• Sturm bound: $128$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(128\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(384))\).

	Total	New	Old
Modular forms	80	8	72
Cusp forms	49	8	41
Eisenstein series	31	0	31

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(16\)	\(1\)	\(15\)		\(9\)	\(1\)	\(8\)		\(7\)	\(0\)	\(7\)
\(+\)	\(-\)	\(-\)		\(20\)	\(3\)	\(17\)		\(12\)	\(3\)	\(9\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(24\)	\(3\)	\(21\)		\(16\)	\(3\)	\(13\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)		\(20\)	\(1\)	\(19\)		\(12\)	\(1\)	\(11\)		\(8\)	\(0\)	\(8\)
Plus space		\(+\)		\(36\)	\(2\)	\(34\)		\(21\)	\(2\)	\(19\)		\(15\)	\(0\)	\(15\)
Minus space		\(-\)		\(44\)	\(6\)	\(38\)		\(28\)	\(6\)	\(22\)		\(16\)	\(0\)	\(16\)

Trace form

\( 8 q + 8 q^{9} + 16 q^{17} + 24 q^{25} + 16 q^{41} - 24 q^{49} - 32 q^{57} - 32 q^{65} - 80 q^{73} + 8 q^{81} - 48 q^{89} + 16 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(384))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3
384.2.a.a	$384$	$2$	384.a	1.a	$1$	$1$	$1$	$3.066$	\(\Q\)	None	✓	✓			384.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-4\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
384.2.a.b	$384$	$2$	384.a	1.a	$1$	$1$	$1$	$3.066$	\(\Q\)	None	✓	✓	✓	✓	384.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
384.2.a.c	$384$	$2$	384.a	1.a	$1$	$1$	$1$	$3.066$	\(\Q\)	None	✓	✓			384.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
384.2.a.d	$384$	$2$	384.a	1.a	$1$	$1$	$1$	$3.066$	\(\Q\)	None	✓	✓			384.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}-2q^{7}+q^{9}+4q^{11}+\cdots\)
384.2.a.e	$384$	$2$	384.a	1.a	$1$	$1$	$1$	$3.066$	\(\Q\)	None	✓	✓	✓	✓	384.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
384.2.a.f	$384$	$2$	384.a	1.a	$1$	$1$	$1$	$3.066$	\(\Q\)	None	✓	✓			384.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
384.2.a.g	$384$	$2$	384.a	1.a	$1$	$1$	$1$	$3.066$	\(\Q\)	None	✓	✓			384.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
384.2.a.h	$384$	$2$	384.a	1.a	$1$	$1$	$1$	$3.066$	\(\Q\)	None	✓	✓			384.2.a.a	$1$	$0$	\(0\)	\(1\)	\(4\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+2q^{7}+q^{9}-4q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(384))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(384)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)