Space of modular forms of level 384, weight 4, and trivial character

• Label: 384.4.a
• Level: $384$
• Weight: $4$
• Character orbit: 384.a
• Rep. character: $\chi_{384}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $16$
• Sturm bound: $256$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(256\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(384))\).

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

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	Dim
\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(14\)
Minus space		\(-\)	\(10\)

Trace form

\( 24 q + 216 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(384))\) 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
384.4.a.a	$384$	$4$	384.a	1.a	$1$	$1$	$1$	$22.657$	\(\Q\)	None	✓	✓			384.4.a.a	$1$	$1$	\(0\)	\(-3\)	\(-8\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-8q^{5}-10q^{7}+9q^{9}+68q^{11}+\cdots\)
384.4.a.b	$384$	$4$	384.a	1.a	$1$	$1$	$1$	$22.657$	\(\Q\)	None	✓	✓			384.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-4\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-4q^{5}+10q^{7}+9q^{9}-4q^{11}+\cdots\)
384.4.a.c	$384$	$4$	384.a	1.a	$1$	$1$	$1$	$22.657$	\(\Q\)	None	✓	✓			384.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(4\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{5}-10q^{7}+9q^{9}-4q^{11}+\cdots\)
384.4.a.d	$384$	$4$	384.a	1.a	$1$	$1$	$1$	$22.657$	\(\Q\)	None	✓	✓			384.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(8\)	\(10\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+8q^{5}+10q^{7}+9q^{9}+68q^{11}+\cdots\)
384.4.a.e	$384$	$4$	384.a	1.a	$1$	$1$	$1$	$22.657$	\(\Q\)	None	✓	✓			384.4.a.a	$1$	$1$	\(0\)	\(3\)	\(-8\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-8q^{5}+10q^{7}+9q^{9}-68q^{11}+\cdots\)
384.4.a.f	$384$	$4$	384.a	1.a	$1$	$1$	$1$	$22.657$	\(\Q\)	None	✓	✓			384.4.a.b	$1$	$1$	\(0\)	\(3\)	\(-4\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{5}-10q^{7}+9q^{9}+4q^{11}+\cdots\)
384.4.a.g	$384$	$4$	384.a	1.a	$1$	$1$	$1$	$22.657$	\(\Q\)	None	✓	✓			384.4.a.b	$1$	$0$	\(0\)	\(3\)	\(4\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+4q^{5}+10q^{7}+9q^{9}+4q^{11}+\cdots\)
384.4.a.h	$384$	$4$	384.a	1.a	$1$	$1$	$1$	$22.657$	\(\Q\)	None	✓	✓			384.4.a.a	$1$	$1$	\(0\)	\(3\)	\(8\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+8q^{5}-10q^{7}+9q^{9}-68q^{11}+\cdots\)
384.4.a.i	$384$	$4$	384.a	1.a	$1$	$2$	$2$	$22.657$	\(\Q(\sqrt{7}) \)	None	✓	✓			384.4.a.i	$1$	$0$	\(0\)	\(-6\)	\(-16\)	\(4\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-8+\beta )q^{5}+(2+3\beta )q^{7}+\cdots\)
384.4.a.j	$384$	$4$	384.a	1.a	$1$	$2$	$2$	$22.657$	\(\Q(\sqrt{15}) \)	None	✓	✓			384.4.a.j	$1$	$0$	\(0\)	\(-6\)	\(-8\)	\(4\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-4+\beta )q^{5}+(2-\beta )q^{7}+9q^{9}+\cdots\)
384.4.a.k	$384$	$4$	384.a	1.a	$1$	$2$	$2$	$22.657$	\(\Q(\sqrt{15}) \)	None	✓	✓	✓		384.4.a.j	$1$	$1$	\(0\)	\(-6\)	\(8\)	\(-4\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(4+\beta )q^{5}+(-2-\beta )q^{7}+9q^{9}+\cdots\)
384.4.a.l	$384$	$4$	384.a	1.a	$1$	$2$	$2$	$22.657$	\(\Q(\sqrt{7}) \)	None	✓	✓			384.4.a.i	$1$	$0$	\(0\)	\(-6\)	\(16\)	\(-4\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(8+\beta )q^{5}+(-2+3\beta )q^{7}+\cdots\)
384.4.a.m	$384$	$4$	384.a	1.a	$1$	$2$	$2$	$22.657$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓		384.4.a.i	$1$	$1$	\(0\)	\(6\)	\(-16\)	\(-4\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-8+\beta )q^{5}+(-2-3\beta )q^{7}+\cdots\)
384.4.a.n	$384$	$4$	384.a	1.a	$1$	$2$	$2$	$22.657$	\(\Q(\sqrt{15}) \)	None	✓	✓			384.4.a.j	$1$	$0$	\(0\)	\(6\)	\(-8\)	\(-4\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-4+\beta )q^{5}+(-2+\beta )q^{7}+\cdots\)
384.4.a.o	$384$	$4$	384.a	1.a	$1$	$2$	$2$	$22.657$	\(\Q(\sqrt{15}) \)	None	✓	✓			384.4.a.j	$1$	$0$	\(0\)	\(6\)	\(8\)	\(4\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(4+\beta )q^{5}+(2+\beta )q^{7}+9q^{9}+\cdots\)
384.4.a.p	$384$	$4$	384.a	1.a	$1$	$2$	$2$	$22.657$	\(\Q(\sqrt{7}) \)	None	✓	✓			384.4.a.i	$1$	$0$	\(0\)	\(6\)	\(16\)	\(4\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(8+\beta )q^{5}+(2-3\beta )q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(384))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(384)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)