Space of modular forms of level 384, weight 8, and trivial character

• Label: 384.8.a
• Level: $384$
• Weight: $8$
• Character orbit: 384.a
• Rep. character: $\chi_{384}(1,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $20$
• Sturm bound: $512$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(512\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(384))\).

	Total	New	Old
Modular forms	464	56	408
Cusp forms	432	56	376
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(120\)	\(15\)	\(105\)		\(112\)	\(15\)	\(97\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)		\(116\)	\(13\)	\(103\)		\(108\)	\(13\)	\(95\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(112\)	\(13\)	\(99\)		\(104\)	\(13\)	\(91\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)		\(116\)	\(15\)	\(101\)		\(108\)	\(15\)	\(93\)		\(8\)	\(0\)	\(8\)
Plus space		\(+\)		\(236\)	\(30\)	\(206\)		\(220\)	\(30\)	\(190\)		\(16\)	\(0\)	\(16\)
Minus space		\(-\)		\(228\)	\(26\)	\(202\)		\(212\)	\(26\)	\(186\)		\(16\)	\(0\)	\(16\)

Trace form

\( 56 q + 40824 q^{9} + 11632 q^{17} + 939496 q^{25} - 1765136 q^{41} + 8632536 q^{49} - 6204384 q^{57} + 4374304 q^{65} - 21180080 q^{73} + 29760696 q^{81} - 28686288 q^{89} + 15808112 q^{97}+O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(384))\) 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
384.8.a.a	$384$	$8$	384.a	1.a	$1$	$1$	$1$	$119.956$	\(\Q\)	None	✓	✓			384.8.a.a	$1$	$0$	\(0\)	\(-27\)	\(-160\)	\(-974\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}-160q^{5}-974q^{7}+3^{6}q^{9}+\cdots\)
384.8.a.b	$384$	$8$	384.a	1.a	$1$	$1$	$1$	$119.956$	\(\Q\)	None	✓	✓			384.8.a.a	$1$	$1$	\(0\)	\(-27\)	\(160\)	\(974\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+160q^{5}+974q^{7}+3^{6}q^{9}+\cdots\)
384.8.a.c	$384$	$8$	384.a	1.a	$1$	$1$	$1$	$119.956$	\(\Q\)	None	✓	✓			384.8.a.a	$1$	$1$	\(0\)	\(27\)	\(-160\)	\(974\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}-160q^{5}+974q^{7}+3^{6}q^{9}+\cdots\)
384.8.a.d	$384$	$8$	384.a	1.a	$1$	$1$	$1$	$119.956$	\(\Q\)	None	✓	✓			384.8.a.a	$1$	$1$	\(0\)	\(27\)	\(160\)	\(-974\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+160q^{5}-974q^{7}+3^{6}q^{9}+\cdots\)
384.8.a.e	$384$	$8$	384.a	1.a	$1$	$2$	$2$	$119.956$	\(\Q(\sqrt{366}) \)	None	✓	✓			384.8.a.e	$1$	$0$	\(0\)	\(-54\)	\(-176\)	\(980\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-88+\beta )q^{5}+(490-5\beta )q^{7}+\cdots\)
384.8.a.f	$384$	$8$	384.a	1.a	$1$	$2$	$2$	$119.956$	\(\Q(\sqrt{366}) \)	None	✓	✓			384.8.a.e	$1$	$1$	\(0\)	\(-54\)	\(176\)	\(-980\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(88+\beta )q^{5}+(-490-5\beta )q^{7}+\cdots\)
384.8.a.g	$384$	$8$	384.a	1.a	$1$	$2$	$2$	$119.956$	\(\Q(\sqrt{366}) \)	None	✓	✓			384.8.a.e	$1$	$1$	\(0\)	\(54\)	\(-176\)	\(-980\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-88+\beta )q^{5}+(-490+5\beta )q^{7}+\cdots\)
384.8.a.h	$384$	$8$	384.a	1.a	$1$	$2$	$2$	$119.956$	\(\Q(\sqrt{366}) \)	None	✓	✓			384.8.a.e	$1$	$1$	\(0\)	\(54\)	\(176\)	\(980\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(88+\beta )q^{5}+(490+5\beta )q^{7}+\cdots\)
384.8.a.i	$384$	$8$	384.a	1.a	$1$	$3$	$3$	$119.956$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			384.8.a.i	$1$	$1$	\(0\)	\(-81\)	\(-308\)	\(6\)		$-$	$+$	$-$	$2^{8}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-103-\beta _{1})q^{5}+(2+\beta _{2})q^{7}+\cdots\)
384.8.a.j	$384$	$8$	384.a	1.a	$1$	$3$	$3$	$119.956$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			384.8.a.i	$1$	$1$	\(0\)	\(-81\)	\(308\)	\(-6\)		$-$	$+$	$-$	$2^{8}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(103+\beta _{1})q^{5}+(-2-\beta _{2})q^{7}+\cdots\)
384.8.a.k	$384$	$8$	384.a	1.a	$1$	$3$	$3$	$119.956$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			384.8.a.i	$1$	$1$	\(0\)	\(81\)	\(-308\)	\(-6\)		$+$	$-$	$-$	$2^{8}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-103-\beta _{1})q^{5}+(-2-\beta _{2})q^{7}+\cdots\)
384.8.a.l	$384$	$8$	384.a	1.a	$1$	$3$	$3$	$119.956$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓			384.8.a.i	$1$	$0$	\(0\)	\(81\)	\(308\)	\(6\)		$-$	$-$	$+$	$2^{8}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(103+\beta _{1})q^{5}+(2+\beta _{2})q^{7}+\cdots\)
384.8.a.m	$384$	$8$	384.a	1.a	$1$	$4$	$4$	$119.956$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		384.8.a.m	$1$	$1$	\(0\)	\(-108\)	\(-336\)	\(-680\)		$-$	$+$	$-$	$2^{15}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-84+\beta _{2})q^{5}+(-170+\cdots)q^{7}+\cdots\)
384.8.a.n	$384$	$8$	384.a	1.a	$1$	$4$	$4$	$119.956$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			384.8.a.n	$1$	$0$	\(0\)	\(-108\)	\(-192\)	\(680\)		$+$	$+$	$+$	$2^{15}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(-48-\beta _{2})q^{5}+(170+\beta _{3})q^{7}+\cdots\)
384.8.a.o	$384$	$8$	384.a	1.a	$1$	$4$	$4$	$119.956$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			384.8.a.n	$1$	$0$	\(0\)	\(-108\)	\(192\)	\(-680\)		$+$	$+$	$+$	$2^{15}\cdot 3$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(48+\beta _{2})q^{5}+(-170-\beta _{3})q^{7}+\cdots\)
384.8.a.p	$384$	$8$	384.a	1.a	$1$	$4$	$4$	$119.956$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			384.8.a.m	$1$	$0$	\(0\)	\(-108\)	\(336\)	\(680\)		$+$	$+$	$+$	$2^{15}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-3^{3}q^{3}+(84-\beta _{2})q^{5}+(170+\beta _{3})q^{7}+\cdots\)
384.8.a.q	$384$	$8$	384.a	1.a	$1$	$4$	$4$	$119.956$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			384.8.a.m	$1$	$0$	\(0\)	\(108\)	\(-336\)	\(680\)		$-$	$-$	$+$	$2^{15}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-84+\beta _{2})q^{5}+(170+\beta _{3})q^{7}+\cdots\)
384.8.a.r	$384$	$8$	384.a	1.a	$1$	$4$	$4$	$119.956$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		384.8.a.n	$1$	$1$	\(0\)	\(108\)	\(-192\)	\(-680\)		$+$	$-$	$-$	$2^{15}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(-48-\beta _{2})q^{5}+(-170+\cdots)q^{7}+\cdots\)
384.8.a.s	$384$	$8$	384.a	1.a	$1$	$4$	$4$	$119.956$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			384.8.a.n	$1$	$0$	\(0\)	\(108\)	\(192\)	\(680\)		$-$	$-$	$+$	$2^{15}\cdot 3$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(48+\beta _{2})q^{5}+(170+\beta _{3})q^{7}+\cdots\)
384.8.a.t	$384$	$8$	384.a	1.a	$1$	$4$	$4$	$119.956$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			384.8.a.m	$1$	$0$	\(0\)	\(108\)	\(336\)	\(-680\)		$-$	$-$	$+$	$2^{15}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+3^{3}q^{3}+(84-\beta _{2})q^{5}+(-170-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(384))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(384)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 14}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 7}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)