Space of modular forms of level 387, weight 8, and trivial character

• Label: 387.8.a
• Level: $387$
• Weight: $8$
• Character orbit: 387.a
• Rep. character: $\chi_{387}(1,\cdot)$
• Character field: $\Q$
• Dimension: $122$
• Newform subspaces: $8$
• Sturm bound: $352$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(352\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(387))\).

	Total	New	Old
Modular forms	312	122	190
Cusp forms	304	122	182
Eisenstein series	8	0	8

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

\(3\)	\(43\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(26\)
\(+\)	\(-\)	$-$	\(22\)
\(-\)	\(+\)	$-$	\(36\)
\(-\)	\(-\)	$+$	\(38\)
Plus space		\(+\)	\(64\)
Minus space		\(-\)	\(58\)

Trace form

\( 122 q - 8 q^{2} + 7692 q^{4} + 530 q^{5} - 1348 q^{7} - 5136 q^{8} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(387))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	43
387.8.a.a	$387$	$8$	387.a	1.a	$1$	$10$	$10$	$120.893$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(122\)	\(-2052\)		$-$	$+$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(37+\beta _{1}+\beta _{2})q^{4}+(13-6\beta _{1}+\cdots)q^{5}+\cdots\)
387.8.a.b	$387$	$8$	387.a	1.a	$1$	$11$	$11$	$120.893$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(24\)	\(0\)	\(752\)	\(-12\)		$-$	$-$	$+$	$2^{6}\cdot 3^{4}\cdot 7$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(54+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.8.a.c	$387$	$8$	387.a	1.a	$1$	$12$	$12$	$120.893$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(0\)	\(766\)	\(-1366\)		$-$	$-$	$+$	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(58+\beta _{2})q^{4}+(2^{6}-\beta _{1}+\cdots)q^{5}+\cdots\)
387.8.a.d	$387$	$8$	387.a	1.a	$1$	$13$	$13$	$120.893$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(-998\)	\(1360\)		$-$	$+$	$-$	$2^{6}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(70+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.8.a.e	$387$	$8$	387.a	1.a	$1$	$13$	$13$	$120.893$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(0\)	\(266\)	\(6\)		$-$	$+$	$-$	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(73-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.8.a.f	$387$	$8$	387.a	1.a	$1$	$15$	$15$	$120.893$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(-15\)	\(0\)	\(-378\)	\(2064\)		$-$	$-$	$+$	$2^{14}\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(84-\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.8.a.g	$387$	$8$	387.a	1.a	$1$	$22$	$22$	$120.893$		None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-2046\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
387.8.a.h	$387$	$8$	387.a	1.a	$1$	$26$	$26$	$120.893$		None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(698\)		$+$	$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(387))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(387)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(129))\)\(^{\oplus 2}\)