Space of modular forms of level 387, weight 6, and trivial character

• Label: 387.6.a
• Level: $387$
• Weight: $6$
• Character orbit: 387.a
• Rep. character: $\chi_{387}(1,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $8$
• Sturm bound: $264$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(264\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(387))\).

	Total	New	Old
Modular forms	224	88	136
Cusp forms	216	88	128
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
\(+\)	\(+\)	$+$	\(16\)
\(+\)	\(-\)	$-$	\(20\)
\(-\)	\(+\)	$-$	\(27\)
\(-\)	\(-\)	$+$	\(25\)
Plus space		\(+\)	\(41\)
Minus space		\(-\)	\(47\)

Trace form

\( 88 q - 4 q^{2} + 1404 q^{4} + 42 q^{5} + 76 q^{7} + 480 q^{8} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(387))\) 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
387.6.a.a	$387$	$6$	387.a	1.a	$1$	$6$	$6$	$62.069$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(56\)	\(-324\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2-\beta _{1}+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
387.6.a.b	$387$	$6$	387.a	1.a	$1$	$8$	$8$	$62.069$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(0\)	\(28\)	\(-30\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(12-\beta _{1}+\beta _{2})q^{4}+(4+\cdots)q^{5}+\cdots\)
387.6.a.c	$387$	$6$	387.a	1.a	$1$	$8$	$8$	$62.069$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(12\)	\(0\)	\(212\)	\(-136\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(15-2\beta _{1}-2\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)
387.6.a.d	$387$	$6$	387.a	1.a	$1$	$9$	$9$	$62.069$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-72\)	\(166\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(17+\beta _{1}+\beta _{2})q^{4}+(-8+\cdots)q^{5}+\cdots\)
387.6.a.e	$387$	$6$	387.a	1.a	$1$	$10$	$10$	$62.069$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(-138\)	\(60\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(20-\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.6.a.f	$387$	$6$	387.a	1.a	$1$	$11$	$11$	$62.069$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(-9\)	\(0\)	\(-44\)	\(264\)		$-$	$+$	$-$	$2^{5}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(21+\beta _{2})q^{4}+(-4+\cdots)q^{5}+\cdots\)
387.6.a.g	$387$	$6$	387.a	1.a	$1$	$16$	$16$	$62.069$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-158\)		$+$	$+$	$+$	$2^{17}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(13+\beta _{2})q^{4}+\beta _{9}q^{5}+(-9+\cdots)q^{7}+\cdots\)
387.6.a.h	$387$	$6$	387.a	1.a	$1$	$20$	$20$	$62.069$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(234\)		$+$	$-$	$-$	$2^{21}\cdot 3^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(19+\beta _{2})q^{4}+(\beta _{1}-\beta _{13}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(387))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(387)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(129))\)\(^{\oplus 2}\)