Space of modular forms of level 387, weight 10, and trivial character

• Label: 387.10.a
• Level: $387$
• Weight: $10$
• Character orbit: 387.a
• Rep. character: $\chi_{387}(1,\cdot)$
• Character field: $\Q$
• Dimension: $158$
• Newform subspaces: $8$
• Sturm bound: $440$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(440\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(387))\).

	Total	New	Old
Modular forms	400	158	242
Cusp forms	392	158	234
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
\(+\)	\(+\)	$+$	\(30\)
\(+\)	\(-\)	$-$	\(34\)
\(-\)	\(+\)	$-$	\(48\)
\(-\)	\(-\)	$+$	\(46\)
Plus space		\(+\)	\(76\)
Minus space		\(-\)	\(82\)

Trace form

\( 158 q + 16 q^{2} + 41388 q^{4} - 3868 q^{5} + 9756 q^{7} + 2496 q^{8} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(387))\) 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
387.10.a.a	$387$	$10$	387.a	1.a	$1$	$13$	$13$	$199.319$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(51\)	\(0\)	\(618\)	\(-14368\)		$-$	$-$	$+$	$2^{12}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(171-4\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.10.a.b	$387$	$10$	387.a	1.a	$1$	$15$	$15$	$199.319$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(-3\)	\(0\)	\(-394\)	\(38\)		$-$	$+$	$-$	$2^{13}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(234-2\beta _{1}+\beta _{2})q^{4}+(-3^{3}+\cdots)q^{5}+\cdots\)
387.10.a.c	$387$	$10$	387.a	1.a	$1$	$15$	$15$	$199.319$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(32\)	\(0\)	\(4717\)	\(-9680\)		$-$	$+$	$-$	$2^{10}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(6^{3}+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.10.a.d	$387$	$10$	387.a	1.a	$1$	$16$	$16$	$199.319$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$1$	$1$	\(-35\)	\(0\)	\(-2894\)	\(9642\)		$-$	$-$	$+$	$2^{15}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(283+\beta _{1}+\beta _{2})q^{4}+\cdots\)
387.10.a.e	$387$	$10$	387.a	1.a	$1$	$17$	$17$	$199.319$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓	$1$	$1$	\(-48\)	\(0\)	\(-4033\)	\(-76\)		$-$	$-$	$+$	$2^{13}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(267-4\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
387.10.a.f	$387$	$10$	387.a	1.a	$1$	$18$	$18$	$199.319$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(19\)	\(0\)	\(-1882\)	\(14444\)		$-$	$+$	$-$	$2^{17}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(322+\beta _{2})q^{4}+(-105+\cdots)q^{5}+\cdots\)
387.10.a.g	$387$	$10$	387.a	1.a	$1$	$30$	$30$	$199.319$		None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4726\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
387.10.a.h	$387$	$10$	387.a	1.a	$1$	$34$	$34$	$199.319$		None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14482\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(387))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(387)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(129))\)\(^{\oplus 2}\)