Space of modular forms of level 318, weight 2, and trivial character

• Label: 318.2.a
• Level: $318$
• Weight: $2$
• Character orbit: 318.a
• Rep. character: $\chi_{318}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $7$
• Sturm bound: $108$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 318 = 2 \cdot 3 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	318.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(108\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(318))\).

	Total	New	Old
Modular forms	58	9	49
Cusp forms	51	9	42
Eisenstein series	7	0	7

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

\(2\)	\(3\)	\(53\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(7\)

Trace form

\( 9 q - q^{2} + q^{3} + 9 q^{4} + 2 q^{5} - q^{6} + 4 q^{7} - q^{8} + 9 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(318))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	53
318.2.a.a	$318$	$2$	318.a	1.a	$1$	$1$	$1$	$2.539$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
318.2.a.b	$318$	$2$	318.a	1.a	$1$	$1$	$1$	$2.539$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(4\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+4q^{5}+q^{6}+q^{7}+\cdots\)
318.2.a.c	$318$	$2$	318.a	1.a	$1$	$1$	$1$	$2.539$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(0\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+5q^{7}-q^{8}+\cdots\)
318.2.a.d	$318$	$2$	318.a	1.a	$1$	$1$	$1$	$2.539$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}-4q^{7}+\cdots\)
318.2.a.e	$318$	$2$	318.a	1.a	$1$	$1$	$1$	$2.539$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
318.2.a.f	$318$	$2$	318.a	1.a	$1$	$2$	$2$	$2.539$	\(\Q(\sqrt{41}) \)	None	✓	$1$	$0$	\(-2\)	\(2\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+\beta q^{5}-q^{6}-q^{8}+\cdots\)
318.2.a.g	$318$	$2$	318.a	1.a	$1$	$2$	$2$	$2.539$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+\beta q^{5}+q^{6}+(1-\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(318))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(318)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(53))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(106))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(159))\)\(^{\oplus 2}\)