Space of modular forms of level 31, weight 2, and trivial character

• Label: 31.2.a
• Level: $31$
• Weight: $2$
• Character orbit: 31.a
• Rep. character: $\chi_{31}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $1$
• Sturm bound: $5$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	31.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(5\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(31))\).

	Total	New	Old
Modular forms	3	3	0
Cusp forms	2	2	0
Eisenstein series	1	1	0

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

\(31\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(-\)		\(3\)	\(3\)	\(0\)		\(2\)	\(2\)	\(0\)		\(1\)	\(1\)	\(0\)

Trace form

2 * q + q^2 - 2 * q^3 - q^4 + 2 * q^5 - 6 * q^6 - 4 * q^7 + 6 * q^9 + q^10 + 4 * q^11 - 4 * q^12 - 2 * q^13 + 3 * q^14 - 2 * q^15 - 3 * q^16 + 6 * q^17 + 13 * q^18 - q^20 - 6 * q^21 + 2 * q^22 - 2 * q^23 + 10 * q^24 - 8 * q^25 - 6 * q^26 - 20 * q^27 + 7 * q^28 + 10 * q^29 - 6 * q^30 + 2 * q^31 - 9 * q^32 - 4 * q^33 - 2 * q^34 - 4 * q^35 + 7 * q^36 - 4 * q^37 - 5 * q^38 + 12 * q^39 + 14 * q^41 + 2 * q^42 - 2 * q^43 - 2 * q^44 + 6 * q^45 + 14 * q^46 - 4 * q^47 + 18 * q^48 + 4 * q^49 - 4 * q^50 + 4 * q^51 - 4 * q^52 - 12 * q^53 - 20 * q^54 + 4 * q^55 - 10 * q^56 + 10 * q^57 - 4 * q^60 - 6 * q^61 + q^62 + 8 * q^63 + 4 * q^64 - 2 * q^65 - 12 * q^66 + 16 * q^67 - 8 * q^68 - 28 * q^69 + 3 * q^70 + 4 * q^71 - 20 * q^72 + 8 * q^73 - 2 * q^74 + 8 * q^75 - 5 * q^76 - 8 * q^77 + 16 * q^78 - 10 * q^79 - 3 * q^80 + 22 * q^81 + 7 * q^82 - 12 * q^83 + 8 * q^84 + 6 * q^85 + 4 * q^86 + 10 * q^89 + 13 * q^90 - 6 * q^91 + 16 * q^92 - 2 * q^93 + 8 * q^94 + 4 * q^96 - 14 * q^97 - 18 * q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		31
31.2.a.a	$31$	$2$	31.a	1.a	$1$	$2$	$2$	$0.248$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	31.2.a.a	$1$	$0$	\(1\)	\(-2\)	\(2\)	\(-4\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-2\beta q^{3}+(-1+\beta )q^{4}+q^{5}+\cdots\)