Space of modular forms of level 31, weight 3, and Character 31.b

• Label: 31.3.b
• Level: $31$
• Weight: $3$
• Character orbit: 31.b
• Rep. character: $\chi_{31}(30,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $2$
• Sturm bound: $8$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	31.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 31 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(8\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(31, [\chi])\).

	Total	New	Old
Modular forms	7	7	0
Cusp forms	5	5	0
Eisenstein series	2	2	0

Trace form

\( 5 q - 2 q^{2} + 6 q^{4} + 4 q^{5} + 16 q^{7} - 31 q^{8} - 7 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(31, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
31.3.b.a	$31$	$3$	31.b	31.b	$2$	$2$	$2$	$0.845$	\(\Q(\sqrt{-26}) \)	None		✓			31.3.b.a	$2$	$0$	\(-2\)	\(0\)	\(4\)	\(16\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{2}+\beta q^{3}-3q^{4}+2q^{5}-\beta q^{6}+\cdots\)
31.3.b.b	$31$	$3$	31.b	31.b	$2$	$3$	$3$	$0.845$	3.3.837.1	\(\Q(\sqrt{-31}) \)	✓	✓	✓		31.3.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{2}q^{2}+(4+\beta _{1}-2\beta _{2})q^{4}+(-3\beta _{1}+\cdots)q^{5}+\cdots\)