Space of modular forms of level 39, weight 4, and Character 39.b

• Label: 39.4.b
• Level: $39$
• Weight: $4$
• Character orbit: 39.b
• Rep. character: $\chi_{39}(25,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $2$
• Sturm bound: $18$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(18\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(39, [\chi])\).

	Total	New	Old
Modular forms	16	8	8
Cusp forms	12	8	4
Eisenstein series	4	0	4

Trace form

\( 8 q - 40 q^{4} + 72 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(39, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
39.4.b.a	$39$	$4$	39.b	13.b	$2$	$4$	$4$	$2.301$	4.0.5054412.1	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-3q^{3}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}+\cdots\)
39.4.b.b	$39$	$4$	39.b	13.b	$2$	$4$	$4$	$2.301$	4.0.1362828.1	None		$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+3q^{3}+(-4+\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(39, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(39, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 2}\)