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 - 68 * q^10 + 36 * q^12 - 84 * q^13 - 264 * q^14 + 372 * q^16 + 312 * q^17 + 244 * q^22 - 168 * q^23 - 64 * q^25 - 432 * q^26 - 432 * q^29 - 324 * q^30 + 984 * q^35 - 360 * q^36 + 24 * q^38 + 180 * q^39 - 316 * q^40 + 1296 * q^42 - 224 * q^43 - 1008 * q^48 - 904 * q^49 - 360 * q^51 + 460 * q^52 + 120 * q^53 + 256 * q^55 + 1728 * q^56 - 1992 * q^61 + 2592 * q^62 + 264 * q^64 + 1272 * q^65 + 1548 * q^66 - 5784 * q^68 - 1224 * q^69 + 840 * q^74 + 72 * q^75 + 600 * q^77 - 936 * q^78 + 4608 * q^79 + 648 * q^81 - 3116 * q^82 - 1008 * q^87 + 3428 * q^88 - 612 * q^90 - 384 * q^91 - 7560 * q^92 + 4996 * q^94 + 1512 * q^95

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
39.4.b.a	$39$	$4$	39.b	13.b	$2$	$4$	$4$	$2.301$	4.0.5054412.1	None		✓			39.4.b.a	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-3q^{3}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}+\cdots\)
39.4.b.b	$39$	$4$	39.b	13.b	$2$	$4$	$4$	$2.301$	4.0.1362828.1	None		✓			39.4.b.b	$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+3q^{3}+(-4+\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)

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

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