Space of modular forms of level 39, weight 2, and Character 39.e

• Label: 39.2.e
• Level: $39$
• Weight: $2$
• Character orbit: 39.e
• Rep. character: $\chi_{39}(16,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $9$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	39.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(9\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(39, [\chi])\).

	Total	New	Old
Modular forms	14	6	8
Cusp forms	6	6	0
Eisenstein series	8	0	8

Trace form

6 * q - 2 * q^2 - q^3 - 4 * q^4 - 8 * q^5 + q^7 + 12 * q^8 - 3 * q^9 - 6 * q^10 + 6 * q^11 + 12 * q^12 - 5 * q^13 - 16 * q^14 + 2 * q^15 - 2 * q^16 + 6 * q^17 + 4 * q^18 - 2 * q^20 - 10 * q^21 + 4 * q^22 + 2 * q^23 - 12 * q^24 - 2 * q^25 + 30 * q^26 + 2 * q^27 + 18 * q^28 - 8 * q^30 + 10 * q^31 - 14 * q^32 + 2 * q^33 + 4 * q^34 + 6 * q^35 - 4 * q^36 - 12 * q^37 - 40 * q^38 - 6 * q^39 - 4 * q^40 - 10 * q^41 + 12 * q^42 - 11 * q^43 - 16 * q^44 + 4 * q^45 + 4 * q^46 + 12 * q^47 - 4 * q^48 + 4 * q^49 + 28 * q^50 + 16 * q^51 - 30 * q^52 + 4 * q^53 - 8 * q^55 + 28 * q^56 + 24 * q^57 + 26 * q^58 + 14 * q^59 - 17 * q^61 + 4 * q^62 + q^63 + 28 * q^64 + 4 * q^65 - 3 * q^67 - 18 * q^68 - 10 * q^69 - 8 * q^70 - 34 * q^71 - 6 * q^72 - 2 * q^73 + 2 * q^74 - 7 * q^75 - 4 * q^76 + 4 * q^77 - 28 * q^78 + 22 * q^79 - 22 * q^80 - 3 * q^81 - 18 * q^82 - 48 * q^83 + 14 * q^84 - 14 * q^85 - 2 * q^87 + 12 * q^88 - 4 * q^89 + 12 * q^90 + 31 * q^91 + 32 * q^92 + 3 * q^93 + 28 * q^94 + 20 * q^95 + 8 * q^96 + 15 * q^97 - 22 * q^98 - 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(39, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
39.2.e.a	$39$	$2$	39.e	13.c	$3$	$2$	$1$	$0.311$	\(\Q(\sqrt{-3}) \)	None		✓			39.2.e.a	$2$	$0$	\(-1\)	\(1\)	\(-2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
39.2.e.b	$39$	$2$	39.e	13.c	$3$	$4$	$2$	$0.311$	\(\Q(\sqrt{-3}, \sqrt{17})\)	None		✓	✓		39.2.e.b	$2$	$0$	\(-1\)	\(-2\)	\(-6\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)