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

• Label: 39.4.j
• Level: $39$
• Weight: $4$
• Character orbit: 39.j
• Rep. character: $\chi_{39}(4,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $16$
• Newform subspaces: $3$
• Sturm bound: $18$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.j (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

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

	Total	New	Old
Modular forms	32	16	16
Cusp forms	24	16	8
Eisenstein series	8	0	8

Trace form

16 * q - 6 * q^3 + 40 * q^4 - 18 * q^7 - 72 * q^9 - 28 * q^10 + 24 * q^11 - 120 * q^12 + 144 * q^14 - 36 * q^15 - 204 * q^16 + 150 * q^17 + 360 * q^19 + 600 * q^20 - 256 * q^22 - 216 * q^23 - 644 * q^25 - 744 * q^26 + 108 * q^27 - 588 * q^28 - 402 * q^29 + 324 * q^30 + 1440 * q^32 - 288 * q^33 + 408 * q^35 + 360 * q^36 + 282 * q^37 - 768 * q^38 + 240 * q^39 + 1864 * q^40 + 1182 * q^41 + 396 * q^42 + 182 * q^43 - 378 * q^45 - 1104 * q^46 - 888 * q^48 + 886 * q^49 - 3096 * q^50 - 360 * q^51 + 2588 * q^52 - 2460 * q^53 - 436 * q^55 - 2568 * q^56 - 204 * q^58 + 1572 * q^59 - 816 * q^61 - 2208 * q^62 + 162 * q^63 + 408 * q^64 + 450 * q^65 + 1944 * q^66 - 126 * q^67 + 984 * q^68 + 288 * q^69 - 504 * q^71 + 3720 * q^74 + 1914 * q^75 - 2856 * q^76 - 264 * q^77 + 1728 * q^78 + 228 * q^79 + 8016 * q^80 - 648 * q^81 - 2356 * q^82 - 2664 * q^84 + 726 * q^85 - 1764 * q^87 + 3136 * q^88 + 4416 * q^89 + 504 * q^90 + 2166 * q^91 - 3936 * q^92 - 3402 * q^93 - 544 * q^94 - 1308 * q^95 - 6090 * q^97 - 6552 * q^98

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.j.a	$39$	$4$	39.j	13.e	$6$	$2$	$1$	$2.301$	\(\Q(\sqrt{-3}) \)	None		✓			39.4.j.a	$2$	$0$	\(0\)	\(3\)	\(0\)	\(18\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(3-3\zeta_{6})q^{3}-8\zeta_{6}q^{4}+(3-6\zeta_{6})q^{5}+\cdots\)
39.4.j.b	$39$	$4$	39.j	13.e	$6$	$4$	$2$	$2.301$	\(\Q(\sqrt{-3}, \sqrt{-17})\)	None		✓			39.4.j.b	$2$	$0$	\(0\)	\(6\)	\(0\)	\(-66\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(3-3\beta _{2})q^{3}+9\beta _{2}q^{4}+(3+\cdots)q^{5}+\cdots\)
39.4.j.c	$39$	$4$	39.j	13.e	$6$	$10$	$5$	$2.301$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓	✓		39.4.j.c	$2$	$0$	\(0\)	\(-15\)	\(0\)	\(30\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{2}-3\beta _{2}q^{3}+(6-6\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)

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

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