Space of modular forms of level 39, weight 3, and Character 39.l

• Label: 39.3.l
• Level: $39$
• Weight: $3$
• Character orbit: 39.l
• Rep. character: $\chi_{39}(7,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $20$
• Newform subspaces: $2$
• Sturm bound: $14$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$39 = 3 \cdot 13$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	39.l (of order $12$ and degree $4$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$13$
Character field:			$\Q(\zeta_{12})$
Newform subspaces:			$2$
Sturm bound:			$14$
Trace bound:			$1$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{3}(39, [\chi])$.

	Total	New	Old
Modular forms	44	20	24
Cusp forms	28	20	8
Eisenstein series	16	0	16

Trace form

20 * q - 4 * q^2 + 20 * q^5 - 18 * q^7 - 48 * q^8 - 30 * q^9 - 12 * q^10 + 8 * q^11 + 4 * q^13 + 64 * q^14 + 24 * q^15 + 24 * q^16 + 12 * q^17 + 24 * q^18 - 90 * q^19 - 140 * q^20 + 66 * q^21 - 52 * q^22 - 48 * q^23 - 36 * q^24 + 92 * q^26 + 68 * q^28 + 64 * q^29 + 14 * q^31 + 272 * q^32 + 36 * q^33 + 240 * q^34 + 136 * q^35 + 74 * q^37 - 120 * q^39 - 216 * q^40 - 304 * q^41 - 108 * q^42 + 6 * q^43 - 328 * q^44 - 48 * q^45 + 156 * q^46 - 292 * q^47 - 144 * q^48 + 66 * q^49 - 212 * q^50 - 464 * q^52 + 112 * q^53 + 44 * q^55 - 36 * q^56 + 144 * q^57 + 116 * q^58 - 112 * q^59 + 528 * q^60 - 52 * q^61 + 516 * q^62 + 72 * q^63 + 728 * q^65 + 48 * q^66 - 144 * q^67 + 192 * q^68 - 256 * q^70 + 164 * q^71 + 72 * q^72 + 162 * q^73 + 220 * q^74 - 210 * q^75 + 580 * q^76 - 468 * q^78 - 64 * q^79 + 44 * q^80 - 90 * q^81 + 24 * q^82 - 40 * q^83 - 96 * q^84 + 228 * q^85 - 1008 * q^86 - 84 * q^87 - 252 * q^88 - 604 * q^89 - 390 * q^91 + 120 * q^92 + 102 * q^93 - 172 * q^94 - 720 * q^95 + 408 * q^96 - 346 * q^97 - 880 * q^98 + 60 * q^99

Decomposition of $S_{3}^{\mathrm{new}}(39, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
39.3.l.a	$39$	$3$	39.l	13.f	$12$	$8$	$2$	$1.063$	8.0.$\cdots$.10	None		✓			39.3.l.a	$2$	$0$	$-2$	$0$	$16$	$14$	$1$	$\mathrm{SU}(2)[C_{12}]$	$q+(-\beta _{1}-\beta _{5})q^{2}+(2\beta _{4}-\beta _{5})q^{3}+(1+\cdots)q^{4}+\cdots$
39.3.l.b	$39$	$3$	39.l	13.f	$12$	$12$	$3$	$1.063$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		✓	✓		39.3.l.b	$2$	$0$	$-2$	$0$	$4$	$-32$	$1$	$\mathrm{SU}(2)[C_{12}]$	$q+(-\beta _{1}-\beta _{2})q^{2}+(\beta _{4}+\beta _{5})q^{3}+(-2+\cdots)q^{4}+\cdots$

Decomposition of $S_{3}^{\mathrm{old}}(39, [\chi])$ into lower level spaces

$S_{3}^{\mathrm{old}}(39, [\chi]) \simeq$ $S_{3}^{\mathrm{new}}(13, [\chi])$$^{\oplus 2}$