Embedded newform 39.4.f.b.5.2

• Label: 39.4.f.b.5.2
• Level: $39$
• Weight: $4$
• Character: 39.5
• Analytic conductor: $2.301$
• Analytic rank: $0$
• Dimension: $20$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30107449022\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial:	\( x^{20} + 1316x^{16} + 520390x^{12} + 64668772x^{8} + 2536036097x^{4} + 8509693504 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{12}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			5.2
Root			\(-3.24982 + 3.24982i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.5
Dual form			39.4.f.b.8.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.24982 - 3.24982i) q^{2} +(-5.00236 - 1.40585i) q^{3} +13.1227i q^{4} +(-2.07291 - 2.07291i) q^{5} +(11.6880 + 20.8255i) q^{6} +(7.93427 + 7.93427i) q^{7} +(16.6478 - 16.6478i) q^{8} +(23.0472 + 14.0651i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 44 q^{6} + 44 q^{7} - 112 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.24982	−	3.24982i	−1.14899	−	1.14899i	−0.986752	−	0.162233i	\(-0.948130\pi\)
							−0.162233	−	0.986752i	\(-0.551870\pi\)
\(3\)	−5.00236	−	1.40585i	−0.962705	−	0.270555i
							\(5\)	−2.07291	−	2.07291i	−0.185407	−	0.185407i	0.608300	−	0.793707i	\(-0.291852\pi\)
−0.793707	+	0.608300i	\(0.791852\pi\)
\(7\)	7.93427	+	7.93427i	0.428410	+	0.428410i	0.888087	−	0.459676i	\(-0.152035\pi\)
							−0.459676	+	0.888087i	\(0.652035\pi\)
\(11\)	−33.2523	+	33.2523i	−0.911451	+	0.911451i	−0.996386	−	0.0849358i	\(-0.972931\pi\)
							0.0849358	+	0.996386i	\(0.472931\pi\)
\(13\)	−45.5555	−	11.0316i	−0.971909	−	0.235356i
							\(17\)	−80.7198			−1.15161			−0.575807	−	0.817586i	\(-0.695312\pi\)
−0.575807	+	0.817586i	\(0.695312\pi\)
\(19\)	−66.4938	+	66.4938i	−0.802880	+	0.802880i	−0.983545	−	0.180665i	\(-0.942175\pi\)
							0.180665	+	0.983545i	\(0.442175\pi\)
\(23\)	131.226			1.18968			0.594838	−	0.803846i	\(-0.297216\pi\)
							0.594838	+	0.803846i	\(0.297216\pi\)
\(29\)			240.056i			1.53715i	0.639763	+	0.768573i	\(0.279033\pi\)
							−0.639763	+	0.768573i	\(0.720967\pi\)
\(31\)	−68.7260	+	68.7260i	−0.398179	+	0.398179i	−0.877590	−	0.479411i	\(-0.840850\pi\)
							0.479411	+	0.877590i	\(0.340850\pi\)
\(37\)	−162.487	−	162.487i	−0.721966	−	0.721966i	0.247039	−	0.969005i	\(-0.420542\pi\)
							−0.969005	+	0.247039i	\(0.920542\pi\)
\(41\)	−217.148	−	217.148i	−0.827143	−	0.827143i	0.159977	−	0.987121i	\(-0.448858\pi\)
							−0.987121	+	0.159977i	\(0.948858\pi\)
\(43\)			235.492i			0.835166i	0.908639	+	0.417583i	\(0.137123\pi\)
							−0.908639	+	0.417583i	\(0.862877\pi\)
\(47\)	38.1141	−	38.1141i	0.118288	−	0.118288i	−0.645485	−	0.763773i	\(-0.723345\pi\)
							0.763773	+	0.645485i	\(0.223345\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.f.b.5.2	✓	20
3.2	odd	2	inner	39.4.f.b.5.9	yes	20
13.8	odd	4	inner	39.4.f.b.8.9	yes	20
39.8	even	4	inner	39.4.f.b.8.2	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.f.b.5.2	✓	20	1.1	even	1	trivial
39.4.f.b.5.9	yes	20	3.2	odd	2	inner
39.4.f.b.8.2	yes	20	39.8	even	4	inner
39.4.f.b.8.9	yes	20	13.8	odd	4	inner