Embedded newform 39.4.f.b.8.10

• Label: 39.4.f.b.8.10
• Level: $39$
• Weight: $4$
• Character: 39.8
• 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			8.10
Root			\(3.62388 + 3.62388i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.8
Dual form			39.4.f.b.5.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.62388 - 3.62388i) q^{2} +(4.19035 + 3.07262i) q^{3} -18.2650i q^{4} +(-10.8970 + 10.8970i) q^{5} +(26.3201 - 4.05051i) q^{6} +(-5.52580 + 5.52580i) q^{7} +(-37.1990 - 37.1990i) q^{8} +(8.11802 + 25.7507i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 44 q^{6} + 44 q^{7} - 112 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.62388	−	3.62388i	1.28123	−	1.28123i	0.341268	−	0.939966i	\(-0.389144\pi\)
							0.939966	−	0.341268i	\(-0.110856\pi\)
\(3\)	4.19035	+	3.07262i	0.806433	+	0.591326i
							\(5\)	−10.8970	+	10.8970i	−0.974661	+	0.974661i	−0.999687	−	0.0250260i	\(-0.992033\pi\)
0.0250260	+	0.999687i	\(0.492033\pi\)
\(7\)	−5.52580	+	5.52580i	−0.298365	+	0.298365i	−0.840373	−	0.542008i	\(-0.817664\pi\)
							0.542008	+	0.840373i	\(0.317664\pi\)
\(11\)	−25.6188	−	25.6188i	−0.702215	−	0.702215i	0.262671	−	0.964885i	\(-0.415397\pi\)
							−0.964885	+	0.262671i	\(0.915397\pi\)
\(13\)	6.01627	−	46.4845i	0.128355	−	0.991728i
							\(17\)	56.1424			0.800972			0.400486	−	0.916303i	\(-0.368841\pi\)
0.400486	+	0.916303i	\(0.368841\pi\)
\(19\)	72.4479	+	72.4479i	0.874773	+	0.874773i	0.992988	−	0.118215i	\(-0.0377171\pi\)
							−0.118215	+	0.992988i	\(0.537717\pi\)
\(23\)	8.33293			0.0755451			0.0377725	−	0.999286i	\(-0.487974\pi\)
							0.0377725	+	0.999286i	\(0.487974\pi\)
\(29\)		−	125.931i		−	0.806375i	−0.915117	−	0.403187i	\(-0.867902\pi\)
							0.915117	−	0.403187i	\(-0.132098\pi\)
\(31\)	−89.8241	−	89.8241i	−0.520415	−	0.520415i	0.397281	−	0.917697i	\(-0.369954\pi\)
							−0.917697	+	0.397281i	\(0.869954\pi\)
\(37\)	−274.724	+	274.724i	−1.22066	+	1.22066i	−0.253261	+	0.967398i	\(0.581503\pi\)
							−0.967398	+	0.253261i	\(0.918497\pi\)
\(41\)	44.2900	−	44.2900i	0.168706	−	0.168706i	−0.617704	−	0.786410i	\(-0.711937\pi\)
							0.786410	+	0.617704i	\(0.211937\pi\)
\(43\)		−	100.272i		−	0.355611i	−0.984066	−	0.177806i	\(-0.943100\pi\)
							0.984066	−	0.177806i	\(-0.0568998\pi\)
\(47\)	−82.5643	−	82.5643i	−0.256239	−	0.256239i	0.567283	−	0.823523i	\(-0.307994\pi\)
							−0.823523	+	0.567283i	\(0.807994\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.f.b.8.10	yes	20
3.2	odd	2	inner	39.4.f.b.8.1	yes	20
13.5	odd	4	inner	39.4.f.b.5.1	✓	20
39.5	even	4	inner	39.4.f.b.5.10	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.f.b.5.1	✓	20	13.5	odd	4	inner
39.4.f.b.5.10	yes	20	39.5	even	4	inner
39.4.f.b.8.1	yes	20	3.2	odd	2	inner
39.4.f.b.8.10	yes	20	1.1	even	1	trivial