Embedded newform 39.4.f.b.5.5

• Label: 39.4.f.b.5.5
• 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			5.5
Root			\(-0.980238 + 0.980238i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.5
Dual form			39.4.f.b.8.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.980238 - 0.980238i) q^{2} +(-2.50090 + 4.55472i) q^{3} -6.07827i q^{4} +(12.5563 + 12.5563i) q^{5} +(6.91619 - 2.01323i) q^{6} +(21.4578 + 21.4578i) q^{7} +(-13.8001 + 13.8001i) q^{8} +(-14.4909 - 22.7818i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 44 q^{6} + 44 q^{7} - 112 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.980238	−	0.980238i	−0.346566	−	0.346566i	0.512263	−	0.858829i	\(-0.328807\pi\)
							−0.858829	+	0.512263i	\(0.828807\pi\)
\(3\)	−2.50090	+	4.55472i	−0.481299	+	0.876556i
							\(5\)	12.5563	+	12.5563i	1.12307	+	1.12307i	0.991278	+	0.131790i	\(0.0420724\pi\)
0.131790	+	0.991278i	\(0.457928\pi\)
\(7\)	21.4578	+	21.4578i	1.15861	+	1.15861i	0.984775	+	0.173836i	\(0.0556164\pi\)
							0.173836	+	0.984775i	\(0.444384\pi\)
\(11\)	−8.34980	+	8.34980i	−0.228869	+	0.228869i	−0.812220	−	0.583351i	\(-0.801741\pi\)
							0.583351	+	0.812220i	\(0.301741\pi\)
\(13\)	29.3793	−	36.5220i	0.626796	−	0.779184i
							\(17\)	−39.5799			−0.564678			−0.282339	−	0.959315i	\(-0.591110\pi\)
−0.282339	+	0.959315i	\(0.591110\pi\)
\(19\)	9.15970	−	9.15970i	0.110599	−	0.110599i	−0.649642	−	0.760241i	\(-0.725081\pi\)
							0.760241	+	0.649642i	\(0.225081\pi\)
\(23\)	2.83931			0.0257408			0.0128704	−	0.999917i	\(-0.495903\pi\)
							0.0128704	+	0.999917i	\(0.495903\pi\)
\(29\)		−	175.398i		−	1.12312i	−0.827435	−	0.561562i	\(-0.810201\pi\)
							0.827435	−	0.561562i	\(-0.189799\pi\)
\(31\)	95.0010	−	95.0010i	0.550409	−	0.550409i	−0.376150	−	0.926559i	\(-0.622752\pi\)
							0.926559	+	0.376150i	\(0.122752\pi\)
\(37\)	−92.4719	−	92.4719i	−0.410873	−	0.410873i	0.471170	−	0.882043i	\(-0.343832\pi\)
							−0.882043	+	0.471170i	\(0.843832\pi\)
\(41\)	−187.117	−	187.117i	−0.712749	−	0.712749i	0.254360	−	0.967110i	\(-0.418135\pi\)
							−0.967110	+	0.254360i	\(0.918135\pi\)
\(43\)		−	52.5720i		−	0.186445i	−0.995645	−	0.0932227i	\(-0.970283\pi\)
							0.995645	−	0.0932227i	\(-0.0297168\pi\)
\(47\)	194.887	−	194.887i	0.604835	−	0.604835i	−0.336756	−	0.941592i	\(-0.609330\pi\)
							0.941592	+	0.336756i	\(0.109330\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.f.b.5.5	✓	20
3.2	odd	2	inner	39.4.f.b.5.6	yes	20
13.8	odd	4	inner	39.4.f.b.8.6	yes	20
39.8	even	4	inner	39.4.f.b.8.5	yes	20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.f.b.5.5	✓	20	1.1	even	1	trivial
39.4.f.b.5.6	yes	20	3.2	odd	2	inner
39.4.f.b.8.5	yes	20	39.8	even	4	inner
39.4.f.b.8.6	yes	20	13.8	odd	4	inner