Embedded newform 39.4.j.c.10.5

• Label: 39.4.j.c.10.5
• Level: $39$
• Weight: $4$
• Character: 39.10
• Analytic conductor: $2.301$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30107449022\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial:	\( x^{10} + 70x^{8} + 1645x^{6} + 14700x^{4} + 44100x^{2} + 27648 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			10.5
Root			\(-5.04537i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.10
Dual form			39.4.j.c.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.36942 - 2.52268i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(8.72787 - 15.1171i) q^{4} +20.1174i q^{5} +(-13.1082 - 7.56805i) q^{6} +(-13.3609 - 7.71395i) q^{7} -47.7076i q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 15 q^{3} + 30 q^{4} + 30 q^{7} - 45 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(14\)	\(28\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.36942	−	2.52268i	1.54482	−	0.891903i	0.546298	−	0.837591i	\(-0.316037\pi\)
							0.998524	−	0.0543124i	\(-0.0172967\pi\)
\(3\)	−1.50000	−	2.59808i	−0.288675	−	0.500000i
							\(5\)			20.1174i			1.79935i	0.436556	+	0.899677i	\(0.356198\pi\)
−0.436556	+	0.899677i	\(0.643802\pi\)
\(7\)	−13.3609	−	7.71395i	−0.721423	−	0.416514i	0.0938530	−	0.995586i	\(-0.470082\pi\)
							−0.815276	+	0.579072i	\(0.803415\pi\)
\(11\)	23.3283	−	13.4686i	0.639432	−	0.369176i	−0.144964	−	0.989437i	\(-0.546307\pi\)
							0.784396	+	0.620261i	\(0.212973\pi\)
\(13\)	−3.96071	+	46.7045i	−0.0845002	+	0.996423i
							\(17\)	11.6167	−	20.1207i	0.165733	−	0.287059i	−0.771182	−	0.636615i	\(-0.780334\pi\)
0.936915	+	0.349556i	\(0.113668\pi\)
\(19\)	−39.0399	−	22.5397i	−0.471388	−	0.272156i	0.245433	−	0.969414i	\(-0.421070\pi\)
							−0.716821	+	0.697258i	\(0.754403\pi\)
\(23\)	−71.0050	−	122.984i	−0.643720	−	1.11496i	−0.984595	−	0.174848i	\(-0.944057\pi\)
							0.340875	−	0.940109i	\(-0.389277\pi\)
\(29\)	−1.14534	−	1.98379i	−0.00733394	−	0.0127028i	0.862335	−	0.506338i	\(-0.169001\pi\)
							−0.869669	+	0.493635i	\(0.835668\pi\)
\(31\)			37.7740i			0.218852i	0.993995	+	0.109426i	\(0.0349012\pi\)
							−0.993995	+	0.109426i	\(0.965099\pi\)
\(37\)	271.793	−	156.920i	1.20764	−	0.697228i	0.245393	−	0.969424i	\(-0.421083\pi\)
							0.962242	+	0.272195i	\(0.0877497\pi\)
\(41\)	5.08201	−	2.93410i	0.0193580	−	0.0111763i	−0.490290	−	0.871559i	\(-0.663109\pi\)
							0.509648	+	0.860383i	\(0.329776\pi\)
\(43\)	−180.449	+	312.547i	−0.639958	+	1.10844i	0.345483	+	0.938425i	\(0.387715\pi\)
							−0.985441	+	0.170015i	\(0.945618\pi\)
\(47\)		−	209.748i		−	0.650956i	−0.945550	−	0.325478i	\(-0.894475\pi\)
							0.945550	−	0.325478i	\(-0.105525\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.j.c.10.5	yes	10
3.2	odd	2		117.4.q.e.10.1		10
4.3	odd	2		624.4.bv.h.49.5		10
13.2	odd	12		507.4.a.r.1.9		10
13.3	even	3		507.4.b.i.337.9		10
13.4	even	6	inner	39.4.j.c.4.5	✓	10
13.10	even	6		507.4.b.i.337.2		10
13.11	odd	12		507.4.a.r.1.2		10
39.2	even	12		1521.4.a.bk.1.2		10
39.11	even	12		1521.4.a.bk.1.9		10
39.17	odd	6		117.4.q.e.82.1		10
52.43	odd	6		624.4.bv.h.433.1		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.c.4.5	✓	10	13.4	even	6	inner
39.4.j.c.10.5	yes	10	1.1	even	1	trivial
117.4.q.e.10.1		10	3.2	odd	2
117.4.q.e.82.1		10	39.17	odd	6
507.4.a.r.1.2		10	13.11	odd	12
507.4.a.r.1.9		10	13.2	odd	12
507.4.b.i.337.2		10	13.10	even	6
507.4.b.i.337.9		10	13.3	even	3
624.4.bv.h.49.5		10	4.3	odd	2
624.4.bv.h.433.1		10	52.43	odd	6
1521.4.a.bk.1.2		10	39.2	even	12
1521.4.a.bk.1.9		10	39.11	even	12