Embedded newform 39.4.e.c.16.1

• Label: 39.4.e.c.16.1
• Level: $39$
• Weight: $4$
• Character: 39.16
• Analytic conductor: $2.301$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.30107449022\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 2^{4}\cdot 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.1
Root			\(2.66520 + 4.61626i\) of defining polynomial
Character	\(\chi\)	\(=\)	39.16
Dual form			39.4.e.c.22.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.66520 - 4.61626i) q^{2} +(1.50000 + 2.59808i) q^{3} +(-10.2065 + 17.6783i) q^{4} -16.4131 q^{5} +(7.99559 - 13.8488i) q^{6} +(-4.83984 + 8.38285i) q^{7} +66.1667 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 12 q^{3} - 22 q^{4} - 12 q^{5} + 6 q^{6} + 14 q^{7} + 108 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.66520	−	4.61626i	−0.942289	−	1.63209i	−0.761090	−	0.648647i	\(-0.775335\pi\)
							−0.181199	−	0.983446i	\(-0.557998\pi\)
\(3\)	1.50000	+	2.59808i	0.288675	+	0.500000i
							\(5\)	−16.4131			−1.46803			−0.734016	−	0.679132i	\(-0.762356\pi\)
−0.734016	+	0.679132i	\(0.762356\pi\)
\(7\)	−4.83984	+	8.38285i	−0.261327	+	0.452631i	−0.966595	−	0.256309i	\(-0.917493\pi\)
							0.705268	+	0.708941i	\(0.250827\pi\)
\(11\)	−13.7941	−	23.8921i	−0.378098	−	0.654884i	0.612688	−	0.790325i	\(-0.290088\pi\)
							−0.990785	+	0.135441i	\(0.956755\pi\)
\(13\)	−37.3033	−	28.3807i	−0.795852	−	0.605491i
							\(17\)	−53.9641	+	93.4685i	−0.769895	+	1.33350i	0.167725	+	0.985834i	\(0.446358\pi\)
−0.937619	+	0.347663i	\(0.886975\pi\)
\(19\)	1.12362	−	1.94616i	0.0135671	−	0.0234989i	−0.859162	−	0.511703i	\(-0.829015\pi\)
							0.872729	+	0.488205i	\(0.162348\pi\)
\(23\)	−20.9045	−	36.2077i	−0.189517	−	0.328253i	0.755572	−	0.655065i	\(-0.227359\pi\)
							−0.945089	+	0.326812i	\(0.894026\pi\)
\(29\)	−30.8106	−	53.3656i	−0.197289	−	0.341715i	0.750359	−	0.661030i	\(-0.229881\pi\)
							−0.947649	+	0.319315i	\(0.896547\pi\)
\(31\)	191.932			1.11200			0.556000	−	0.831182i	\(-0.312335\pi\)
							0.556000	+	0.831182i	\(0.312335\pi\)
\(37\)	−49.2118	−	85.2373i	−0.218659	−	0.378728i	0.735740	−	0.677265i	\(-0.236835\pi\)
							−0.954398	+	0.298537i	\(0.903501\pi\)
\(41\)	15.3726	+	26.6261i	0.0585561	+	0.101422i	0.893817	−	0.448431i	\(-0.148017\pi\)
							−0.835261	+	0.549853i	\(0.814684\pi\)
\(43\)	−119.163	+	206.396i	−0.422608	+	0.731978i	−0.996194	−	0.0871672i	\(-0.972219\pi\)
							0.573586	+	0.819145i	\(0.305552\pi\)
\(47\)	−511.482			−1.58739			−0.793695	−	0.608316i	\(-0.791845\pi\)
							−0.793695	+	0.608316i	\(0.791845\pi\)