Embedded newform 729.2.e.t.163.1

• Label: 729.2.e.t.163.1
• Level: $729$
• Weight: $2$
• Character: 729.163
• Analytic conductor: $5.821$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.e (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.82109430735\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 18x^{10} + 105x^{8} + 266x^{6} + 306x^{4} + 132x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			163.1
Root			\(0.0878222i\) of defining polynomial
Character	\(\chi\)	\(=\)	729.163
Dual form			729.2.e.t.568.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.162544 + 0.0591613i) q^{2} +(-1.50917 + 1.26634i) q^{4} +(-0.648847 - 3.67980i) q^{5} +(2.32226 + 1.94861i) q^{7} +(0.343364 - 0.594724i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} - 3 q^{4} + 6 q^{5} + 6 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

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\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.162544	+	0.0591613i	−0.114936	+	0.0418333i	−0.398848	−	0.917017i	\(-0.630590\pi\)
							0.283912	+	0.958850i	\(0.408368\pi\)
\(3\)		0			0
							\(5\)	−0.648847	−	3.67980i	−0.290173	−	1.64565i	−0.686198	−	0.727415i	\(-0.740722\pi\)
0.396025	−	0.918240i	\(-0.370389\pi\)
\(7\)	2.32226	+	1.94861i	0.877733	+	0.736505i	0.965712	−	0.259617i	\(-0.0835964\pi\)
							−0.0879791	+	0.996122i	\(0.528041\pi\)
\(11\)	−0.432678	+	2.45384i	−0.130457	+	0.739861i	0.847458	+	0.530862i	\(0.178132\pi\)
							−0.977916	+	0.208999i	\(0.932979\pi\)
\(13\)	0.718995	+	0.261693i	0.199413	+	0.0725805i	0.439796	−	0.898098i	\(-0.355051\pi\)
							−0.240383	+	0.970678i	\(0.577273\pi\)
\(17\)	−2.31139	−	4.00345i	−0.560595	−	0.970979i	−0.997445	−	0.0714442i	\(-0.977239\pi\)
							0.436850	−	0.899534i	\(-0.356094\pi\)
\(19\)	0.305922	−	0.529872i	0.0701833	−	0.121561i	−0.828798	−	0.559548i	\(-0.810975\pi\)
							0.898982	+	0.437987i	\(0.144308\pi\)
\(23\)	4.99796	−	4.19379i	1.04215	−	0.874465i	0.0499011	−	0.998754i	\(-0.484109\pi\)
							0.992246	+	0.124289i	\(0.0396650\pi\)
\(29\)	6.15583	−	2.24054i	1.14311	−	0.416057i	0.300073	−	0.953916i	\(-0.402989\pi\)
							0.843035	+	0.537859i	\(0.180767\pi\)
\(31\)	5.01792	−	4.21053i	0.901245	−	0.756234i	−0.0691887	−	0.997604i	\(-0.522041\pi\)
							0.970433	+	0.241370i	\(0.0775966\pi\)
\(37\)	−2.47984	−	4.29522i	−0.407684	−	0.706129i	0.586946	−	0.809626i	\(-0.300330\pi\)
							−0.994630	+	0.103497i	\(0.966997\pi\)
\(41\)	−4.94301	−	1.79911i	−0.771968	−	0.280973i	−0.0741488	−	0.997247i	\(-0.523624\pi\)
							−0.697819	+	0.716274i	\(0.745846\pi\)
\(43\)	0.967320	−	5.48594i	0.147515	−	0.836599i	−0.817798	−	0.575505i	\(-0.804806\pi\)
							0.965313	−	0.261094i	\(-0.0840832\pi\)
\(47\)	0.848483	+	0.711962i	0.123764	+	0.103850i	0.702569	−	0.711616i	\(-0.252036\pi\)
							−0.578805	+	0.815466i	\(0.696481\pi\)