Embedded newform 729.2.e.j.649.1

• Label: 729.2.e.j.649.1
• Level: $729$
• Weight: $2$
• Character: 729.649
• 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			649.1
Root			\(-3.10658i\) of defining polynomial
Character	\(\chi\)	\(=\)	729.649
Dual form			729.2.e.j.82.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.369007 + 2.09274i) q^{2} +(-2.36403 - 0.860436i) q^{4} +(-1.58643 + 1.33117i) q^{5} +(-4.55626 + 1.65834i) q^{7} +(0.547989 - 0.949144i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.369007	+	2.09274i	−0.260928	+	1.47979i	0.519458	+	0.854496i	\(0.326134\pi\)
							−0.780386	+	0.625298i	\(0.784977\pi\)
\(3\)		0			0
							\(5\)	−1.58643	+	1.33117i	−0.709474	+	0.595319i	−0.924452	−	0.381300i	\(-0.875477\pi\)
0.214977	+	0.976619i	\(0.431032\pi\)
\(7\)	−4.55626	+	1.65834i	−1.72211	+	0.626795i	−0.998019	−	0.0629144i	\(-0.979960\pi\)
							−0.724086	+	0.689709i	\(0.757738\pi\)
\(11\)	3.17869	+	2.66724i	0.958412	+	0.804203i	0.980694	−	0.195549i	\(-0.0626487\pi\)
							−0.0222820	+	0.999752i	\(0.507093\pi\)
\(13\)	−0.211159	−	1.19754i	−0.0585649	−	0.332138i	0.941422	−	0.337230i	\(-0.109490\pi\)
							−0.999987	+	0.00509231i	\(0.998379\pi\)
\(17\)	−1.18182	−	2.04697i	−0.286633	−	0.496463i	0.686371	−	0.727252i	\(-0.259203\pi\)
							−0.973004	+	0.230789i	\(0.925869\pi\)
\(19\)	0.919003	−	1.59176i	0.210834	−	0.365175i	−0.741142	−	0.671348i	\(-0.765715\pi\)
							0.951976	+	0.306174i	\(0.0990488\pi\)
\(23\)	4.04403	+	1.47191i	0.843239	+	0.306914i	0.727281	−	0.686340i	\(-0.240784\pi\)
							0.115958	+	0.993254i	\(0.463006\pi\)
\(29\)	0.517788	−	2.93652i	0.0961507	−	0.545298i	−0.898238	−	0.439510i	\(-0.855152\pi\)
							0.994389	−	0.105788i	\(-0.0337366\pi\)
\(31\)	−1.38472	−	0.503996i	−0.248703	−	0.0905204i	0.214661	−	0.976689i	\(-0.431135\pi\)
							−0.463364	+	0.886168i	\(0.653358\pi\)
\(37\)	−4.48554	−	7.76918i	−0.737418	−	1.27725i	−0.953654	−	0.300905i	\(-0.902711\pi\)
							0.216236	−	0.976341i	\(-0.430622\pi\)
\(41\)	0.392536	+	2.22618i	0.0613038	+	0.347671i	0.999996	+	0.00291413i	\(0.000927599\pi\)
							−0.938692	+	0.344757i	\(0.887961\pi\)
\(43\)	−4.20164	−	3.52560i	−0.640745	−	0.537649i	0.263502	−	0.964659i	\(-0.415122\pi\)
							−0.904247	+	0.427010i	\(0.859567\pi\)
\(47\)	−6.74994	+	2.45678i	−0.984579	+	0.358358i	−0.783619	−	0.621242i	\(-0.786628\pi\)
							−0.200960	+	0.979599i	\(0.564406\pi\)