Embedded newform 729.2.e.j.568.1

• Label: 729.2.e.j.568.1
• Level: $729$
• Weight: $2$
• Character: 729.568
• 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			568.1
Root			\(-1.91182i\) of defining polynomial
Character	\(\chi\)	\(=\)	729.568
Dual form			729.2.e.j.163.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.30957 - 0.840614i) q^{2} +(3.09538 + 2.59733i) q^{4} +(0.534882 - 3.03347i) q^{5} +(-2.03666 + 1.70896i) q^{7} +(-2.50784 - 4.34371i) q^{8} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.30957	−	0.840614i	−1.63311	−	0.594404i	−0.647295	−	0.762239i	\(-0.724100\pi\)
							−0.985815	+	0.167836i	\(0.946322\pi\)
\(3\)		0			0
							\(5\)	0.534882	−	3.03347i	0.239207	−	1.35661i	−0.594365	−	0.804196i	\(-0.702596\pi\)
0.833571	−	0.552412i	\(-0.186292\pi\)
\(7\)	−2.03666	+	1.70896i	−0.769784	+	0.645926i	−0.940654	−	0.339368i	\(-0.889787\pi\)
							0.170870	+	0.985294i	\(0.445342\pi\)
\(11\)	0.596367	+	3.38217i	0.179811	+	1.01976i	0.932442	+	0.361319i	\(0.117673\pi\)
							−0.752631	+	0.658443i	\(0.771216\pi\)
\(13\)	3.14229	−	1.14370i	0.871515	−	0.317205i	0.132734	−	0.991152i	\(-0.457624\pi\)
							0.738780	+	0.673946i	\(0.235402\pi\)
\(17\)	1.28641	−	2.22813i	0.312000	−	0.540400i	−0.666795	−	0.745241i	\(-0.732334\pi\)
							0.978795	+	0.204841i	\(0.0656678\pi\)
\(19\)	1.04838	+	1.81585i	0.240515	+	0.416585i	0.960861	−	0.277030i	\(-0.0893503\pi\)
							−0.720346	+	0.693615i	\(0.756017\pi\)
\(23\)	−0.409408	−	0.343534i	−0.0853674	−	0.0716318i	0.599105	−	0.800670i	\(-0.295523\pi\)
							−0.684473	+	0.729038i	\(0.739968\pi\)
\(29\)	2.37826	+	0.865617i	0.441632	+	0.160741i	0.553259	−	0.833009i	\(-0.313384\pi\)
							−0.111627	+	0.993750i	\(0.535606\pi\)
\(31\)	5.90980	+	4.95891i	1.06143	+	0.890647i	0.994249	−	0.107093i	\(-0.0341544\pi\)
							0.0671832	+	0.997741i	\(0.478599\pi\)
\(37\)	5.14783	−	8.91631i	0.846298	−	1.46583i	−0.0381907	−	0.999270i	\(-0.512159\pi\)
							0.884489	−	0.466561i	\(-0.154507\pi\)
\(41\)	−4.59040	+	1.67077i	−0.716901	+	0.260931i	−0.674610	−	0.738175i	\(-0.735688\pi\)
							−0.0422913	+	0.999105i	\(0.513466\pi\)
\(43\)	0.476055	+	2.69984i	0.0725977	+	0.411722i	0.999350	+	0.0360490i	\(0.0114772\pi\)
							−0.926752	+	0.375673i	\(0.877412\pi\)
\(47\)	4.33428	−	3.63689i	0.632219	−	0.530495i	−0.269398	−	0.963029i	\(-0.586825\pi\)
							0.901618	+	0.432534i	\(0.142380\pi\)