Embedded newform 729.2.k.a.4.16

• Label: 729.2.k.a.4.16
• Level: $729$
• Weight: $2$
• Character: 729.4
• Analytic conductor: $5.821$
• Analytic rank: $0$
• Dimension: $12960$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	729.k (of order \(243\), degree \(162\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.82109430735\)
Analytic rank:	\(0\)
Dimension:	\(12960\)
Relative dimension:	\(80\) over \(\Q(\zeta_{243})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{243}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	729.4
Dual form			729.2.k.a.547.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90902 - 0.0246819i) q^{2} +(0.0146033 - 1.73199i) q^{3} +(1.64441 + 0.0425285i) q^{4} +(-2.27162 - 0.696109i) q^{5} +(-0.0706267 + 3.30604i) q^{6} +(0.132583 - 0.330612i) q^{7} +(0.677327 + 0.0262834i) q^{8} +(-2.99957 - 0.0505857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12960 q - 162 q^{2} - 162 q^{3} - 162 q^{4} - 162 q^{5} - 162 q^{6} - 162 q^{7} - 162 q^{8} - 162 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{243}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90902	−	0.0246819i	−1.34988	−	0.0174527i	−0.665401	−	0.746486i	\(-0.731740\pi\)
							−0.684478	+	0.729033i	\(0.739970\pi\)
\(3\)	0.0146033	−	1.73199i	0.00843124	−	0.999964i
							\(5\)	−2.27162	−	0.696109i	−1.01590	−	0.311310i	−0.259726	−	0.965682i	\(-0.583632\pi\)
−0.756172	+	0.654373i	\(0.772933\pi\)
\(7\)	0.132583	−	0.330612i	0.0501115	−	0.124960i	−0.900988	−	0.433844i	\(-0.857157\pi\)
							0.951100	+	0.308884i	\(0.0999554\pi\)
\(11\)	−4.51004	−	2.56515i	−1.35983	−	0.773420i	−0.371891	−	0.928276i	\(-0.621291\pi\)
							−0.987937	+	0.154856i	\(0.950509\pi\)
\(13\)	1.99430	+	4.46372i	0.553118	+	1.23801i	0.947703	+	0.319154i	\(0.103399\pi\)
							−0.394584	+	0.918860i	\(0.629111\pi\)
\(17\)	0.761861	+	0.346308i	0.184778	+	0.0839921i	0.504063	−	0.863667i	\(-0.331838\pi\)
							−0.319285	+	0.947659i	\(0.603443\pi\)
\(19\)	−0.376229	−	0.548556i	−0.0863130	−	0.125847i	0.779114	−	0.626882i	\(-0.215669\pi\)
							−0.865427	+	0.501034i	\(0.832953\pi\)
\(23\)	0.0415526	−	0.126769i	0.00866432	−	0.0264333i	−0.945830	−	0.324662i	\(-0.894750\pi\)
							0.954495	+	0.298228i	\(0.0963957\pi\)
\(29\)	−4.04583	+	2.58668i	−0.751292	+	0.480335i	−0.857804	−	0.513977i	\(-0.828172\pi\)
							0.106512	+	0.994311i	\(0.466032\pi\)
\(31\)	1.19820	+	4.41462i	0.215203	+	0.792889i	0.987499	+	0.157626i	\(0.0503841\pi\)
							−0.772296	+	0.635263i	\(0.780892\pi\)
\(37\)	−5.74445	−	1.12817i	−0.944382	−	0.185471i	−0.303259	−	0.952908i	\(-0.598075\pi\)
							−0.641124	+	0.767438i	\(0.721531\pi\)
\(41\)	8.11419	−	3.07269i	1.26722	−	0.479874i	0.373140	−	0.927775i	\(-0.378281\pi\)
							0.894083	+	0.447901i	\(0.147828\pi\)
\(43\)	6.88896	−	3.02457i	1.05056	−	0.461243i	0.196001	−	0.980604i	\(-0.437204\pi\)
							0.854555	+	0.519361i	\(0.173830\pi\)
\(47\)	0.700712	−	2.58169i	0.102209	−	0.376578i	−0.895434	−	0.445195i	\(-0.853134\pi\)
							0.997643	+	0.0686171i	\(0.0218587\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.k.a.4.16	✓	12960
729.547	even	243	inner	729.2.k.a.547.16	yes	12960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.k.a.4.16	✓	12960	1.1	even	1	trivial
729.2.k.a.547.16	yes	12960	729.547	even	243	inner