Embedded newform 729.2.k.a.4.10

• Label: 729.2.k.a.4.10
• 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			4.10
Character	\(\chi\)	\(=\)	729.4
Dual form			729.2.k.a.547.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.29202 - 0.0296337i) q^{2} +(-1.03658 - 1.38762i) q^{3} +(3.25315 + 0.0841346i) q^{4} +(3.65994 + 1.12154i) q^{5} +(2.33475 + 3.21117i) q^{6} +(-0.556114 + 1.38674i) q^{7} +(-2.87282 - 0.111478i) q^{8} +(-0.850984 + 2.87677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12960 q - 162 q^{2} - 162 q^{3} - 162 q^{4} - 162 q^{5} - 162 q^{6} - 162 q^{7} - 162 q^{8} - 162 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.29202	−	0.0296337i	−1.62070	−	0.0209542i	−0.802778	−	0.596278i	\(-0.796646\pi\)
							−0.817926	+	0.575324i	\(0.804876\pi\)
\(3\)	−1.03658	−	1.38762i	−0.598472	−	0.801143i
							\(5\)	3.65994	+	1.12154i	1.63677	+	0.501569i	0.969880	−	0.243584i	\(-0.0783230\pi\)
0.666894	+	0.745153i	\(0.267623\pi\)
\(7\)	−0.556114	+	1.38674i	−0.210191	+	0.524140i	−0.995478	−	0.0949925i	\(-0.969717\pi\)
							0.785287	+	0.619133i	\(0.212516\pi\)
\(11\)	−3.20888	−	1.82509i	−0.967512	−	0.550285i	−0.0730015	−	0.997332i	\(-0.523258\pi\)
							−0.894511	+	0.447046i	\(0.852476\pi\)
\(13\)	−1.41919	−	3.17649i	−0.393611	−	0.880998i	−0.996533	−	0.0831981i	\(-0.973487\pi\)
							0.602922	−	0.797800i	\(-0.294003\pi\)
\(17\)	1.56997	+	0.713639i	0.380774	+	0.173083i	0.595047	−	0.803691i	\(-0.297133\pi\)
							−0.214273	+	0.976774i	\(0.568738\pi\)
\(19\)	−3.85413	−	5.61945i	−0.884197	−	1.28919i	−0.956477	−	0.291806i	\(-0.905744\pi\)
							0.0722803	−	0.997384i	\(-0.476972\pi\)
\(23\)	−1.69479	+	5.17050i	−0.353388	+	1.07812i	0.605863	+	0.795569i	\(0.292828\pi\)
							−0.959251	+	0.282554i	\(0.908818\pi\)
\(29\)	5.36459	−	3.42982i	0.996180	−	0.636902i	0.0636413	−	0.997973i	\(-0.479729\pi\)
							0.932538	+	0.361071i	\(0.117589\pi\)
\(31\)	−2.84278	−	10.4739i	−0.510578	−	1.88116i	−0.471383	−	0.881929i	\(-0.656245\pi\)
							−0.0391949	−	0.999232i	\(-0.512479\pi\)
\(37\)	−10.1592	−	1.99520i	−1.67016	−	0.328010i	−0.733887	−	0.679271i	\(-0.762296\pi\)
							−0.936277	+	0.351262i	\(0.885753\pi\)
\(41\)	2.77699	−	1.05159i	0.433693	−	0.164231i	−0.127642	−	0.991820i	\(-0.540741\pi\)
							0.561335	+	0.827589i	\(0.310288\pi\)
\(43\)	10.6392	−	4.67109i	1.62246	−	0.712335i	0.624804	−	0.780781i	\(-0.285179\pi\)
							0.997655	+	0.0684464i	\(0.0218042\pi\)
\(47\)	1.51873	−	5.59557i	0.221529	−	0.816198i	−0.763788	−	0.645467i	\(-0.776663\pi\)
							0.985318	−	0.170731i	\(-0.0546129\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.k.a.4.10	✓	12960
729.547	even	243	inner	729.2.k.a.547.10	yes	12960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.k.a.4.10	✓	12960	1.1	even	1	trivial
729.2.k.a.547.10	yes	12960	729.547	even	243	inner