Embedded newform 729.2.k.a.4.19

• Label: 729.2.k.a.4.19
• 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			4.19
Character	\(\chi\)	\(=\)	729.4
Dual form			729.2.k.a.547.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68750 - 0.0218178i) q^{2} +(-0.499964 + 1.65832i) q^{3} +(0.847837 + 0.0219272i) q^{4} +(0.0353662 + 0.0108375i) q^{5} +(0.879869 - 2.78751i) q^{6} +(-1.23579 + 3.08161i) q^{7} +(1.94249 + 0.0753776i) q^{8} +(-2.50007 - 1.65820i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12960 q - 162 q^{2} - 162 q^{3} - 162 q^{4} - 162 q^{5} - 162 q^{6} - 162 q^{7} - 162 q^{8} - 162 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68750	−	0.0218178i	−1.19324	−	0.0154275i	−0.586246	−	0.810133i	\(-0.699395\pi\)
							−0.606995	+	0.794706i	\(0.707625\pi\)
\(3\)	−0.499964	+	1.65832i	−0.288654	+	0.957433i
							\(5\)	0.0353662	+	0.0108375i	0.0158162	+	0.00484669i	0.300888	−	0.953660i	\(-0.402717\pi\)
−0.285072	+	0.958506i	\(0.592017\pi\)
\(7\)	−1.23579	+	3.08161i	−0.467086	+	1.16474i	0.489164	+	0.872192i	\(0.337302\pi\)
							−0.956251	+	0.292549i	\(0.905497\pi\)
\(11\)	−1.55843	−	0.886380i	−0.469885	−	0.267253i	0.241048	−	0.970513i	\(-0.422509\pi\)
							−0.710933	+	0.703260i	\(0.751727\pi\)
\(13\)	−1.61511	−	3.61500i	−0.447950	−	1.00262i	−0.987055	−	0.160383i	\(-0.948727\pi\)
							0.539105	−	0.842238i	\(-0.318763\pi\)
\(17\)	−2.25299	−	1.02411i	−0.546429	−	0.248383i	0.121520	−	0.992589i	\(-0.461223\pi\)
							−0.667950	+	0.744206i	\(0.732828\pi\)
\(19\)	3.01767	+	4.39988i	0.692302	+	1.00940i	0.998352	+	0.0573857i	\(0.0182765\pi\)
							−0.306050	+	0.952015i	\(0.599007\pi\)
\(23\)	0.500513	−	1.52697i	0.104364	−	0.318396i	−0.884642	−	0.466271i	\(-0.845597\pi\)
							0.989006	+	0.147875i	\(0.0472433\pi\)
\(29\)	1.72071	−	1.10013i	0.319528	−	0.204288i	−0.369125	−	0.929380i	\(-0.620343\pi\)
							0.688652	+	0.725092i	\(0.258203\pi\)
\(31\)	1.29438	+	4.76898i	0.232477	+	0.856534i	0.981094	+	0.193531i	\(0.0619941\pi\)
							−0.748617	+	0.663003i	\(0.769282\pi\)
\(37\)	−8.92589	−	1.75299i	−1.46741	−	0.288189i	−0.605740	−	0.795663i	\(-0.707123\pi\)
							−0.861667	+	0.507473i	\(0.830580\pi\)
\(41\)	5.48293	−	2.07628i	0.856290	−	0.324261i	0.113298	−	0.993561i	\(-0.463859\pi\)
							0.742992	+	0.669300i	\(0.233406\pi\)
\(43\)	−9.84639	+	4.32302i	−1.50156	+	0.659255i	−0.980901	−	0.194509i	\(-0.937689\pi\)
							−0.520660	+	0.853764i	\(0.674314\pi\)
\(47\)	2.14880	−	7.91697i	0.313434	−	1.15481i	−0.616582	−	0.787291i	\(-0.711483\pi\)
							0.930016	−	0.367519i	\(-0.119793\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.k.a.4.19	✓	12960
729.547	even	243	inner	729.2.k.a.547.19	yes	12960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.k.a.4.19	✓	12960	1.1	even	1	trivial
729.2.k.a.547.19	yes	12960	729.547	even	243	inner