Embedded newform 729.2.k.a.4.11

• Label: 729.2.k.a.4.11
• 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			4.11
Character	\(\chi\)	\(=\)	729.4
Dual form			729.2.k.a.547.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.27606 - 0.0294274i) q^{2} +(-1.57284 - 0.725368i) q^{3} +(3.18025 + 0.0822491i) q^{4} +(-3.48668 - 1.06845i) q^{5} +(3.55854 + 1.69727i) q^{6} +(0.216487 - 0.539838i) q^{7} +(-2.68694 - 0.104265i) q^{8} +(1.94768 + 2.28178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12960 q - 162 q^{2} - 162 q^{3} - 162 q^{4} - 162 q^{5} - 162 q^{6} - 162 q^{7} - 162 q^{8} - 162 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.27606	−	0.0294274i	−1.60942	−	0.0208083i	−0.797035	−	0.603934i	\(-0.793599\pi\)
							−0.812382	+	0.583125i	\(0.801830\pi\)
\(3\)	−1.57284	−	0.725368i	−0.908082	−	0.418791i
							\(5\)	−3.48668	−	1.06845i	−1.55929	−	0.477826i	−0.610047	−	0.792365i	\(-0.708849\pi\)
−0.949244	+	0.314540i	\(0.898150\pi\)
\(7\)	0.216487	−	0.539838i	0.0818243	−	0.204040i	−0.881613	−	0.471972i	\(-0.843542\pi\)
							0.963438	+	0.267933i	\(0.0863405\pi\)
\(11\)	−0.775368	−	0.441001i	−0.233782	−	0.132967i	0.373010	−	0.927827i	\(-0.378326\pi\)
							−0.606792	+	0.794861i	\(0.707544\pi\)
\(13\)	−1.68337	−	3.76779i	−0.466883	−	1.04500i	−0.982173	−	0.187982i	\(-0.939805\pi\)
							0.515290	−	0.857016i	\(-0.327684\pi\)
\(17\)	−6.45469	−	2.93401i	−1.56549	−	0.711603i	−0.571453	−	0.820635i	\(-0.693620\pi\)
							−0.994038	+	0.109032i	\(0.965225\pi\)
\(19\)	−2.98512	−	4.35242i	−0.684834	−	0.998513i	−0.998795	−	0.0490770i	\(-0.984372\pi\)
							0.313961	−	0.949436i	\(-0.398344\pi\)
\(23\)	1.64220	−	5.01006i	0.342423	−	1.04467i	−0.622608	−	0.782534i	\(-0.713927\pi\)
							0.965031	−	0.262137i	\(-0.0844271\pi\)
\(29\)	−6.32489	+	4.04378i	−1.17450	+	0.750912i	−0.973502	−	0.228680i	\(-0.926559\pi\)
							−0.201000	+	0.979591i	\(0.564419\pi\)
\(31\)	−1.11447	−	4.10614i	−0.200166	−	0.737485i	−0.991944	−	0.126677i	\(-0.959569\pi\)
							0.791778	−	0.610808i	\(-0.209155\pi\)
\(37\)	3.07552	+	0.604013i	0.505613	+	0.0992992i	0.439013	−	0.898481i	\(-0.355328\pi\)
							0.0665999	+	0.997780i	\(0.478785\pi\)
\(41\)	−0.830224	+	0.314390i	−0.129659	+	0.0490995i	−0.418118	−	0.908393i	\(-0.637310\pi\)
							0.288458	+	0.957492i	\(0.406857\pi\)
\(43\)	−9.55271	+	4.19408i	−1.45678	+	0.639592i	−0.971986	−	0.235039i	\(-0.924478\pi\)
							−0.484789	+	0.874631i	\(0.661104\pi\)
\(47\)	1.89759	−	6.99145i	0.276792	−	1.01981i	−0.680985	−	0.732297i	\(-0.738448\pi\)
							0.957778	−	0.287510i	\(-0.0928275\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.k.a.4.11	✓	12960
729.547	even	243	inner	729.2.k.a.547.11	yes	12960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.k.a.4.11	✓	12960	1.1	even	1	trivial
729.2.k.a.547.11	yes	12960	729.547	even	243	inner