Embedded newform 729.2.k.a.4.6

• Label: 729.2.k.a.4.6
• 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			4.6
Character	\(\chi\)	\(=\)	729.4
Dual form			729.2.k.a.547.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.48775 - 0.0321643i) q^{2} +(0.111243 + 1.72847i) q^{3} +(4.18852 + 0.108326i) q^{4} +(2.61631 + 0.801736i) q^{5} +(-0.221150 - 4.30359i) q^{6} +(0.558840 - 1.39354i) q^{7} +(-5.44433 - 0.211265i) q^{8} +(-2.97525 + 0.384562i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12960 q - 162 q^{2} - 162 q^{3} - 162 q^{4} - 162 q^{5} - 162 q^{6} - 162 q^{7} - 162 q^{8} - 162 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48775	−	0.0321643i	−1.75910	−	0.0227436i	−0.873402	−	0.487000i	\(-0.838091\pi\)
							−0.885701	+	0.464256i	\(0.846322\pi\)
\(3\)	0.111243	+	1.72847i	0.0642263	+	0.997935i
							\(5\)	2.61631	+	0.801736i	1.17005	+	0.358547i	0.816766	−	0.576969i	\(-0.195765\pi\)
0.353283	+	0.935516i	\(0.385065\pi\)
\(7\)	0.558840	−	1.39354i	0.211222	−	0.526709i	−0.784392	−	0.620265i	\(-0.787025\pi\)
							0.995614	+	0.0935559i	\(0.0298234\pi\)
\(11\)	3.15263	+	1.79310i	0.950553	+	0.540640i	0.889221	−	0.457478i	\(-0.151247\pi\)
							0.0613320	+	0.998117i	\(0.480465\pi\)
\(13\)	−1.89074	−	4.23194i	−0.524397	−	1.17373i	−0.961622	−	0.274377i	\(-0.911528\pi\)
							0.437225	−	0.899352i	\(-0.355961\pi\)
\(17\)	7.11213	+	3.23286i	1.72494	+	0.784083i	0.994921	+	0.100661i	\(0.0320958\pi\)
							0.730023	+	0.683422i	\(0.239509\pi\)
\(19\)	3.02456	+	4.40991i	0.693881	+	1.01170i	0.998249	+	0.0591493i	\(0.0188388\pi\)
							−0.304368	+	0.952555i	\(0.598445\pi\)
\(23\)	0.788145	−	2.40449i	0.164340	−	0.501371i	−0.834422	−	0.551126i	\(-0.814198\pi\)
							0.998761	+	0.0497558i	\(0.0158443\pi\)
\(29\)	−3.87709	+	2.47880i	−0.719958	+	0.460301i	−0.846998	−	0.531596i	\(-0.821592\pi\)
							0.127040	+	0.991898i	\(0.459452\pi\)
\(31\)	−2.21573	−	8.16359i	−0.397957	−	1.46622i	−0.826621	−	0.562759i	\(-0.809740\pi\)
							0.428664	−	0.903464i	\(-0.358985\pi\)
\(37\)	6.29290	+	1.23588i	1.03455	+	0.203178i	0.681034	−	0.732252i	\(-0.261531\pi\)
							0.353512	+	0.935430i	\(0.384987\pi\)
\(41\)	0.249606	−	0.0945212i	0.0389819	−	0.0147617i	−0.334572	−	0.942370i	\(-0.608592\pi\)
							0.373554	+	0.927608i	\(0.378139\pi\)
\(43\)	4.02908	−	1.76895i	0.614429	−	0.269763i	−0.0714895	−	0.997441i	\(-0.522775\pi\)
							0.685918	+	0.727679i	\(0.259401\pi\)
\(47\)	−0.976517	+	3.59786i	−0.142440	+	0.524801i	0.857507	+	0.514472i	\(0.172012\pi\)
							−0.999947	+	0.0103290i	\(0.996712\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.2.k.a.4.6	✓	12960
729.547	even	243	inner	729.2.k.a.547.6	yes	12960

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
729.2.k.a.4.6	✓	12960	1.1	even	1	trivial
729.2.k.a.547.6	yes	12960	729.547	even	243	inner