Embedded newform 547.2.o.a.4.1

• Label: 547.2.o.a.4.1
• Level: $547$
• Weight: $2$
• Character: 547.4
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $6480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.o (of order \(273\), degree \(144\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(6480\)
Relative dimension:	\(45\) over \(\Q(\zeta_{273})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{273}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.82015 - 0.0324547i) q^{2} +(-0.0795262 + 0.348427i) q^{3} +(5.95271 + 0.137028i) q^{4} +(0.114891 - 0.215879i) q^{5} +(0.235584 - 0.980034i) q^{6} +(-0.421000 - 1.03743i) q^{7} +(-11.1458 - 0.384938i) q^{8} +(2.58783 + 1.24623i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{273}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.82015	−	0.0324547i	−1.99415	−	0.0229490i	−0.996203	−	0.0870637i	\(-0.972252\pi\)
							−0.997943	+	0.0641147i	\(0.979578\pi\)
\(3\)	−0.0795262	+	0.348427i	−0.0459145	+	0.201164i	−0.992683	−	0.120751i	\(-0.961470\pi\)
							0.946768	+	0.321916i	\(0.104327\pi\)
\(5\)	0.114891	−	0.215879i	0.0513810	−	0.0965442i	−0.855756	−	0.517379i	\(-0.826908\pi\)
							0.907137	+	0.420835i	\(0.138263\pi\)
\(7\)	−0.421000	−	1.03743i	−0.159123	−	0.392110i	0.826068	−	0.563570i	\(-0.190573\pi\)
							−0.985191	+	0.171461i	\(0.945151\pi\)
\(11\)	0.169700	+	0.176676i	0.0511664	+	0.0532698i	0.746294	−	0.665617i	\(-0.231831\pi\)
							−0.695127	+	0.718887i	\(0.744652\pi\)
\(13\)	3.67499	+	0.553916i	1.01926	+	0.153629i	0.637353	−	0.770572i	\(-0.280029\pi\)
							0.381907	+	0.924201i	\(0.375267\pi\)
\(17\)	1.18161	−	7.01553i	0.286582	−	1.70152i	−0.355333	−	0.934740i	\(-0.615632\pi\)
							0.641914	−	0.766776i	\(-0.278140\pi\)
\(19\)	−2.98793	+	1.61231i	−0.685479	+	0.369890i	−0.780127	−	0.625621i	\(-0.784846\pi\)
							0.0946482	+	0.995511i	\(0.469827\pi\)
\(23\)	−2.49148	−	0.583767i	−0.519510	−	0.121724i	−0.0398966	−	0.999204i	\(-0.512703\pi\)
							−0.479613	+	0.877480i	\(0.659223\pi\)
\(29\)	2.51771	+	1.43435i	0.467526	+	0.266352i	0.710521	−	0.703676i	\(-0.248459\pi\)
							−0.242994	+	0.970028i	\(0.578130\pi\)
\(31\)	−4.58048	−	1.64729i	−0.822679	−	0.295862i	−0.106972	−	0.994262i	\(-0.534115\pi\)
							−0.715708	+	0.698400i	\(0.753896\pi\)
\(37\)	1.98006	−	1.59777i	0.325520	−	0.262672i	−0.451330	−	0.892357i	\(-0.649050\pi\)
							0.776851	+	0.629685i	\(0.216816\pi\)
\(41\)	1.53217	+	2.65379i	0.239284	+	0.414452i	0.960509	−	0.278249i	\(-0.0897539\pi\)
							−0.721225	+	0.692701i	\(0.756421\pi\)
\(43\)	0.809375	−	8.24813i	0.123428	−	1.25783i	−0.709603	−	0.704602i	\(-0.751126\pi\)
							0.833031	−	0.553226i	\(-0.186603\pi\)
\(47\)	8.47408	−	0.684099i	1.23607	−	0.0997861i	0.554905	−	0.831914i	\(-0.312755\pi\)
							0.681167	+	0.732128i	\(0.261473\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.1	✓	6480
547.137	even	273	inner	547.2.o.a.137.1	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.1	✓	6480	1.1	even	1	trivial
547.2.o.a.137.1	yes	6480	547.137	even	273	inner