Embedded newform 547.2.o.a.4.14

• Label: 547.2.o.a.4.14
• 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			4.14
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30259 - 0.0149904i) q^{2} +(-0.660348 + 2.89317i) q^{3} +(-0.302966 - 0.00697409i) q^{4} +(0.484844 - 0.911015i) q^{5} +(0.903530 - 3.75871i) q^{6} +(-1.50505 - 3.70874i) q^{7} +(2.99833 + 0.103552i) q^{8} +(-5.23149 - 2.51935i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30259	−	0.0149904i	−0.921067	−	0.0105998i	−0.450319	−	0.892868i	\(-0.648690\pi\)
							−0.470748	+	0.882268i	\(0.656016\pi\)
\(3\)	−0.660348	+	2.89317i	−0.381252	+	1.67038i	0.312311	+	0.949980i	\(0.398897\pi\)
							−0.693564	+	0.720395i	\(0.743960\pi\)
\(5\)	0.484844	−	0.911015i	0.216829	−	0.407418i	−0.750527	−	0.660839i	\(-0.770200\pi\)
							0.967356	+	0.253421i	\(0.0815558\pi\)
\(7\)	−1.50505	−	3.70874i	−0.568856	−	1.40177i	−0.890532	−	0.454921i	\(-0.849667\pi\)
							0.321676	−	0.946850i	\(-0.395754\pi\)
\(11\)	4.17748	+	4.34922i	1.25956	+	1.31134i	0.930163	+	0.367146i	\(0.119665\pi\)
							0.329394	+	0.944193i	\(0.393156\pi\)
\(13\)	3.30939	+	0.498811i	0.917861	+	0.138345i	0.590948	−	0.806709i	\(-0.298754\pi\)
							0.326912	+	0.945055i	\(0.393992\pi\)
\(17\)	−0.132367	+	0.785902i	−0.0321038	+	0.190609i	−0.997547	−	0.0700058i	\(-0.977698\pi\)
							0.965443	+	0.260615i	\(0.0839253\pi\)
\(19\)	1.46916	−	0.792771i	0.337049	−	0.181874i	−0.297569	−	0.954700i	\(-0.596176\pi\)
							0.634618	+	0.772826i	\(0.281158\pi\)
\(23\)	1.32633	+	0.310767i	0.276560	+	0.0647994i	0.364094	−	0.931362i	\(-0.381379\pi\)
							−0.0875343	+	0.996162i	\(0.527899\pi\)
\(29\)	−2.94406	−	1.67724i	−0.546698	−	0.311456i	0.196526	−	0.980499i	\(-0.437034\pi\)
							−0.743224	+	0.669043i	\(0.766704\pi\)
\(31\)	−4.53246	−	1.63002i	−0.814054	−	0.292760i	−0.101910	−	0.994794i	\(-0.532495\pi\)
							−0.712144	+	0.702034i	\(0.752276\pi\)
\(37\)	−0.0216091	+	0.0174370i	−0.00355252	+	0.00286663i	−0.629753	−	0.776796i	\(-0.716844\pi\)
							0.626200	+	0.779662i	\(0.284609\pi\)
\(41\)	2.89602	+	5.01605i	0.452282	+	0.783375i	0.998527	−	0.0542498i	\(-0.0172767\pi\)
							−0.546245	+	0.837625i	\(0.683943\pi\)
\(43\)	−0.311340	+	3.17278i	−0.0474788	+	0.483844i	0.941629	+	0.336652i	\(0.109295\pi\)
							−0.989108	+	0.147192i	\(0.952976\pi\)
\(47\)	13.0884	−	1.05661i	1.90914	−	0.154122i	0.931410	−	0.363973i	\(-0.118580\pi\)
							0.977733	+	0.209851i	\(0.0672978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.14	✓	6480
547.137	even	273	inner	547.2.o.a.137.14	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.14	✓	6480	1.1	even	1	trivial
547.2.o.a.137.14	yes	6480	547.137	even	273	inner