Embedded newform 547.2.o.a.4.13

• Label: 547.2.o.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37136 - 0.0157819i) q^{2} +(0.455404 - 1.99525i) q^{3} +(-0.119088 - 0.00274133i) q^{4} +(-0.496687 + 0.933269i) q^{5} +(-0.656012 + 2.72903i) q^{6} +(0.132038 + 0.325368i) q^{7} +(2.90454 + 0.100313i) q^{8} +(-1.07074 - 0.515640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37136	−	0.0157819i	−0.969699	−	0.0111595i	−0.474785	−	0.880102i	\(-0.657474\pi\)
							−0.494914	+	0.868942i	\(0.664800\pi\)
\(3\)	0.455404	−	1.99525i	0.262927	−	1.15196i	−0.655131	−	0.755515i	\(-0.727387\pi\)
							0.918059	−	0.396445i	\(-0.129756\pi\)
\(5\)	−0.496687	+	0.933269i	−0.222125	+	0.417371i	−0.968809	−	0.247809i	\(-0.920290\pi\)
							0.746684	+	0.665179i	\(0.231645\pi\)
\(7\)	0.132038	+	0.325368i	0.0499058	+	0.122977i	0.949518	−	0.313711i	\(-0.101572\pi\)
							−0.899613	+	0.436689i	\(0.856151\pi\)
\(11\)	0.249485	+	0.259742i	0.0752226	+	0.0783150i	0.757750	−	0.652545i	\(-0.226299\pi\)
							−0.682527	+	0.730860i	\(0.739119\pi\)
\(13\)	2.70982	+	0.408440i	0.751569	+	0.113281i	0.513645	−	0.858003i	\(-0.328295\pi\)
							0.237924	+	0.971284i	\(0.423533\pi\)
\(17\)	0.171912	−	1.02069i	0.0416948	−	0.247554i	−0.957464	−	0.288551i	\(-0.906826\pi\)
							0.999159	+	0.0409973i	\(0.0130535\pi\)
\(19\)	−0.153460	+	0.0828083i	−0.0352062	+	0.0189975i	−0.492390	−	0.870375i	\(-0.663877\pi\)
							0.457184	+	0.889372i	\(0.348858\pi\)
\(23\)	7.82623	+	1.83373i	1.63188	+	0.382358i	0.940409	−	0.340044i	\(-0.110442\pi\)
							0.691473	+	0.722403i	\(0.256962\pi\)
\(29\)	−6.27863	−	3.57696i	−1.16591	−	0.664225i	−0.215893	−	0.976417i	\(-0.569266\pi\)
							−0.950019	+	0.312192i	\(0.898937\pi\)
\(31\)	1.19682	+	0.430414i	0.214954	+	0.0773046i	0.443676	−	0.896187i	\(-0.353674\pi\)
							−0.228721	+	0.973492i	\(0.573454\pi\)
\(37\)	7.12842	−	5.75213i	1.17190	−	0.945645i	0.172678	−	0.984978i	\(-0.444758\pi\)
							0.999226	+	0.0393333i	\(0.0125234\pi\)
\(41\)	−1.72771	−	2.99247i	−0.269822	−	0.467346i	0.698994	−	0.715128i	\(-0.253632\pi\)
							−0.968816	+	0.247782i	\(0.920298\pi\)
\(43\)	0.900165	−	9.17335i	0.137274	−	1.39892i	−0.639330	−	0.768933i	\(-0.720788\pi\)
							0.776603	−	0.629990i	\(-0.216941\pi\)
\(47\)	−6.52365	+	0.526643i	−0.951572	+	0.0768188i	−0.546494	−	0.837463i	\(-0.684038\pi\)
							−0.405078	+	0.914282i	\(0.632756\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.13	✓	6480
547.137	even	273	inner	547.2.o.a.137.13	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.13	✓	6480	1.1	even	1	trivial
547.2.o.a.137.13	yes	6480	547.137	even	273	inner