Embedded newform 547.2.o.a.4.12

• Label: 547.2.o.a.4.12
• 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			4.12
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48739 - 0.0171172i) q^{2} +(-0.358018 + 1.56858i) q^{3} +(0.212570 + 0.00489324i) q^{4} +(1.55631 - 2.92429i) q^{5} +(0.559362 - 2.32696i) q^{6} +(-0.286333 - 0.705581i) q^{7} +(2.65712 + 0.0917681i) q^{8} +(0.370646 + 0.178494i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48739	−	0.0171172i	−1.05174	−	0.0121037i	−0.516085	−	0.856537i	\(-0.672611\pi\)
							−0.535660	+	0.844434i	\(0.679937\pi\)
\(3\)	−0.358018	+	1.56858i	−0.206702	+	0.905619i	0.760042	+	0.649873i	\(0.225178\pi\)
							−0.966744	+	0.255746i	\(0.917679\pi\)
\(5\)	1.55631	−	2.92429i	0.696004	−	1.30778i	−0.245045	−	0.969512i	\(-0.578803\pi\)
							0.941049	−	0.338271i	\(-0.109842\pi\)
\(7\)	−0.286333	−	0.705581i	−0.108224	−	0.266684i	0.862852	−	0.505457i	\(-0.168676\pi\)
							−0.971075	+	0.238773i	\(0.923255\pi\)
\(11\)	−1.81023	−	1.88465i	−0.545806	−	0.568244i	0.389976	−	0.920825i	\(-0.372483\pi\)
							−0.935781	+	0.352581i	\(0.885304\pi\)
\(13\)	−3.95628	−	0.596314i	−1.09728	−	0.165388i	−0.424640	−	0.905362i	\(-0.639599\pi\)
							−0.672635	+	0.739974i	\(0.734838\pi\)
\(17\)	0.346163	−	2.05527i	0.0839568	−	0.498476i	−0.912116	−	0.409933i	\(-0.865552\pi\)
							0.996072	−	0.0885424i	\(-0.0282209\pi\)
\(19\)	−2.33329	+	1.25906i	−0.535294	+	0.288849i	−0.720034	−	0.693938i	\(-0.755874\pi\)
							0.184740	+	0.982787i	\(0.440856\pi\)
\(23\)	−3.53575	−	0.828443i	−0.737254	−	0.172742i	−0.157483	−	0.987522i	\(-0.550338\pi\)
							−0.579771	+	0.814779i	\(0.696858\pi\)
\(29\)	3.74701	+	2.13468i	0.695802	+	0.396401i	0.801498	−	0.597998i	\(-0.204037\pi\)
							−0.105696	+	0.994398i	\(0.533707\pi\)
\(31\)	−7.64688	−	2.75007i	−1.37342	−	0.493926i	−0.455339	−	0.890318i	\(-0.650482\pi\)
							−0.918082	+	0.396391i	\(0.870262\pi\)
\(37\)	6.18185	−	4.98832i	1.01629	−	0.820074i	0.0325125	−	0.999471i	\(-0.489649\pi\)
							0.983777	+	0.179397i	\(0.0574147\pi\)
\(41\)	−3.86068	−	6.68690i	−0.602938	−	1.04432i	−0.992374	−	0.123265i	\(-0.960663\pi\)
							0.389436	−	0.921053i	\(-0.372670\pi\)
\(43\)	0.607581	−	6.19169i	0.0926552	−	0.944225i	−0.829785	−	0.558084i	\(-0.811537\pi\)
							0.922440	−	0.386141i	\(-0.126192\pi\)
\(47\)	0.124689	−	0.0100659i	0.0181878	−	0.00146827i	−0.0713693	−	0.997450i	\(-0.522737\pi\)
							0.0895571	+	0.995982i	\(0.471455\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.12	✓	6480
547.137	even	273	inner	547.2.o.a.137.12	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.12	✓	6480	1.1	even	1	trivial
547.2.o.a.137.12	yes	6480	547.137	even	273	inner