Embedded newform 547.2.o.a.4.9

• Label: 547.2.o.a.4.9
• 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			4.9
Character	\(\chi\)	\(=\)	547.4
Dual form			547.2.o.a.137.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95847 - 0.0225384i) q^{2} +(0.189262 - 0.829210i) q^{3} +(1.83561 + 0.0422546i) q^{4} +(0.0364523 - 0.0684934i) q^{5} +(-0.389352 + 1.61971i) q^{6} +(1.35863 + 3.34794i) q^{7} +(0.320834 + 0.0110806i) q^{8} +(2.05114 + 0.987776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6480 q - 143 q^{2} - 94 q^{3} - 189 q^{4} - 141 q^{5} - 114 q^{6} - 154 q^{7} - 130 q^{8} - 1178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95847	−	0.0225384i	−1.38484	−	0.0159370i	−0.684120	−	0.729369i	\(-0.739814\pi\)
							−0.700724	+	0.713432i	\(0.747140\pi\)
\(3\)	0.189262	−	0.829210i	0.109270	−	0.478745i	−0.890450	−	0.455082i	\(-0.849610\pi\)
							0.999720	−	0.0236629i	\(-0.00753284\pi\)
\(5\)	0.0364523	−	0.0684934i	0.0163020	−	0.0306312i	−0.874484	−	0.485055i	\(-0.838800\pi\)
							0.890786	+	0.454424i	\(0.150155\pi\)
\(7\)	1.35863	+	3.34794i	0.513516	+	1.26540i	0.933730	+	0.357977i	\(0.116534\pi\)
							−0.420215	+	0.907425i	\(0.638045\pi\)
\(11\)	−3.41775	−	3.55826i	−1.03049	−	1.07286i	−0.997302	−	0.0734066i	\(-0.976613\pi\)
							−0.0331898	−	0.999449i	\(-0.510567\pi\)
\(13\)	4.19753	+	0.632676i	1.16419	+	0.175473i	0.702575	−	0.711610i	\(-0.252033\pi\)
							0.461611	+	0.887082i	\(0.347272\pi\)
\(17\)	−0.899167	+	5.33861i	−0.218080	+	1.29480i	0.634792	+	0.772683i	\(0.281086\pi\)
							−0.852872	+	0.522120i	\(0.825141\pi\)
\(19\)	−6.01202	+	3.24413i	−1.37925	+	0.744255i	−0.984641	−	0.174594i	\(-0.944139\pi\)
							−0.394610	+	0.918849i	\(0.629120\pi\)
\(23\)	−7.76432	−	1.81922i	−1.61897	−	0.379334i	−0.682726	−	0.730675i	\(-0.739206\pi\)
							−0.936248	+	0.351341i	\(0.885726\pi\)
\(29\)	5.05101	+	2.87758i	0.937950	+	0.534354i	0.885689	−	0.464280i	\(-0.153687\pi\)
							0.0522613	+	0.998633i	\(0.483357\pi\)
\(31\)	0.458829	+	0.165010i	0.0824082	+	0.0296366i	0.379293	−	0.925277i	\(-0.376167\pi\)
							−0.296884	+	0.954913i	\(0.595948\pi\)
\(37\)	2.68788	−	2.16893i	0.441884	−	0.356569i	−0.381197	−	0.924494i	\(-0.624488\pi\)
							0.823081	+	0.567925i	\(0.192253\pi\)
\(41\)	3.72040	+	6.44392i	0.581029	+	1.00637i	0.995358	+	0.0962442i	\(0.0306830\pi\)
							−0.414329	+	0.910127i	\(0.635984\pi\)
\(43\)	−0.343626	+	3.50181i	−0.0524025	+	0.534020i	0.932531	+	0.361089i	\(0.117595\pi\)
							−0.984934	+	0.172931i	\(0.944676\pi\)
\(47\)	2.28768	−	0.184681i	0.333693	−	0.0269384i	0.0875150	−	0.996163i	\(-0.472107\pi\)
							0.246178	+	0.969225i	\(0.420825\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.o.a.4.9	✓	6480
547.137	even	273	inner	547.2.o.a.137.9	yes	6480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.o.a.4.9	✓	6480	1.1	even	1	trivial
547.2.o.a.137.9	yes	6480	547.137	even	273	inner