Embedded newform 547.2.f.a.46.6

• Label: 547.2.f.a.46.6
• Level: $547$
• Weight: $2$
• Character: 547.46
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $528$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.f (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(528\)
Relative dimension:	\(44\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			46.6
Character	\(\chi\)	\(=\)	547.46
Dual form			547.2.f.a.440.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.255736 + 2.10617i) q^{2} +1.99966 q^{3} +(-2.42868 - 0.598615i) q^{4} +(0.984637 + 2.59628i) q^{5} +(-0.511385 + 4.21164i) q^{6} +(0.216641 + 1.78420i) q^{7} +(0.377195 - 0.994582i) q^{8} +0.998658 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 528 q - 8 q^{2} - 46 q^{3} - 50 q^{4} - 9 q^{5} - 21 q^{6} - 3 q^{7} + 14 q^{8} + 466 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.255736	+	2.10617i	−0.180832	+	1.48929i	0.566071	+	0.824356i	\(0.308463\pi\)
							−0.746904	+	0.664932i	\(0.768460\pi\)
\(3\)	1.99966			1.15451			0.577253	−	0.816565i	\(-0.304124\pi\)
							0.577253	+	0.816565i	\(0.304124\pi\)
\(5\)	0.984637	+	2.59628i	0.440343	+	1.16109i	0.953127	+	0.302572i	\(0.0978452\pi\)
							−0.512783	+	0.858518i	\(0.671386\pi\)
\(7\)	0.216641	+	1.78420i	0.0818825	+	0.674363i	0.974633	+	0.223811i	\(0.0718497\pi\)
							−0.892750	+	0.450552i	\(0.851227\pi\)
\(11\)	1.96772	−	2.85073i	0.593289	−	0.859528i	−0.405214	−	0.914222i	\(-0.632803\pi\)
							0.998503	+	0.0546938i	\(0.0174183\pi\)
\(13\)	−5.77553			−1.60184			−0.800922	−	0.598768i	\(-0.795657\pi\)
							−0.800922	+	0.598768i	\(0.795657\pi\)
\(17\)	1.77285	+	2.56841i	0.429979	+	0.622932i	0.976758	−	0.214345i	\(-0.0687617\pi\)
							−0.546779	+	0.837277i	\(0.684146\pi\)
\(19\)	6.40398	−	3.36107i	1.46917	−	0.771082i	0.475115	−	0.879924i	\(-0.342407\pi\)
							0.994059	+	0.108842i	\(0.0347143\pi\)
\(23\)	2.17698	−	1.92863i	0.453931	−	0.402148i	−0.404937	−	0.914345i	\(-0.632706\pi\)
							0.858868	+	0.512196i	\(0.171168\pi\)
\(29\)	5.68424	+	5.03580i	1.05554	+	0.935125i	0.998000	−	0.0632187i	\(-0.0201366\pi\)
							0.0575378	+	0.998343i	\(0.481675\pi\)
\(31\)	−6.17357	−	3.24014i	−1.10881	−	0.581946i	−0.192046	−	0.981386i	\(-0.561512\pi\)
							−0.916759	+	0.399440i	\(0.869205\pi\)
\(37\)	5.12280	+	4.53840i	0.842183	+	0.746109i	0.969327	−	0.245775i	\(-0.0790425\pi\)
							−0.127144	+	0.991884i	\(0.540581\pi\)
\(41\)	3.30349			0.515919			0.257960	−	0.966156i	\(-0.416950\pi\)
							0.257960	+	0.966156i	\(0.416950\pi\)
\(43\)	−5.24478	−	1.29272i	−0.799821	−	0.197138i	−0.181837	−	0.983329i	\(-0.558204\pi\)
							−0.617984	+	0.786191i	\(0.712050\pi\)
\(47\)	−3.78919	−	3.35693i	−0.552710	−	0.489658i	0.339916	−	0.940456i	\(-0.389601\pi\)
							−0.892626	+	0.450797i	\(0.851140\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.f.a.46.6	✓	528
547.440	even	13	inner	547.2.f.a.440.6	yes	528

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.f.a.46.6	✓	528	1.1	even	1	trivial
547.2.f.a.440.6	yes	528	547.440	even	13	inner