Embedded newform 547.2.e.a.9.18

• Label: 547.2.e.a.9.18
• Level: $547$
• Weight: $2$
• Character: 547.9
• Analytic conductor: $4.368$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	547.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.36781699056\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(44\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			9.18
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.183655 + 0.804646i) q^{2} +(-2.23554 - 1.07658i) q^{3} +(1.18821 + 0.572213i) q^{4} +(2.30110 + 2.88549i) q^{5} +(1.27683 - 1.60110i) q^{6} +(0.712284 - 0.343018i) q^{7} +(-1.70783 + 2.14155i) q^{8} +(1.96815 + 2.46798i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.183655	+	0.804646i	−0.129864	+	0.568971i	0.867566	+	0.497322i	\(0.165683\pi\)
							−0.997430	+	0.0716488i	\(0.977174\pi\)
\(3\)	−2.23554	−	1.07658i	−1.29069	−	0.621563i	−0.342575	−	0.939491i	\(-0.611299\pi\)
							−0.948115	+	0.317927i	\(0.897013\pi\)
\(5\)	2.30110	+	2.88549i	1.02908	+	1.29043i	0.956083	+	0.293096i	\(0.0946856\pi\)
							0.0729992	+	0.997332i	\(0.476743\pi\)
\(7\)	0.712284	−	0.343018i	0.269218	−	0.129649i	−0.294405	−	0.955681i	\(-0.595121\pi\)
							0.563623	+	0.826032i	\(0.309407\pi\)
\(11\)	1.90334			0.573878			0.286939	−	0.957949i	\(-0.407362\pi\)
							0.286939	+	0.957949i	\(0.407362\pi\)
\(13\)	0.496214	+	2.17406i	0.137625	+	0.602975i	0.995953	+	0.0898735i	\(0.0286463\pi\)
							−0.858328	+	0.513101i	\(0.828497\pi\)
\(17\)	−1.75132	−	2.19608i	−0.424757	−	0.532628i	0.522698	−	0.852518i	\(-0.324925\pi\)
							−0.947455	+	0.319890i	\(0.896354\pi\)
\(19\)	0.170139	+	0.745430i	0.0390327	+	0.171013i	0.990687	−	0.136157i	\(-0.0434751\pi\)
							−0.951655	+	0.307170i	\(0.900618\pi\)
\(23\)	−1.68939	−	7.40170i	−0.352262	−	1.54336i	−0.771945	−	0.635690i	\(-0.780716\pi\)
							0.419683	−	0.907671i	\(-0.362141\pi\)
\(29\)	−3.41703	+	4.28482i	−0.634527	+	0.795671i	−0.990307	−	0.138899i	\(-0.955644\pi\)
							0.355780	+	0.934570i	\(0.384215\pi\)
\(31\)	5.75480	+	2.77137i	1.03359	+	0.497752i	0.872206	−	0.489139i	\(-0.162689\pi\)
							0.161387	+	0.986891i	\(0.448403\pi\)
\(37\)	6.04903	+	7.58525i	0.994455	+	1.24701i	0.968931	+	0.247333i	\(0.0795542\pi\)
							0.0255246	+	0.999674i	\(0.491874\pi\)
\(41\)	8.68664			1.35662			0.678312	−	0.734774i	\(-0.262712\pi\)
							0.678312	+	0.734774i	\(0.262712\pi\)
\(43\)	−6.30948	+	3.03849i	−0.962187	+	0.463365i	−0.847943	−	0.530088i	\(-0.822159\pi\)
							−0.114244	+	0.993453i	\(0.536445\pi\)
\(47\)	−8.84265			−1.28983			−0.644916	−	0.764253i	\(-0.723108\pi\)
							−0.644916	+	0.764253i	\(0.723108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.18	✓	264
547.304	even	7	inner	547.2.e.a.304.18	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.18	✓	264	1.1	even	1	trivial
547.2.e.a.304.18	yes	264	547.304	even	7	inner