Embedded newform 547.2.e.a.9.8

• Label: 547.2.e.a.9.8
• 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			9.8
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.455062 + 1.99376i) q^{2} +(1.77328 + 0.853969i) q^{3} +(-1.96604 - 0.946797i) q^{4} +(2.51219 + 3.15019i) q^{5} +(-2.50956 + 3.14689i) q^{6} +(2.12663 - 1.02413i) q^{7} +(0.232245 - 0.291225i) q^{8} +(0.544806 + 0.683165i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.455062	+	1.99376i	−0.321777	+	1.40980i	0.512611	+	0.858621i	\(0.328678\pi\)
							−0.834388	+	0.551177i	\(0.814179\pi\)
\(3\)	1.77328	+	0.853969i	1.02381	+	0.493039i	0.868951	−	0.494898i	\(-0.164795\pi\)
							0.154855	+	0.987937i	\(0.450509\pi\)
\(5\)	2.51219	+	3.15019i	1.12349	+	1.40881i	0.900971	+	0.433879i	\(0.142855\pi\)
							0.222516	+	0.974929i	\(0.428573\pi\)
\(7\)	2.12663	−	1.02413i	0.803791	−	0.387085i	0.0135706	−	0.999908i	\(-0.495680\pi\)
							0.790221	+	0.612822i	\(0.209966\pi\)
\(11\)	0.713010			0.214980			0.107490	−	0.994206i	\(-0.465719\pi\)
							0.107490	+	0.994206i	\(0.465719\pi\)
\(13\)	−0.876716	−	3.84114i	−0.243157	−	1.06534i	−0.938124	−	0.346299i	\(-0.887438\pi\)
							0.694967	−	0.719042i	\(-0.255419\pi\)
\(17\)	0.909181	+	1.14008i	0.220509	+	0.276509i	0.879765	−	0.475409i	\(-0.157700\pi\)
							−0.659256	+	0.751919i	\(0.729129\pi\)
\(19\)	0.669391	+	2.93280i	0.153569	+	0.672829i	0.991831	+	0.127561i	\(0.0407150\pi\)
							−0.838262	+	0.545268i	\(0.816428\pi\)
\(23\)	−1.66189	−	7.28120i	−0.346527	−	1.51824i	−0.785002	−	0.619493i	\(-0.787338\pi\)
							0.438475	−	0.898743i	\(-0.355519\pi\)
\(29\)	4.87733	−	6.11598i	0.905698	−	1.13571i	−0.0845537	−	0.996419i	\(-0.526946\pi\)
							0.990252	−	0.139290i	\(-0.0444821\pi\)
\(31\)	0.314783	+	0.151592i	0.0565368	+	0.0272267i	0.461939	−	0.886912i	\(-0.347154\pi\)
							−0.405402	+	0.914139i	\(0.632868\pi\)
\(37\)	−1.37485	−	1.72400i	−0.226024	−	0.283425i	0.655869	−	0.754875i	\(-0.272302\pi\)
							−0.881893	+	0.471450i	\(0.843731\pi\)
\(41\)	−7.45975			−1.16502			−0.582509	−	0.812824i	\(-0.697929\pi\)
							−0.582509	+	0.812824i	\(0.697929\pi\)
\(43\)	−8.04805	+	3.87573i	−1.22732	+	0.591044i	−0.931340	−	0.364151i	\(-0.881359\pi\)
							−0.295976	+	0.955195i	\(0.595645\pi\)
\(47\)	0.499574			0.0728704			0.0364352	−	0.999336i	\(-0.488400\pi\)
							0.0364352	+	0.999336i	\(0.488400\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.8	✓	264
547.304	even	7	inner	547.2.e.a.304.8	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.8	✓	264	1.1	even	1	trivial
547.2.e.a.304.8	yes	264	547.304	even	7	inner