Embedded newform 547.2.e.a.9.14

• Label: 547.2.e.a.9.14
• 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			9.14
Character	\(\chi\)	\(=\)	547.9
Dual form			547.2.e.a.304.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.264233 + 1.15768i) q^{2} +(-2.41010 - 1.16064i) q^{3} +(0.531537 + 0.255975i) q^{4} +(-2.54037 - 3.18553i) q^{5} +(1.98048 - 2.48344i) q^{6} +(-1.02886 + 0.495474i) q^{7} +(-1.91751 + 2.40448i) q^{8} +(2.59103 + 3.24905i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 2 q^{2} - q^{3} - 44 q^{4} - 3 q^{5} - 10 q^{6} - 18 q^{7} - q^{8} - 35 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.264233	+	1.15768i	−0.186841	+	0.818602i	0.791428	+	0.611262i	\(0.209338\pi\)
							−0.978269	+	0.207340i	\(0.933519\pi\)
\(3\)	−2.41010	−	1.16064i	−1.39147	−	0.670098i	−0.420061	−	0.907496i	\(-0.637991\pi\)
							−0.971413	+	0.237398i	\(0.923706\pi\)
\(5\)	−2.54037	−	3.18553i	−1.13609	−	1.42461i	−0.890351	−	0.455275i	\(-0.849541\pi\)
							−0.245738	−	0.969336i	\(-0.579030\pi\)
\(7\)	−1.02886	+	0.495474i	−0.388873	+	0.187272i	−0.618096	−	0.786102i	\(-0.712096\pi\)
							0.229223	+	0.973374i	\(0.426382\pi\)
\(11\)	1.24087			0.374138			0.187069	−	0.982347i	\(-0.440101\pi\)
							0.187069	+	0.982347i	\(0.440101\pi\)
\(13\)	−1.39622	−	6.11725i	−0.387243	−	1.69662i	−0.674094	−	0.738646i	\(-0.735466\pi\)
							0.286851	−	0.957975i	\(-0.407391\pi\)
\(17\)	1.10370	+	1.38400i	0.267687	+	0.335669i	0.897448	−	0.441120i	\(-0.145419\pi\)
							−0.629761	+	0.776789i	\(0.716847\pi\)
\(19\)	1.15939	+	5.07961i	0.265982	+	1.16534i	0.914642	+	0.404264i	\(0.132472\pi\)
							−0.648660	+	0.761078i	\(0.724670\pi\)
\(23\)	0.280315	+	1.22814i	0.0584498	+	0.256085i	0.995708	−	0.0925509i	\(-0.0295021\pi\)
							−0.937258	+	0.348636i	\(0.886645\pi\)
\(29\)	−5.38710	+	6.75521i	−1.00036	+	1.25441i	−0.0334105	+	0.999442i	\(0.510637\pi\)
							−0.966949	+	0.254970i	\(0.917935\pi\)
\(31\)	5.27310	+	2.53939i	0.947077	+	0.456088i	0.842660	−	0.538445i	\(-0.180988\pi\)
							0.104417	+	0.994534i	\(0.466702\pi\)
\(37\)	3.37176	+	4.22805i	0.554314	+	0.695088i	0.977495	−	0.210957i	\(-0.0676580\pi\)
							−0.423181	+	0.906045i	\(0.639087\pi\)
\(41\)	−4.69679			−0.733515			−0.366758	−	0.930317i	\(-0.619532\pi\)
							−0.366758	+	0.930317i	\(0.619532\pi\)
\(43\)	8.67980	−	4.17997i	1.32366	−	0.637440i	0.367426	−	0.930053i	\(-0.380239\pi\)
							0.956231	+	0.292613i	\(0.0945249\pi\)
\(47\)	−3.34823			−0.488389			−0.244195	−	0.969726i	\(-0.578524\pi\)
							−0.244195	+	0.969726i	\(0.578524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.2.e.a.9.14	✓	264
547.304	even	7	inner	547.2.e.a.304.14	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.e.a.9.14	✓	264	1.1	even	1	trivial
547.2.e.a.304.14	yes	264	547.304	even	7	inner