Embedded newform 201.4.f.a.164.47

• Label: 201.4.f.a.164.47
• Level: $201$
• Weight: $4$
• Character: 201.164
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			164.47
Character	\(\chi\)	\(=\)	201.164
Dual form			201.4.f.a.38.47

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38906 + 2.40592i) q^{2} +(3.61547 - 3.73207i) q^{3} +(0.141034 - 0.244278i) q^{4} -10.9301 q^{5} +(14.0012 + 3.51448i) q^{6} +(2.00045 + 1.15496i) q^{7} +23.0086 q^{8} +(-0.856687 - 26.9864i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38906	+	2.40592i	0.491106	+	0.850621i	0.999948	−	0.0102393i	\(-0.00325933\pi\)
							−0.508841	+	0.860860i	\(0.669926\pi\)
\(3\)	3.61547	−	3.73207i	0.695798	−	0.718237i
							\(5\)	−10.9301			−0.977619			−0.488810	−	0.872391i	\(-0.662569\pi\)
−0.488810	+	0.872391i	\(0.662569\pi\)
\(7\)	2.00045	+	1.15496i	0.108014	+	0.0623620i	0.553034	−	0.833159i	\(-0.313470\pi\)
							−0.445020	+	0.895521i	\(0.646803\pi\)
\(11\)	18.3727	−	31.8225i	0.503599	−	0.872259i	−0.496392	−	0.868098i	\(-0.665342\pi\)
							0.999991	−	0.00416065i	\(-0.00132438\pi\)
\(13\)	−16.5662	+	9.56448i	−0.353433	+	0.204055i	−0.666196	−	0.745777i	\(-0.732079\pi\)
							0.312763	+	0.949831i	\(0.398745\pi\)
\(17\)	107.618	−	62.1332i	1.53536	−	0.886442i	0.536261	−	0.844052i	\(-0.319836\pi\)
							0.999101	−	0.0423901i	\(-0.0134972\pi\)
\(19\)	−27.5352	−	47.6924i	−0.332474	−	0.575863i	0.650522	−	0.759487i	\(-0.274550\pi\)
							−0.982996	+	0.183625i	\(0.941217\pi\)
\(23\)	100.451	−	57.9956i	0.910676	−	0.525779i	0.0300276	−	0.999549i	\(-0.490440\pi\)
							0.880649	+	0.473770i	\(0.157107\pi\)
\(29\)	−152.435	−	88.0084i	−0.976086	−	0.563543i	−0.0749996	−	0.997184i	\(-0.523896\pi\)
							−0.901086	+	0.433640i	\(0.857229\pi\)
\(31\)	−7.32049	−	4.22648i	−0.0424128	−	0.0244871i	0.478644	−	0.878009i	\(-0.341129\pi\)
							−0.521056	+	0.853522i	\(0.674462\pi\)
\(37\)	172.841	+	299.369i	0.767968	+	1.33016i	0.938663	+	0.344836i	\(0.112065\pi\)
							−0.170695	+	0.985324i	\(0.554601\pi\)
\(41\)	−200.399	+	347.102i	−0.763344	+	1.32215i	0.177773	+	0.984071i	\(0.443111\pi\)
							−0.941118	+	0.338080i	\(0.890223\pi\)
\(43\)			97.5694i			0.346028i	0.984919	+	0.173014i	\(0.0553506\pi\)
							−0.984919	+	0.173014i	\(0.944649\pi\)
\(47\)	316.170	+	182.541i	0.981238	+	0.566518i	0.902644	−	0.430389i	\(-0.141624\pi\)
							0.0785941	+	0.996907i	\(0.474957\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.164.47	yes	132
3.2	odd	2	inner	201.4.f.a.164.20	yes	132
67.38	odd	6	inner	201.4.f.a.38.20	✓	132
201.38	even	6	inner	201.4.f.a.38.47	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.20	✓	132	67.38	odd	6	inner
201.4.f.a.38.47	yes	132	201.38	even	6	inner
201.4.f.a.164.20	yes	132	3.2	odd	2	inner
201.4.f.a.164.47	yes	132	1.1	even	1	trivial