Embedded newform 201.4.f.a.38.13

• Label: 201.4.f.a.38.13
• Level: $201$
• Weight: $4$
• Character: 201.38
• 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			38.13
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.01114 + 3.48339i) q^{2} +(-5.13832 - 0.773071i) q^{3} +(-4.08936 - 7.08298i) q^{4} -12.8586 q^{5} +(13.0268 - 16.3441i) q^{6} +(-3.82711 + 2.20958i) q^{7} +0.718856 q^{8} +(25.8047 + 7.94458i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.01114	+	3.48339i	−0.711045	+	1.23157i	0.253420	+	0.967356i	\(0.418444\pi\)
							−0.964465	+	0.264210i	\(0.914889\pi\)
\(3\)	−5.13832	−	0.773071i	−0.988871	−	0.148778i
							\(5\)	−12.8586			−1.15011			−0.575055	−	0.818114i	\(-0.695020\pi\)
−0.575055	+	0.818114i	\(0.695020\pi\)
\(7\)	−3.82711	+	2.20958i	−0.206645	+	0.119306i	−0.599751	−	0.800187i	\(-0.704734\pi\)
							0.393106	+	0.919493i	\(0.371400\pi\)
\(11\)	−31.0054	−	53.7029i	−0.849862	−	1.47200i	−0.881331	−	0.472499i	\(-0.843352\pi\)
							0.0314691	−	0.999505i	\(-0.489981\pi\)
\(13\)	−39.6823	−	22.9106i	−0.846607	−	0.488789i	0.0128973	−	0.999917i	\(-0.495895\pi\)
							−0.859505	+	0.511128i	\(0.829228\pi\)
\(17\)	54.8797	+	31.6848i	0.782958	+	0.452041i	0.837478	−	0.546472i	\(-0.184029\pi\)
							−0.0545197	+	0.998513i	\(0.517363\pi\)
\(19\)	−54.9873	+	95.2407i	−0.663944	+	1.14999i	0.315626	+	0.948884i	\(0.397786\pi\)
							−0.979570	+	0.201102i	\(0.935548\pi\)
\(23\)	99.7156	+	57.5708i	0.904006	+	0.521928i	0.878498	−	0.477747i	\(-0.158546\pi\)
							0.0255080	+	0.999675i	\(0.491880\pi\)
\(29\)	−77.2296	+	44.5886i	−0.494524	+	0.285513i	−0.726449	−	0.687220i	\(-0.758831\pi\)
							0.231926	+	0.972734i	\(0.425497\pi\)
\(31\)	−259.503	+	149.824i	−1.50349	+	0.868039i	−0.503496	+	0.863998i	\(0.667953\pi\)
							−0.999992	+	0.00404136i	\(0.998714\pi\)
\(37\)	146.707	−	254.104i	0.651851	−	1.12904i	−0.330823	−	0.943693i	\(-0.607326\pi\)
							0.982673	−	0.185346i	\(-0.0593405\pi\)
\(41\)	56.8097	+	98.3972i	0.216395	+	0.374806i	0.953703	−	0.300750i	\(-0.0972369\pi\)
							−0.737308	+	0.675556i	\(0.763904\pi\)
\(43\)		−	54.1044i		−	0.191880i	−0.995387	−	0.0959400i	\(-0.969414\pi\)
							0.995387	−	0.0959400i	\(-0.0305857\pi\)
\(47\)	386.520	−	223.157i	1.19957	−	0.692571i	0.239110	−	0.970993i	\(-0.423144\pi\)
							0.960459	+	0.278421i	\(0.0898111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.13	✓	132
3.2	odd	2	inner	201.4.f.a.38.54	yes	132
67.30	odd	6	inner	201.4.f.a.164.54	yes	132
201.164	even	6	inner	201.4.f.a.164.13	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.13	✓	132	1.1	even	1	trivial
201.4.f.a.38.54	yes	132	3.2	odd	2	inner
201.4.f.a.164.13	yes	132	201.164	even	6	inner
201.4.f.a.164.54	yes	132	67.30	odd	6	inner