Embedded newform 201.2.p.b.2.11

• Label: 201.2.p.b.2.11
• Level: $201$
• Weight: $2$
• Character: 201.2
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $400$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(400\)
Relative dimension:	\(20\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.00573694 - 0.120433i) q^{2} +(0.311708 + 1.70377i) q^{3} +(1.97647 + 0.188730i) q^{4} +(-1.79917 + 2.07635i) q^{5} +(0.206979 - 0.0277656i) q^{6} +(0.198599 + 0.385227i) q^{7} +(0.0683861 - 0.475635i) q^{8} +(-2.80568 + 1.06216i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.00573694	−	0.120433i	0.00405663	−	0.0851592i	−0.995931	−	0.0901182i	\(-0.971276\pi\)
							0.999988	+	0.00495908i	\(0.00157853\pi\)
\(3\)	0.311708	+	1.70377i	0.179965	+	0.983673i
							\(5\)	−1.79917	+	2.07635i	−0.804614	+	0.928574i	−0.998625	−	0.0524252i	\(-0.983305\pi\)
0.194011	+	0.980999i	\(0.437850\pi\)
\(7\)	0.198599	+	0.385227i	0.0750632	+	0.145602i	0.923381	−	0.383886i	\(-0.125414\pi\)
							−0.848317	+	0.529488i	\(0.822384\pi\)
\(11\)	−0.588008	−	1.69894i	−0.177291	−	0.512249i	0.820985	−	0.570950i	\(-0.193425\pi\)
							−0.998276	+	0.0587017i	\(0.981304\pi\)
\(13\)	−0.108599	−	0.138095i	−0.0301200	−	0.0383006i	0.770760	−	0.637126i	\(-0.219877\pi\)
							−0.800880	+	0.598825i	\(0.795634\pi\)
\(17\)	−0.192945	−	2.02061i	−0.0467959	−	0.490069i	−0.988241	−	0.152903i	\(-0.951138\pi\)
							0.941445	−	0.337166i	\(-0.109468\pi\)
\(19\)	2.29445	+	1.18287i	0.526383	+	0.271369i	0.700869	−	0.713290i	\(-0.252796\pi\)
							−0.174485	+	0.984660i	\(0.555826\pi\)
\(23\)	5.30694	−	1.28745i	1.10657	−	0.268452i	0.359468	−	0.933157i	\(-0.382958\pi\)
							0.747106	+	0.664705i	\(0.231443\pi\)
\(29\)	6.53136	+	3.77088i	1.21284	+	0.700235i	0.963377	−	0.268149i	\(-0.0864120\pi\)
							0.249465	+	0.968384i	\(0.419745\pi\)
\(31\)	0.867775	−	1.10347i	0.155857	−	0.198188i	−0.701853	−	0.712321i	\(-0.747644\pi\)
							0.857710	+	0.514133i	\(0.171886\pi\)
\(37\)	−3.86975	−	6.70261i	−0.636183	−	1.10190i	−0.986263	−	0.165182i	\(-0.947179\pi\)
							0.350080	−	0.936720i	\(-0.386154\pi\)
\(41\)	2.90135	+	4.07438i	0.453115	+	0.636311i	0.976432	−	0.215827i	\(-0.0692448\pi\)
							−0.523316	+	0.852138i	\(0.675305\pi\)
\(43\)	−8.83396	−	4.03434i	−1.34717	−	0.615231i	−0.394402	−	0.918938i	\(-0.629048\pi\)
							−0.952765	+	0.303708i	\(0.901775\pi\)
\(47\)	2.13335	+	2.23740i	0.311182	+	0.326358i	0.860689	−	0.509131i	\(-0.170033\pi\)
							−0.549508	+	0.835489i	\(0.685185\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.b.2.11	yes	400
3.2	odd	2	inner	201.2.p.b.2.10	✓	400
67.34	odd	66	inner	201.2.p.b.101.10	yes	400
201.101	even	66	inner	201.2.p.b.101.11	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.b.2.10	✓	400	3.2	odd	2	inner
201.2.p.b.2.11	yes	400	1.1	even	1	trivial
201.2.p.b.101.10	yes	400	67.34	odd	66	inner
201.2.p.b.101.11	yes	400	201.101	even	66	inner