Embedded newform 201.2.i.a.25.3

• Label: 201.2.i.a.25.3
• Level: $201$
• Weight: $2$
• Character: 201.25
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $50$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(50\)
Relative dimension:	\(5\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			25.3
Character	\(\chi\)	\(=\)	201.25
Dual form			201.2.i.a.193.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0345454 + 0.240269i) q^{2} +(0.654861 - 0.755750i) q^{3} +(1.86245 - 0.546865i) q^{4} +(0.0517218 - 0.0332396i) q^{5} +(0.204205 + 0.131235i) q^{6} +(-0.587961 - 4.08936i) q^{7} +(0.397409 + 0.870204i) q^{8} +(-0.142315 - 0.989821i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} + 5 q^{3} - 6 q^{4} + 2 q^{5} - 9 q^{6} + 2 q^{7} + 27 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0345454	+	0.240269i	0.0244273	+	0.169896i	0.998383	−	0.0568420i	\(-0.0181031\pi\)
							−0.973956	+	0.226738i	\(0.927194\pi\)
\(3\)	0.654861	−	0.755750i	0.378084	−	0.436332i
							\(5\)	0.0517218	−	0.0332396i	0.0231307	−	0.0148652i	−0.529024	−	0.848607i	\(-0.677442\pi\)
0.552155	+	0.833741i	\(0.313806\pi\)
\(7\)	−0.587961	−	4.08936i	−0.222228	−	1.54563i	−0.729582	−	0.683893i	\(-0.760285\pi\)
							0.507354	−	0.861738i	\(-0.330624\pi\)
\(11\)	−3.49302	+	2.24483i	−1.05318	+	0.676841i	−0.948213	−	0.317635i	\(-0.897111\pi\)
							−0.104971	+	0.994475i	\(0.533475\pi\)
\(13\)	−2.09709	+	4.59199i	−0.581628	+	1.27359i	0.358742	+	0.933437i	\(0.383206\pi\)
							−0.940371	+	0.340152i	\(0.889522\pi\)
\(17\)	6.79186	+	1.99427i	1.64727	+	0.483681i	0.968155	−	0.250352i	\(-0.0805465\pi\)
							0.679113	+	0.734034i	\(0.262365\pi\)
\(19\)	0.543724	−	3.78168i	0.124739	−	0.867577i	−0.827335	−	0.561709i	\(-0.810144\pi\)
							0.952074	−	0.305868i	\(-0.0989467\pi\)
\(23\)	−2.50552	+	2.89153i	−0.522438	+	0.602925i	−0.954240	−	0.299043i	\(-0.903332\pi\)
							0.431802	+	0.901968i	\(0.357878\pi\)
\(29\)	2.91488			0.541280			0.270640	−	0.962681i	\(-0.412765\pi\)
							0.270640	+	0.962681i	\(0.412765\pi\)
\(31\)	1.46880	+	3.21623i	0.263804	+	0.577651i	0.994462	−	0.105092i	\(-0.0335138\pi\)
							−0.730658	+	0.682744i	\(0.760787\pi\)
\(37\)	7.30035			1.20017			0.600085	−	0.799936i	\(-0.295134\pi\)
							0.600085	+	0.799936i	\(0.295134\pi\)
\(41\)	−10.6442	−	3.12543i	−1.66235	−	0.488110i	−0.690425	−	0.723404i	\(-0.742576\pi\)
							−0.971924	+	0.235294i	\(0.924395\pi\)
\(43\)	−2.12072	−	0.622701i	−0.323407	−	0.0949609i	0.115999	−	0.993249i	\(-0.462993\pi\)
							−0.439406	+	0.898288i	\(0.644811\pi\)
\(47\)	−7.24279	+	8.35862i	−1.05647	+	1.21923i	−0.0815504	+	0.996669i	\(0.525987\pi\)
							−0.974919	+	0.222561i	\(0.928558\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.i.a.25.3	✓	50
3.2	odd	2		603.2.u.d.226.3		50
67.59	even	11	inner	201.2.i.a.193.3	yes	50
201.59	odd	22		603.2.u.d.595.3		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.i.a.25.3	✓	50	1.1	even	1	trivial
201.2.i.a.193.3	yes	50	67.59	even	11	inner
603.2.u.d.226.3		50	3.2	odd	2
603.2.u.d.595.3		50	201.59	odd	22