Embedded newform 201.2.j.a.5.13

• Label: 201.2.j.a.5.13
• Level: $201$
• Weight: $2$
• Character: 201.5
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(20\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.13
Character	\(\chi\)	\(=\)	201.5
Dual form			201.2.j.a.161.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.422797 - 0.487934i) q^{2} +(-1.61689 - 0.621033i) q^{3} +(0.225308 + 1.56705i) q^{4} +(-1.17382 + 0.344663i) q^{5} +(-0.986637 + 0.526362i) q^{6} +(2.44605 + 2.11952i) q^{7} +(1.94615 + 1.25071i) q^{8} +(2.22864 + 2.00828i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 11 q^{3} - 34 q^{4} - 7 q^{6} - 22 q^{7} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.422797	−	0.487934i	0.298963	−	0.345021i	−0.586316	−	0.810082i	\(-0.699422\pi\)
							0.885278	+	0.465061i	\(0.153968\pi\)
\(3\)	−1.61689	−	0.621033i	−0.933509	−	0.358553i
							\(5\)	−1.17382	+	0.344663i	−0.524946	+	0.154138i	−0.533459	−	0.845826i	\(-0.679108\pi\)
0.00851328	+	0.999964i	\(0.497290\pi\)
\(7\)	2.44605	+	2.11952i	0.924521	+	0.801102i	0.980335	−	0.197339i	\(-0.0632299\pi\)
							−0.0558139	+	0.998441i	\(0.517775\pi\)
\(11\)	0.398272	−	0.116943i	0.120084	−	0.0352597i	−0.221139	−	0.975242i	\(-0.570977\pi\)
							0.341222	+	0.939983i	\(0.389159\pi\)
\(13\)	0.261305	+	0.406599i	0.0724730	+	0.112770i	0.875598	−	0.483040i	\(-0.160467\pi\)
							−0.803125	+	0.595810i	\(0.796831\pi\)
\(17\)	3.72444	+	0.535493i	0.903308	+	0.129876i	0.578290	−	0.815832i	\(-0.303720\pi\)
							0.325019	+	0.945708i	\(0.394629\pi\)
\(19\)	0.610142	+	0.704141i	0.139976	+	0.161541i	0.821409	−	0.570340i	\(-0.193188\pi\)
							−0.681433	+	0.731881i	\(0.738643\pi\)
\(23\)	−5.29383	+	2.41761i	−1.10384	+	0.504107i	−0.882130	−	0.471006i	\(-0.843891\pi\)
							−0.221710	+	0.975113i	\(0.571164\pi\)
\(29\)		−	3.04202i		−	0.564888i	−0.959284	−	0.282444i	\(-0.908855\pi\)
							0.959284	−	0.282444i	\(-0.0911452\pi\)
\(31\)	3.37725	−	5.25511i	0.606572	−	0.943845i	−0.393131	−	0.919482i	\(-0.628608\pi\)
							0.999703	−	0.0243623i	\(-0.00775553\pi\)
\(37\)	9.29615			1.52828			0.764139	−	0.645052i	\(-0.223164\pi\)
							0.764139	+	0.645052i	\(0.223164\pi\)
\(41\)	0.592240	−	4.11912i	0.0924924	−	0.643299i	−0.889856	−	0.456241i	\(-0.849196\pi\)
							0.982349	−	0.187058i	\(-0.0598953\pi\)
\(43\)	−2.27668	−	0.327337i	−0.347190	−	0.0499184i	−0.0334861	−	0.999439i	\(-0.510661\pi\)
							−0.313704	+	0.949521i	\(0.601570\pi\)
\(47\)	−8.54576	+	3.90272i	−1.24653	+	0.569270i	−0.925840	−	0.377915i	\(-0.876641\pi\)
							−0.320687	+	0.947185i	\(0.603914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.j.a.5.13	yes	200
3.2	odd	2	inner	201.2.j.a.5.8	✓	200
67.27	odd	22	inner	201.2.j.a.161.8	yes	200
201.161	even	22	inner	201.2.j.a.161.13	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.j.a.5.8	✓	200	3.2	odd	2	inner
201.2.j.a.5.13	yes	200	1.1	even	1	trivial
201.2.j.a.161.8	yes	200	67.27	odd	22	inner
201.2.j.a.161.13	yes	200	201.161	even	22	inner