Embedded newform 201.2.p.b.2.1

• Label: 201.2.p.b.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.131747 + 2.76571i) q^{2} +(-1.12721 - 1.31507i) q^{3} +(-5.64086 - 0.538637i) q^{4} +(-0.779337 + 0.899403i) q^{5} +(3.78560 - 2.94428i) q^{6} +(-0.762358 - 1.47877i) q^{7} +(1.44479 - 10.0487i) q^{8} +(-0.458797 + 2.96471i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.131747	+	2.76571i	−0.0931593	+	1.95565i	0.157357	+	0.987542i	\(0.449703\pi\)
							−0.250517	+	0.968112i	\(0.580600\pi\)
\(3\)	−1.12721	−	1.31507i	−0.650795	−	0.759254i
							\(5\)	−0.779337	+	0.899403i	−0.348530	+	0.402225i	−0.902764	−	0.430136i	\(-0.858466\pi\)
0.554234	+	0.832361i	\(0.313011\pi\)
\(7\)	−0.762358	−	1.47877i	−0.288144	−	0.558922i	0.699687	−	0.714449i	\(-0.253323\pi\)
							−0.987832	+	0.155527i	\(0.950292\pi\)
\(11\)	−1.23371	−	3.56457i	−0.371977	−	1.07476i	−0.963324	−	0.268341i	\(-0.913525\pi\)
							0.591347	−	0.806418i	\(-0.298597\pi\)
\(13\)	0.212061	+	0.269657i	0.0588151	+	0.0747895i	0.814571	−	0.580064i	\(-0.196973\pi\)
							−0.755756	+	0.654854i	\(0.772730\pi\)
\(17\)	−0.0743024	−	0.778129i	−0.0180210	−	0.188724i	0.981979	−	0.188990i	\(-0.0605214\pi\)
							−1.00000		0.000265966i	\(0.999915\pi\)
\(19\)	−6.68232	−	3.44498i	−1.53303	−	0.790332i	−0.534533	−	0.845148i	\(-0.679512\pi\)
							−0.998496	+	0.0548159i	\(0.982543\pi\)
\(23\)	−7.15614	+	1.73606i	−1.49216	+	0.361994i	−0.897150	−	0.441726i	\(-0.854366\pi\)
							−0.595008	+	0.803720i	\(0.702851\pi\)
\(29\)	−4.11457	−	2.37555i	−0.764057	−	0.441128i	0.0666937	−	0.997773i	\(-0.478755\pi\)
							−0.830750	+	0.556645i	\(0.812088\pi\)
\(31\)	−2.32926	+	2.96189i	−0.418347	+	0.531971i	−0.948857	−	0.315708i	\(-0.897758\pi\)
							0.530510	+	0.847679i	\(0.322001\pi\)
\(37\)	−1.22213	−	2.11679i	−0.200917	−	0.347998i	0.747907	−	0.663803i	\(-0.231059\pi\)
							−0.948824	+	0.315805i	\(0.897725\pi\)
\(41\)	2.88543	+	4.05202i	0.450629	+	0.632820i	0.975927	−	0.218099i	\(-0.0699854\pi\)
							−0.525298	+	0.850919i	\(0.676046\pi\)
\(43\)	−0.202014	−	0.0922565i	−0.0308068	−	0.0140690i	0.399952	−	0.916536i	\(-0.369027\pi\)
							−0.430759	+	0.902467i	\(0.641754\pi\)
\(47\)	−1.00411	−	1.05308i	−0.146464	−	0.153607i	0.646415	−	0.762986i	\(-0.276267\pi\)
							−0.792879	+	0.609379i	\(0.791419\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.b.2.1	✓	400
3.2	odd	2	inner	201.2.p.b.2.20	yes	400
67.34	odd	66	inner	201.2.p.b.101.20	yes	400
201.101	even	66	inner	201.2.p.b.101.1	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.b.2.1	✓	400	1.1	even	1	trivial
201.2.p.b.2.20	yes	400	3.2	odd	2	inner
201.2.p.b.101.1	yes	400	201.101	even	66	inner
201.2.p.b.101.20	yes	400	67.34	odd	66	inner