Embedded newform 201.2.p.b.2.16

• Label: 201.2.p.b.2.16
• 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			2.16
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0888646 - 1.86550i) q^{2} +(0.830708 + 1.51984i) q^{3} +(-1.48124 - 0.141441i) q^{4} +(1.38993 - 1.60406i) q^{5} +(2.90908 - 1.41462i) q^{6} +(1.29209 + 2.50631i) q^{7} +(0.136090 - 0.946526i) q^{8} +(-1.61985 + 2.52509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0888646	−	1.86550i	0.0628368	−	1.31911i	−0.718732	−	0.695287i	\(-0.755277\pi\)
							0.781569	−	0.623819i	\(-0.214420\pi\)
\(3\)	0.830708	+	1.51984i	0.479610	+	0.877482i
							\(5\)	1.38993	−	1.60406i	0.621594	−	0.717358i	−0.354415	−	0.935088i	\(-0.615320\pi\)
0.976009	+	0.217730i	\(0.0698654\pi\)
\(7\)	1.29209	+	2.50631i	0.488365	+	0.947296i	0.996304	+	0.0859027i	\(0.0273774\pi\)
							−0.507938	+	0.861393i	\(0.669592\pi\)
\(11\)	1.50511	+	4.34873i	0.453807	+	1.31119i	0.907508	+	0.420036i	\(0.137982\pi\)
							−0.453700	+	0.891154i	\(0.649896\pi\)
\(13\)	−3.21050	−	4.08248i	−0.890431	−	1.13228i	−0.990530	−	0.137298i	\(-0.956158\pi\)
							0.100098	−	0.994978i	\(-0.468084\pi\)
\(17\)	−0.768314	−	8.04614i	−0.186343	−	1.95148i	−0.290400	−	0.956905i	\(-0.593788\pi\)
							0.104056	−	0.994571i	\(-0.466818\pi\)
\(19\)	1.82000	+	0.938278i	0.417538	+	0.215256i	0.654176	−	0.756342i	\(-0.273015\pi\)
							−0.236639	+	0.971598i	\(0.576046\pi\)
\(23\)	−2.55721	+	0.620371i	−0.533214	+	0.129356i	−0.493321	−	0.869848i	\(-0.664217\pi\)
							−0.0398936	+	0.999204i	\(0.512702\pi\)
\(29\)	−2.61245	−	1.50830i	−0.485119	−	0.280084i	0.237428	−	0.971405i	\(-0.423696\pi\)
							−0.722547	+	0.691321i	\(0.757029\pi\)
\(31\)	−3.61274	+	4.59397i	−0.648867	+	0.825101i	−0.993523	−	0.113628i	\(-0.963753\pi\)
							0.344657	+	0.938729i	\(0.387995\pi\)
\(37\)	5.46463	+	9.46502i	0.898380	+	1.55604i	0.829565	+	0.558409i	\(0.188588\pi\)
							0.0688140	+	0.997630i	\(0.478078\pi\)
\(41\)	−1.22751	−	1.72380i	−0.191705	−	0.269212i	0.707525	−	0.706688i	\(-0.249812\pi\)
							−0.899230	+	0.437477i	\(0.855872\pi\)
\(43\)	−6.47502	−	2.95704i	−0.987431	−	0.450945i	−0.144815	−	0.989459i	\(-0.546259\pi\)
							−0.842617	+	0.538514i	\(0.818986\pi\)
\(47\)	3.79235	+	3.97730i	0.553171	+	0.580150i	0.939645	−	0.342151i	\(-0.111155\pi\)
							−0.386474	+	0.922301i	\(0.626307\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.b.2.16	yes	400
3.2	odd	2	inner	201.2.p.b.2.5	✓	400
67.34	odd	66	inner	201.2.p.b.101.5	yes	400
201.101	even	66	inner	201.2.p.b.101.16	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.b.2.5	✓	400	3.2	odd	2	inner
201.2.p.b.2.16	yes	400	1.1	even	1	trivial
201.2.p.b.101.5	yes	400	67.34	odd	66	inner
201.2.p.b.101.16	yes	400	201.101	even	66	inner