Embedded newform 201.2.p.b.2.5

• Label: 201.2.p.b.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0888646 + 1.86550i) q^{2} +(-1.52053 + 0.829459i) q^{3} +(-1.48124 - 0.141441i) q^{4} +(-1.38993 + 1.60406i) q^{5} +(-1.41223 - 2.91025i) q^{6} +(1.29209 + 2.50631i) q^{7} +(-0.136090 + 0.946526i) q^{8} +(1.62400 - 2.52243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

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	\(0.244723\pi\)
							−0.781569	+	0.623819i	\(0.785580\pi\)
\(3\)	−1.52053	+	0.829459i	−0.877876	+	0.478888i
							\(5\)	−1.38993	+	1.60406i	−0.621594	+	0.717358i	−0.976009	−	0.217730i	\(-0.930135\pi\)
0.354415	+	0.935088i	\(0.384680\pi\)
\(7\)	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	\(-0.862018\pi\)
							0.453700	−	0.891154i	\(-0.350104\pi\)
\(13\)	−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	\(0.406212\pi\)
							−0.104056	+	0.994571i	\(0.533182\pi\)
\(19\)	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	0.0398936	−	0.999204i	\(-0.487298\pi\)
							0.493321	+	0.869848i	\(0.335783\pi\)
\(29\)	2.61245	+	1.50830i	0.485119	+	0.280084i	0.722547	−	0.691321i	\(-0.242971\pi\)
							−0.237428	+	0.971405i	\(0.576304\pi\)
\(31\)	−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	0.899230	−	0.437477i	\(-0.144128\pi\)
							−0.707525	+	0.706688i	\(0.750188\pi\)
\(43\)	−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	0.386474	−	0.922301i	\(-0.373693\pi\)
							−0.939645	+	0.342151i	\(0.888845\pi\)

(See \(a_n\) instead)

Twists

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

    

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