Embedded newform 201.2.p.b.2.13

• Label: 201.2.p.b.2.13
• 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			2.13
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0485141 - 1.01844i) q^{2} +(-1.63340 - 0.576190i) q^{3} +(0.956085 + 0.0912951i) q^{4} +(0.397858 - 0.459152i) q^{5} +(-0.666056 + 1.63556i) q^{6} +(-1.04530 - 2.02760i) q^{7} +(0.429568 - 2.98771i) q^{8} +(2.33601 + 1.88230i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0485141	−	1.01844i	0.0343046	−	0.720143i	−0.914558	−	0.404454i	\(-0.867461\pi\)
							0.948863	−	0.315689i	\(-0.102236\pi\)
\(3\)	−1.63340	−	0.576190i	−0.943046	−	0.332664i
							\(5\)	0.397858	−	0.459152i	0.177927	−	0.205339i	−0.659780	−	0.751459i	\(-0.729350\pi\)
0.837707	+	0.546120i	\(0.183896\pi\)
\(7\)	−1.04530	−	2.02760i	−0.395086	−	0.766359i	0.604456	−	0.796639i	\(-0.293391\pi\)
							−0.999541	+	0.0302793i	\(0.990360\pi\)
\(11\)	−0.238795	−	0.689952i	−0.0719993	−	0.208028i	0.903259	−	0.429095i	\(-0.141167\pi\)
							−0.975259	+	0.221067i	\(0.929046\pi\)
\(13\)	−3.03993	−	3.86559i	−0.843125	−	1.07212i	−0.996470	−	0.0839482i	\(-0.973247\pi\)
							0.153345	−	0.988173i	\(-0.450995\pi\)
\(17\)	0.112892	+	1.18226i	0.0273804	+	0.286741i	0.998784	+	0.0493054i	\(0.0157008\pi\)
							−0.971403	+	0.237435i	\(0.923693\pi\)
\(19\)	−1.27107	−	0.655283i	−0.291604	−	0.150332i	0.306218	−	0.951962i	\(-0.400936\pi\)
							−0.597821	+	0.801629i	\(0.703967\pi\)
\(23\)	2.42783	−	0.588985i	0.506237	−	0.122812i	0.0254970	−	0.999675i	\(-0.491883\pi\)
							0.480740	+	0.876863i	\(0.340368\pi\)
\(29\)	4.74036	+	2.73685i	0.880263	+	0.508220i	0.870745	−	0.491735i	\(-0.163637\pi\)
							0.00951782	+	0.999955i	\(0.496970\pi\)
\(31\)	3.03533	−	3.85973i	0.545161	−	0.693228i	−0.432925	−	0.901430i	\(-0.642519\pi\)
							0.978086	+	0.208202i	\(0.0667610\pi\)
\(37\)	1.90846	+	3.30555i	0.313749	+	0.543429i	0.979171	−	0.203039i	\(-0.0650818\pi\)
							−0.665422	+	0.746467i	\(0.731748\pi\)
\(41\)	4.67268	+	6.56187i	0.729751	+	1.02479i	0.998109	+	0.0614721i	\(0.0195795\pi\)
							−0.268358	+	0.963319i	\(0.586481\pi\)
\(43\)	3.11663	+	1.42332i	0.475282	+	0.217054i	0.638629	−	0.769515i	\(-0.279502\pi\)
							−0.163347	+	0.986569i	\(0.552229\pi\)
\(47\)	4.71459	+	4.94452i	0.687693	+	0.721232i	0.971339	−	0.237697i	\(-0.0763926\pi\)
							−0.283646	+	0.958929i	\(0.591544\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.p.b.2.13	yes	400
3.2	odd	2	inner	201.2.p.b.2.8	✓	400
67.34	odd	66	inner	201.2.p.b.101.8	yes	400
201.101	even	66	inner	201.2.p.b.101.13	yes	400

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.p.b.2.8	✓	400	3.2	odd	2	inner
201.2.p.b.2.13	yes	400	1.1	even	1	trivial
201.2.p.b.101.8	yes	400	67.34	odd	66	inner
201.2.p.b.101.13	yes	400	201.101	even	66	inner