Embedded newform 201.2.p.b.2.8

• Label: 201.2.p.b.2.8
• 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			2.8
Character	\(\chi\)	\(=\)	201.2
Dual form			201.2.p.b.101.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0485141 + 1.01844i) q^{2} +(1.68562 + 0.398362i) q^{3} +(0.956085 + 0.0912951i) q^{4} +(-0.397858 + 0.459152i) q^{5} +(-0.487483 + 1.69737i) q^{6} +(-1.04530 - 2.02760i) q^{7} +(-0.429568 + 2.98771i) q^{8} +(2.68262 + 1.34297i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 400 q - 22 q^{3} - 20 q^{4} - 20 q^{6} - 38 q^{7} - 30 q^{9}+O(q^{10}) \)

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	\(0.132539\pi\)
							−0.948863	+	0.315689i	\(0.897764\pi\)
\(3\)	1.68562	+	0.398362i	0.973192	+	0.229994i
							\(5\)	−0.397858	+	0.459152i	−0.177927	+	0.205339i	−0.837707	−	0.546120i	\(-0.816104\pi\)
0.659780	+	0.751459i	\(0.270650\pi\)
\(7\)	−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	0.975259	−	0.221067i	\(-0.0709538\pi\)
							−0.903259	+	0.429095i	\(0.858833\pi\)
\(13\)	−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	\(-0.984299\pi\)
							0.971403	−	0.237435i	\(-0.0763068\pi\)
\(19\)	−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	−0.480740	−	0.876863i	\(-0.659632\pi\)
							−0.0254970	+	0.999675i	\(0.508117\pi\)
\(29\)	−4.74036	−	2.73685i	−0.880263	−	0.508220i	−0.00951782	−	0.999955i	\(-0.503030\pi\)
							−0.870745	+	0.491735i	\(0.836363\pi\)
\(31\)	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	\(-0.980420\pi\)
							0.268358	−	0.963319i	\(-0.413519\pi\)
\(43\)	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	0.283646	−	0.958929i	\(-0.408456\pi\)
							−0.971339	+	0.237697i	\(0.923607\pi\)

(See \(a_n\) instead)

Twists

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

    

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