Embedded newform 201.2.j.a.5.11

• Label: 201.2.j.a.5.11
• Level: $201$
• Weight: $2$
• Character: 201.5
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(20\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	201.5
Dual form			201.2.j.a.161.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.333078 - 0.384392i) q^{2} +(1.65359 - 0.515399i) q^{3} +(0.247813 + 1.72358i) q^{4} +(-2.02247 + 0.593851i) q^{5} +(0.352659 - 0.807295i) q^{6} +(0.616666 + 0.534344i) q^{7} +(1.60083 + 1.02879i) q^{8} +(2.46873 - 1.70452i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 11 q^{3} - 34 q^{4} - 7 q^{6} - 22 q^{7} + 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.333078	−	0.384392i	0.235521	−	0.271806i	−0.625669	−	0.780089i	\(-0.715174\pi\)
							0.861190	+	0.508282i	\(0.169719\pi\)
\(3\)	1.65359	−	0.515399i	0.954701	−	0.297566i
							\(5\)	−2.02247	+	0.593851i	−0.904476	+	0.265578i	−0.700714	−	0.713442i	\(-0.747135\pi\)
−0.203762	+	0.979020i	\(0.565317\pi\)
\(7\)	0.616666	+	0.534344i	0.233078	+	0.201963i	0.763567	−	0.645729i	\(-0.223446\pi\)
							−0.530489	+	0.847692i	\(0.677992\pi\)
\(11\)	5.10828	−	1.49993i	1.54020	−	0.452245i	0.602049	−	0.798459i	\(-0.294351\pi\)
							0.938155	+	0.346214i	\(0.112533\pi\)
\(13\)	0.0224038	+	0.0348610i	0.00621370	+	0.00966871i	0.844346	−	0.535798i	\(-0.179989\pi\)
							−0.838133	+	0.545466i	\(0.816353\pi\)
\(17\)	−7.28287	−	1.04712i	−1.76636	−	0.253964i	−0.818920	−	0.573908i	\(-0.805427\pi\)
							−0.947436	+	0.319945i	\(0.896336\pi\)
\(19\)	−2.77765	−	3.20558i	−0.637237	−	0.735410i	0.341647	−	0.939828i	\(-0.389015\pi\)
							−0.978884	+	0.204418i	\(0.934470\pi\)
\(23\)	−0.595782	+	0.272085i	−0.124229	+	0.0567336i	−0.476560	−	0.879142i	\(-0.658116\pi\)
							0.352331	+	0.935876i	\(0.385389\pi\)
\(29\)			1.24850i			0.231841i	0.993258	+	0.115921i	\(0.0369818\pi\)
							−0.993258	+	0.115921i	\(0.963018\pi\)
\(31\)	−0.189598	+	0.295021i	−0.0340529	+	0.0529873i	−0.857863	−	0.513879i	\(-0.828208\pi\)
							0.823810	+	0.566866i	\(0.191844\pi\)
\(37\)	−8.65866			−1.42348			−0.711738	−	0.702445i	\(-0.752092\pi\)
							−0.711738	+	0.702445i	\(0.752092\pi\)
\(41\)	0.952760	−	6.62659i	0.148796	−	1.03490i	−0.769398	−	0.638769i	\(-0.779444\pi\)
							0.918194	−	0.396130i	\(-0.129647\pi\)
\(43\)	2.95167	+	0.424386i	0.450125	+	0.0647183i	0.363650	−	0.931536i	\(-0.381530\pi\)
							0.0864755	+	0.996254i	\(0.472440\pi\)
\(47\)	10.6285	−	4.85387i	1.55033	−	0.708010i	0.557791	−	0.829981i	\(-0.311649\pi\)
							0.992534	+	0.121971i	\(0.0389216\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.j.a.5.11	yes	200
3.2	odd	2	inner	201.2.j.a.5.10	✓	200
67.27	odd	22	inner	201.2.j.a.161.10	yes	200
201.161	even	22	inner	201.2.j.a.161.11	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.j.a.5.10	✓	200	3.2	odd	2	inner
201.2.j.a.5.11	yes	200	1.1	even	1	trivial
201.2.j.a.161.10	yes	200	67.27	odd	22	inner
201.2.j.a.161.11	yes	200	201.161	even	22	inner