Embedded newform 201.4.p.a.2.20

• Label: 201.4.p.a.2.20
• Level: $201$
• Weight: $4$
• Character: 201.2
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $1320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(1320\)
Relative dimension:	\(66\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			2.20
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.119268 + 2.50375i) q^{2} +(4.20022 - 3.05911i) q^{3} +(1.70924 + 0.163213i) q^{4} +(-3.68755 + 4.25566i) q^{5} +(7.15829 + 10.8812i) q^{6} +(5.98781 + 11.6147i) q^{7} +(-3.46630 + 24.1086i) q^{8} +(8.28370 - 25.6979i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.119268	+	2.50375i	−0.0421677	+	0.885209i	0.874370	+	0.485260i	\(0.161275\pi\)
							−0.916537	+	0.399949i	\(0.869028\pi\)
\(3\)	4.20022	−	3.05911i	0.808333	−	0.588726i
							\(5\)	−3.68755	+	4.25566i	−0.329825	+	0.380638i	−0.896306	−	0.443437i	\(-0.853759\pi\)
0.566481	+	0.824075i	\(0.308305\pi\)
\(7\)	5.98781	+	11.6147i	0.323311	+	0.627136i	0.993343	−	0.115196i	\(-0.0367498\pi\)
							−0.670032	+	0.742333i	\(0.733719\pi\)
\(11\)	14.9978	+	43.3332i	0.411090	+	1.18777i	0.940616	+	0.339474i	\(0.110249\pi\)
							−0.529525	+	0.848294i	\(0.677630\pi\)
\(13\)	−27.0142	−	34.3514i	−0.576339	−	0.732874i	0.407247	−	0.913318i	\(-0.366489\pi\)
							−0.983585	+	0.180444i	\(0.942247\pi\)
\(17\)	4.37205	+	45.7862i	0.0623752	+	0.653222i	0.971554	+	0.236817i	\(0.0761042\pi\)
							−0.909179	+	0.416405i	\(0.863290\pi\)
\(19\)	30.9061	+	15.9332i	0.373176	+	0.192386i	0.634604	−	0.772837i	\(-0.281163\pi\)
							−0.261428	+	0.965223i	\(0.584193\pi\)
\(23\)	115.256	−	27.9607i	1.04489	−	0.253488i	0.323653	−	0.946176i	\(-0.395089\pi\)
							0.721237	+	0.692688i	\(0.243574\pi\)
\(29\)	30.6484	+	17.6949i	0.196250	+	0.113305i	0.594905	−	0.803796i	\(-0.297190\pi\)
							−0.398655	+	0.917101i	\(0.630523\pi\)
\(31\)	67.5159	−	85.8534i	0.391168	−	0.497411i	−0.550087	−	0.835107i	\(-0.685405\pi\)
							0.941255	+	0.337697i	\(0.109648\pi\)
\(37\)	104.378	+	180.789i	0.463775	+	0.803282i	0.999145	−	0.0413352i	\(-0.0131611\pi\)
							−0.535370	+	0.844618i	\(0.679828\pi\)
\(41\)	62.4727	+	87.7306i	0.237966	+	0.334176i	0.916238	−	0.400635i	\(-0.131210\pi\)
							−0.678272	+	0.734811i	\(0.737271\pi\)
\(43\)	−348.494	−	159.152i	−1.23593	−	0.564428i	−0.313127	−	0.949711i	\(-0.601377\pi\)
							−0.922799	+	0.385283i	\(0.874104\pi\)
\(47\)	−292.097	−	306.343i	−0.906526	−	0.950737i	0.0924618	−	0.995716i	\(-0.470526\pi\)
							−0.998988	+	0.0449790i	\(0.985678\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.20	✓	1320
3.2	odd	2	inner	201.4.p.a.2.47	yes	1320
67.34	odd	66	inner	201.4.p.a.101.47	yes	1320
201.101	even	66	inner	201.4.p.a.101.20	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.20	✓	1320	1.1	even	1	trivial
201.4.p.a.2.47	yes	1320	3.2	odd	2	inner
201.4.p.a.101.20	yes	1320	201.101	even	66	inner
201.4.p.a.101.47	yes	1320	67.34	odd	66	inner