Embedded newform 201.4.f.a.38.3

• Label: 201.4.f.a.38.3
• Level: $201$
• Weight: $4$
• Character: 201.38
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(66\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.3
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60270 + 4.50801i) q^{2} +(-3.38102 - 3.94572i) q^{3} +(-9.54810 - 16.5378i) q^{4} +1.95562 q^{5} +(26.5871 - 4.97217i) q^{6} +(6.62765 - 3.82647i) q^{7} +57.7601 q^{8} +(-4.13736 + 26.6811i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60270	+	4.50801i	−0.920194	+	1.59382i	−0.121079	+	0.992643i	\(0.538636\pi\)
							−0.799114	+	0.601179i	\(0.794698\pi\)
\(3\)	−3.38102	−	3.94572i	−0.650678	−	0.759353i
							\(5\)	1.95562			0.174916			0.0874579	−	0.996168i	\(-0.472126\pi\)
0.0874579	+	0.996168i	\(0.472126\pi\)
\(7\)	6.62765	−	3.82647i	0.357859	−	0.206610i	−0.310282	−	0.950645i	\(-0.600423\pi\)
							0.668141	+	0.744034i	\(0.267090\pi\)
\(11\)	−6.48086	−	11.2252i	−0.177641	−	0.307683i	0.763431	−	0.645889i	\(-0.223513\pi\)
							−0.941072	+	0.338206i	\(0.890180\pi\)
\(13\)	27.1129	+	15.6536i	0.578443	+	0.333964i	0.760514	−	0.649321i	\(-0.224947\pi\)
							−0.182071	+	0.983285i	\(0.558280\pi\)
\(17\)	15.2593	+	8.80997i	0.217702	+	0.125690i	0.604886	−	0.796312i	\(-0.293219\pi\)
							−0.387184	+	0.922002i	\(0.626552\pi\)
\(19\)	−0.862830	+	1.49447i	−0.0104183	+	0.0180449i	−0.871188	−	0.490950i	\(-0.836650\pi\)
							0.860769	+	0.508995i	\(0.169983\pi\)
\(23\)	−147.345	−	85.0697i	−1.33581	−	0.771228i	−0.349624	−	0.936890i	\(-0.613691\pi\)
							−0.986183	+	0.165662i	\(0.947024\pi\)
\(29\)	−77.7869	+	44.9103i	−0.498092	+	0.287574i	−0.727925	−	0.685656i	\(-0.759515\pi\)
							0.229833	+	0.973230i	\(0.426182\pi\)
\(31\)	62.0850	−	35.8448i	0.359703	−	0.207675i	−0.309247	−	0.950982i	\(-0.600077\pi\)
							0.668950	+	0.743307i	\(0.266744\pi\)
\(37\)	57.1960	−	99.0664i	0.254134	−	0.440173i	−0.710526	−	0.703671i	\(-0.751543\pi\)
							0.964660	+	0.263498i	\(0.0848762\pi\)
\(41\)	90.6745	+	157.053i	0.345390	+	0.598232i	0.985424	−	0.170114i	\(-0.0544134\pi\)
							−0.640035	+	0.768346i	\(0.721080\pi\)
\(43\)			70.6093i			0.250415i	0.992131	+	0.125207i	\(0.0399596\pi\)
							−0.992131	+	0.125207i	\(0.960040\pi\)
\(47\)	−363.880	+	210.086i	−1.12931	+	0.652005i	−0.943760	−	0.330631i	\(-0.892739\pi\)
							−0.185546	+	0.982636i	\(0.559405\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.3	✓	132
3.2	odd	2	inner	201.4.f.a.38.64	yes	132
67.30	odd	6	inner	201.4.f.a.164.64	yes	132
201.164	even	6	inner	201.4.f.a.164.3	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.3	✓	132	1.1	even	1	trivial
201.4.f.a.38.64	yes	132	3.2	odd	2	inner
201.4.f.a.164.3	yes	132	201.164	even	6	inner
201.4.f.a.164.64	yes	132	67.30	odd	6	inner