Embedded newform 201.2.f.b.38.20

• Label: 201.2.f.b.38.20
• Level: $201$
• Weight: $2$
• Character: 201.38
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.20
Character	\(\chi\)	\(=\)	201.38
Dual form			201.2.f.b.164.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35950 - 2.35473i) q^{2} +(-0.494616 + 1.65993i) q^{3} +(-2.69650 - 4.67047i) q^{4} +3.45591 q^{5} +(3.23624 + 3.42136i) q^{6} +(0.770900 - 0.445079i) q^{7} -9.22558 q^{8} +(-2.51071 - 1.64205i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{4} - 2 q^{6} - 6 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35950	−	2.35473i	0.961314	−	1.66504i	0.242106	−	0.970250i	\(-0.422162\pi\)
							0.719208	−	0.694795i	\(-0.244505\pi\)
\(3\)	−0.494616	+	1.65993i	−0.285567	+	0.958359i
							\(5\)	3.45591			1.54553			0.772766	−	0.634691i	\(-0.218873\pi\)
0.772766	+	0.634691i	\(0.218873\pi\)
\(7\)	0.770900	−	0.445079i	0.291373	−	0.168224i	−0.347188	−	0.937796i	\(-0.612863\pi\)
							0.638561	+	0.769571i	\(0.279530\pi\)
\(11\)	1.16058	+	2.01018i	0.349928	+	0.606093i	0.986236	−	0.165342i	\(-0.0528726\pi\)
							−0.636308	+	0.771435i	\(0.719539\pi\)
\(13\)	−2.43397	−	1.40525i	−0.675062	−	0.389747i	0.122930	−	0.992415i	\(-0.460771\pi\)
							−0.797992	+	0.602668i	\(0.794104\pi\)
\(17\)	−2.55015	−	1.47233i	−0.618501	−	0.357092i	0.157784	−	0.987474i	\(-0.449565\pi\)
							−0.776285	+	0.630382i	\(0.782898\pi\)
\(19\)	−2.80139	+	4.85216i	−0.642684	+	1.11316i	0.342148	+	0.939646i	\(0.388846\pi\)
							−0.984831	+	0.173515i	\(0.944488\pi\)
\(23\)	6.63675	+	3.83173i	1.38386	+	0.798971i	0.992614	−	0.121315i	\(-0.0387112\pi\)
							0.391245	+	0.920287i	\(0.372045\pi\)
\(29\)	0.503511	−	0.290702i	0.0934997	−	0.0539821i	−0.452521	−	0.891754i	\(-0.649475\pi\)
							0.546021	+	0.837772i	\(0.316142\pi\)
\(31\)	−0.699382	+	0.403789i	−0.125613	+	0.0725226i	−0.561490	−	0.827484i	\(-0.689772\pi\)
							0.435877	+	0.900006i	\(0.356438\pi\)
\(37\)	−2.66518	+	4.61622i	−0.438153	+	0.758903i	−0.997547	−	0.0699990i	\(-0.977700\pi\)
							0.559394	+	0.828902i	\(0.311034\pi\)
\(41\)	0.521010	+	0.902417i	0.0813682	+	0.140934i	0.903838	−	0.427875i	\(-0.140738\pi\)
							−0.822470	+	0.568809i	\(0.807404\pi\)
\(43\)		−	3.32336i		−	0.506807i	−0.967361	−	0.253404i	\(-0.918450\pi\)
							0.967361	−	0.253404i	\(-0.0815500\pi\)
\(47\)	−2.03074	+	1.17245i	−0.296214	+	0.171019i	−0.640741	−	0.767757i	\(-0.721373\pi\)
							0.344527	+	0.938776i	\(0.388039\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.f.b.38.20	yes	40
3.2	odd	2	inner	201.2.f.b.38.1	✓	40
67.30	odd	6	inner	201.2.f.b.164.1	yes	40
201.164	even	6	inner	201.2.f.b.164.20	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.f.b.38.1	✓	40	3.2	odd	2	inner
201.2.f.b.38.20	yes	40	1.1	even	1	trivial
201.2.f.b.164.1	yes	40	67.30	odd	6	inner
201.2.f.b.164.20	yes	40	201.164	even	6	inner