Embedded newform 201.2.f.b.38.18

• Label: 201.2.f.b.38.18
• 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			38.18
Character	\(\chi\)	\(=\)	201.38
Dual form			201.2.f.b.164.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.22221 - 2.11693i) q^{2} +(-1.40479 - 1.01319i) q^{3} +(-1.98761 - 3.44264i) q^{4} -0.554650 q^{5} +(-3.86182 + 1.73551i) q^{6} +(0.273771 - 0.158062i) q^{7} -4.82826 q^{8} +(0.946874 + 2.84665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{4} - 2 q^{6} - 6 q^{7} + 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.22221	−	2.11693i	0.864235	−	1.49690i	−0.00357031	−	0.999994i	\(-0.501136\pi\)
							0.867805	−	0.496905i	\(-0.165530\pi\)
\(3\)	−1.40479	−	1.01319i	−0.811056	−	0.584968i
							\(5\)	−0.554650			−0.248047			−0.124024	−	0.992279i	\(-0.539580\pi\)
−0.124024	+	0.992279i	\(0.539580\pi\)
\(7\)	0.273771	−	0.158062i	0.103476	−	0.0597418i	−0.447369	−	0.894349i	\(-0.647639\pi\)
							0.550845	+	0.834608i	\(0.314306\pi\)
\(11\)	−2.42671	−	4.20319i	−0.731681	−	1.26731i	−0.956164	−	0.292831i	\(-0.905403\pi\)
							0.224483	−	0.974478i	\(-0.427931\pi\)
\(13\)	4.34339	+	2.50766i	1.20464	+	0.695499i	0.961583	−	0.274513i	\(-0.0885168\pi\)
							0.243056	+	0.970012i	\(0.421850\pi\)
\(17\)	−1.69341	−	0.977692i	−0.410713	−	0.237125i	0.280383	−	0.959888i	\(-0.409538\pi\)
							−0.691096	+	0.722763i	\(0.742872\pi\)
\(19\)	1.70071	−	2.94571i	0.390169	−	0.675792i	−0.602303	−	0.798268i	\(-0.705750\pi\)
							0.992471	+	0.122476i	\(0.0390834\pi\)
\(23\)	4.07208	+	2.35102i	0.849088	+	0.490221i	0.860343	−	0.509716i	\(-0.170249\pi\)
							−0.0112553	+	0.999937i	\(0.503583\pi\)
\(29\)	4.07773	−	2.35428i	0.757216	−	0.437179i	−0.0710791	−	0.997471i	\(-0.522644\pi\)
							0.828295	+	0.560292i	\(0.189311\pi\)
\(31\)	−0.0759515	+	0.0438506i	−0.0136413	+	0.00787580i	−0.506805	−	0.862061i	\(-0.669174\pi\)
							0.493164	+	0.869936i	\(0.335840\pi\)
\(37\)	2.84485	−	4.92742i	0.467690	−	0.810063i	−0.531628	−	0.846978i	\(-0.678420\pi\)
							0.999318	+	0.0369148i	\(0.0117530\pi\)
\(41\)	6.23388	+	10.7974i	0.973569	+	1.68627i	0.684580	+	0.728938i	\(0.259986\pi\)
							0.288989	+	0.957332i	\(0.406681\pi\)
\(43\)			2.81401i			0.429132i	0.976710	+	0.214566i	\(0.0688337\pi\)
							−0.976710	+	0.214566i	\(0.931166\pi\)
\(47\)	4.42357	−	2.55395i	0.645244	−	0.372532i	−0.141388	−	0.989954i	\(-0.545156\pi\)
							0.786632	+	0.617422i	\(0.211823\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.f.b.38.18	yes	40
3.2	odd	2	inner	201.2.f.b.38.3	✓	40
67.30	odd	6	inner	201.2.f.b.164.3	yes	40
201.164	even	6	inner	201.2.f.b.164.18	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.f.b.38.3	✓	40	3.2	odd	2	inner
201.2.f.b.38.18	yes	40	1.1	even	1	trivial
201.2.f.b.164.3	yes	40	67.30	odd	6	inner
201.2.f.b.164.18	yes	40	201.164	even	6	inner