Embedded newform 201.4.f.a.38.11

• Label: 201.4.f.a.38.11
• 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			38.11
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.07228 + 3.58929i) q^{2} +(3.94053 + 3.38707i) q^{3} +(-4.58865 - 7.94778i) q^{4} -6.98175 q^{5} +(-20.3230 + 7.12474i) q^{6} +(-12.0511 + 6.95772i) q^{7} +4.87942 q^{8} +(4.05552 + 26.6937i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.07228	+	3.58929i	−0.732660	+	1.26900i	0.223082	+	0.974800i	\(0.428388\pi\)
							−0.955742	+	0.294205i	\(0.904945\pi\)
\(3\)	3.94053	+	3.38707i	0.758355	+	0.651842i
							\(5\)	−6.98175			−0.624467			−0.312233	−	0.950005i	\(-0.601077\pi\)
−0.312233	+	0.950005i	\(0.601077\pi\)
\(7\)	−12.0511	+	6.95772i	−0.650699	+	0.375681i	−0.788724	−	0.614747i	\(-0.789258\pi\)
							0.138025	+	0.990429i	\(0.455925\pi\)
\(11\)	24.4487	+	42.3464i	0.670143	+	1.16072i	0.977863	+	0.209244i	\(0.0671004\pi\)
							−0.307721	+	0.951477i	\(0.599566\pi\)
\(13\)	−62.5272	−	36.1001i	−1.33399	−	0.770182i	−0.348086	−	0.937463i	\(-0.613168\pi\)
							−0.985909	+	0.167280i	\(0.946501\pi\)
\(17\)	39.7960	+	22.9763i	0.567762	+	0.327798i	0.756255	−	0.654277i	\(-0.227027\pi\)
							−0.188493	+	0.982075i	\(0.560360\pi\)
\(19\)	47.5402	−	82.3421i	0.574025	−	0.994240i	−0.422122	−	0.906539i	\(-0.638715\pi\)
							0.996147	−	0.0877013i	\(-0.0279521\pi\)
\(23\)	−122.210	−	70.5582i	−1.10794	−	0.639669i	−0.169645	−	0.985505i	\(-0.554262\pi\)
							−0.938295	+	0.345836i	\(0.887595\pi\)
\(29\)	143.292	−	82.7297i	0.917540	−	0.529742i	0.0346906	−	0.999398i	\(-0.488955\pi\)
							0.882849	+	0.469656i	\(0.155622\pi\)
\(31\)	−163.479	+	94.3846i	−0.947151	+	0.546838i	−0.892195	−	0.451651i	\(-0.850835\pi\)
							−0.0549562	+	0.998489i	\(0.517502\pi\)
\(37\)	122.678	−	212.485i	0.545087	−	0.944118i	−0.453515	−	0.891249i	\(-0.649830\pi\)
							0.998601	−	0.0528692i	\(-0.0168366\pi\)
\(41\)	11.4325	+	19.8016i	0.0435476	+	0.0754267i	0.886978	−	0.461812i	\(-0.152801\pi\)
							−0.843430	+	0.537239i	\(0.819467\pi\)
\(43\)		−	230.990i		−	0.819200i	−0.912265	−	0.409600i	\(-0.865668\pi\)
							0.912265	−	0.409600i	\(-0.134332\pi\)
\(47\)	−144.490	+	83.4213i	−0.448426	+	0.258899i	−0.707165	−	0.707048i	\(-0.750026\pi\)
							0.258739	+	0.965947i	\(0.416693\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.11	✓	132
3.2	odd	2	inner	201.4.f.a.38.56	yes	132
67.30	odd	6	inner	201.4.f.a.164.56	yes	132
201.164	even	6	inner	201.4.f.a.164.11	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.11	✓	132	1.1	even	1	trivial
201.4.f.a.38.56	yes	132	3.2	odd	2	inner
201.4.f.a.164.11	yes	132	201.164	even	6	inner
201.4.f.a.164.56	yes	132	67.30	odd	6	inner