Embedded newform 201.4.f.a.38.12

• Label: 201.4.f.a.38.12
• 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			38.12
Character	\(\chi\)	\(=\)	201.38
Dual form			201.4.f.a.164.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.03028 + 3.51655i) q^{2} +(0.528423 - 5.16921i) q^{3} +(-4.24406 - 7.35093i) q^{4} +13.6688 q^{5} +(17.1049 + 12.3532i) q^{6} +(-13.4260 + 7.75148i) q^{7} +1.98205 q^{8} +(-26.4415 - 5.46307i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.03028	+	3.51655i	−0.717812	+	1.24329i	0.244053	+	0.969762i	\(0.421523\pi\)
							−0.961865	+	0.273525i	\(0.911810\pi\)
\(3\)	0.528423	−	5.16921i	0.101695	−	0.994816i
							\(5\)	13.6688			1.22258			0.611288	−	0.791408i	\(-0.290652\pi\)
0.611288	+	0.791408i	\(0.290652\pi\)
\(7\)	−13.4260	+	7.75148i	−0.724934	+	0.418541i	−0.816566	−	0.577252i	\(-0.804125\pi\)
							0.0916323	+	0.995793i	\(0.470792\pi\)
\(11\)	0.494089	+	0.855787i	0.0135430	+	0.0234572i	0.872717	−	0.488226i	\(-0.162356\pi\)
							−0.859174	+	0.511683i	\(0.829022\pi\)
\(13\)	52.6856	+	30.4180i	1.12403	+	0.648957i	0.942426	−	0.334415i	\(-0.108539\pi\)
							0.181601	+	0.983372i	\(0.441872\pi\)
\(17\)	42.3288	+	24.4386i	0.603897	+	0.348660i	0.770573	−	0.637352i	\(-0.219970\pi\)
							−0.166676	+	0.986012i	\(0.553303\pi\)
\(19\)	9.05289	−	15.6801i	0.109309	−	0.189329i	−0.806181	−	0.591668i	\(-0.798469\pi\)
							0.915491	+	0.402339i	\(0.131803\pi\)
\(23\)	178.361	+	102.977i	1.61699	+	0.933570i	0.987693	+	0.156405i	\(0.0499905\pi\)
							0.629297	+	0.777165i	\(0.283343\pi\)
\(29\)	228.167	−	131.732i	1.46102	−	0.843518i	0.461957	−	0.886902i	\(-0.347147\pi\)
							0.999058	+	0.0433842i	\(0.0138140\pi\)
\(31\)	−128.752	+	74.3350i	−0.745953	+	0.430676i	−0.824230	−	0.566255i	\(-0.808392\pi\)
							0.0782767	+	0.996932i	\(0.475058\pi\)
\(37\)	173.344	−	300.241i	0.770207	−	1.33404i	−0.167243	−	0.985916i	\(-0.553486\pi\)
							0.937450	−	0.348121i	\(-0.113180\pi\)
\(41\)	76.1288	+	131.859i	0.289983	+	0.502266i	0.973805	−	0.227383i	\(-0.0730169\pi\)
							−0.683822	+	0.729649i	\(0.739684\pi\)
\(43\)			424.233i			1.50453i	0.658860	+	0.752266i	\(0.271039\pi\)
							−0.658860	+	0.752266i	\(0.728961\pi\)
\(47\)	−67.8358	+	39.1650i	−0.210529	+	0.121549i	−0.601557	−	0.798830i	\(-0.705453\pi\)
							0.391028	+	0.920379i	\(0.372119\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.f.a.38.12	✓	132
3.2	odd	2	inner	201.4.f.a.38.55	yes	132
67.30	odd	6	inner	201.4.f.a.164.55	yes	132
201.164	even	6	inner	201.4.f.a.164.12	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.f.a.38.12	✓	132	1.1	even	1	trivial
201.4.f.a.38.55	yes	132	3.2	odd	2	inner
201.4.f.a.164.12	yes	132	201.164	even	6	inner
201.4.f.a.164.55	yes	132	67.30	odd	6	inner