Embedded newform 201.3.n.b.7.4

• Label: 201.3.n.b.7.4
• Level: $201$
• Weight: $3$
• Character: 201.7
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.n (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(12\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	201.7
Dual form			201.3.n.b.115.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.738773 - 1.43302i) q^{2} +(-0.487975 + 1.66189i) q^{3} +(0.812467 - 1.14095i) q^{4} +(1.60508 + 1.39081i) q^{5} +(2.74203 - 0.528482i) q^{6} +(-7.37536 - 0.351332i) q^{7} +(-8.61857 - 1.23916i) q^{8} +(-2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 34 q^{4} - 33 q^{6} - 21 q^{7} - 33 q^{8} + 72 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{23}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.738773	−	1.43302i	−0.369387	−	0.716510i	0.628745	−	0.777612i	\(-0.283569\pi\)
							−0.998132	+	0.0611015i	\(0.980539\pi\)
\(3\)	−0.487975	+	1.66189i	−0.162658	+	0.553964i
							\(5\)	1.60508	+	1.39081i	0.321016	+	0.278162i	0.800429	−	0.599428i	\(-0.204605\pi\)
−0.479413	+	0.877589i	\(0.659151\pi\)
\(7\)	−7.37536	−	0.351332i	−1.05362	−	0.0501903i	−0.486385	−	0.873744i	\(-0.661685\pi\)
							−0.567238	+	0.823554i	\(0.691988\pi\)
\(11\)	1.99724	−	10.3627i	0.181567	−	0.942061i	−0.770793	−	0.637086i	\(-0.780140\pi\)
							0.952360	−	0.304975i	\(-0.0986483\pi\)
\(13\)	7.40353	−	18.4931i	0.569502	−	1.42255i	−0.311870	−	0.950125i	\(-0.600955\pi\)
							0.881372	−	0.472423i	\(-0.156620\pi\)
\(17\)	−3.05265	−	4.28685i	−0.179568	−	0.252168i	0.714975	−	0.699150i	\(-0.246438\pi\)
							−0.894543	+	0.446982i	\(0.852499\pi\)
\(19\)	−0.273139	−	5.73390i	−0.0143758	−	0.301784i	−0.994592	−	0.103856i	\(-0.966882\pi\)
							0.980217	−	0.197928i	\(-0.0634212\pi\)
\(23\)	−26.0180	+	24.8081i	−1.13122	+	1.07861i	−0.135076	+	0.990835i	\(0.543128\pi\)
							−0.996140	+	0.0877770i	\(0.972024\pi\)
\(29\)	−11.0141	−	19.0770i	−0.379797	−	0.657828i	0.611235	−	0.791449i	\(-0.290673\pi\)
							−0.991033	+	0.133621i	\(0.957340\pi\)
\(31\)	−19.6360	−	49.0485i	−0.633421	−	1.58221i	−0.802530	−	0.596611i	\(-0.796513\pi\)
							0.169110	−	0.985597i	\(-0.445911\pi\)
\(37\)	7.48861	−	12.9706i	0.202395	−	0.350558i	−0.746905	−	0.664931i	\(-0.768461\pi\)
							0.949300	+	0.314373i	\(0.101794\pi\)
\(41\)	5.91850	+	61.9813i	0.144354	+	1.51174i	0.719962	+	0.694013i	\(0.244159\pi\)
							−0.575609	+	0.817725i	\(0.695235\pi\)
\(43\)	50.6616	+	23.1364i	1.17818	+	0.538055i	0.905622	−	0.424086i	\(-0.139404\pi\)
							0.272554	+	0.962141i	\(0.412132\pi\)
\(47\)	20.3814	+	84.0134i	0.433648	+	1.78752i	0.598413	+	0.801187i	\(0.295798\pi\)
							−0.164766	+	0.986333i	\(0.552687\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.n.b.7.4	✓	240
67.48	odd	66	inner	201.3.n.b.115.4	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.n.b.7.4	✓	240	1.1	even	1	trivial
201.3.n.b.115.4	yes	240	67.48	odd	66	inner