Embedded newform 201.4.i.a.22.8

• Label: 201.4.i.a.22.8
• Level: $201$
• Weight: $4$
• Character: 201.22
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $170$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(170\)
Relative dimension:	\(17\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			22.8
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.a.64.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.550733 + 0.161710i) q^{2} +(-0.426945 - 2.96946i) q^{3} +(-6.45287 + 4.14701i) q^{4} +(0.539245 - 1.18078i) q^{5} +(0.715324 + 1.56634i) q^{6} +(-23.0466 + 6.76708i) q^{7} +(5.89023 - 6.79769i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} - 51 q^{3} - 82 q^{4} + 4 q^{5} - 27 q^{6} - 16 q^{7} - 177 q^{8} - 153 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.550733	+	0.161710i	−0.194713	+	0.0571730i	−0.377635	−	0.925954i	\(-0.623263\pi\)
							0.182922	+	0.983127i	\(0.441444\pi\)
\(3\)	−0.426945	−	2.96946i	−0.0821655	−	0.571474i
							\(5\)	0.539245	−	1.18078i	0.0482315	−	0.105612i	−0.883982	−	0.467521i	\(-0.845147\pi\)
0.932214	+	0.361908i	\(0.117875\pi\)
\(7\)	−23.0466	+	6.76708i	−1.24440	+	0.365388i	−0.836666	−	0.547714i	\(-0.815498\pi\)
							−0.407732	+	0.913102i	\(0.633680\pi\)
\(11\)	19.0821	−	41.7839i	0.523042	−	1.14530i	−0.445232	−	0.895415i	\(-0.646879\pi\)
							0.968275	−	0.249888i	\(-0.0803938\pi\)
\(13\)	52.3846	+	60.4550i	1.11761	+	1.28979i	0.952847	+	0.303451i	\(0.0981389\pi\)
							0.164758	+	0.986334i	\(0.447316\pi\)
\(17\)	55.0549	+	35.3816i	0.785457	+	0.504782i	0.870841	−	0.491566i	\(-0.163575\pi\)
							−0.0853836	+	0.996348i	\(0.527212\pi\)
\(19\)	−49.7220	−	14.5997i	−0.600369	−	0.176284i	−0.0325950	−	0.999469i	\(-0.510377\pi\)
							−0.567774	+	0.823184i	\(0.692195\pi\)
\(23\)	−14.2266	−	98.9485i	−0.128977	−	0.897052i	−0.946855	−	0.321661i	\(-0.895759\pi\)
							0.817878	−	0.575391i	\(-0.195150\pi\)
\(29\)	194.141			1.24314			0.621570	−	0.783359i	\(-0.286495\pi\)
							0.621570	+	0.783359i	\(0.286495\pi\)
\(31\)	156.613	−	180.741i	0.907373	−	1.04716i	−0.0913084	−	0.995823i	\(-0.529105\pi\)
							0.998681	−	0.0513413i	\(-0.0163496\pi\)
\(37\)	86.8786			0.386020			0.193010	−	0.981197i	\(-0.438175\pi\)
							0.193010	+	0.981197i	\(0.438175\pi\)
\(41\)	239.305	+	153.792i	0.911540	+	0.585811i	0.910191	−	0.414188i	\(-0.135934\pi\)
							0.00134833	+	0.999999i	\(0.499571\pi\)
\(43\)	−168.349	−	108.192i	−0.597047	−	0.383699i	0.206933	−	0.978355i	\(-0.433652\pi\)
							−0.803980	+	0.594656i	\(0.797288\pi\)
\(47\)	53.5777	+	372.641i	0.166279	+	1.15650i	0.886492	+	0.462743i	\(0.153135\pi\)
							−0.720213	+	0.693753i	\(0.755956\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.a.22.8	✓	170
67.64	even	11	inner	201.4.i.a.64.8	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.a.22.8	✓	170	1.1	even	1	trivial
201.4.i.a.64.8	yes	170	67.64	even	11	inner