Embedded newform 189.2.v.a.22.1

• Label: 189.2.v.a.22.1
• Level: $189$
• Weight: $2$
• Character: 189.22
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.v (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(9\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			22.1
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88664 + 1.58308i) q^{2} +(0.279041 - 1.70943i) q^{3} +(0.705975 - 4.00378i) q^{4} +(-1.28676 + 0.468343i) q^{5} +(2.17970 + 3.66681i) q^{6} +(0.173648 + 0.984808i) q^{7} +(2.54355 + 4.40555i) q^{8} +(-2.84427 - 0.954001i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 3 q^{3} - 3 q^{5} - 27 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88664	+	1.58308i	−1.33406	+	1.11941i	−0.350944	+	0.936396i	\(0.614139\pi\)
							−0.983111	+	0.183009i	\(0.941416\pi\)
\(3\)	0.279041	−	1.70943i	0.161105	−	0.986937i
							\(5\)	−1.28676	+	0.468343i	−0.575457	+	0.209449i	−0.613321	−	0.789834i	\(-0.710167\pi\)
0.0378642	+	0.999283i	\(0.487945\pi\)
\(7\)	0.173648	+	0.984808i	0.0656328	+	0.372222i
							\(11\)	−3.59139	−	1.30716i	−1.08284	−	0.394123i	−0.261879	−	0.965101i	\(-0.584342\pi\)
−0.820965	+	0.570978i	\(0.806564\pi\)
\(13\)	−3.54356	−	2.97340i	−0.982808	−	0.824673i	0.00170302	−	0.999999i	\(-0.499458\pi\)
							−0.984511	+	0.175325i	\(0.943902\pi\)
\(17\)	−1.43245	+	2.48108i	−0.347420	+	0.601750i	−0.985790	−	0.167980i	\(-0.946276\pi\)
							0.638370	+	0.769730i	\(0.279609\pi\)
\(19\)	−2.13911	−	3.70505i	−0.490746	−	0.849997i	0.509197	−	0.860650i	\(-0.329942\pi\)
							−0.999943	+	0.0106528i	\(0.996609\pi\)
\(23\)	1.00499	−	5.69960i	0.209555	−	1.18845i	−0.680553	−	0.732699i	\(-0.738260\pi\)
							0.890108	−	0.455749i	\(-0.150629\pi\)
\(29\)	5.29520	−	4.44320i	0.983295	−	0.825082i	−0.00128845	−	0.999999i	\(-0.500410\pi\)
							0.984583	+	0.174917i	\(0.0559657\pi\)
\(31\)	−0.0783192	+	0.444170i	−0.0140665	+	0.0797753i	−0.991033	−	0.133618i	\(-0.957340\pi\)
							0.976966	+	0.213393i	\(0.0684516\pi\)
\(37\)	−3.07675	+	5.32908i	−0.505814	+	0.876096i	0.494163	+	0.869369i	\(0.335475\pi\)
							−0.999977	+	0.00672691i	\(0.997859\pi\)
\(41\)	−8.69548	−	7.29638i	−1.35801	−	1.13950i	−0.976592	−	0.215098i	\(-0.930993\pi\)
							−0.381414	−	0.924404i	\(-0.624563\pi\)
\(43\)	5.96967	+	2.17278i	0.910367	+	0.331346i	0.754399	−	0.656416i	\(-0.227928\pi\)
							0.155968	+	0.987762i	\(0.450150\pi\)
\(47\)	0.807676	+	4.58056i	0.117812	+	0.668143i	0.985320	+	0.170720i	\(0.0546093\pi\)
							−0.867508	+	0.497423i	\(0.834280\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.a.22.1	✓	54
3.2	odd	2		567.2.v.b.442.9		54
27.4	even	9		5103.2.a.i.1.1		27
27.11	odd	18		567.2.v.b.127.9		54
27.16	even	9	inner	189.2.v.a.43.1	yes	54
27.23	odd	18		5103.2.a.f.1.27		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.a.22.1	✓	54	1.1	even	1	trivial
189.2.v.a.43.1	yes	54	27.16	even	9	inner
567.2.v.b.127.9		54	27.11	odd	18
567.2.v.b.442.9		54	3.2	odd	2
5103.2.a.f.1.27		27	27.23	odd	18
5103.2.a.i.1.1		27	27.4	even	9