Embedded newform 189.2.v.a.22.6

• Label: 189.2.v.a.22.6
• 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			22.6
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.a.43.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.786541 - 0.659987i) q^{2} +(-0.244375 - 1.71472i) q^{3} +(-0.164231 + 0.931402i) q^{4} +(3.56827 - 1.29874i) q^{5} +(-1.32391 - 1.18742i) q^{6} +(0.173648 + 0.984808i) q^{7} +(1.51229 + 2.61937i) q^{8} +(-2.88056 + 0.838073i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 3 q^{3} - 3 q^{5} - 27 q^{8} - 9 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.786541	−	0.659987i	0.556169	−	0.466681i	−0.320855	−	0.947128i	\(-0.603970\pi\)
							0.877024	+	0.480447i	\(0.159526\pi\)
\(3\)	−0.244375	−	1.71472i	−0.141090	−	0.989997i
							\(5\)	3.56827	−	1.29874i	1.59578	−	0.580816i	0.617221	−	0.786790i	\(-0.288258\pi\)
0.978557	+	0.205974i	\(0.0660362\pi\)
\(7\)	0.173648	+	0.984808i	0.0656328	+	0.372222i
							\(11\)	−1.53871	−	0.560043i	−0.463937	−	0.168859i	0.0994668	−	0.995041i	\(-0.468286\pi\)
−0.563404	+	0.826182i	\(0.690509\pi\)
\(13\)	−4.81098	−	4.03689i	−1.33433	−	1.11963i	−0.983045	−	0.183362i	\(-0.941302\pi\)
							−0.351281	−	0.936270i	\(-0.614254\pi\)
\(17\)	0.424366	−	0.735023i	0.102924	−	0.178269i	−0.809964	−	0.586479i	\(-0.800514\pi\)
							0.912888	+	0.408210i	\(0.133847\pi\)
\(19\)	2.19397	+	3.80006i	0.503330	+	0.871794i	0.999993	+	0.00384981i	\(0.00122544\pi\)
							−0.496662	+	0.867944i	\(0.665441\pi\)
\(23\)	−0.517102	+	2.93263i	−0.107823	+	0.611496i	0.882232	+	0.470815i	\(0.156040\pi\)
							−0.990055	+	0.140681i	\(0.955071\pi\)
\(29\)	−5.54559	+	4.65331i	−1.02979	+	0.864097i	−0.990826	−	0.135144i	\(-0.956850\pi\)
							−0.0389647	+	0.999241i	\(0.512406\pi\)
\(31\)	−0.831089	+	4.71334i	−0.149268	+	0.846541i	0.814573	+	0.580062i	\(0.196971\pi\)
							−0.963841	+	0.266479i	\(0.914140\pi\)
\(37\)	4.25038	−	7.36187i	0.698758	−	1.21028i	−0.270140	−	0.962821i	\(-0.587070\pi\)
							0.968897	−	0.247463i	\(-0.0795967\pi\)
\(41\)	3.46354	+	2.90626i	0.540915	+	0.453881i	0.871851	−	0.489772i	\(-0.162920\pi\)
							−0.330936	+	0.943653i	\(0.607364\pi\)
\(43\)	−7.40891	−	2.69662i	−1.12985	−	0.411231i	−0.291611	−	0.956537i	\(-0.594191\pi\)
							−0.838237	+	0.545306i	\(0.816414\pi\)
\(47\)	0.0270638	+	0.153487i	0.00394767	+	0.0223883i	0.986718	−	0.162443i	\(-0.0519373\pi\)
							−0.982770	+	0.184831i	\(0.940826\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.a.22.6	✓	54
3.2	odd	2		567.2.v.b.442.4		54
27.4	even	9		5103.2.a.i.1.16		27
27.11	odd	18		567.2.v.b.127.4		54
27.16	even	9	inner	189.2.v.a.43.6	yes	54
27.23	odd	18		5103.2.a.f.1.12		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.a.22.6	✓	54	1.1	even	1	trivial
189.2.v.a.43.6	yes	54	27.16	even	9	inner
567.2.v.b.127.4		54	27.11	odd	18
567.2.v.b.442.4		54	3.2	odd	2
5103.2.a.f.1.12		27	27.23	odd	18
5103.2.a.i.1.16		27	27.4	even	9