Embedded newform 189.2.v.b.43.9

• Label: 189.2.v.b.43.9
• Level: $189$
• Weight: $2$
• Character: 189.43
• 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			43.9
Character	\(\chi\)	\(=\)	189.43
Dual form			189.2.v.b.22.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.81332 + 1.52156i) q^{2} +(-1.25490 + 1.19383i) q^{3} +(0.625702 + 3.54853i) q^{4} +(-0.550279 - 0.200285i) q^{5} +(-4.09202 + 0.255396i) q^{6} +(-0.173648 + 0.984808i) q^{7} +(-1.89757 + 3.28669i) q^{8} +(0.149542 - 2.99627i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{2}{9}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.81332	+	1.52156i	1.28221	+	1.07590i	0.992935	+	0.118658i	\(0.0378593\pi\)
							0.289277	+	0.957245i	\(0.406585\pi\)
\(3\)	−1.25490	+	1.19383i	−0.724516	+	0.689258i
							\(5\)	−0.550279	−	0.200285i	−0.246092	−	0.0895702i	0.216029	−	0.976387i	\(-0.430689\pi\)
−0.462121	+	0.886817i	\(0.652912\pi\)
\(7\)	−0.173648	+	0.984808i	−0.0656328	+	0.372222i
							\(11\)	3.14941	−	1.14629i	0.949582	−	0.345620i	0.179639	−	0.983733i	\(-0.442507\pi\)
0.769943	+	0.638113i	\(0.220285\pi\)
\(13\)	−2.37797	+	1.99535i	−0.659530	+	0.553412i	−0.909946	−	0.414727i	\(-0.863877\pi\)
							0.250416	+	0.968138i	\(0.419433\pi\)
\(17\)	1.59231	+	2.75797i	0.386193	+	0.668905i	0.991934	−	0.126756i	\(-0.0404567\pi\)
							−0.605741	+	0.795662i	\(0.707123\pi\)
\(19\)	3.06009	−	5.30023i	0.702032	−	1.21596i	−0.265720	−	0.964050i	\(-0.585610\pi\)
							0.967752	−	0.251905i	\(-0.0810570\pi\)
\(23\)	0.631504	+	3.58144i	0.131678	+	0.746781i	0.977116	+	0.212708i	\(0.0682282\pi\)
							−0.845438	+	0.534073i	\(0.820661\pi\)
\(29\)	3.57775	+	3.00208i	0.664371	+	0.557473i	0.911393	−	0.411537i	\(-0.135008\pi\)
							−0.247023	+	0.969010i	\(0.579452\pi\)
\(31\)	−1.23977	−	7.03108i	−0.222669	−	1.26282i	−0.867091	−	0.498149i	\(-0.834013\pi\)
							0.644422	−	0.764670i	\(-0.277098\pi\)
\(37\)	−3.08518	−	5.34369i	−0.507201	−	0.878498i	−0.999965	−	0.00833494i	\(-0.997347\pi\)
							0.492764	−	0.870163i	\(-0.335986\pi\)
\(41\)	−3.74683	+	3.14396i	−0.585156	+	0.491004i	−0.886636	−	0.462468i	\(-0.846964\pi\)
							0.301480	+	0.953473i	\(0.402519\pi\)
\(43\)	−1.56252	+	0.568712i	−0.238282	+	0.0867277i	−0.458401	−	0.888745i	\(-0.651578\pi\)
							0.220119	+	0.975473i	\(0.429355\pi\)
\(47\)	1.91617	−	10.8671i	0.279501	−	1.58513i	−0.444788	−	0.895636i	\(-0.646721\pi\)
							0.724290	−	0.689496i	\(-0.242168\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.b.43.9	yes	54
3.2	odd	2		567.2.v.a.127.1		54
27.5	odd	18		567.2.v.a.442.1		54
27.7	even	9		5103.2.a.g.1.25		27
27.20	odd	18		5103.2.a.h.1.3		27
27.22	even	9	inner	189.2.v.b.22.9	✓	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.22.9	✓	54	27.22	even	9	inner
189.2.v.b.43.9	yes	54	1.1	even	1	trivial
567.2.v.a.127.1		54	3.2	odd	2
567.2.v.a.442.1		54	27.5	odd	18
5103.2.a.g.1.25		27	27.7	even	9
5103.2.a.h.1.3		27	27.20	odd	18