Embedded newform 189.2.v.b.22.9

• Label: 189.2.v.b.22.9
• 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			22.9
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.b.43.9

$q$-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 \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.22.9	✓	54	1.1	even	1	trivial
189.2.v.b.43.9	yes	54	27.16	even	9	inner
567.2.v.a.127.1		54	27.11	odd	18
567.2.v.a.442.1		54	3.2	odd	2
5103.2.a.g.1.25		27	27.4	even	9
5103.2.a.h.1.3		27	27.23	odd	18