Embedded newform 189.2.u.a.4.13

• Label: 189.2.u.a.4.13
• Level: $189$
• Weight: $2$
• Character: 189.4
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.13
Character	\(\chi\)	\(=\)	189.4
Dual form			189.2.u.a.142.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0570934 + 0.323793i) q^{2} +(-0.691412 - 1.58806i) q^{3} +(1.77780 - 0.647067i) q^{4} +(1.85642 - 0.675680i) q^{5} +(0.474729 - 0.314543i) q^{6} +(-2.33644 - 1.24140i) q^{7} +(0.639805 + 1.10817i) q^{8} +(-2.04390 + 2.19601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0570934	+	0.323793i	0.0403712	+	0.228956i	0.998317	−	0.0579928i	\(-0.0184701\pi\)
							−0.957946	+	0.286949i	\(0.907359\pi\)
\(3\)	−0.691412	−	1.58806i	−0.399187	−	0.916870i
							\(5\)	1.85642	−	0.675680i	0.830214	−	0.302173i	0.108267	−	0.994122i	\(-0.465470\pi\)
0.721947	+	0.691949i	\(0.243248\pi\)
\(7\)	−2.33644	−	1.24140i	−0.883090	−	0.469204i
							\(11\)	−3.32802	−	1.21130i	−1.00344	−	0.365221i	−0.212527	−	0.977155i	\(-0.568169\pi\)
−0.790908	+	0.611935i	\(0.790392\pi\)
\(13\)	3.07096	−	1.11774i	0.851730	−	0.310004i	0.120984	−	0.992654i	\(-0.461395\pi\)
							0.730745	+	0.682650i	\(0.239173\pi\)
\(17\)	3.33900	+	5.78332i	0.809827	+	1.40266i	0.912983	+	0.407997i	\(0.133772\pi\)
							−0.103156	+	0.994665i	\(0.532894\pi\)
\(19\)	3.31187	−	5.73632i	0.759795	−	1.31600i	−0.183161	−	0.983083i	\(-0.558633\pi\)
							0.942955	−	0.332920i	\(-0.108034\pi\)
\(23\)	−1.04852	+	5.94644i	−0.218631	+	1.23992i	0.655863	+	0.754880i	\(0.272305\pi\)
							−0.874493	+	0.485037i	\(0.838806\pi\)
\(29\)	−4.03401	−	1.46826i	−0.749098	−	0.272649i	−0.0608717	−	0.998146i	\(-0.519388\pi\)
							−0.688226	+	0.725496i	\(0.741610\pi\)
\(31\)	−3.97516	+	1.44684i	−0.713961	+	0.259860i	−0.673360	−	0.739315i	\(-0.735149\pi\)
							−0.0406009	+	0.999175i	\(0.512927\pi\)
\(37\)	2.07250			0.340717			0.170358	−	0.985382i	\(-0.445507\pi\)
							0.170358	+	0.985382i	\(0.445507\pi\)
\(41\)	−0.663863	+	0.241626i	−0.103678	+	0.0377357i	−0.393338	−	0.919394i	\(-0.628680\pi\)
							0.289660	+	0.957130i	\(0.406458\pi\)
\(43\)	1.66178	+	9.42445i	0.253420	+	1.43722i	0.800096	+	0.599871i	\(0.204782\pi\)
							−0.546677	+	0.837344i	\(0.684107\pi\)
\(47\)	7.26473	+	2.64414i	1.05967	+	0.385688i	0.812304	−	0.583234i	\(-0.198213\pi\)
							0.247365	+	0.968922i	\(0.420435\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.u.a.4.13	✓	132
3.2	odd	2		567.2.u.a.550.10		132
7.2	even	3		189.2.w.a.58.13	yes	132
21.2	odd	6		567.2.w.a.226.10		132
27.7	even	9		189.2.w.a.88.13	yes	132
27.20	odd	18		567.2.w.a.424.10		132
189.128	odd	18		567.2.u.a.100.10		132
189.142	even	9	inner	189.2.u.a.142.13	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.4.13	✓	132	1.1	even	1	trivial
189.2.u.a.142.13	yes	132	189.142	even	9	inner
189.2.w.a.58.13	yes	132	7.2	even	3
189.2.w.a.88.13	yes	132	27.7	even	9
567.2.u.a.100.10		132	189.128	odd	18
567.2.u.a.550.10		132	3.2	odd	2
567.2.w.a.226.10		132	21.2	odd	6
567.2.w.a.424.10		132	27.20	odd	18