Embedded newform 189.2.v.b.22.7

• Label: 189.2.v.b.22.7
• 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			22.7
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.b.43.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19955 - 1.00654i) q^{2} +(1.37519 - 1.05302i) q^{3} +(0.0784988 - 0.445189i) q^{4} +(-0.920293 + 0.334959i) q^{5} +(0.589706 - 2.64734i) q^{6} +(-0.173648 - 0.984808i) q^{7} +(1.21197 + 2.09919i) q^{8} +(0.782305 - 2.89620i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19955	−	1.00654i	0.848212	−	0.711734i	−0.111183	−	0.993800i	\(-0.535464\pi\)
							0.959395	+	0.282066i	\(0.0910197\pi\)
\(3\)	1.37519	−	1.05302i	0.793967	−	0.607960i
							\(5\)	−0.920293	+	0.334959i	−0.411568	+	0.149798i	−0.539502	−	0.841984i	\(-0.681387\pi\)
0.127935	+	0.991783i	\(0.459165\pi\)
\(7\)	−0.173648	−	0.984808i	−0.0656328	−	0.372222i
							\(11\)	−0.683043	−	0.248607i	−0.205945	−	0.0749579i	0.236988	−	0.971513i	\(-0.423840\pi\)
−0.442933	+	0.896555i	\(0.646062\pi\)
\(13\)	−1.50165	−	1.26003i	−0.416483	−	0.349471i	0.410340	−	0.911932i	\(-0.365410\pi\)
							−0.826823	+	0.562462i	\(0.809854\pi\)
\(17\)	−3.15448	+	5.46372i	−0.765074	+	1.32515i	0.175133	+	0.984545i	\(0.443964\pi\)
							−0.940207	+	0.340602i	\(0.889369\pi\)
\(19\)	−0.224384	−	0.388644i	−0.0514772	−	0.0891611i	0.839139	−	0.543918i	\(-0.183060\pi\)
							−0.890616	+	0.454757i	\(0.849726\pi\)
\(23\)	−1.39262	+	7.89794i	−0.290381	+	1.64683i	0.395023	+	0.918671i	\(0.370737\pi\)
							−0.685404	+	0.728163i	\(0.740375\pi\)
\(29\)	3.48127	−	2.92113i	0.646456	−	0.542441i	−0.259538	−	0.965733i	\(-0.583570\pi\)
							0.905993	+	0.423292i	\(0.139126\pi\)
\(31\)	0.814166	−	4.61737i	0.146229	−	0.829304i	−0.820144	−	0.572157i	\(-0.806107\pi\)
							0.966372	−	0.257146i	\(-0.0827822\pi\)
\(37\)	3.63874	−	6.30248i	0.598205	−	1.03612i	−0.394881	−	0.918732i	\(-0.629214\pi\)
							0.993086	−	0.117390i	\(-0.0374526\pi\)
\(41\)	−1.96285	−	1.64702i	−0.306545	−	0.257222i	0.476517	−	0.879165i	\(-0.341899\pi\)
							−0.783062	+	0.621943i	\(0.786343\pi\)
\(43\)	10.0140	+	3.64478i	1.52711	+	0.555824i	0.962913	−	0.269813i	\(-0.0869619\pi\)
							0.564202	+	0.825637i	\(0.309184\pi\)
\(47\)	−0.999232	−	5.66693i	−0.145753	−	0.826606i	−0.966759	−	0.255688i	\(-0.917698\pi\)
							0.821006	−	0.570919i	\(-0.193413\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.b.22.7	✓	54
3.2	odd	2		567.2.v.a.442.3		54
27.4	even	9		5103.2.a.g.1.21		27
27.11	odd	18		567.2.v.a.127.3		54
27.16	even	9	inner	189.2.v.b.43.7	yes	54
27.23	odd	18		5103.2.a.h.1.7		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.22.7	✓	54	1.1	even	1	trivial
189.2.v.b.43.7	yes	54	27.16	even	9	inner
567.2.v.a.127.3		54	27.11	odd	18
567.2.v.a.442.3		54	3.2	odd	2
5103.2.a.g.1.21		27	27.4	even	9
5103.2.a.h.1.7		27	27.23	odd	18