Embedded newform 189.2.w.a.184.12

• Label: 189.2.w.a.184.12
• Level: $189$
• Weight: $2$
• Character: 189.184
• 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.w (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			184.12
Character	\(\chi\)	\(=\)	189.184
Dual form			189.2.w.a.151.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0119703 + 0.0678870i) q^{2} +(-0.127266 - 1.72737i) q^{3} +(1.87492 - 0.682415i) q^{4} +(0.115724 - 0.656302i) q^{5} +(0.115743 - 0.0293169i) q^{6} +(-1.45670 - 2.20863i) q^{7} +(0.137705 + 0.238512i) q^{8} +(-2.96761 + 0.439670i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0119703	+	0.0678870i	0.00846429	+	0.0480034i	0.988747	−	0.149595i	\(-0.0477970\pi\)
							−0.980283	+	0.197598i	\(0.936686\pi\)
\(3\)	−0.127266	−	1.72737i	−0.0734769	−	0.997297i
							\(5\)	0.115724	−	0.656302i	0.0517533	−	0.293507i	−0.947935	−	0.318463i	\(-0.896833\pi\)
0.999689	+	0.0249556i	\(0.00794445\pi\)
\(7\)	−1.45670	−	2.20863i	−0.550580	−	0.834782i
							\(11\)	0.981742	+	5.56773i	0.296006	+	1.67873i	0.663084	+	0.748545i	\(0.269247\pi\)
−0.367078	+	0.930190i	\(0.619642\pi\)
\(13\)	−2.75045	−	2.30790i	−0.762838	−	0.640097i	0.176026	−	0.984386i	\(-0.443676\pi\)
							−0.938864	+	0.344289i	\(0.888120\pi\)
\(17\)	4.02377			0.975907			0.487954	−	0.872870i	\(-0.337744\pi\)
							0.487954	+	0.872870i	\(0.337744\pi\)
\(19\)	3.80320			0.872515			0.436257	−	0.899822i	\(-0.356304\pi\)
							0.436257	+	0.899822i	\(0.356304\pi\)
\(23\)	−2.90660	−	2.43893i	−0.606068	−	0.508551i	0.287322	−	0.957834i	\(-0.407235\pi\)
							−0.893390	+	0.449283i	\(0.851680\pi\)
\(29\)	−2.12893	+	1.78638i	−0.395332	+	0.331723i	−0.818686	−	0.574242i	\(-0.805297\pi\)
							0.423354	+	0.905964i	\(0.360853\pi\)
\(31\)	−1.19218	+	0.433918i	−0.214122	+	0.0779340i	−0.446854	−	0.894607i	\(-0.647456\pi\)
							0.232732	+	0.972541i	\(0.425233\pi\)
\(37\)	3.61219	+	6.25651i	0.593841	+	1.02856i	0.993709	+	0.111990i	\(0.0357226\pi\)
							−0.399868	+	0.916573i	\(0.630944\pi\)
\(41\)	7.35765	+	6.17380i	1.14907	+	0.964186i	0.999697	−	0.0245949i	\(-0.00782960\pi\)
							0.149374	+	0.988781i	\(0.452274\pi\)
\(43\)	−1.42842	−	0.519902i	−0.217832	−	0.0792843i	0.230799	−	0.973001i	\(-0.425866\pi\)
							−0.448631	+	0.893717i	\(0.648088\pi\)
\(47\)	−9.15358	−	3.33163i	−1.33519	−	0.485969i	−0.426894	−	0.904302i	\(-0.640392\pi\)
							−0.908294	+	0.418333i	\(0.862614\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.w.a.184.12	yes	132
3.2	odd	2		567.2.w.a.37.11		132
7.4	even	3		189.2.u.a.130.11	yes	132
21.11	odd	6		567.2.u.a.361.12		132
27.11	odd	18		567.2.u.a.289.12		132
27.16	even	9		189.2.u.a.16.11	✓	132
189.11	odd	18		567.2.w.a.46.11		132
189.151	even	9	inner	189.2.w.a.151.12	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.16.11	✓	132	27.16	even	9
189.2.u.a.130.11	yes	132	7.4	even	3
189.2.w.a.151.12	yes	132	189.151	even	9	inner
189.2.w.a.184.12	yes	132	1.1	even	1	trivial
567.2.u.a.289.12		132	27.11	odd	18
567.2.u.a.361.12		132	21.11	odd	6
567.2.w.a.37.11		132	3.2	odd	2
567.2.w.a.46.11		132	189.11	odd	18