Embedded newform 189.2.u.a.4.9

• Label: 189.2.u.a.4.9
• 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			4.9
Character	\(\chi\)	\(=\)	189.4
Dual form			189.2.u.a.142.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.165988 - 0.941364i) q^{2} +(1.70872 - 0.283332i) q^{3} +(1.02077 - 0.371530i) q^{4} +(-2.75749 + 1.00364i) q^{5} +(-0.550345 - 1.56150i) q^{6} +(0.0940141 - 2.64408i) q^{7} +(-1.47507 - 2.55489i) q^{8} +(2.83945 - 0.968269i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.165988	−	0.941364i	−0.117371	−	0.665645i	−0.985549	−	0.169392i	\(-0.945820\pi\)
							0.868178	−	0.496254i	\(-0.165291\pi\)
\(3\)	1.70872	−	0.283332i	0.986530	−	0.163582i
							\(5\)	−2.75749	+	1.00364i	−1.23319	+	0.448843i	−0.874688	−	0.484687i	\(-0.838934\pi\)
−0.358499	+	0.933530i	\(0.616711\pi\)
\(7\)	0.0940141	−	2.64408i	0.0355340	−	0.999368i
							\(11\)	1.43006	+	0.520500i	0.431180	+	0.156937i	0.548487	−	0.836159i	\(-0.315204\pi\)
−0.117307	+	0.993096i	\(0.537426\pi\)
\(13\)	1.16189	−	0.422892i	0.322249	−	0.117289i	−0.175830	−	0.984420i	\(-0.556261\pi\)
							0.498079	+	0.867131i	\(0.334039\pi\)
\(17\)	2.85926	+	4.95239i	0.693473	+	1.20113i	0.970693	+	0.240325i	\(0.0772539\pi\)
							−0.277219	+	0.960807i	\(0.589413\pi\)
\(19\)	−2.37888	+	4.12034i	−0.545752	+	0.945270i	0.452807	+	0.891608i	\(0.350423\pi\)
							−0.998559	+	0.0536616i	\(0.982911\pi\)
\(23\)	−0.469032	+	2.66001i	−0.0977999	+	0.554651i	0.896053	+	0.443946i	\(0.146422\pi\)
							−0.993853	+	0.110705i	\(0.964689\pi\)
\(29\)	−3.71574	−	1.35242i	−0.689996	−	0.251138i	−0.0268623	−	0.999639i	\(-0.508552\pi\)
							−0.663133	+	0.748501i	\(0.730774\pi\)
\(31\)	−6.85655	+	2.49558i	−1.23147	+	0.448219i	−0.874102	−	0.485743i	\(-0.838549\pi\)
							−0.357371	+	0.933963i	\(0.616327\pi\)
\(37\)	1.19678			0.196749			0.0983746	−	0.995149i	\(-0.468636\pi\)
							0.0983746	+	0.995149i	\(0.468636\pi\)
\(41\)	−1.59504	+	0.580547i	−0.249103	+	0.0906661i	−0.463554	−	0.886069i	\(-0.653426\pi\)
							0.214451	+	0.976735i	\(0.431204\pi\)
\(43\)	2.22210	+	12.6022i	0.338867	+	1.92181i	0.385087	+	0.922880i	\(0.374171\pi\)
							−0.0462199	+	0.998931i	\(0.514717\pi\)
\(47\)	0.399825	+	0.145524i	0.0583204	+	0.0212269i	0.371016	−	0.928627i	\(-0.379010\pi\)
							−0.312695	+	0.949854i	\(0.601232\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.u.a.4.9	✓	132
3.2	odd	2		567.2.u.a.550.14		132
7.2	even	3		189.2.w.a.58.9	yes	132
21.2	odd	6		567.2.w.a.226.14		132
27.7	even	9		189.2.w.a.88.9	yes	132
27.20	odd	18		567.2.w.a.424.14		132
189.128	odd	18		567.2.u.a.100.14		132
189.142	even	9	inner	189.2.u.a.142.9	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.4.9	✓	132	1.1	even	1	trivial
189.2.u.a.142.9	yes	132	189.142	even	9	inner
189.2.w.a.58.9	yes	132	7.2	even	3
189.2.w.a.88.9	yes	132	27.7	even	9
567.2.u.a.100.14		132	189.128	odd	18
567.2.u.a.550.14		132	3.2	odd	2
567.2.w.a.226.14		132	21.2	odd	6
567.2.w.a.424.14		132	27.20	odd	18