Embedded newform 189.2.v.b.22.8

• Label: 189.2.v.b.22.8
• 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			22.8
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.b.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.35330 - 1.13555i) q^{2} +(0.722934 + 1.57397i) q^{3} +(0.194644 - 1.10388i) q^{4} +(0.893231 - 0.325110i) q^{5} +(2.76567 + 1.30912i) q^{6} +(-0.173648 - 0.984808i) q^{7} +(0.776506 + 1.34495i) q^{8} +(-1.95473 + 2.27575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.35330	−	1.13555i	0.956928	−	0.802958i	−0.0235223	−	0.999723i	\(-0.507488\pi\)
							0.980451	+	0.196765i	\(0.0630436\pi\)
\(3\)	0.722934	+	1.57397i	0.417386	+	0.908729i
							\(5\)	0.893231	−	0.325110i	0.399465	−	0.145393i	−0.134472	−	0.990917i	\(-0.542934\pi\)
0.533938	+	0.845524i	\(0.320712\pi\)
\(7\)	−0.173648	−	0.984808i	−0.0656328	−	0.372222i
							\(11\)	−5.42626	−	1.97500i	−1.63608	−	0.595484i	−0.649733	−	0.760163i	\(-0.725119\pi\)
−0.986347	+	0.164678i	\(0.947341\pi\)
\(13\)	0.425846	+	0.357327i	0.118108	+	0.0991047i	0.699929	−	0.714212i	\(-0.253215\pi\)
							−0.581821	+	0.813317i	\(0.697659\pi\)
\(17\)	3.49190	−	6.04815i	0.846910	−	1.46689i	−0.0370415	−	0.999314i	\(-0.511793\pi\)
							0.883952	−	0.467578i	\(-0.154873\pi\)
\(19\)	−1.27047	−	2.20053i	−0.291467	−	0.504835i	0.682690	−	0.730708i	\(-0.260810\pi\)
							−0.974157	+	0.225873i	\(0.927477\pi\)
\(23\)	0.689784	−	3.91196i	0.143830	−	0.815700i	−0.824469	−	0.565907i	\(-0.808526\pi\)
							0.968299	−	0.249793i	\(-0.0803626\pi\)
\(29\)	−5.70629	+	4.78815i	−1.05963	+	0.889136i	−0.994073	−	0.108712i	\(-0.965327\pi\)
							−0.0655581	+	0.997849i	\(0.520883\pi\)
\(31\)	−0.930589	+	5.27763i	−0.167139	+	0.947891i	0.779693	+	0.626162i	\(0.215375\pi\)
							−0.946832	+	0.321729i	\(0.895736\pi\)
\(37\)	−2.89192	+	5.00895i	−0.475428	+	0.823465i	−0.999604	−	0.0281446i	\(-0.991040\pi\)
							0.524176	+	0.851610i	\(0.324373\pi\)
\(41\)	5.74878	+	4.82380i	0.897809	+	0.753351i	0.969761	−	0.244057i	\(-0.0784784\pi\)
							−0.0719521	+	0.997408i	\(0.522923\pi\)
\(43\)	4.78497	+	1.74159i	0.729701	+	0.265590i	0.680038	−	0.733176i	\(-0.261963\pi\)
							0.0496629	+	0.998766i	\(0.484185\pi\)
\(47\)	−1.72916	−	9.80654i	−0.252224	−	1.43043i	−0.803100	−	0.595844i	\(-0.796818\pi\)
							0.550876	−	0.834587i	\(-0.314294\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.b.22.8	✓	54
3.2	odd	2		567.2.v.a.442.2		54
27.4	even	9		5103.2.a.g.1.22		27
27.11	odd	18		567.2.v.a.127.2		54
27.16	even	9	inner	189.2.v.b.43.8	yes	54
27.23	odd	18		5103.2.a.h.1.6		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.22.8	✓	54	1.1	even	1	trivial
189.2.v.b.43.8	yes	54	27.16	even	9	inner
567.2.v.a.127.2		54	27.11	odd	18
567.2.v.a.442.2		54	3.2	odd	2
5103.2.a.g.1.22		27	27.4	even	9
5103.2.a.h.1.6		27	27.23	odd	18