Embedded newform 189.2.ba.a.101.22

• Label: 189.2.ba.a.101.22
• Level: $189$
• Weight: $2$
• Character: 189.101
• 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.ba (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			101.22
Character	\(\chi\)	\(=\)	189.101
Dual form			189.2.ba.a.131.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.75048 - 2.08614i) q^{2} +(-1.10271 + 1.33567i) q^{3} +(-0.940509 - 5.33389i) q^{4} +(2.08375 - 1.74847i) q^{5} +(0.856127 + 4.63849i) q^{6} +(-0.122613 + 2.64291i) q^{7} +(-8.05677 - 4.65158i) q^{8} +(-0.568048 - 2.94573i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.75048	−	2.08614i	1.23778	−	1.47513i	0.411946	−	0.911208i	\(-0.364849\pi\)
							0.825831	−	0.563917i	\(-0.190706\pi\)
\(3\)	−1.10271	+	1.33567i	−0.636652	+	0.771152i
							\(5\)	2.08375	−	1.74847i	0.931881	−	0.781941i	−0.0442728	−	0.999019i	\(-0.514097\pi\)
0.976154	+	0.217078i	\(0.0696526\pi\)
\(7\)	−0.122613	+	2.64291i	−0.0463432	+	0.998926i
							\(11\)	0.0911635	−	0.108644i	0.0274868	−	0.0327575i	−0.752126	−	0.659019i	\(-0.770971\pi\)
0.779613	+	0.626262i	\(0.215416\pi\)
\(13\)	−0.695830	+	1.91178i	−0.192988	+	0.530231i	−0.998013	−	0.0630103i	\(-0.979930\pi\)
							0.805024	+	0.593242i	\(0.202152\pi\)
\(17\)	4.00161			0.970532			0.485266	−	0.874367i	\(-0.338723\pi\)
							0.485266	+	0.874367i	\(0.338723\pi\)
\(19\)			3.04328i			0.698176i	0.937090	+	0.349088i	\(0.113509\pi\)
							−0.937090	+	0.349088i	\(0.886491\pi\)
\(23\)	−0.449297	+	1.23443i	−0.0936849	+	0.257397i	−0.977680	−	0.210099i	\(-0.932621\pi\)
							0.883995	+	0.467496i	\(0.154844\pi\)
\(29\)	2.12744	+	5.84511i	0.395056	+	1.08541i	0.964662	+	0.263491i	\(0.0848738\pi\)
							−0.569605	+	0.821918i	\(0.692904\pi\)
\(31\)	−4.18959	+	0.738737i	−0.752472	+	0.132681i	−0.536712	−	0.843765i	\(-0.680334\pi\)
							−0.215760	+	0.976446i	\(0.569223\pi\)
\(37\)	4.69031	−	8.12386i	0.771083	−	1.33555i	−0.165888	−	0.986145i	\(-0.553049\pi\)
							0.936970	−	0.349409i	\(-0.113618\pi\)
\(41\)	0.303455	+	0.110449i	0.0473917	+	0.0172492i	0.365607	−	0.930769i	\(-0.380861\pi\)
							−0.318216	+	0.948018i	\(0.603084\pi\)
\(43\)	0.643705	−	3.65063i	0.0981640	−	0.556716i	−0.895568	−	0.444925i	\(-0.853230\pi\)
							0.993732	−	0.111791i	\(-0.0356586\pi\)
\(47\)	−1.15909	+	6.57353i	−0.169071	+	0.958847i	0.775697	+	0.631106i	\(0.217399\pi\)
							−0.944767	+	0.327741i	\(0.893713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.ba.a.101.22	✓	132
3.2	odd	2		567.2.ba.a.143.1		132
7.5	odd	6		189.2.bd.a.47.1	yes	132
21.5	even	6		567.2.bd.a.467.22		132
27.4	even	9		567.2.bd.a.17.22		132
27.23	odd	18		189.2.bd.a.185.1	yes	132
189.131	even	18	inner	189.2.ba.a.131.22	yes	132
189.166	odd	18		567.2.ba.a.341.1		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.101.22	✓	132	1.1	even	1	trivial
189.2.ba.a.131.22	yes	132	189.131	even	18	inner
189.2.bd.a.47.1	yes	132	7.5	odd	6
189.2.bd.a.185.1	yes	132	27.23	odd	18
567.2.ba.a.143.1		132	3.2	odd	2
567.2.ba.a.341.1		132	189.166	odd	18
567.2.bd.a.17.22		132	27.4	even	9
567.2.bd.a.467.22		132	21.5	even	6