Embedded newform 189.2.ba.a.5.1

• Label: 189.2.ba.a.5.1
• Level: $189$
• Weight: $2$
• Character: 189.5
• 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			5.1
Character	\(\chi\)	\(=\)	189.5
Dual form			189.2.ba.a.38.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.51634 + 0.443699i) q^{2} +(-1.35749 + 1.07574i) q^{3} +(4.25573 - 1.54896i) q^{4} +(-0.181827 + 1.03119i) q^{5} +(2.93860 - 3.30926i) q^{6} +(2.00134 - 1.73050i) q^{7} +(-5.59594 + 3.23082i) q^{8} +(0.685547 - 2.92062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.51634	+	0.443699i	−1.77932	+	0.313743i	−0.964127	−	0.265441i	\(-0.914482\pi\)
							−0.815197	+	0.579184i	\(0.803371\pi\)
\(3\)	−1.35749	+	1.07574i	−0.783746	+	0.621081i
							\(5\)	−0.181827	+	1.03119i	−0.0813155	+	0.461163i	0.916775	+	0.399403i	\(0.130783\pi\)
−0.998091	+	0.0617604i	\(0.980329\pi\)
\(7\)	2.00134	−	1.73050i	0.756436	−	0.654067i
							\(11\)	4.05020	−	0.714160i	1.22118	−	0.215327i	0.474348	−	0.880337i	\(-0.342684\pi\)
0.746834	+	0.665010i	\(0.231573\pi\)
\(13\)	−2.92808	+	3.48955i	−0.812103	+	0.967827i	−0.999897	−	0.0143818i	\(-0.995422\pi\)
							0.187793	+	0.982209i	\(0.439866\pi\)
\(17\)	3.48283			0.844710			0.422355	−	0.906431i	\(-0.361204\pi\)
							0.422355	+	0.906431i	\(0.361204\pi\)
\(19\)			5.54678i			1.27252i	0.771475	+	0.636260i	\(0.219519\pi\)
							−0.771475	+	0.636260i	\(0.780481\pi\)
\(23\)	−3.90274	+	4.65110i	−0.813777	+	0.969822i	−0.999919	−	0.0127010i	\(-0.995957\pi\)
							0.186142	+	0.982523i	\(0.440401\pi\)
\(29\)	−1.57154	−	1.87289i	−0.291828	−	0.347787i	0.600132	−	0.799901i	\(-0.295115\pi\)
							−0.891960	+	0.452114i	\(0.850670\pi\)
\(31\)	0.362555	+	0.996113i	0.0651169	+	0.178907i	0.967984	−	0.251013i	\(-0.0807638\pi\)
							−0.902867	+	0.429920i	\(0.858542\pi\)
\(37\)	−1.64217	−	2.84431i	−0.269970	−	0.467602i	0.698884	−	0.715235i	\(-0.253681\pi\)
							−0.968854	+	0.247633i	\(0.920347\pi\)
\(41\)	−0.164894	−	0.138363i	−0.0257522	−	0.0216086i	0.629821	−	0.776741i	\(-0.283128\pi\)
							−0.655573	+	0.755132i	\(0.727573\pi\)
\(43\)	9.06382	+	3.29896i	1.38222	+	0.503087i	0.922850	−	0.385159i	\(-0.125853\pi\)
							0.459369	+	0.888246i	\(0.348076\pi\)
\(47\)	−0.827325	−	0.301122i	−0.120678	−	0.0439231i	0.280975	−	0.959715i	\(-0.409342\pi\)
							−0.401653	+	0.915792i	\(0.631564\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.ba.a.5.1	✓	132
3.2	odd	2		567.2.ba.a.530.22		132
7.3	odd	6		189.2.bd.a.59.22	yes	132
21.17	even	6		567.2.bd.a.206.1		132
27.11	odd	18		189.2.bd.a.173.22	yes	132
27.16	even	9		567.2.bd.a.278.1		132
189.38	even	18	inner	189.2.ba.a.38.1	yes	132
189.178	odd	18		567.2.ba.a.521.22		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.5.1	✓	132	1.1	even	1	trivial
189.2.ba.a.38.1	yes	132	189.38	even	18	inner
189.2.bd.a.59.22	yes	132	7.3	odd	6
189.2.bd.a.173.22	yes	132	27.11	odd	18
567.2.ba.a.521.22		132	189.178	odd	18
567.2.ba.a.530.22		132	3.2	odd	2
567.2.bd.a.206.1		132	21.17	even	6
567.2.bd.a.278.1		132	27.16	even	9