Embedded newform 189.2.bd.a.185.12

• Label: 189.2.bd.a.185.12
• Level: $189$
• Weight: $2$
• Character: 189.185
• 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.bd (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			185.12
Character	\(\chi\)	\(=\)	189.185
Dual form			189.2.bd.a.47.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0159182 - 0.00280680i) q^{2} +(-0.402650 + 1.68460i) q^{3} +(-1.87914 + 0.683951i) q^{4} +(3.75147 - 1.36542i) q^{5} +(-0.00168110 + 0.0279459i) q^{6} +(0.157938 + 2.64103i) q^{7} +(-0.0559891 + 0.0323253i) q^{8} +(-2.67575 - 1.35661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 18 q^{6} - 6 q^{7} - 18 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0159182	−	0.00280680i	0.0112558	−	0.00198471i	−0.168017	−	0.985784i	\(-0.553736\pi\)
							0.179273	+	0.983799i	\(0.442625\pi\)
\(3\)	−0.402650	+	1.68460i	−0.232470	+	0.972604i
							\(5\)	3.75147	−	1.36542i	1.67771	−	0.610636i	0.684715	−	0.728811i	\(-0.259927\pi\)
0.992993	+	0.118175i	\(0.0377045\pi\)
\(7\)	0.157938	+	2.64103i	0.0596949	+	0.998217i
							\(11\)	−1.76436	+	4.84753i	−0.531973	+	1.46158i	0.324746	+	0.945801i	\(0.394721\pi\)
−0.856719	+	0.515783i	\(0.827501\pi\)
\(13\)	−0.220791	−	0.606618i	−0.0612364	−	0.168246i	0.905301	−	0.424770i	\(-0.139645\pi\)
							−0.966538	+	0.256524i	\(0.917423\pi\)
\(17\)	1.69383	+	2.93380i	0.410815	+	0.711552i	0.994979	−	0.100084i	\(-0.0319111\pi\)
							−0.584165	+	0.811635i	\(0.698578\pi\)
\(19\)	0.328436	+	0.189623i	0.0753485	+	0.0435025i	0.537201	−	0.843454i	\(-0.319482\pi\)
							−0.461852	+	0.886957i	\(0.652815\pi\)
\(23\)	−0.993035	−	0.175099i	−0.207062	−	0.0365106i	0.0691549	−	0.997606i	\(-0.477970\pi\)
							−0.276217	+	0.961095i	\(0.589081\pi\)
\(29\)	1.97120	−	5.41582i	0.366042	−	1.00569i	−0.610810	−	0.791777i	\(-0.709156\pi\)
							0.976852	−	0.213916i	\(-0.0686218\pi\)
\(31\)	−2.20741	−	6.06480i	−0.396462	−	1.08927i	−0.963995	−	0.265920i	\(-0.914324\pi\)
							0.567533	−	0.823351i	\(-0.307898\pi\)
\(37\)	6.16814			1.01404			0.507018	−	0.861936i	\(-0.330748\pi\)
							0.507018	+	0.861936i	\(0.330748\pi\)
\(41\)	0.266252	−	0.0969077i	0.0415816	−	0.0151344i	−0.321146	−	0.947030i	\(-0.604068\pi\)
							0.362727	+	0.931895i	\(0.381846\pi\)
\(43\)	−1.45052	−	8.22633i	−0.221203	−	1.25450i	−0.869812	−	0.493383i	\(-0.835760\pi\)
							0.648610	−	0.761121i	\(-0.275351\pi\)
\(47\)	6.91258	+	2.51597i	1.00830	+	0.366992i	0.792781	−	0.609507i	\(-0.208632\pi\)
							0.215522	+	0.976499i	\(0.430855\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.bd.a.185.12	yes	132
3.2	odd	2		567.2.bd.a.17.11		132
7.5	odd	6		189.2.ba.a.131.11	yes	132
21.5	even	6		567.2.ba.a.341.12		132
27.7	even	9		567.2.ba.a.143.12		132
27.20	odd	18		189.2.ba.a.101.11	✓	132
189.47	even	18	inner	189.2.bd.a.47.12	yes	132
189.61	odd	18		567.2.bd.a.467.11		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.101.11	✓	132	27.20	odd	18
189.2.ba.a.131.11	yes	132	7.5	odd	6
189.2.bd.a.47.12	yes	132	189.47	even	18	inner
189.2.bd.a.185.12	yes	132	1.1	even	1	trivial
567.2.ba.a.143.12		132	27.7	even	9
567.2.ba.a.341.12		132	21.5	even	6
567.2.bd.a.17.11		132	3.2	odd	2
567.2.bd.a.467.11		132	189.61	odd	18