Embedded newform 189.2.ba.a.131.11

• Label: 189.2.ba.a.131.11
• Level: $189$
• Weight: $2$
• Character: 189.131
• 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			131.11
Character	\(\chi\)	\(=\)	189.131
Dual form			189.2.ba.a.101.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0103898 - 0.0123821i) q^{2} +(1.25758 + 1.19100i) q^{3} +(0.347251 - 1.96936i) q^{4} +(3.05823 + 2.56616i) q^{5} +(0.00168110 - 0.0279459i) q^{6} +(-2.57348 - 0.614149i) q^{7} +(-0.0559891 + 0.0323253i) q^{8} +(0.163018 + 2.99557i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0103898	−	0.0123821i	−0.00734672	−	0.00875548i	0.762359	−	0.647155i	\(-0.224041\pi\)
							−0.769705	+	0.638399i	\(0.779597\pi\)
\(3\)	1.25758	+	1.19100i	0.726065	+	0.687627i
							\(5\)	3.05823	+	2.56616i	1.36768	+	1.14762i	0.973526	+	0.228575i	\(0.0734066\pi\)
0.394154	+	0.919045i	\(0.371038\pi\)
\(7\)	−2.57348	−	0.614149i	−0.972686	−	0.232127i
							\(11\)	−3.31590	−	3.95174i	−0.999782	−	1.19149i	−0.981461	−	0.191662i	\(-0.938612\pi\)
−0.0183214	−	0.999832i	\(-0.505832\pi\)
\(13\)	0.220791	+	0.606618i	0.0612364	+	0.168246i	0.966538	−	0.256524i	\(-0.0825773\pi\)
							−0.905301	+	0.424770i	\(0.860355\pi\)
\(17\)	3.38766			0.821629			0.410815	−	0.911719i	\(-0.365244\pi\)
							0.410815	+	0.911719i	\(0.365244\pi\)
\(19\)			0.379246i			0.0870049i	0.999053	+	0.0435025i	\(0.0138516\pi\)
							−0.999053	+	0.0435025i	\(0.986148\pi\)
\(23\)	0.344877	+	0.947543i	0.0719119	+	0.197576i	0.970441	−	0.241337i	\(-0.0775858\pi\)
							−0.898530	+	0.438913i	\(0.855364\pi\)
\(29\)	1.97120	−	5.41582i	0.366042	−	1.00569i	−0.610810	−	0.791777i	\(-0.709156\pi\)
							0.976852	−	0.213916i	\(-0.0686218\pi\)
\(31\)	−6.35598	−	1.12073i	−1.14157	−	0.201289i	−0.429276	−	0.903173i	\(-0.641231\pi\)
							−0.712291	+	0.701884i	\(0.752342\pi\)
\(37\)	−3.08407	−	5.34176i	−0.507018	−	0.878181i	−0.999967	−	0.00812261i	\(-0.997414\pi\)
							0.492949	−	0.870058i	\(-0.335919\pi\)
\(41\)	−0.266252	+	0.0969077i	−0.0415816	+	0.0151344i	−0.362727	−	0.931895i	\(-0.618154\pi\)
							0.321146	+	0.947030i	\(0.395932\pi\)
\(43\)	−1.45052	−	8.22633i	−0.221203	−	1.25450i	−0.869812	−	0.493383i	\(-0.835760\pi\)
							0.648610	−	0.761121i	\(-0.275351\pi\)
\(47\)	1.27739	+	7.24446i	0.186327	+	1.05671i	0.924239	+	0.381815i	\(0.124701\pi\)
							−0.737912	+	0.674897i	\(0.764188\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.101.11	✓	132	189.101	even	18	inner
189.2.ba.a.131.11	yes	132	1.1	even	1	trivial
189.2.bd.a.47.12	yes	132	27.20	odd	18
189.2.bd.a.185.12	yes	132	7.3	odd	6
567.2.ba.a.143.12		132	189.115	odd	18
567.2.ba.a.341.12		132	3.2	odd	2
567.2.bd.a.17.11		132	21.17	even	6
567.2.bd.a.467.11		132	27.7	even	9