Embedded newform 189.2.v.b.22.4

• Label: 189.2.v.b.22.4
• 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			22.4
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.b.43.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.461316 + 0.387090i) q^{2} +(1.33551 + 1.10291i) q^{3} +(-0.284323 + 1.61247i) q^{4} +(2.68051 - 0.975627i) q^{5} +(-1.04302 + 0.00817329i) q^{6} +(-0.173648 - 0.984808i) q^{7} +(-1.09521 - 1.89697i) q^{8} +(0.567181 + 2.94590i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.461316	+	0.387090i	−0.326199	+	0.273714i	−0.791149	−	0.611623i	\(-0.790517\pi\)
							0.464950	+	0.885337i	\(0.346072\pi\)
\(3\)	1.33551	+	1.10291i	0.771058	+	0.636765i
							\(5\)	2.68051	−	0.975627i	1.19876	−	0.436313i	0.335970	−	0.941873i	\(-0.390936\pi\)
0.862792	+	0.505559i	\(0.168714\pi\)
\(7\)	−0.173648	−	0.984808i	−0.0656328	−	0.372222i
							\(11\)	2.58460	+	0.940717i	0.779286	+	0.283637i	0.700875	−	0.713284i	\(-0.252793\pi\)
0.0784112	+	0.996921i	\(0.475015\pi\)
\(13\)	−5.14461	−	4.31684i	−1.42686	−	1.19728i	−0.947541	−	0.319635i	\(-0.896440\pi\)
							−0.479318	−	0.877641i	\(-0.659116\pi\)
\(17\)	−2.24341	+	3.88570i	−0.544107	+	0.942422i	0.454555	+	0.890719i	\(0.349798\pi\)
							−0.998662	+	0.0517031i	\(0.983535\pi\)
\(19\)	0.387203	+	0.670656i	0.0888305	+	0.153859i	0.907017	−	0.421094i	\(-0.138354\pi\)
							−0.818187	+	0.574953i	\(0.805020\pi\)
\(23\)	0.127329	−	0.722116i	0.0265498	−	0.150572i	−0.968651	−	0.248426i	\(-0.920087\pi\)
							0.995201	+	0.0978542i	\(0.0311979\pi\)
\(29\)	4.52455	−	3.79655i	0.840188	−	0.705002i	−0.117418	−	0.993083i	\(-0.537462\pi\)
							0.957606	+	0.288081i	\(0.0930172\pi\)
\(31\)	1.33749	−	7.58528i	0.240220	−	1.36236i	−0.591116	−	0.806587i	\(-0.701312\pi\)
							0.831336	−	0.555770i	\(-0.187577\pi\)
\(37\)	−4.13831	+	7.16777i	−0.680334	+	1.17837i	0.294545	+	0.955638i	\(0.404832\pi\)
							−0.974879	+	0.222736i	\(0.928501\pi\)
\(41\)	−0.257675	−	0.216215i	−0.0402421	−	0.0337671i	0.622445	−	0.782664i	\(-0.286140\pi\)
							−0.662687	+	0.748897i	\(0.730584\pi\)
\(43\)	−6.60663	−	2.40462i	−1.00750	−	0.366701i	−0.215029	−	0.976608i	\(-0.568985\pi\)
							−0.792473	+	0.609907i	\(0.791207\pi\)
\(47\)	−2.14455	−	12.1623i	−0.312814	−	1.77406i	−0.584221	−	0.811594i	\(-0.698600\pi\)
							0.271407	−	0.962465i	\(-0.412511\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.b.22.4	✓	54
3.2	odd	2		567.2.v.a.442.6		54
27.4	even	9		5103.2.a.g.1.12		27
27.11	odd	18		567.2.v.a.127.6		54
27.16	even	9	inner	189.2.v.b.43.4	yes	54
27.23	odd	18		5103.2.a.h.1.16		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.22.4	✓	54	1.1	even	1	trivial
189.2.v.b.43.4	yes	54	27.16	even	9	inner
567.2.v.a.127.6		54	27.11	odd	18
567.2.v.a.442.6		54	3.2	odd	2
5103.2.a.g.1.12		27	27.4	even	9
5103.2.a.h.1.16		27	27.23	odd	18