Embedded newform 189.2.v.b.22.2

• Label: 189.2.v.b.22.2
• 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			22.2
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.b.43.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67142 + 1.40248i) q^{2} +(-1.72870 - 0.107685i) q^{3} +(0.479372 - 2.71866i) q^{4} +(-2.82618 + 1.02865i) q^{5} +(3.04040 - 2.24449i) q^{6} +(-0.173648 - 0.984808i) q^{7} +(0.829763 + 1.43719i) q^{8} +(2.97681 + 0.372309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67142	+	1.40248i	−1.18187	+	0.991706i	−0.181905	+	0.983316i	\(0.558226\pi\)
							−0.999965	+	0.00839031i	\(0.997329\pi\)
\(3\)	−1.72870	−	0.107685i	−0.998065	−	0.0621718i
							\(5\)	−2.82618	+	1.02865i	−1.26391	+	0.460025i	−0.885078	−	0.465442i	\(-0.845895\pi\)
−0.378829	+	0.925467i	\(0.623673\pi\)
\(7\)	−0.173648	−	0.984808i	−0.0656328	−	0.372222i
							\(11\)	1.22699	+	0.446587i	0.369951	+	0.134651i	0.520304	−	0.853981i	\(-0.325819\pi\)
−0.150353	+	0.988632i	\(0.548041\pi\)
\(13\)	−0.374438	−	0.314191i	−0.103851	−	0.0871410i	0.589384	−	0.807853i	\(-0.299371\pi\)
							−0.693235	+	0.720712i	\(0.743815\pi\)
\(17\)	3.32864	−	5.76537i	0.807313	−	1.39831i	−0.107405	−	0.994215i	\(-0.534254\pi\)
							0.914718	−	0.404092i	\(-0.132413\pi\)
\(19\)	0.478257	+	0.828366i	0.109720	+	0.190040i	0.915657	−	0.401961i	\(-0.131671\pi\)
							−0.805937	+	0.592001i	\(0.798338\pi\)
\(23\)	−0.513759	+	2.91367i	−0.107126	+	0.607543i	0.883224	+	0.468952i	\(0.155368\pi\)
							−0.990350	+	0.138591i	\(0.955743\pi\)
\(29\)	5.24112	−	4.39783i	0.973252	−	0.816656i	−0.00980542	−	0.999952i	\(-0.503121\pi\)
							0.983058	+	0.183296i	\(0.0586768\pi\)
\(31\)	1.08429	−	6.14930i	0.194744	−	1.10445i	−0.718039	−	0.696002i	\(-0.754960\pi\)
							0.912783	−	0.408444i	\(-0.133929\pi\)
\(37\)	2.43630	−	4.21979i	0.400524	−	0.693729i	−0.593265	−	0.805007i	\(-0.702161\pi\)
							0.993789	+	0.111279i	\(0.0354946\pi\)
\(41\)	−3.17692	−	2.66575i	−0.496151	−	0.416320i	0.360074	−	0.932924i	\(-0.382752\pi\)
							−0.856225	+	0.516604i	\(0.827196\pi\)
\(43\)	−10.4383	−	3.79924i	−1.59183	−	0.579378i	−0.614095	−	0.789232i	\(-0.710479\pi\)
							−0.977733	+	0.209854i	\(0.932701\pi\)
\(47\)	−2.19327	−	12.4387i	−0.319921	−	1.81436i	−0.543193	−	0.839608i	\(-0.682785\pi\)
							0.223272	−	0.974756i	\(-0.428326\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.b.22.2	✓	54
3.2	odd	2		567.2.v.a.442.8		54
27.4	even	9		5103.2.a.g.1.5		27
27.11	odd	18		567.2.v.a.127.8		54
27.16	even	9	inner	189.2.v.b.43.2	yes	54
27.23	odd	18		5103.2.a.h.1.23		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.22.2	✓	54	1.1	even	1	trivial
189.2.v.b.43.2	yes	54	27.16	even	9	inner
567.2.v.a.127.8		54	27.11	odd	18
567.2.v.a.442.8		54	3.2	odd	2
5103.2.a.g.1.5		27	27.4	even	9
5103.2.a.h.1.23		27	27.23	odd	18