Embedded newform 189.2.v.a.22.9

• Label: 189.2.v.a.22.9
• 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			22.9
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.a.43.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.00215 - 1.68000i) q^{2} +(1.58061 + 0.708280i) q^{3} +(0.838893 - 4.75760i) q^{4} +(-3.29720 + 1.20008i) q^{5} +(4.35453 - 1.23735i) q^{6} +(0.173648 + 0.984808i) q^{7} +(-3.69957 - 6.40784i) q^{8} +(1.99668 + 2.23903i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 3 q^{3} - 3 q^{5} - 27 q^{8} - 9 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.00215	−	1.68000i	1.41573	−	1.18794i	0.462151	−	0.886801i	\(-0.347078\pi\)
							0.953580	−	0.301139i	\(-0.0973667\pi\)
\(3\)	1.58061	+	0.708280i	0.912568	+	0.408925i
							\(5\)	−3.29720	+	1.20008i	−1.47455	+	0.536693i	−0.949333	−	0.314273i	\(-0.898239\pi\)
−0.525220	+	0.850966i	\(0.676017\pi\)
\(7\)	0.173648	+	0.984808i	0.0656328	+	0.372222i
							\(11\)	−0.308948	−	0.112448i	−0.0931514	−	0.0339044i	0.295024	−	0.955490i	\(-0.404672\pi\)
−0.388175	+	0.921585i	\(0.626895\pi\)
\(13\)	−1.82041	−	1.52750i	−0.504890	−	0.423653i	0.354437	−	0.935080i	\(-0.384673\pi\)
							−0.859327	+	0.511427i	\(0.829117\pi\)
\(17\)	−1.28196	+	2.22042i	−0.310921	+	0.538531i	−0.978562	−	0.205952i	\(-0.933971\pi\)
							0.667641	+	0.744483i	\(0.267304\pi\)
\(19\)	2.50633	+	4.34109i	0.574991	+	0.995914i	0.996043	+	0.0888772i	\(0.0283279\pi\)
							−0.421051	+	0.907037i	\(0.638339\pi\)
\(23\)	0.785944	−	4.45731i	0.163881	−	0.929414i	−0.786330	−	0.617806i	\(-0.788022\pi\)
							0.950211	−	0.311607i	\(-0.100867\pi\)
\(29\)	−7.49985	+	6.29312i	−1.39269	+	1.16860i	−0.428448	+	0.903566i	\(0.640940\pi\)
							−0.964238	+	0.265036i	\(0.914616\pi\)
\(31\)	1.70905	−	9.69250i	0.306954	−	1.74082i	−0.307207	−	0.951643i	\(-0.599394\pi\)
							0.614161	−	0.789181i	\(-0.289495\pi\)
\(37\)	1.70374	−	2.95097i	0.280094	−	0.485137i	−0.691314	−	0.722555i	\(-0.742968\pi\)
							0.971408	+	0.237418i	\(0.0763011\pi\)
\(41\)	−5.18169	−	4.34795i	−0.809244	−	0.679036i	0.141183	−	0.989983i	\(-0.454909\pi\)
							−0.950427	+	0.310947i	\(0.899354\pi\)
\(43\)	0.843380	+	0.306965i	0.128614	+	0.0468118i	0.405525	−	0.914084i	\(-0.367089\pi\)
							−0.276911	+	0.960896i	\(0.589311\pi\)
\(47\)	−0.875085	−	4.96285i	−0.127644	−	0.723907i	−0.979702	−	0.200459i	\(-0.935757\pi\)
							0.852058	−	0.523448i	\(-0.175354\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.a.22.9	✓	54
3.2	odd	2		567.2.v.b.442.1		54
27.4	even	9		5103.2.a.i.1.25		27
27.11	odd	18		567.2.v.b.127.1		54
27.16	even	9	inner	189.2.v.a.43.9	yes	54
27.23	odd	18		5103.2.a.f.1.3		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.a.22.9	✓	54	1.1	even	1	trivial
189.2.v.a.43.9	yes	54	27.16	even	9	inner
567.2.v.b.127.1		54	27.11	odd	18
567.2.v.b.442.1		54	3.2	odd	2
5103.2.a.f.1.3		27	27.23	odd	18
5103.2.a.i.1.25		27	27.4	even	9