Embedded newform 189.2.v.b.22.3

• Label: 189.2.v.b.22.3
• 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			22.3
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.b.43.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18344 + 0.993024i) q^{2} +(0.993131 - 1.41905i) q^{3} +(0.0671374 - 0.380755i) q^{4} +(2.98981 - 1.08820i) q^{5} +(0.233835 + 2.66556i) q^{6} +(-0.173648 - 0.984808i) q^{7} +(-1.24623 - 2.15853i) q^{8} +(-1.02738 - 2.81860i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18344	+	0.993024i	−0.836819	+	0.702174i	−0.956846	−	0.290596i	\(-0.906146\pi\)
							0.120027	+	0.992771i	\(0.461702\pi\)
\(3\)	0.993131	−	1.41905i	0.573385	−	0.819286i
							\(5\)	2.98981	−	1.08820i	1.33709	−	0.486659i	0.428192	−	0.903688i	\(-0.359151\pi\)
0.908893	+	0.417028i	\(0.136928\pi\)
\(7\)	−0.173648	−	0.984808i	−0.0656328	−	0.372222i
							\(11\)	−3.89325	−	1.41703i	−1.17386	−	0.427250i	−0.319830	−	0.947475i	\(-0.603626\pi\)
−0.854029	+	0.520225i	\(0.825848\pi\)
\(13\)	4.11465	+	3.45260i	1.14120	+	0.957580i	0.999477	−	0.0323267i	\(-0.0102917\pi\)
							0.141722	+	0.989907i	\(0.454736\pi\)
\(17\)	2.60857	−	4.51818i	0.632671	−	1.09582i	−0.354332	−	0.935120i	\(-0.615292\pi\)
							0.987003	−	0.160699i	\(-0.0513748\pi\)
\(19\)	1.20154	+	2.08113i	0.275652	+	0.477443i	0.970299	−	0.241907i	\(-0.0777729\pi\)
							−0.694647	+	0.719350i	\(0.744440\pi\)
\(23\)	−1.54942	+	8.78722i	−0.323077	+	1.83226i	0.199778	+	0.979841i	\(0.435978\pi\)
							−0.522855	+	0.852421i	\(0.675133\pi\)
\(29\)	3.31484	−	2.78148i	0.615550	−	0.516508i	−0.280851	−	0.959751i	\(-0.590617\pi\)
							0.896401	+	0.443243i	\(0.146172\pi\)
\(31\)	−0.428098	+	2.42787i	−0.0768887	+	0.436058i	0.921925	+	0.387368i	\(0.126616\pi\)
							−0.998814	+	0.0486897i	\(0.984495\pi\)
\(37\)	−2.98636	+	5.17253i	−0.490955	+	0.850359i	−0.999946	−	0.0104130i	\(-0.996685\pi\)
							0.508991	+	0.860772i	\(0.330019\pi\)
\(41\)	−1.98629	−	1.66670i	−0.310207	−	0.260295i	0.474370	−	0.880325i	\(-0.342676\pi\)
							−0.784578	+	0.620031i	\(0.787120\pi\)
\(43\)	−4.92273	−	1.79173i	−0.750709	−	0.273236i	−0.0618047	−	0.998088i	\(-0.519686\pi\)
							−0.688904	+	0.724853i	\(0.741908\pi\)
\(47\)	1.30831	+	7.41982i	0.190837	+	1.08229i	0.918223	+	0.396063i	\(0.129624\pi\)
							−0.727386	+	0.686229i	\(0.759265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.b.22.3	✓	54
3.2	odd	2		567.2.v.a.442.7		54
27.4	even	9		5103.2.a.g.1.8		27
27.11	odd	18		567.2.v.a.127.7		54
27.16	even	9	inner	189.2.v.b.43.3	yes	54
27.23	odd	18		5103.2.a.h.1.20		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.22.3	✓	54	1.1	even	1	trivial
189.2.v.b.43.3	yes	54	27.16	even	9	inner
567.2.v.a.127.7		54	27.11	odd	18
567.2.v.a.442.7		54	3.2	odd	2
5103.2.a.g.1.8		27	27.4	even	9
5103.2.a.h.1.20		27	27.23	odd	18