Embedded newform 189.2.w.a.25.19

• Label: 189.2.w.a.25.19
• Level: $189$
• Weight: $2$
• Character: 189.25
• 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.w (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			25.19
Character	\(\chi\)	\(=\)	189.25
Dual form			189.2.w.a.121.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.90772 - 0.694353i) q^{2} +(0.482660 - 1.66344i) q^{3} +(1.62518 - 1.36369i) q^{4} +(-2.38896 - 0.869509i) q^{5} +(-0.234236 - 3.50852i) q^{6} +(1.31108 + 2.29806i) q^{7} +(0.123353 - 0.213653i) q^{8} +(-2.53408 - 1.60575i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.90772	−	0.694353i	1.34896	−	0.490982i	0.436338	−	0.899783i	\(-0.356275\pi\)
							0.912624	+	0.408801i	\(0.134053\pi\)
\(3\)	0.482660	−	1.66344i	0.278664	−	0.960389i
							\(5\)	−2.38896	−	0.869509i	−1.06837	−	0.388856i	−0.252806	−	0.967517i	\(-0.581353\pi\)
−0.815568	+	0.578661i	\(0.803576\pi\)
\(7\)	1.31108	+	2.29806i	0.495543	+	0.868584i
							\(11\)	5.38280	−	1.95918i	1.62298	−	0.590715i	0.639030	−	0.769182i	\(-0.279336\pi\)
0.983946	+	0.178467i	\(0.0571137\pi\)
\(13\)	0.494719	+	2.80569i	0.137210	+	0.778159i	0.973295	+	0.229558i	\(0.0737282\pi\)
							−0.836084	+	0.548601i	\(0.815161\pi\)
\(17\)	−4.83742			−1.17325			−0.586624	−	0.809860i	\(-0.699543\pi\)
							−0.586624	+	0.809860i	\(0.699543\pi\)
\(19\)	0.555703			0.127487			0.0637435	−	0.997966i	\(-0.479696\pi\)
							0.0637435	+	0.997966i	\(0.479696\pi\)
\(23\)	0.476702	+	2.70351i	0.0993992	+	0.563721i	0.993310	+	0.115477i	\(0.0368395\pi\)
							−0.893911	+	0.448245i	\(0.852049\pi\)
\(29\)	0.164033	−	0.930279i	0.0304602	−	0.172749i	−0.965783	−	0.259353i	\(-0.916491\pi\)
							0.996243	+	0.0866047i	\(0.0276017\pi\)
\(31\)	3.82842	−	3.21243i	0.687605	−	0.576969i	−0.230612	−	0.973046i	\(-0.574073\pi\)
							0.918217	+	0.396077i	\(0.129629\pi\)
\(37\)	−4.08057	+	7.06775i	−0.670841	+	1.16193i	0.306824	+	0.951766i	\(0.400734\pi\)
							−0.977666	+	0.210165i	\(0.932600\pi\)
\(41\)	−1.92050	−	10.8917i	−0.299932	−	1.70100i	−0.646452	−	0.762955i	\(-0.723748\pi\)
							0.346520	−	0.938043i	\(-0.387363\pi\)
\(43\)	−4.80243	−	4.02972i	−0.732364	−	0.614526i	0.198411	−	0.980119i	\(-0.436422\pi\)
							−0.930775	+	0.365593i	\(0.880866\pi\)
\(47\)	−2.28401	−	1.91651i	−0.333157	−	0.279552i	0.460828	−	0.887490i	\(-0.347553\pi\)
							−0.793985	+	0.607938i	\(0.791997\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.w.a.25.19	yes	132
3.2	odd	2		567.2.w.a.235.4		132
7.2	even	3		189.2.u.a.79.4	yes	132
21.2	odd	6		567.2.u.a.478.19		132
27.13	even	9		189.2.u.a.67.4	✓	132
27.14	odd	18		567.2.u.a.172.19		132
189.121	even	9	inner	189.2.w.a.121.19	yes	132
189.149	odd	18		567.2.w.a.415.4		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.67.4	✓	132	27.13	even	9
189.2.u.a.79.4	yes	132	7.2	even	3
189.2.w.a.25.19	yes	132	1.1	even	1	trivial
189.2.w.a.121.19	yes	132	189.121	even	9	inner
567.2.u.a.172.19		132	27.14	odd	18
567.2.u.a.478.19		132	21.2	odd	6
567.2.w.a.235.4		132	3.2	odd	2
567.2.w.a.415.4		132	189.149	odd	18