Embedded newform 189.2.w.a.25.1

• Label: 189.2.w.a.25.1
• 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			25.1
Character	\(\chi\)	\(=\)	189.25
Dual form			189.2.w.a.121.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.58171 + 0.939667i) q^{2} +(-1.71864 + 0.215106i) q^{3} +(4.25019 - 3.56633i) q^{4} +(0.335317 + 0.122045i) q^{5} +(4.23491 - 2.17029i) q^{6} +(-0.921726 + 2.48000i) q^{7} +(-4.87420 + 8.44237i) q^{8} +(2.90746 - 0.739379i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.58171	+	0.939667i	−1.82555	+	0.664445i	−0.831497	+	0.555529i	\(0.812516\pi\)
							−0.994051	+	0.108916i	\(0.965262\pi\)
\(3\)	−1.71864	+	0.215106i	−0.992258	+	0.124191i
							\(5\)	0.335317	+	0.122045i	0.149958	+	0.0545804i	0.415909	−	0.909406i	\(-0.363464\pi\)
−0.265950	+	0.963987i	\(0.585686\pi\)
\(7\)	−0.921726	+	2.48000i	−0.348380	+	0.937353i
							\(11\)	−1.67405	+	0.609306i	−0.504746	+	0.183713i	−0.581827	−	0.813312i	\(-0.697662\pi\)
0.0770813	+	0.997025i	\(0.475440\pi\)
\(13\)	−0.143143	−	0.811807i	−0.0397009	−	0.225155i	0.958502	−	0.285087i	\(-0.0920226\pi\)
							−0.998202	+	0.0599325i	\(0.980911\pi\)
\(17\)	−1.85116			−0.448971			−0.224486	−	0.974477i	\(-0.572070\pi\)
							−0.224486	+	0.974477i	\(0.572070\pi\)
\(19\)	−4.90094			−1.12435			−0.562176	−	0.827018i	\(-0.690036\pi\)
							−0.562176	+	0.827018i	\(0.690036\pi\)
\(23\)	−1.13525	−	6.43834i	−0.236717	−	1.34249i	−0.838968	−	0.544180i	\(-0.816841\pi\)
							0.602252	−	0.798306i	\(-0.294270\pi\)
\(29\)	−0.600809	+	3.40736i	−0.111567	+	0.632730i	0.876825	+	0.480809i	\(0.159657\pi\)
							−0.988393	+	0.151921i	\(0.951454\pi\)
\(31\)	−7.06750	+	5.93034i	−1.26936	+	1.06512i	−0.274742	+	0.961518i	\(0.588593\pi\)
							−0.994619	+	0.103602i	\(0.966963\pi\)
\(37\)	−2.49443	+	4.32047i	−0.410081	+	0.710281i	−0.994898	−	0.100884i	\(-0.967833\pi\)
							0.584817	+	0.811165i	\(0.301166\pi\)
\(41\)	−0.733977	−	4.16259i	−0.114628	−	0.650087i	−0.986934	−	0.161126i	\(-0.948487\pi\)
							0.872306	−	0.488961i	\(-0.162624\pi\)
\(43\)	3.83244	+	3.21580i	0.584442	+	0.490405i	0.886402	−	0.462916i	\(-0.153197\pi\)
							−0.301961	+	0.953320i	\(0.597641\pi\)
\(47\)	−7.02871	−	5.89779i	−1.02524	−	0.860281i	−0.0349659	−	0.999389i	\(-0.511132\pi\)
							−0.990277	+	0.139107i	\(0.955577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.w.a.25.1	yes	132
3.2	odd	2		567.2.w.a.235.22		132
7.2	even	3		189.2.u.a.79.22	yes	132
21.2	odd	6		567.2.u.a.478.1		132
27.13	even	9		189.2.u.a.67.22	✓	132
27.14	odd	18		567.2.u.a.172.1		132
189.121	even	9	inner	189.2.w.a.121.1	yes	132
189.149	odd	18		567.2.w.a.415.22		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.67.22	✓	132	27.13	even	9
189.2.u.a.79.22	yes	132	7.2	even	3
189.2.w.a.25.1	yes	132	1.1	even	1	trivial
189.2.w.a.121.1	yes	132	189.121	even	9	inner
567.2.u.a.172.1		132	27.14	odd	18
567.2.u.a.478.1		132	21.2	odd	6
567.2.w.a.235.22		132	3.2	odd	2
567.2.w.a.415.22		132	189.149	odd	18