Embedded newform 189.2.u.a.4.4

• Label: 189.2.u.a.4.4
• Level: $189$
• Weight: $2$
• Character: 189.4
• 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.u (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			4.4
Character	\(\chi\)	\(=\)	189.4
Dual form			189.2.u.a.142.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.324131 - 1.83824i) q^{2} +(0.743955 - 1.56414i) q^{3} +(-1.39467 + 0.507617i) q^{4} +(1.37479 - 0.500384i) q^{5} +(-3.11640 - 0.860580i) q^{6} +(-2.42382 + 1.06071i) q^{7} +(-0.481419 - 0.833843i) q^{8} +(-1.89306 - 2.32730i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.324131	−	1.83824i	−0.229195	−	1.29983i	−0.854501	−	0.519450i	\(-0.826137\pi\)
							0.625306	−	0.780380i	\(-0.284974\pi\)
\(3\)	0.743955	−	1.56414i	0.429523	−	0.903056i
							\(5\)	1.37479	−	0.500384i	0.614827	−	0.223779i	−0.0157870	−	0.999875i	\(-0.505025\pi\)
0.630614	+	0.776097i	\(0.282803\pi\)
\(7\)	−2.42382	+	1.06071i	−0.916117	+	0.400910i
							\(11\)	4.29056	+	1.56163i	1.29365	+	0.470851i	0.894924	−	0.446219i	\(-0.147230\pi\)
0.398727	+	0.917069i	\(0.369452\pi\)
\(13\)	0.925783	−	0.336957i	0.256766	−	0.0934552i	−0.210430	−	0.977609i	\(-0.567486\pi\)
							0.467196	+	0.884154i	\(0.345264\pi\)
\(17\)	0.620234	+	1.07428i	0.150429	+	0.260550i	0.931385	−	0.364035i	\(-0.118601\pi\)
							−0.780956	+	0.624586i	\(0.785268\pi\)
\(19\)	−0.718027	+	1.24366i	−0.164727	+	0.285315i	−0.936558	−	0.350512i	\(-0.886008\pi\)
							0.771832	+	0.635827i	\(0.219341\pi\)
\(23\)	0.165636	−	0.939366i	0.0345374	−	0.195871i	−0.962657	−	0.270723i	\(-0.912737\pi\)
							0.997195	+	0.0748517i	\(0.0238483\pi\)
\(29\)	5.18835	+	1.88840i	0.963452	+	0.350668i	0.775385	−	0.631488i	\(-0.217556\pi\)
							0.188067	+	0.982156i	\(0.439778\pi\)
\(31\)	6.28413	−	2.28724i	1.12866	−	0.410800i	0.290857	−	0.956767i	\(-0.406060\pi\)
							0.837807	+	0.545967i	\(0.183837\pi\)
\(37\)	−2.88764			−0.474725			−0.237363	−	0.971421i	\(-0.576283\pi\)
							−0.237363	+	0.971421i	\(0.576283\pi\)
\(41\)	11.1576	−	4.06104i	1.74253	−	0.634227i	0.743135	−	0.669141i	\(-0.233338\pi\)
							0.999390	+	0.0349137i	\(0.0111156\pi\)
\(43\)	0.559660	+	3.17399i	0.0853473	+	0.484029i	0.997281	+	0.0736913i	\(0.0234780\pi\)
							−0.911934	+	0.410337i	\(0.865411\pi\)
\(47\)	−10.6381	−	3.87197i	−1.55173	−	0.564785i	−0.582909	−	0.812537i	\(-0.698086\pi\)
							−0.968823	+	0.247753i	\(0.920308\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.u.a.4.4	✓	132
3.2	odd	2		567.2.u.a.550.19		132
7.2	even	3		189.2.w.a.58.4	yes	132
21.2	odd	6		567.2.w.a.226.19		132
27.7	even	9		189.2.w.a.88.4	yes	132
27.20	odd	18		567.2.w.a.424.19		132
189.128	odd	18		567.2.u.a.100.19		132
189.142	even	9	inner	189.2.u.a.142.4	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.4.4	✓	132	1.1	even	1	trivial
189.2.u.a.142.4	yes	132	189.142	even	9	inner
189.2.w.a.58.4	yes	132	7.2	even	3
189.2.w.a.88.4	yes	132	27.7	even	9
567.2.u.a.100.19		132	189.128	odd	18
567.2.u.a.550.19		132	3.2	odd	2
567.2.w.a.226.19		132	21.2	odd	6
567.2.w.a.424.19		132	27.20	odd	18