Embedded newform 189.2.u.a.79.10

• Label: 189.2.u.a.79.10
• Level: $189$
• Weight: $2$
• Character: 189.79
• 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			79.10
Character	\(\chi\)	\(=\)	189.79
Dual form			189.2.u.a.67.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.372124 - 0.312249i) q^{2} +(-0.750052 + 1.56122i) q^{3} +(-0.306320 - 1.73722i) q^{4} +(-0.266880 - 1.51355i) q^{5} +(0.766603 - 0.346766i) q^{6} +(-2.38579 - 1.14369i) q^{7} +(-0.914231 + 1.58349i) q^{8} +(-1.87484 - 2.34200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.372124	−	0.312249i	−0.263131	−	0.220793i	0.501671	−	0.865059i	\(-0.332719\pi\)
							−0.764802	+	0.644265i	\(0.777163\pi\)
\(3\)	−0.750052	+	1.56122i	−0.433043	+	0.901373i
							\(5\)	−0.266880	−	1.51355i	−0.119352	−	0.676881i	−0.984503	−	0.175369i	\(-0.943888\pi\)
0.865150	−	0.501512i	\(-0.167223\pi\)
\(7\)	−2.38579	−	1.14369i	−0.901742	−	0.432274i
							\(11\)	−0.379279	+	2.15100i	−0.114357	+	0.648551i	0.872709	+	0.488240i	\(0.162361\pi\)
−0.987067	+	0.160311i	\(0.948750\pi\)
\(13\)	−1.09206	−	6.19341i	−0.302884	−	1.71774i	−0.633300	−	0.773906i	\(-0.718300\pi\)
							0.330416	−	0.943835i	\(-0.392811\pi\)
\(17\)	2.64457	−	4.58053i	0.641403	−	1.11094i	−0.343717	−	0.939073i	\(-0.611686\pi\)
							0.985120	−	0.171869i	\(-0.0549806\pi\)
\(19\)	−1.16718	−	2.02162i	−0.267771	−	0.463792i	0.700515	−	0.713637i	\(-0.252954\pi\)
							−0.968286	+	0.249845i	\(0.919620\pi\)
\(23\)	−2.20637	+	1.85136i	−0.460060	+	0.386036i	−0.843153	−	0.537674i	\(-0.819303\pi\)
							0.383093	+	0.923710i	\(0.374859\pi\)
\(29\)	−1.64020	+	9.30205i	−0.304578	+	1.72735i	0.320907	+	0.947111i	\(0.396012\pi\)
							−0.625485	+	0.780237i	\(0.715099\pi\)
\(31\)	0.463969	+	2.63130i	0.0833313	+	0.472595i	0.997704	+	0.0677222i	\(0.0215732\pi\)
							−0.914373	+	0.404873i	\(0.867316\pi\)
\(37\)	0.458521			0.0753804			0.0376902	−	0.999289i	\(-0.488000\pi\)
							0.0376902	+	0.999289i	\(0.488000\pi\)
\(41\)	0.513695	+	2.91331i	0.0802256	+	0.454982i	0.998285	+	0.0585394i	\(0.0186443\pi\)
							−0.918059	+	0.396443i	\(0.870245\pi\)
\(43\)	−4.87748	−	4.09269i	−0.743809	−	0.624130i	0.190049	−	0.981775i	\(-0.439135\pi\)
							−0.933858	+	0.357645i	\(0.883580\pi\)
\(47\)	1.15593	−	6.55559i	0.168609	−	0.956232i	−0.776655	−	0.629926i	\(-0.783085\pi\)
							0.945264	−	0.326305i	\(-0.105804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.u.a.79.10	yes	132
3.2	odd	2		567.2.u.a.478.13		132
7.4	even	3		189.2.w.a.25.13	yes	132
21.11	odd	6		567.2.w.a.235.10		132
27.13	even	9		189.2.w.a.121.13	yes	132
27.14	odd	18		567.2.w.a.415.10		132
189.67	even	9	inner	189.2.u.a.67.10	✓	132
189.95	odd	18		567.2.u.a.172.13		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.67.10	✓	132	189.67	even	9	inner
189.2.u.a.79.10	yes	132	1.1	even	1	trivial
189.2.w.a.25.13	yes	132	7.4	even	3
189.2.w.a.121.13	yes	132	27.13	even	9
567.2.u.a.172.13		132	189.95	odd	18
567.2.u.a.478.13		132	3.2	odd	2
567.2.w.a.235.10		132	21.11	odd	6
567.2.w.a.415.10		132	27.14	odd	18