Embedded newform 189.2.v.b.85.2

• Label: 189.2.v.b.85.2
• Level: $189$
• Weight: $2$
• Character: 189.85
• 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			85.2
Character	\(\chi\)	\(=\)	189.85
Dual form			189.2.v.b.169.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78358 - 0.649172i) q^{2} +(0.189618 + 1.72164i) q^{3} +(1.22766 + 1.03013i) q^{4} +(0.611519 - 3.46809i) q^{5} +(0.779441 - 3.19379i) q^{6} +(-0.766044 + 0.642788i) q^{7} +(0.377143 + 0.653231i) q^{8} +(-2.92809 + 0.652907i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 3 q^{3} - 3 q^{5} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78358	−	0.649172i	−1.26119	−	0.459034i	−0.377019	−	0.926206i	\(-0.623051\pi\)
							−0.884167	+	0.467172i	\(0.845273\pi\)
\(3\)	0.189618	+	1.72164i	0.109476	+	0.993989i
							\(5\)	0.611519	−	3.46809i	0.273479	−	1.55098i	−0.470271	−	0.882522i	\(-0.655844\pi\)
0.743751	−	0.668457i	\(-0.233045\pi\)
\(7\)	−0.766044	+	0.642788i	−0.289538	+	0.242951i
							\(11\)	−0.564924	−	3.20384i	−0.170331	−	0.965995i	−0.943396	−	0.331668i	\(-0.892388\pi\)
0.773065	−	0.634327i	\(-0.218723\pi\)
\(13\)	4.32141	−	1.57287i	1.19854	−	0.436235i	0.335828	−	0.941923i	\(-0.390984\pi\)
							0.862716	+	0.505689i	\(0.168762\pi\)
\(17\)	3.71062	−	6.42698i	0.899958	−	1.55877i	0.0724117	−	0.997375i	\(-0.476930\pi\)
							0.827546	−	0.561398i	\(-0.189736\pi\)
\(19\)	0.192882	+	0.334081i	0.0442501	+	0.0766434i	0.887302	−	0.461189i	\(-0.152577\pi\)
							−0.843052	+	0.537832i	\(0.819243\pi\)
\(23\)	0.530577	+	0.445207i	0.110633	+	0.0928321i	0.696426	−	0.717629i	\(-0.254772\pi\)
							−0.585793	+	0.810461i	\(0.699217\pi\)
\(29\)	0.989773	+	0.360248i	0.183796	+	0.0668964i	0.432279	−	0.901740i	\(-0.357710\pi\)
							−0.248483	+	0.968636i	\(0.579932\pi\)
\(31\)	4.37245	+	3.66892i	0.785315	+	0.658958i	0.944581	−	0.328278i	\(-0.106468\pi\)
							−0.159266	+	0.987236i	\(0.550913\pi\)
\(37\)	3.46664	−	6.00440i	0.569912	−	0.987117i	−0.426662	−	0.904411i	\(-0.640311\pi\)
							0.996574	−	0.0827055i	\(-0.0263561\pi\)
\(41\)	−8.26917	+	3.00973i	−1.29143	+	0.470041i	−0.894195	−	0.447677i	\(-0.852251\pi\)
							−0.397232	+	0.917718i	\(0.630029\pi\)
\(43\)	−0.197980	−	1.12280i	−0.0301917	−	0.171226i	0.965984	−	0.258604i	\(-0.0832624\pi\)
							−0.996175	+	0.0873779i	\(0.972151\pi\)
\(47\)	−8.54960	+	7.17396i	−1.24709	+	1.04643i	−0.250151	+	0.968207i	\(0.580480\pi\)
							−0.996936	+	0.0782233i	\(0.975075\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.b.85.2	✓	54
3.2	odd	2		567.2.v.a.64.8		54
27.7	even	9	inner	189.2.v.b.169.2	yes	54
27.13	even	9		5103.2.a.g.1.23		27
27.14	odd	18		5103.2.a.h.1.5		27
27.20	odd	18		567.2.v.a.505.8		54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.b.85.2	✓	54	1.1	even	1	trivial
189.2.v.b.169.2	yes	54	27.7	even	9	inner
567.2.v.a.64.8		54	3.2	odd	2
567.2.v.a.505.8		54	27.20	odd	18
5103.2.a.g.1.23		27	27.13	even	9
5103.2.a.h.1.5		27	27.14	odd	18