Embedded newform 189.2.v.a.85.2

• Label: 189.2.v.a.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.a.169.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.17921 - 0.793168i) q^{2} +(-1.72903 - 0.102202i) q^{3} +(2.58775 + 2.17138i) q^{4} +(0.443161 - 2.51329i) q^{5} +(3.68686 + 1.59413i) q^{6} +(0.766044 - 0.642788i) q^{7} +(-1.59792 - 2.76768i) q^{8} +(2.97911 + 0.353421i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 3 q^{3} - 3 q^{5} - 27 q^{8} - 9 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.17921	−	0.793168i	−1.54093	−	0.560854i	−0.574665	−	0.818389i	\(-0.694868\pi\)
							−0.966269	+	0.257534i	\(0.917090\pi\)
\(3\)	−1.72903	−	0.102202i	−0.998258	−	0.0590064i
							\(5\)	0.443161	−	2.51329i	0.198188	−	1.12398i	−0.709618	−	0.704587i	\(-0.751132\pi\)
0.907806	−	0.419391i	\(-0.137756\pi\)
\(7\)	0.766044	−	0.642788i	0.289538	−	0.242951i
							\(11\)	−0.776717	−	4.40498i	−0.234189	−	1.32815i	−0.844315	−	0.535847i	\(-0.819992\pi\)
0.610126	−	0.792304i	\(-0.291119\pi\)
\(13\)	−3.79638	+	1.38177i	−1.05293	+	0.383233i	−0.809766	−	0.586753i	\(-0.800406\pi\)
							−0.243159	+	0.969986i	\(0.578184\pi\)
\(17\)	−3.83523	+	6.64281i	−0.930180	+	1.61112i	−0.147169	+	0.989111i	\(0.547016\pi\)
							−0.783011	+	0.622008i	\(0.786317\pi\)
\(19\)	−1.85220	−	3.20810i	−0.424923	−	0.735989i	0.571490	−	0.820609i	\(-0.306366\pi\)
							−0.996413	+	0.0846203i	\(0.973032\pi\)
\(23\)	−2.63630	−	2.21211i	−0.549706	−	0.461258i	0.325136	−	0.945667i	\(-0.394590\pi\)
							−0.874841	+	0.484410i	\(0.839034\pi\)
\(29\)	−4.86584	−	1.77102i	−0.903565	−	0.328871i	−0.151884	−	0.988398i	\(-0.548534\pi\)
							−0.751680	+	0.659528i	\(0.770756\pi\)
\(31\)	1.10711	+	0.928979i	0.198844	+	0.166850i	0.736773	−	0.676140i	\(-0.236349\pi\)
							−0.537929	+	0.842990i	\(0.680793\pi\)
\(37\)	0.0540049	−	0.0935393i	0.00887836	−	0.0153778i	−0.861552	−	0.507669i	\(-0.830507\pi\)
							0.870430	+	0.492291i	\(0.163841\pi\)
\(41\)	−0.384547	+	0.139964i	−0.0600561	+	0.0218586i	−0.371874	−	0.928283i	\(-0.621285\pi\)
							0.311817	+	0.950142i	\(0.399062\pi\)
\(43\)	−1.74698	−	9.90762i	−0.266412	−	1.51090i	−0.764983	−	0.644050i	\(-0.777253\pi\)
							0.498571	−	0.866849i	\(-0.333858\pi\)
\(47\)	5.45009	−	4.57317i	0.794978	−	0.667066i	−0.151994	−	0.988381i	\(-0.548570\pi\)
							0.946972	+	0.321316i	\(0.104125\pi\)

(See \(a_n\) instead)

Twists

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

    

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