Embedded newform 189.2.v.a.22.7

• Label: 189.2.v.a.22.7
• Level: $189$
• Weight: $2$
• Character: 189.22
• 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			22.7
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.a.43.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.969537 - 0.813538i) q^{2} +(-0.430185 + 1.67778i) q^{3} +(-0.0691389 + 0.392106i) q^{4} +(-0.855069 + 0.311220i) q^{5} +(0.947857 + 1.97664i) q^{6} +(0.173648 + 0.984808i) q^{7} +(1.51760 + 2.62856i) q^{8} +(-2.62988 - 1.44351i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 3 q^{3} - 3 q^{5} - 27 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.969537	−	0.813538i	0.685566	−	0.575258i	−0.232061	−	0.972701i	\(-0.574547\pi\)
							0.917627	+	0.397443i	\(0.130102\pi\)
\(3\)	−0.430185	+	1.67778i	−0.248367	+	0.968666i
							\(5\)	−0.855069	+	0.311220i	−0.382398	+	0.139182i	−0.526065	−	0.850445i	\(-0.676333\pi\)
0.143666	+	0.989626i	\(0.454111\pi\)
\(7\)	0.173648	+	0.984808i	0.0656328	+	0.372222i
							\(11\)	5.60170	+	2.03885i	1.68898	+	0.614737i	0.994496	−	0.104776i	\(-0.0334126\pi\)
0.694479	+	0.719513i	\(0.255635\pi\)
\(13\)	−2.54437	−	2.13498i	−0.705683	−	0.592138i	0.217701	−	0.976015i	\(-0.430144\pi\)
							−0.923384	+	0.383877i	\(0.874589\pi\)
\(17\)	0.518461	−	0.898001i	0.125745	−	0.217797i	−0.796279	−	0.604930i	\(-0.793201\pi\)
							0.922024	+	0.387133i	\(0.126534\pi\)
\(19\)	−1.02967	−	1.78343i	−0.236222	−	0.409148i	0.723405	−	0.690424i	\(-0.242576\pi\)
							−0.959627	+	0.281276i	\(0.909242\pi\)
\(23\)	0.554384	−	3.14407i	0.115597	−	0.655584i	−0.870856	−	0.491539i	\(-0.836435\pi\)
							0.986453	−	0.164045i	\(-0.0524543\pi\)
\(29\)	5.59972	−	4.69872i	1.03984	−	0.872531i	0.0478528	−	0.998854i	\(-0.484762\pi\)
							0.991989	+	0.126323i	\(0.0403177\pi\)
\(31\)	1.21241	−	6.87590i	0.217755	−	1.23495i	−0.658308	−	0.752749i	\(-0.728727\pi\)
							0.876062	−	0.482198i	\(-0.160162\pi\)
\(37\)	1.93726	−	3.35543i	0.318483	−	0.551630i	−0.661688	−	0.749779i	\(-0.730160\pi\)
							0.980172	+	0.198149i	\(0.0634931\pi\)
\(41\)	4.36990	+	3.66678i	0.682463	+	0.572654i	0.916725	−	0.399519i	\(-0.130823\pi\)
							−0.234262	+	0.972174i	\(0.575267\pi\)
\(43\)	−4.17077	−	1.51804i	−0.636036	−	0.231498i	0.00382025	−	0.999993i	\(-0.498784\pi\)
							−0.639856	+	0.768494i	\(0.721006\pi\)
\(47\)	1.09019	+	6.18276i	0.159020	+	0.901849i	0.955018	+	0.296549i	\(0.0958357\pi\)
							−0.795997	+	0.605300i	\(0.793053\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.a.22.7	✓	54
3.2	odd	2		567.2.v.b.442.3		54
27.4	even	9		5103.2.a.i.1.18		27
27.11	odd	18		567.2.v.b.127.3		54
27.16	even	9	inner	189.2.v.a.43.7	yes	54
27.23	odd	18		5103.2.a.f.1.10		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.a.22.7	✓	54	1.1	even	1	trivial
189.2.v.a.43.7	yes	54	27.16	even	9	inner
567.2.v.b.127.3		54	27.11	odd	18
567.2.v.b.442.3		54	3.2	odd	2
5103.2.a.f.1.10		27	27.23	odd	18
5103.2.a.i.1.18		27	27.4	even	9