Embedded newform 189.2.v.a.22.3

• Label: 189.2.v.a.22.3
• 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			22.3
Character	\(\chi\)	\(=\)	189.22
Dual form			189.2.v.a.43.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09798 + 0.921318i) q^{2} +(-1.39977 - 1.02012i) q^{3} +(0.00944557 - 0.0535685i) q^{4} +(1.35272 - 0.492351i) q^{5} +(2.47678 - 0.169559i) q^{6} +(0.173648 + 0.984808i) q^{7} +(-1.39433 - 2.41506i) q^{8} +(0.918713 + 2.85587i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 3 q^{3} - 3 q^{5} - 27 q^{8} - 9 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09798	+	0.921318i	−0.776392	+	0.651470i	−0.942337	−	0.334665i	\(-0.891377\pi\)
							0.165945	+	0.986135i	\(0.446932\pi\)
\(3\)	−1.39977	−	1.02012i	−0.808158	−	0.588966i
							\(5\)	1.35272	−	0.492351i	0.604956	−	0.220186i	−0.0213385	−	0.999772i	\(-0.506793\pi\)
0.626295	+	0.779586i	\(0.284571\pi\)
\(7\)	0.173648	+	0.984808i	0.0656328	+	0.372222i
							\(11\)	1.39413	+	0.507422i	0.420346	+	0.152993i	0.543529	−	0.839391i	\(-0.317088\pi\)
−0.123183	+	0.992384i	\(0.539310\pi\)
\(13\)	3.79855	+	3.18736i	1.05353	+	0.884015i	0.993460	−	0.114180i	\(-0.0364240\pi\)
							0.0600674	+	0.998194i	\(0.480868\pi\)
\(17\)	−3.76596	+	6.52284i	−0.913380	+	1.58202i	−0.104125	+	0.994564i	\(0.533204\pi\)
							−0.809256	+	0.587457i	\(0.800129\pi\)
\(19\)	4.13067	+	7.15453i	0.947641	+	1.64136i	0.750376	+	0.661012i	\(0.229873\pi\)
							0.197265	+	0.980350i	\(0.436794\pi\)
\(23\)	1.31455	−	7.45520i	0.274103	−	1.55452i	−0.467693	−	0.883891i	\(-0.654915\pi\)
							0.741796	−	0.670625i	\(-0.233974\pi\)
\(29\)	0.675850	−	0.567105i	0.125502	−	0.105309i	−0.577876	−	0.816124i	\(-0.696118\pi\)
							0.703378	+	0.710816i	\(0.251674\pi\)
\(31\)	1.22342	−	6.93837i	0.219733	−	1.24617i	−0.652769	−	0.757557i	\(-0.726393\pi\)
							0.872502	−	0.488611i	\(-0.162496\pi\)
\(37\)	0.822178	−	1.42405i	0.135165	−	0.234113i	−0.790495	−	0.612468i	\(-0.790177\pi\)
							0.925661	+	0.378355i	\(0.123510\pi\)
\(41\)	7.19412	+	6.03659i	1.12353	+	0.942756i	0.998778	−	0.0494312i	\(-0.0157409\pi\)
							0.124756	+	0.992187i	\(0.460185\pi\)
\(43\)	−1.81964	−	0.662296i	−0.277493	−	0.100999i	0.199525	−	0.979893i	\(-0.436060\pi\)
							−0.477018	+	0.878894i	\(0.658282\pi\)
\(47\)	−0.0880328	−	0.499259i	−0.0128409	−	0.0728244i	0.977714	−	0.209942i	\(-0.0673274\pi\)
							−0.990555	+	0.137117i	\(0.956216\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.v.a.22.3	✓	54
3.2	odd	2		567.2.v.b.442.7		54
27.4	even	9		5103.2.a.i.1.7		27
27.11	odd	18		567.2.v.b.127.7		54
27.16	even	9	inner	189.2.v.a.43.3	yes	54
27.23	odd	18		5103.2.a.f.1.21		27

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.v.a.22.3	✓	54	1.1	even	1	trivial
189.2.v.a.43.3	yes	54	27.16	even	9	inner
567.2.v.b.127.7		54	27.11	odd	18
567.2.v.b.442.7		54	3.2	odd	2
5103.2.a.f.1.21		27	27.23	odd	18
5103.2.a.i.1.7		27	27.4	even	9