Embedded newform 189.2.u.a.130.5

• Label: 189.2.u.a.130.5
• Level: $189$
• Weight: $2$
• Character: 189.130
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $132$
• 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			130.5
Character	\(\chi\)	\(=\)	189.130
Dual form			189.2.u.a.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.82816 - 0.665398i) q^{2} +(-0.461596 + 1.66941i) q^{3} +(1.36734 + 1.14734i) q^{4} +(2.57005 + 2.15653i) q^{5} +(1.95469 - 2.74481i) q^{6} +(-1.50198 - 2.17808i) q^{7} +(0.209199 + 0.362343i) q^{8} +(-2.57386 - 1.54119i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.82816	−	0.665398i	−1.29271	−	0.470507i	−0.398093	−	0.917345i	\(-0.630328\pi\)
							−0.894615	+	0.446838i	\(0.852550\pi\)
\(3\)	−0.461596	+	1.66941i	−0.266502	+	0.963834i
							\(5\)	2.57005	+	2.15653i	1.14936	+	0.964427i	0.999704	−	0.0243109i	\(-0.00773915\pi\)
0.149655	+	0.988738i	\(0.452184\pi\)
\(7\)	−1.50198	−	2.17808i	−0.567696	−	0.823239i
							\(11\)	−3.43544	+	2.88268i	−1.03582	+	0.869160i	−0.991533	−	0.129858i	\(-0.958548\pi\)
−0.0442922	+	0.999019i	\(0.514103\pi\)
\(13\)	2.58000	+	2.16488i	0.715564	+	0.600430i	0.926154	−	0.377145i	\(-0.123094\pi\)
							−0.210590	+	0.977574i	\(0.567538\pi\)
\(17\)	1.98260	+	3.43397i	0.480852	+	0.832860i	0.999759	−	0.0219708i	\(-0.00699407\pi\)
							−0.518907	+	0.854831i	\(0.673661\pi\)
\(19\)	−2.48870	+	4.31055i	−0.570946	+	0.988908i	0.425523	+	0.904948i	\(0.360090\pi\)
							−0.996469	+	0.0839604i	\(0.973243\pi\)
\(23\)	−2.08727	+	0.759706i	−0.435227	+	0.158410i	−0.550336	−	0.834943i	\(-0.685500\pi\)
							0.115109	+	0.993353i	\(0.463278\pi\)
\(29\)	−3.26643	+	2.74086i	−0.606561	+	0.508965i	−0.893547	−	0.448970i	\(-0.851791\pi\)
							0.286986	+	0.957935i	\(0.407347\pi\)
\(31\)	−1.78723	−	1.49966i	−0.320995	−	0.269347i	0.468024	−	0.883716i	\(-0.344966\pi\)
							−0.789019	+	0.614369i	\(0.789411\pi\)
\(37\)	−2.99931			−0.493083			−0.246541	−	0.969132i	\(-0.579294\pi\)
							−0.246541	+	0.969132i	\(0.579294\pi\)
\(41\)	0.219856	+	0.184481i	0.0343358	+	0.0288112i	0.659794	−	0.751446i	\(-0.270644\pi\)
							−0.625458	+	0.780258i	\(0.715088\pi\)
\(43\)	9.46658	+	3.44555i	1.44364	+	0.525442i	0.940807	−	0.338942i	\(-0.110069\pi\)
							0.502833	+	0.864384i	\(0.332291\pi\)
\(47\)	4.14512	−	3.47817i	0.604628	−	0.507343i	−0.288302	−	0.957540i	\(-0.593091\pi\)
							0.892929	+	0.450197i	\(0.148646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.u.a.130.5	yes	132
3.2	odd	2		567.2.u.a.361.18		132
7.2	even	3		189.2.w.a.184.18	yes	132
21.2	odd	6		567.2.w.a.37.5		132
27.11	odd	18		567.2.w.a.46.5		132
27.16	even	9		189.2.w.a.151.18	yes	132
189.16	even	9	inner	189.2.u.a.16.5	✓	132
189.65	odd	18		567.2.u.a.289.18		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.16.5	✓	132	189.16	even	9	inner
189.2.u.a.130.5	yes	132	1.1	even	1	trivial
189.2.w.a.151.18	yes	132	27.16	even	9
189.2.w.a.184.18	yes	132	7.2	even	3
567.2.u.a.289.18		132	189.65	odd	18
567.2.u.a.361.18		132	3.2	odd	2
567.2.w.a.37.5		132	21.2	odd	6
567.2.w.a.46.5		132	27.11	odd	18