Embedded newform 189.2.bd.a.47.17

• Label: 189.2.bd.a.47.17
• Level: $189$
• Weight: $2$
• Character: 189.47
• Analytic conductor: $1.509$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	189.bd (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.50917259820\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			47.17
Character	\(\chi\)	\(=\)	189.47
Dual form			189.2.bd.a.185.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.65422 + 0.291684i) q^{2} +(-0.549790 + 1.64248i) q^{3} +(0.771989 + 0.280981i) q^{4} +(3.52188 + 1.28186i) q^{5} +(-1.38856 + 2.55666i) q^{6} +(-2.17331 - 1.50888i) q^{7} +(-1.71431 - 0.989760i) q^{8} +(-2.39546 - 1.80604i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 18 q^{6} - 6 q^{7} - 18 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.65422	+	0.291684i	1.16971	+	0.206252i	0.724566	−	0.689206i	\(-0.242040\pi\)
							0.445147	+	0.895458i	\(0.353152\pi\)
\(3\)	−0.549790	+	1.64248i	−0.317422	+	0.948285i
							\(5\)	3.52188	+	1.28186i	1.57503	+	0.573266i	0.974117	−	0.226045i	\(-0.0725796\pi\)
0.600918	+	0.799311i	\(0.294802\pi\)
\(7\)	−2.17331	−	1.50888i	−0.821435	−	0.570302i
							\(11\)	1.63515	+	4.49253i	0.493015	+	1.35455i	0.897908	+	0.440184i	\(0.145087\pi\)
−0.404893	+	0.914364i	\(0.632691\pi\)
\(13\)	1.10865	−	3.04598i	0.307483	−	0.844804i	−0.685662	−	0.727920i	\(-0.740487\pi\)
							0.993146	−	0.116884i	\(-0.0372905\pi\)
\(17\)	1.93415	−	3.35005i	0.469101	−	0.812507i	−0.530275	−	0.847826i	\(-0.677911\pi\)
							0.999376	+	0.0353188i	\(0.0112446\pi\)
\(19\)	−1.31749	+	0.760652i	−0.302252	+	0.174506i	−0.643454	−	0.765485i	\(-0.722499\pi\)
							0.341202	+	0.939990i	\(0.389166\pi\)
\(23\)	−0.358638	+	0.0632375i	−0.0747811	+	0.0131859i	−0.210914	−	0.977505i	\(-0.567644\pi\)
							0.136132	+	0.990691i	\(0.456533\pi\)
\(29\)	−2.29492	−	6.30525i	−0.426157	−	1.17086i	−0.948127	−	0.317892i	\(-0.897025\pi\)
							0.521970	−	0.852964i	\(-0.325197\pi\)
\(31\)	−0.240377	+	0.660432i	−0.0431731	+	0.118617i	−0.959406	−	0.282030i	\(-0.908992\pi\)
							0.916233	+	0.400647i	\(0.131215\pi\)
\(37\)	−5.50394			−0.904841			−0.452421	−	0.891805i	\(-0.649439\pi\)
							−0.452421	+	0.891805i	\(0.649439\pi\)
\(41\)	−0.793900	−	0.288956i	−0.123986	−	0.0451273i	0.279282	−	0.960209i	\(-0.409904\pi\)
							−0.403268	+	0.915082i	\(0.632126\pi\)
\(43\)	0.0136704	−	0.0775288i	0.00208472	−	0.0118230i	−0.983748	−	0.179557i	\(-0.942534\pi\)
							0.985832	+	0.167734i	\(0.0536448\pi\)
\(47\)	3.34124	−	1.21611i	0.487369	−	0.177388i	−0.0866351	−	0.996240i	\(-0.527611\pi\)
							0.574004	+	0.818852i	\(0.305389\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.bd.a.47.17	yes	132
3.2	odd	2		567.2.bd.a.467.6		132
7.3	odd	6		189.2.ba.a.101.6	✓	132
21.17	even	6		567.2.ba.a.143.17		132
27.4	even	9		567.2.ba.a.341.17		132
27.23	odd	18		189.2.ba.a.131.6	yes	132
189.31	odd	18		567.2.bd.a.17.6		132
189.185	even	18	inner	189.2.bd.a.185.17	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.101.6	✓	132	7.3	odd	6
189.2.ba.a.131.6	yes	132	27.23	odd	18
189.2.bd.a.47.17	yes	132	1.1	even	1	trivial
189.2.bd.a.185.17	yes	132	189.185	even	18	inner
567.2.ba.a.143.17		132	21.17	even	6
567.2.ba.a.341.17		132	27.4	even	9
567.2.bd.a.17.6		132	189.31	odd	18
567.2.bd.a.467.6		132	3.2	odd	2