Embedded newform 189.2.ba.a.5.17

• Label: 189.2.ba.a.5.17
• Level: $189$
• Weight: $2$
• Character: 189.5
• 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.ba (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			5.17
Character	\(\chi\)	\(=\)	189.5
Dual form			189.2.ba.a.38.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.72891 - 0.304854i) q^{2} +(1.65859 - 0.499066i) q^{3} +(1.01682 - 0.370091i) q^{4} +(-0.0612092 + 0.347134i) q^{5} +(2.71542 - 1.36847i) q^{6} +(-2.64149 + 0.150064i) q^{7} +(-1.39560 + 0.805749i) q^{8} +(2.50187 - 1.65549i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.72891	−	0.304854i	1.22253	−	0.215564i	0.475113	−	0.879925i	\(-0.342407\pi\)
							0.747412	+	0.664360i	\(0.231296\pi\)
\(3\)	1.65859	−	0.499066i	0.957590	−	0.288136i
							\(5\)	−0.0612092	+	0.347134i	−0.0273736	+	0.155243i	−0.995431	−	0.0954863i	\(-0.969559\pi\)
0.968057	+	0.250730i	\(0.0806705\pi\)
\(7\)	−2.64149	+	0.150064i	−0.998390	+	0.0567187i
							\(11\)	0.991951	−	0.174908i	0.299084	−	0.0527366i	−0.0220923	−	0.999756i	\(-0.507033\pi\)
0.321177	+	0.947019i	\(0.395922\pi\)
\(13\)	−0.216492	+	0.258005i	−0.0600440	+	0.0715576i	−0.795230	−	0.606307i	\(-0.792650\pi\)
							0.735186	+	0.677865i	\(0.237095\pi\)
\(17\)	−5.22261			−1.26667			−0.633334	−	0.773879i	\(-0.718314\pi\)
							−0.633334	+	0.773879i	\(0.718314\pi\)
\(19\)		−	0.554345i		−	0.127176i	−0.997976	−	0.0635878i	\(-0.979746\pi\)
							0.997976	−	0.0635878i	\(-0.0202543\pi\)
\(23\)	1.52594	−	1.81855i	0.318181	−	0.379194i	−0.583120	−	0.812386i	\(-0.698168\pi\)
							0.901302	+	0.433192i	\(0.142613\pi\)
\(29\)	0.750003	+	0.893819i	0.139272	+	0.165978i	0.831172	−	0.556015i	\(-0.187670\pi\)
							−0.691900	+	0.721994i	\(0.743226\pi\)
\(31\)	−0.597247	−	1.64092i	−0.107269	−	0.294718i	0.874432	−	0.485148i	\(-0.161234\pi\)
							−0.981701	+	0.190429i	\(0.939012\pi\)
\(37\)	3.07080	+	5.31879i	0.504837	+	0.874403i	0.999984	+	0.00559410i	\(0.00178067\pi\)
							−0.495148	+	0.868809i	\(0.664886\pi\)
\(41\)	−8.82053	−	7.40131i	−1.37754	−	1.15589i	−0.970114	−	0.242650i	\(-0.921983\pi\)
							−0.407422	−	0.913240i	\(-0.633572\pi\)
\(43\)	5.78827	+	2.10676i	0.882703	+	0.321278i	0.743300	−	0.668958i	\(-0.233259\pi\)
							0.139403	+	0.990236i	\(0.455482\pi\)
\(47\)	5.19006	+	1.88903i	0.757047	+	0.275543i	0.691568	−	0.722311i	\(-0.256920\pi\)
							0.0654793	+	0.997854i	\(0.479142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.ba.a.5.17	✓	132
3.2	odd	2		567.2.ba.a.530.6		132
7.3	odd	6		189.2.bd.a.59.6	yes	132
21.17	even	6		567.2.bd.a.206.17		132
27.11	odd	18		189.2.bd.a.173.6	yes	132
27.16	even	9		567.2.bd.a.278.17		132
189.38	even	18	inner	189.2.ba.a.38.17	yes	132
189.178	odd	18		567.2.ba.a.521.6		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.5.17	✓	132	1.1	even	1	trivial
189.2.ba.a.38.17	yes	132	189.38	even	18	inner
189.2.bd.a.59.6	yes	132	7.3	odd	6
189.2.bd.a.173.6	yes	132	27.11	odd	18
567.2.ba.a.521.6		132	189.178	odd	18
567.2.ba.a.530.6		132	3.2	odd	2
567.2.bd.a.206.17		132	21.17	even	6
567.2.bd.a.278.17		132	27.16	even	9