Embedded newform 189.2.bd.a.185.22

• Label: 189.2.bd.a.185.22
• Level: $189$
• Weight: $2$
• Character: 189.185
• 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			185.22
Character	\(\chi\)	\(=\)	189.185
Dual form			189.2.bd.a.47.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.50283 - 0.441316i) q^{2} +(-0.707460 + 1.58098i) q^{3} +(4.19001 - 1.52504i) q^{4} +(-0.437341 + 0.159179i) q^{5} +(-1.07294 + 4.26914i) q^{6} +(-1.48170 + 2.19193i) q^{7} +(5.41196 - 3.12460i) q^{8} +(-1.99900 - 2.23696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 18 q^{6} - 6 q^{7} - 18 q^{8} - 15 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.50283	−	0.441316i	1.76977	−	0.312058i	0.808669	−	0.588263i	\(-0.200188\pi\)
							0.961099	+	0.276205i	\(0.0890770\pi\)
\(3\)	−0.707460	+	1.58098i	−0.408452	+	0.912780i
							\(5\)	−0.437341	+	0.159179i	−0.195585	+	0.0711870i	−0.437955	−	0.898997i	\(-0.644297\pi\)
0.242371	+	0.970184i	\(0.422075\pi\)
\(7\)	−1.48170	+	2.19193i	−0.560031	+	0.828471i
							\(11\)	0.845970	−	2.32428i	0.255069	−	0.700798i	−0.744384	−	0.667751i	\(-0.767257\pi\)
0.999454	−	0.0330462i	\(-0.0105208\pi\)
\(13\)	−2.13324	−	5.86102i	−0.591653	−	1.62555i	−0.767436	−	0.641125i	\(-0.778468\pi\)
							0.175783	−	0.984429i	\(-0.443754\pi\)
\(17\)	0.336582	+	0.582978i	0.0816332	+	0.141393i	0.903952	−	0.427635i	\(-0.140653\pi\)
							−0.822318	+	0.569028i	\(0.807320\pi\)
\(19\)	0.628166	+	0.362672i	0.144111	+	0.0832027i	0.570322	−	0.821421i	\(-0.306818\pi\)
							−0.426211	+	0.904624i	\(0.640152\pi\)
\(23\)	3.18900	+	0.562306i	0.664952	+	0.117249i	0.495929	−	0.868363i	\(-0.334828\pi\)
							0.169023	+	0.985612i	\(0.445939\pi\)
\(29\)	−1.84864	+	5.07909i	−0.343284	+	0.943164i	0.641151	+	0.767415i	\(0.278457\pi\)
							−0.984435	+	0.175750i	\(0.943765\pi\)
\(31\)	2.36874	+	6.50807i	0.425439	+	1.16888i	0.948552	+	0.316620i	\(0.102548\pi\)
							−0.523114	+	0.852263i	\(0.675230\pi\)
\(37\)	−7.40866			−1.21798			−0.608988	−	0.793179i	\(-0.708424\pi\)
							−0.608988	+	0.793179i	\(0.708424\pi\)
\(41\)	3.54917	−	1.29179i	0.554287	−	0.201744i	−0.0496629	−	0.998766i	\(-0.515815\pi\)
							0.603950	+	0.797022i	\(0.293592\pi\)
\(43\)	−1.41606	−	8.03089i	−0.215948	−	1.22470i	−0.879254	−	0.476352i	\(-0.841959\pi\)
							0.663307	−	0.748348i	\(-0.269152\pi\)
\(47\)	−8.57711	−	3.12181i	−1.25110	−	0.455363i	−0.370327	−	0.928901i	\(-0.620754\pi\)
							−0.880773	+	0.473538i	\(0.842977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.bd.a.185.22	yes	132
3.2	odd	2		567.2.bd.a.17.1		132
7.5	odd	6		189.2.ba.a.131.1	yes	132
21.5	even	6		567.2.ba.a.341.22		132
27.7	even	9		567.2.ba.a.143.22		132
27.20	odd	18		189.2.ba.a.101.1	✓	132
189.47	even	18	inner	189.2.bd.a.47.22	yes	132
189.61	odd	18		567.2.bd.a.467.1		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.101.1	✓	132	27.20	odd	18
189.2.ba.a.131.1	yes	132	7.5	odd	6
189.2.bd.a.47.22	yes	132	189.47	even	18	inner
189.2.bd.a.185.22	yes	132	1.1	even	1	trivial
567.2.ba.a.143.22		132	27.7	even	9
567.2.ba.a.341.22		132	21.5	even	6
567.2.bd.a.17.1		132	3.2	odd	2
567.2.bd.a.467.1		132	189.61	odd	18