Embedded newform 189.2.bd.a.185.8

• Label: 189.2.bd.a.185.8
• 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			185.8
Character	\(\chi\)	\(=\)	189.185
Dual form			189.2.bd.a.47.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.910598 + 0.160563i) q^{2} +(-1.05237 + 1.37569i) q^{3} +(-1.07598 + 0.391623i) q^{4} +(-0.473927 + 0.172495i) q^{5} +(0.737399 - 1.42167i) q^{6} +(-2.46079 - 0.971863i) q^{7} +(2.51844 - 1.45402i) q^{8} +(-0.785044 - 2.89546i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} + 18 q^{6} - 6 q^{7} - 18 q^{8} - 15 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.910598	+	0.160563i	−0.643890	+	0.113535i	−0.486052	−	0.873930i	\(-0.661564\pi\)
							−0.157838	+	0.987465i	\(0.550452\pi\)
\(3\)	−1.05237	+	1.37569i	−0.607585	+	0.794255i
							\(5\)	−0.473927	+	0.172495i	−0.211947	+	0.0771423i	−0.445812	−	0.895127i	\(-0.647085\pi\)
0.233865	+	0.972269i	\(0.424863\pi\)
\(7\)	−2.46079	−	0.971863i	−0.930091	−	0.367330i
							\(11\)	1.66156	−	4.56510i	0.500979	−	1.37643i	−0.389341	−	0.921094i	\(-0.627297\pi\)
0.890320	−	0.455335i	\(-0.150481\pi\)
\(13\)	1.60911	+	4.42100i	0.446287	+	1.22616i	0.935290	+	0.353882i	\(0.115138\pi\)
							−0.489003	+	0.872282i	\(0.662639\pi\)
\(17\)	−2.38696	−	4.13434i	−0.578923	−	1.00272i	−0.995603	−	0.0936714i	\(-0.970140\pi\)
							0.416680	−	0.909053i	\(-0.363194\pi\)
\(19\)	−5.51758	−	3.18557i	−1.26582	−	0.730821i	−0.291625	−	0.956533i	\(-0.594196\pi\)
							−0.974194	+	0.225712i	\(0.927529\pi\)
\(23\)	−3.53298	−	0.622960i	−0.736677	−	0.129896i	−0.207294	−	0.978279i	\(-0.566466\pi\)
							−0.529383	+	0.848383i	\(0.677577\pi\)
\(29\)	0.997075	−	2.73944i	0.185152	−	0.508701i	−0.812039	−	0.583604i	\(-0.801642\pi\)
							0.997191	+	0.0749022i	\(0.0238645\pi\)
\(31\)	−0.268468	−	0.737610i	−0.0482183	−	0.132479i	0.913246	−	0.407409i	\(-0.133568\pi\)
							−0.961464	+	0.274930i	\(0.911345\pi\)
\(37\)	−5.47451			−0.900004			−0.450002	−	0.893028i	\(-0.648577\pi\)
							−0.450002	+	0.893028i	\(0.648577\pi\)
\(41\)	−5.44849	+	1.98309i	−0.850912	+	0.309707i	−0.730412	−	0.683007i	\(-0.760672\pi\)
							−0.120500	+	0.992713i	\(0.538450\pi\)
\(43\)	−1.36729	−	7.75429i	−0.208510	−	1.18252i	−0.891820	−	0.452390i	\(-0.850571\pi\)
							0.683310	−	0.730128i	\(-0.260540\pi\)
\(47\)	7.22256	+	2.62880i	1.05352	+	0.383449i	0.809989	−	0.586445i	\(-0.199473\pi\)
							0.243529	+	0.969894i	\(0.421695\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.bd.a.185.8	yes	132
3.2	odd	2		567.2.bd.a.17.15		132
7.5	odd	6		189.2.ba.a.131.15	yes	132
21.5	even	6		567.2.ba.a.341.8		132
27.7	even	9		567.2.ba.a.143.8		132
27.20	odd	18		189.2.ba.a.101.15	✓	132
189.47	even	18	inner	189.2.bd.a.47.8	yes	132
189.61	odd	18		567.2.bd.a.467.15		132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.ba.a.101.15	✓	132	27.20	odd	18
189.2.ba.a.131.15	yes	132	7.5	odd	6
189.2.bd.a.47.8	yes	132	189.47	even	18	inner
189.2.bd.a.185.8	yes	132	1.1	even	1	trivial
567.2.ba.a.143.8		132	27.7	even	9
567.2.ba.a.341.8		132	21.5	even	6
567.2.bd.a.17.15		132	3.2	odd	2
567.2.bd.a.467.15		132	189.61	odd	18