Embedded newform 189.2.w.a.184.7

• Label: 189.2.w.a.184.7
• Level: $189$
• Weight: $2$
• Character: 189.184
• 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.w (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			184.7
Character	\(\chi\)	\(=\)	189.184
Dual form			189.2.w.a.151.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.199381 - 1.13075i) q^{2} +(1.60659 + 0.647204i) q^{3} +(0.640547 - 0.233140i) q^{4} +(-0.0295570 + 0.167626i) q^{5} +(0.411501 - 1.94569i) q^{6} +(-1.82867 - 1.91206i) q^{7} +(-1.53953 - 2.66654i) q^{8} +(2.16225 + 2.07958i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(136\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.199381	−	1.13075i	−0.140984	−	0.799560i	−0.970504	−	0.241083i	\(-0.922497\pi\)
							0.829520	−	0.558476i	\(-0.188614\pi\)
\(3\)	1.60659	+	0.647204i	0.927564	+	0.373664i
							\(5\)	−0.0295570	+	0.167626i	−0.0132183	+	0.0749647i	−0.990703	−	0.136040i	\(-0.956562\pi\)
0.977485	+	0.211005i	\(0.0676735\pi\)
\(7\)	−1.82867	−	1.91206i	−0.691173	−	0.722689i
							\(11\)	−0.0250224	−	0.141909i	−0.00754455	−	0.0427873i	0.980803	−	0.195001i	\(-0.0624710\pi\)
−0.988348	+	0.152214i	\(0.951360\pi\)
\(13\)	4.82298	+	4.04696i	1.33765	+	1.12242i	0.982222	+	0.187723i	\(0.0601106\pi\)
							0.355432	+	0.934702i	\(0.384334\pi\)
\(17\)	−3.38088			−0.819983			−0.409992	−	0.912089i	\(-0.634468\pi\)
							−0.409992	+	0.912089i	\(0.634468\pi\)
\(19\)	−1.68694			−0.387010			−0.193505	−	0.981099i	\(-0.561986\pi\)
							−0.193505	+	0.981099i	\(0.561986\pi\)
\(23\)	−3.64166	−	3.05571i	−0.759338	−	0.637160i	0.178616	−	0.983919i	\(-0.442838\pi\)
							−0.937954	+	0.346758i	\(0.887282\pi\)
\(29\)	−7.64234	+	6.41269i	−1.41915	+	1.19081i	−0.467353	+	0.884071i	\(0.654792\pi\)
							−0.951795	+	0.306735i	\(0.900763\pi\)
\(31\)	−2.01924	+	0.734943i	−0.362666	+	0.132000i	−0.516925	−	0.856031i	\(-0.672923\pi\)
							0.154259	+	0.988030i	\(0.450701\pi\)
\(37\)	−2.27659	−	3.94317i	−0.374269	−	0.648254i	0.615948	−	0.787787i	\(-0.288773\pi\)
							−0.990217	+	0.139533i	\(0.955440\pi\)
\(41\)	4.05191	+	3.39995i	0.632801	+	0.530983i	0.901798	−	0.432158i	\(-0.142248\pi\)
							−0.268997	+	0.963141i	\(0.586692\pi\)
\(43\)	−4.02809	−	1.46610i	−0.614278	−	0.223579i	0.0160961	−	0.999870i	\(-0.494876\pi\)
							−0.630374	+	0.776292i	\(0.717098\pi\)
\(47\)	9.05842	+	3.29700i	1.32131	+	0.480916i	0.903877	−	0.427793i	\(-0.140709\pi\)
							0.417430	+	0.908709i	\(0.362931\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.2.w.a.184.7	yes	132
3.2	odd	2		567.2.w.a.37.16		132
7.4	even	3		189.2.u.a.130.16	yes	132
21.11	odd	6		567.2.u.a.361.7		132
27.11	odd	18		567.2.u.a.289.7		132
27.16	even	9		189.2.u.a.16.16	✓	132
189.11	odd	18		567.2.w.a.46.16		132
189.151	even	9	inner	189.2.w.a.151.7	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.u.a.16.16	✓	132	27.16	even	9
189.2.u.a.130.16	yes	132	7.4	even	3
189.2.w.a.151.7	yes	132	189.151	even	9	inner
189.2.w.a.184.7	yes	132	1.1	even	1	trivial
567.2.u.a.289.7		132	27.11	odd	18
567.2.u.a.361.7		132	21.11	odd	6
567.2.w.a.37.16		132	3.2	odd	2
567.2.w.a.46.16		132	189.11	odd	18