Embedded newform 162.4.g.a.13.1

• Label: 162.4.g.a.13.1
• Level: $162$
• Weight: $4$
• Character: 162.13
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $234$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.55830942093\)
Analytic rank:	\(0\)
Dimension:	\(234\)
Relative dimension:	\(13\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	162.13
Dual form			162.4.g.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.78727 + 0.897598i) q^{2} +(-5.19043 + 0.243791i) q^{3} +(2.38863 + 3.20849i) q^{4} +(0.946763 + 3.16241i) q^{5} +(-9.49550 - 4.22320i) q^{6} +(-13.1374 - 30.4558i) q^{7} +(1.38919 + 7.87846i) q^{8} +(26.8811 - 2.53076i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 234 q + 36 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{4}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.78727	+	0.897598i	0.631894	+	0.317349i
							\(3\)	−5.19043	+	0.243791i	−0.998899	+	0.0469175i
\(5\)	0.946763	+	3.16241i	0.0846810	+	0.282854i								0.989833	−	0.142234i	\(-0.0454284\pi\)
							−0.905152	+	0.425088i	\(0.860243\pi\)
\(7\)	−13.1374	−	30.4558i	−0.709351	−	1.64446i	−0.763403	−	0.645923i	\(-0.776473\pi\)
							0.0540517	−	0.998538i	\(-0.482786\pi\)
\(11\)	40.8954	−	9.69239i	1.12095	−	0.265670i	0.371958	−	0.928250i	\(-0.378687\pi\)
							0.748991	+	0.662580i	\(0.230539\pi\)
\(13\)	36.4911	+	24.0006i	0.778523	+	0.512043i	0.875485	−	0.483245i	\(-0.160542\pi\)
							−0.0969618	+	0.995288i	\(0.530912\pi\)
\(17\)	104.807	−	38.1466i	1.49526	−	0.544230i	0.540432	−	0.841388i	\(-0.318261\pi\)
							0.954828	+	0.297157i	\(0.0960386\pi\)
\(19\)	−79.4813	−	28.9288i	−0.959697	−	0.349301i	−0.185783	−	0.982591i	\(-0.559482\pi\)
							−0.773915	+	0.633290i	\(0.781704\pi\)
\(23\)	−39.5682	+	91.7293i	−0.358719	+	0.831604i	0.639316	+	0.768944i	\(0.279218\pi\)
							−0.998035	+	0.0626598i	\(0.980042\pi\)
\(29\)	−17.1173	−	293.893i	−0.109607	−	1.88188i	−0.388343	−	0.921515i	\(-0.626952\pi\)
							0.278736	−	0.960368i	\(-0.410085\pi\)
\(31\)	275.724	−	32.2275i	1.59746	−	0.186717i	0.729727	−	0.683739i	\(-0.239647\pi\)
							0.867738	+	0.497022i	\(0.165573\pi\)
\(37\)	35.9076	+	30.1301i	0.159545	+	0.133874i	0.719065	−	0.694942i	\(-0.244570\pi\)
							−0.559520	+	0.828817i	\(0.689015\pi\)
\(41\)	118.965	−	59.7465i	0.453152	−	0.227581i	−0.207558	−	0.978223i	\(-0.566552\pi\)
							0.660710	+	0.750641i	\(0.270255\pi\)
\(43\)	−242.937	−	257.498i	−0.861571	−	0.913212i	0.135415	−	0.990789i	\(-0.456763\pi\)
							−0.996986	+	0.0775769i	\(0.975282\pi\)
\(47\)	−65.3595	−	7.63943i	−0.202844	−	0.0237091i	0.0140641	−	0.999901i	\(-0.495523\pi\)
							−0.216908	+	0.976192i	\(0.569597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.a.13.1	✓	234
81.25	even	27	inner	162.4.g.a.25.1	yes	234

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.a.13.1	✓	234	1.1	even	1	trivial
162.4.g.a.25.1	yes	234	81.25	even	27	inner