Embedded newform 162.3.h.a.5.9

• Label: 162.3.h.a.5.9
• Level: $162$
• Weight: $3$
• Character: 162.5
• Analytic conductor: $4.414$
• Analytic rank: $0$
• Dimension: $324$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	162.h (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.41418028264\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(18\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			5.9
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326140 - 1.37609i) q^{2} +(2.89210 + 0.797352i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(-7.40578 + 5.51340i) q^{5} +(0.154002 - 4.23984i) q^{6} +(-1.07376 + 0.706223i) q^{7} +(1.81808 + 2.16670i) q^{8} +(7.72846 + 4.61204i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{23}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.326140	−	1.37609i	−0.163070	−	0.688047i
							\(3\)	2.89210	+	0.797352i	0.964033	+	0.265784i
\(5\)	−7.40578	+	5.51340i	−1.48116	+	1.10268i								−0.511181	+	0.859473i	\(0.670792\pi\)
							−0.969974	+	0.243207i	\(0.921801\pi\)
\(7\)	−1.07376	+	0.706223i	−0.153394	+	0.100889i	−0.623886	−	0.781515i	\(-0.714447\pi\)
							0.470492	+	0.882404i	\(0.344077\pi\)
\(11\)	−1.07678	+	9.21247i	−0.0978894	+	0.837497i	0.851706	+	0.524020i	\(0.175568\pi\)
							−0.949596	+	0.313478i	\(0.898506\pi\)
\(13\)	4.35535	+	14.5479i	0.335027	+	1.11907i	0.945116	+	0.326734i	\(0.105948\pi\)
							−0.610090	+	0.792332i	\(0.708867\pi\)
\(17\)	0.471048	+	0.0830584i	0.0277087	+	0.00488579i	0.187485	−	0.982267i	\(-0.439966\pi\)
							−0.159777	+	0.987153i	\(0.551077\pi\)
\(19\)	−3.61111	−	20.4796i	−0.190059	−	1.07788i	−0.919282	−	0.393600i	\(-0.871230\pi\)
							0.729223	−	0.684276i	\(-0.239882\pi\)
\(23\)	−15.9635	+	24.2713i	−0.694065	+	1.05527i	0.300763	+	0.953699i	\(0.402759\pi\)
							−0.994828	+	0.101575i	\(0.967612\pi\)
\(29\)	−7.55899	+	7.13154i	−0.260655	+	0.245915i	−0.805464	−	0.592645i	\(-0.798084\pi\)
							0.544809	+	0.838560i	\(0.316602\pi\)
\(31\)	−0.914886	+	15.7080i	−0.0295124	+	0.506709i	0.950868	+	0.309596i	\(0.100194\pi\)
							−0.980381	+	0.197113i	\(0.936843\pi\)
\(37\)	61.2478	−	22.2924i	1.65535	−	0.602497i	0.665725	−	0.746197i	\(-0.268122\pi\)
							0.989621	+	0.143700i	\(0.0459001\pi\)
\(41\)	8.54541	−	36.0559i	0.208425	−	0.879413i	−0.763817	−	0.645433i	\(-0.776677\pi\)
							0.972241	−	0.233980i	\(-0.0751750\pi\)
\(43\)	−12.7320	−	29.5161i	−0.296093	−	0.686420i	0.703615	−	0.710581i	\(-0.251568\pi\)
							−0.999708	+	0.0241609i	\(0.992309\pi\)
\(47\)	10.0825	−	0.587238i	0.214521	−	0.0124944i	0.0494523	−	0.998776i	\(-0.484252\pi\)
							0.165069	+	0.986282i	\(0.447215\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.9	✓	324
81.65	odd	54	inner	162.3.h.a.65.9	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.9	✓	324	1.1	even	1	trivial
162.3.h.a.65.9	yes	324	81.65	odd	54	inner