Embedded newform 162.3.h.a.5.16

• Label: 162.3.h.a.5.16
• 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			5.16
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.326140 + 1.37609i) q^{2} +(2.18959 + 2.05078i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(-1.03753 + 0.772415i) q^{5} +(-2.10794 + 3.68193i) q^{6} +(-4.52918 + 2.97889i) q^{7} +(-1.81808 - 2.16670i) q^{8} +(0.588642 + 8.98073i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	2.18959	+	2.05078i	0.729865	+	0.683592i
\(5\)	−1.03753	+	0.772415i	−0.207507	+	0.154483i								−0.695895	−	0.718143i	\(-0.744992\pi\)
							0.488389	+	0.872626i	\(0.337585\pi\)
\(7\)	−4.52918	+	2.97889i	−0.647025	+	0.425555i	−0.830154	−	0.557534i	\(-0.811748\pi\)
							0.183129	+	0.983089i	\(0.441377\pi\)
\(11\)	−0.852172	+	7.29080i	−0.0774702	+	0.662800i	0.897586	+	0.440840i	\(0.145319\pi\)
							−0.975056	+	0.221960i	\(0.928755\pi\)
\(13\)	0.717336	+	2.39607i	0.0551797	+	0.184313i	0.981136	−	0.193317i	\(-0.0619244\pi\)
							−0.925957	+	0.377630i	\(0.876739\pi\)
\(17\)	12.8216	+	2.26080i	0.754212	+	0.132988i	0.537519	−	0.843252i	\(-0.319362\pi\)
							0.216693	+	0.976240i	\(0.430473\pi\)
\(19\)	−0.163680	−	0.928274i	−0.00861473	−	0.0488565i	0.980197	−	0.198023i	\(-0.0634521\pi\)
							−0.988812	+	0.149166i	\(0.952341\pi\)
\(23\)	23.9389	−	36.3973i	1.04082	−	1.58249i	0.252137	−	0.967692i	\(-0.418867\pi\)
							0.788684	−	0.614799i	\(-0.210763\pi\)
\(29\)	10.5679	−	9.97029i	0.364410	−	0.343803i	−0.482157	−	0.876085i	\(-0.660147\pi\)
							0.846567	+	0.532281i	\(0.178665\pi\)
\(31\)	−0.520156	+	8.93074i	−0.0167792	+	0.288088i	0.979472	+	0.201581i	\(0.0646078\pi\)
							−0.996251	+	0.0865078i	\(0.972429\pi\)
\(37\)	18.2424	−	6.63969i	0.493038	−	0.179451i	−0.0835222	−	0.996506i	\(-0.526617\pi\)
							0.576560	+	0.817055i	\(0.304395\pi\)
\(41\)	2.27520	−	9.59983i	0.0554927	−	0.234142i	−0.938230	−	0.346012i	\(-0.887536\pi\)
							0.993723	+	0.111870i	\(0.0356839\pi\)
\(43\)	2.47852	+	5.74586i	0.0576401	+	0.133625i	0.944602	−	0.328217i	\(-0.106448\pi\)
							−0.886962	+	0.461842i	\(0.847189\pi\)
\(47\)	64.0157	−	3.72849i	1.36204	−	0.0793296i	0.638504	−	0.769619i	\(-0.279554\pi\)
							0.723534	+	0.690289i	\(0.242517\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.16	✓	324
81.65	odd	54	inner	162.3.h.a.65.16	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.16	✓	324	1.1	even	1	trivial
162.3.h.a.65.16	yes	324	81.65	odd	54	inner