Embedded newform 162.3.h.a.5.8

• Label: 162.3.h.a.5.8
• 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			5.8
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326140 - 1.37609i) q^{2} +(2.56706 - 1.55248i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(-0.00733898 + 0.00546367i) q^{5} +(-2.97357 - 3.02620i) q^{6} +(2.72886 - 1.79480i) q^{7} +(1.81808 + 2.16670i) q^{8} +(4.17964 - 7.97061i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	2.56706	−	1.55248i	0.855688	−	0.517492i
\(5\)	−0.00733898	+	0.00546367i	−0.00146780	+	0.00109273i								−0.597892	−	0.801576i	\(-0.703995\pi\)
							0.596424	+	0.802669i	\(0.296588\pi\)
\(7\)	2.72886	−	1.79480i	0.389837	−	0.256400i	−0.339427	−	0.940632i	\(-0.610233\pi\)
							0.729264	+	0.684233i	\(0.239863\pi\)
\(11\)	1.72906	−	14.7930i	0.157187	−	1.34482i	−0.652423	−	0.757855i	\(-0.726248\pi\)
							0.809610	−	0.586968i	\(-0.199678\pi\)
\(13\)	−1.56207	−	5.21766i	−0.120159	−	0.401359i	0.876655	−	0.481119i	\(-0.159770\pi\)
							−0.996814	+	0.0797606i	\(0.974584\pi\)
\(17\)	−6.21509	−	1.09589i	−0.365593	−	0.0644640i	−0.0121661	−	0.999926i	\(-0.503873\pi\)
							−0.353427	+	0.935462i	\(0.614984\pi\)
\(19\)	4.07160	+	23.0912i	0.214295	+	1.21533i	0.882126	+	0.471014i	\(0.156112\pi\)
							−0.667831	+	0.744313i	\(0.732777\pi\)
\(23\)	6.78406	−	10.3147i	0.294959	−	0.448464i	−0.657356	−	0.753580i	\(-0.728325\pi\)
							0.952315	+	0.305117i	\(0.0986955\pi\)
\(29\)	2.05855	−	1.94214i	0.0709845	−	0.0669704i	−0.649932	−	0.759992i	\(-0.725203\pi\)
							0.720916	+	0.693022i	\(0.243721\pi\)
\(31\)	−0.593780	+	10.1948i	−0.0191542	+	0.328865i	0.975096	+	0.221783i	\(0.0711877\pi\)
							−0.994250	+	0.107082i	\(0.965849\pi\)
\(37\)	8.05693	−	2.93248i	0.217755	−	0.0792563i	−0.230839	−	0.972992i	\(-0.574147\pi\)
							0.448594	+	0.893736i	\(0.351925\pi\)
\(41\)	2.32118	−	9.79384i	0.0566142	−	0.238874i	−0.937380	−	0.348309i	\(-0.886756\pi\)
							0.993994	+	0.109435i	\(0.0349041\pi\)
\(43\)	19.9851	+	46.3306i	0.464769	+	1.07746i	0.975952	+	0.217985i	\(0.0699482\pi\)
							−0.511183	+	0.859472i	\(0.670793\pi\)
\(47\)	63.1002	−	3.67517i	1.34256	−	0.0781950i	0.628241	−	0.778019i	\(-0.283775\pi\)
							0.714316	+	0.699824i	\(0.246738\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.8	✓	324
81.65	odd	54	inner	162.3.h.a.65.8	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.8	✓	324	1.1	even	1	trivial
162.3.h.a.65.8	yes	324	81.65	odd	54	inner