Embedded newform 162.3.h.a.5.6

• Label: 162.3.h.a.5.6
• 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			5.6
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.326140 - 1.37609i) q^{2} +(0.243053 - 2.99014i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(-2.26822 + 1.68862i) q^{5} +(-4.19398 + 0.640740i) q^{6} +(-8.30694 + 5.46356i) q^{7} +(1.81808 + 2.16670i) q^{8} +(-8.88185 - 1.45353i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	0.243053	−	2.99014i	0.0810178	−	0.996713i
\(5\)	−2.26822	+	1.68862i	−0.453643	+	0.337725i								−0.799608	−	0.600523i	\(-0.794959\pi\)
							0.345964	+	0.938248i	\(0.387552\pi\)
\(7\)	−8.30694	+	5.46356i	−1.18671	+	0.780509i	−0.980214	−	0.197940i	\(-0.936575\pi\)
							−0.206491	+	0.978448i	\(0.566205\pi\)
\(11\)	−1.30129	+	11.1332i	−0.118299	+	1.01211i	0.795502	+	0.605951i	\(0.207207\pi\)
							−0.913801	+	0.406162i	\(0.866867\pi\)
\(13\)	−4.43168	−	14.8028i	−0.340898	−	1.13868i	−0.940901	−	0.338682i	\(-0.890019\pi\)
							0.600002	−	0.799998i	\(-0.295166\pi\)
\(17\)	−4.36022	−	0.768825i	−0.256484	−	0.0452250i	0.0439277	−	0.999035i	\(-0.486013\pi\)
							−0.300411	+	0.953810i	\(0.597124\pi\)
\(19\)	−1.93069	−	10.9495i	−0.101616	−	0.576290i	−0.992518	−	0.122096i	\(-0.961038\pi\)
							0.890903	−	0.454194i	\(-0.150073\pi\)
\(23\)	3.87583	−	5.89292i	0.168515	−	0.256214i	−0.741351	−	0.671118i	\(-0.765814\pi\)
							0.909865	+	0.414904i	\(0.136185\pi\)
\(29\)	−27.5564	+	25.9981i	−0.950220	+	0.896487i	−0.994731	−	0.102522i	\(-0.967309\pi\)
							0.0445104	+	0.999009i	\(0.485827\pi\)
\(31\)	3.19250	−	54.8131i	0.102984	−	1.76816i	−0.412208	−	0.911090i	\(-0.635242\pi\)
							0.515192	−	0.857075i	\(-0.327721\pi\)
\(37\)	−37.2076	+	13.5425i	−1.00561	+	0.366013i	−0.791746	−	0.610850i	\(-0.790828\pi\)
							−0.213866	+	0.976863i	\(0.568605\pi\)
\(41\)	−9.88313	+	41.7002i	−0.241052	+	1.01708i	0.709055	+	0.705154i	\(0.249122\pi\)
							−0.950107	+	0.311925i	\(0.899026\pi\)
\(43\)	−27.3467	−	63.3968i	−0.635970	−	1.47435i	−0.865431	−	0.501027i	\(-0.832956\pi\)
							0.229461	−	0.973318i	\(-0.426304\pi\)
\(47\)	62.3782	−	3.63312i	1.32720	−	0.0773003i	0.620158	−	0.784477i	\(-0.287068\pi\)
							0.707037	+	0.707176i	\(0.250031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.6	✓	324
81.65	odd	54	inner	162.3.h.a.65.6	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.6	✓	324	1.1	even	1	trivial
162.3.h.a.65.6	yes	324	81.65	odd	54	inner