Embedded newform 162.3.h.a.5.12

• Label: 162.3.h.a.5.12
• 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			5.12
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.326140 + 1.37609i) q^{2} +(-2.21854 - 2.01943i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(2.97271 - 2.21310i) q^{5} +(2.05537 - 3.71153i) q^{6} +(-8.93104 + 5.87404i) q^{7} +(-1.81808 - 2.16670i) q^{8} +(0.843823 + 8.96036i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	−2.21854	−	2.01943i	−0.739513	−	0.673143i
\(5\)	2.97271	−	2.21310i	0.594542	−	0.442620i								−0.257364	−	0.966314i	\(-0.582854\pi\)
							0.851906	+	0.523694i	\(0.175447\pi\)
\(7\)	−8.93104	+	5.87404i	−1.27586	+	0.839149i	−0.992780	−	0.119947i	\(-0.961728\pi\)
							−0.283083	+	0.959095i	\(0.591357\pi\)
\(11\)	−1.77098	+	15.1517i	−0.160998	+	1.37743i	0.635179	+	0.772365i	\(0.280926\pi\)
							−0.796178	+	0.605063i	\(0.793148\pi\)
\(13\)	4.78178	+	15.9723i	0.367829	+	1.22864i	0.919005	+	0.394246i	\(0.128994\pi\)
							−0.551175	+	0.834389i	\(0.685820\pi\)
\(17\)	−10.6405	−	1.87621i	−0.625911	−	0.110365i	−0.148309	−	0.988941i	\(-0.547383\pi\)
							−0.477603	+	0.878576i	\(0.658494\pi\)
\(19\)	−5.69767	−	32.3131i	−0.299878	−	1.70069i	−0.646688	−	0.762755i	\(-0.723846\pi\)
							0.346810	−	0.937935i	\(-0.387265\pi\)
\(23\)	−19.5064	+	29.6580i	−0.848104	+	1.28948i	0.107315	+	0.994225i	\(0.465775\pi\)
							−0.955419	+	0.295255i	\(0.904596\pi\)
\(29\)	−14.7526	+	13.9183i	−0.508709	+	0.479942i	−0.897271	−	0.441480i	\(-0.854453\pi\)
							0.388562	+	0.921423i	\(0.372972\pi\)
\(31\)	1.11649	−	19.1694i	0.0360158	−	0.618368i	−0.931215	−	0.364470i	\(-0.881250\pi\)
							0.967231	−	0.253898i	\(-0.0817127\pi\)
\(37\)	52.1313	−	18.9742i	1.40895	−	0.512817i	0.478132	−	0.878288i	\(-0.341314\pi\)
							0.930823	+	0.365471i	\(0.119092\pi\)
\(41\)	6.39904	−	26.9997i	0.156074	−	0.658528i	−0.837606	−	0.546275i	\(-0.816045\pi\)
							0.993680	−	0.112253i	\(-0.0358067\pi\)
\(43\)	16.3061	+	37.8018i	0.379212	+	0.879112i	0.995777	+	0.0918076i	\(0.0292645\pi\)
							−0.616565	+	0.787304i	\(0.711476\pi\)
\(47\)	−39.3092	+	2.28950i	−0.836366	+	0.0487128i	−0.470981	−	0.882143i	\(-0.656100\pi\)
							−0.365386	+	0.930856i	\(0.619063\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.12	✓	324
81.65	odd	54	inner	162.3.h.a.65.12	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.12	✓	324	1.1	even	1	trivial
162.3.h.a.65.12	yes	324	81.65	odd	54	inner