Embedded newform 162.3.h.a.5.18

• Label: 162.3.h.a.5.18
• 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			5.18
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.326140 + 1.37609i) q^{2} +(2.88671 - 0.816639i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(6.99459 - 5.20728i) q^{5} +(2.06524 + 3.70604i) q^{6} +(-4.06903 + 2.67625i) q^{7} +(-1.81808 - 2.16670i) q^{8} +(7.66620 - 4.71480i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	2.88671	−	0.816639i	0.962237	−	0.272213i
\(5\)	6.99459	−	5.20728i	1.39892	−	1.04146i								0.407155	−	0.913359i	\(-0.366521\pi\)
							0.991762	−	0.128096i	\(-0.0408867\pi\)
\(7\)	−4.06903	+	2.67625i	−0.581291	+	0.382321i	−0.805832	−	0.592144i	\(-0.798282\pi\)
							0.224542	+	0.974464i	\(0.427911\pi\)
\(11\)	−0.175095	+	1.49803i	−0.0159177	+	0.136185i	−0.998860	−	0.0477273i	\(-0.984802\pi\)
							0.982943	+	0.183912i	\(0.0588762\pi\)
\(13\)	−4.03892	−	13.4909i	−0.310686	−	1.03777i	−0.960641	−	0.277792i	\(-0.910397\pi\)
							0.649955	−	0.759973i	\(-0.274788\pi\)
\(17\)	3.91777	+	0.690808i	0.230457	+	0.0406358i	0.287684	−	0.957725i	\(-0.407115\pi\)
							−0.0572270	+	0.998361i	\(0.518226\pi\)
\(19\)	5.84384	+	33.1421i	0.307571	+	1.74432i	0.611151	+	0.791514i	\(0.290707\pi\)
							−0.303580	+	0.952806i	\(0.598182\pi\)
\(23\)	−21.1782	+	32.1999i	−0.920791	+	1.39999i	−0.00427724	+	0.999991i	\(0.501361\pi\)
							−0.916514	+	0.400004i	\(0.869009\pi\)
\(29\)	−31.6339	+	29.8451i	−1.09083	+	1.02914i	−0.0913812	+	0.995816i	\(0.529128\pi\)
							−0.999445	+	0.0333250i	\(0.989390\pi\)
\(31\)	0.170096	−	2.92044i	0.00548698	−	0.0942078i	−0.994453	−	0.105184i	\(-0.966457\pi\)
							0.999940	+	0.0109762i	\(0.00349389\pi\)
\(37\)	12.9051	−	4.69706i	0.348786	−	0.126948i	−0.161686	−	0.986842i	\(-0.551693\pi\)
							0.510472	+	0.859895i	\(0.329471\pi\)
\(41\)	−3.69932	+	15.6087i	−0.0902273	+	0.380699i	−0.999359	−	0.0358107i	\(-0.988599\pi\)
							0.909131	+	0.416510i	\(0.136747\pi\)
\(43\)	−25.5187	−	59.1590i	−0.593458	−	1.37579i	−0.904940	−	0.425538i	\(-0.860085\pi\)
							0.311483	−	0.950252i	\(-0.399174\pi\)
\(47\)	29.9107	−	1.74210i	0.636397	−	0.0370659i	0.263086	−	0.964772i	\(-0.415260\pi\)
							0.373311	+	0.927706i	\(0.378223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.18	✓	324
81.65	odd	54	inner	162.3.h.a.65.18	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.18	✓	324	1.1	even	1	trivial
162.3.h.a.65.18	yes	324	81.65	odd	54	inner