Embedded newform 162.3.h.a.5.13

• Label: 162.3.h.a.5.13
• 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			5.13
Character	\(\chi\)	\(=\)	162.5
Dual form			162.3.h.a.65.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.326140 + 1.37609i) q^{2} +(-1.18720 + 2.75509i) q^{3} +(-1.78727 + 0.897598i) q^{4} +(-3.75904 + 2.79850i) q^{5} +(-4.17846 - 0.735156i) q^{6} +(1.03380 - 0.679940i) q^{7} +(-1.81808 - 2.16670i) q^{8} +(-6.18109 - 6.54172i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

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
							\(3\)	−1.18720	+	2.75509i	−0.395735	+	0.918365i
\(5\)	−3.75904	+	2.79850i	−0.751808	+	0.559701i								−0.903429	−	0.428738i	\(-0.858958\pi\)
							0.151620	+	0.988439i	\(0.451551\pi\)
\(7\)	1.03380	−	0.679940i	0.147685	−	0.0971343i	−0.473516	−	0.880785i	\(-0.657015\pi\)
							0.621201	+	0.783651i	\(0.286645\pi\)
\(11\)	−0.479947	+	4.10621i	−0.0436315	+	0.373292i	0.953728	+	0.300671i	\(0.0972105\pi\)
							−0.997360	+	0.0726211i	\(0.976864\pi\)
\(13\)	−1.23625	−	4.12937i	−0.0950963	−	0.317644i	0.897184	−	0.441656i	\(-0.145609\pi\)
							−0.992281	+	0.124012i	\(0.960424\pi\)
\(17\)	−7.03846	−	1.24107i	−0.414027	−	0.0730042i	−0.0372453	−	0.999306i	\(-0.511858\pi\)
							−0.376782	+	0.926302i	\(0.622969\pi\)
\(19\)	−1.18798	−	6.73736i	−0.0625252	−	0.354598i	−0.999979	−	0.00648804i	\(-0.997935\pi\)
							0.937454	−	0.348110i	\(-0.113176\pi\)
\(23\)	−18.9862	+	28.8671i	−0.825487	+	1.25509i	0.138823	+	0.990317i	\(0.455668\pi\)
							−0.964310	+	0.264775i	\(0.914702\pi\)
\(29\)	−8.94383	+	8.43806i	−0.308408	+	0.290968i	−0.824847	−	0.565356i	\(-0.808739\pi\)
							0.516439	+	0.856324i	\(0.327257\pi\)
\(31\)	−0.301108	+	5.16983i	−0.00971317	+	0.166769i	0.989963	+	0.141326i	\(0.0451366\pi\)
							−0.999676	+	0.0254427i	\(0.991900\pi\)
\(37\)	−13.6953	+	4.98470i	−0.370145	+	0.134722i	−0.520393	−	0.853927i	\(-0.674215\pi\)
							0.150249	+	0.988648i	\(0.451992\pi\)
\(41\)	−17.2211	+	72.6617i	−0.420028	+	1.77224i	0.191994	+	0.981396i	\(0.438505\pi\)
							−0.612022	+	0.790841i	\(0.709643\pi\)
\(43\)	−15.2354	−	35.3195i	−0.354311	−	0.821384i	−0.998406	−	0.0564391i	\(-0.982025\pi\)
							0.644095	−	0.764945i	\(-0.277234\pi\)
\(47\)	5.69788	−	0.331864i	0.121231	−	0.00706093i	0.00257817	−	0.999997i	\(-0.499179\pi\)
							0.118653	+	0.992936i	\(0.462142\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.h.a.5.13	✓	324
81.65	odd	54	inner	162.3.h.a.65.13	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.h.a.5.13	✓	324	1.1	even	1	trivial
162.3.h.a.65.13	yes	324	81.65	odd	54	inner