Embedded newform 162.2.g.b.157.3

• Label: 162.2.g.b.157.3
• Level: $162$
• Weight: $2$
• Character: 162.157
• Analytic conductor: $1.294$
• Analytic rank: $0$
• Dimension: $90$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	162.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.29357651274\)
Analytic rank:	\(0\)
Dimension:	\(90\)
Relative dimension:	\(5\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			157.3
Character	\(\chi\)	\(=\)	162.157
Dual form			162.2.g.b.97.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.973045 - 0.230616i) q^{2} +(0.190350 - 1.72156i) q^{3} +(0.893633 - 0.448799i) q^{4} +(-1.06995 - 1.43719i) q^{5} +(-0.211800 - 1.71905i) q^{6} +(-0.983264 + 0.646703i) q^{7} +(0.766044 - 0.642788i) q^{8} +(-2.92753 - 0.655396i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 9 q^{6}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{25}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.973045	−	0.230616i	0.688047	−	0.163070i
							\(3\)	0.190350	−	1.72156i	0.109898	−	0.993943i
\(5\)	−1.06995	−	1.43719i	−0.478495	−	0.642731i								0.495685	−	0.868502i	\(-0.334917\pi\)
							−0.974180	+	0.225772i	\(0.927510\pi\)
\(7\)	−0.983264	+	0.646703i	−0.371639	+	0.244431i	−0.721565	−	0.692347i	\(-0.756577\pi\)
							0.349926	+	0.936777i	\(0.386207\pi\)
\(11\)	3.61528	+	0.422566i	1.09005	+	0.127408i	0.642077	−	0.766640i	\(-0.278073\pi\)
							0.447971	+	0.894048i	\(0.352147\pi\)
\(13\)	1.84593	+	6.16584i	0.511969	+	1.71010i	0.687999	+	0.725711i	\(0.258489\pi\)
							−0.176030	+	0.984385i	\(0.556326\pi\)
\(17\)	0.461322	−	2.61629i	0.111887	−	0.634543i	−0.876358	−	0.481661i	\(-0.840034\pi\)
							0.988245	−	0.152882i	\(-0.0488553\pi\)
\(19\)	−0.128292	−	0.727580i	−0.0294322	−	0.166918i	0.966549	−	0.256483i	\(-0.0825637\pi\)
							−0.995981	+	0.0895643i	\(0.971453\pi\)
\(23\)	3.51483	+	2.31174i	0.732893	+	0.482031i	0.860280	−	0.509822i	\(-0.170289\pi\)
							−0.127387	+	0.991853i	\(0.540659\pi\)
\(29\)	−1.71661	−	1.81950i	−0.318767	−	0.337873i	0.548100	−	0.836413i	\(-0.315351\pi\)
							−0.866867	+	0.498540i	\(0.833870\pi\)
\(31\)	−0.374563	+	6.43100i	−0.0672735	+	1.15504i	0.780666	+	0.624949i	\(0.214880\pi\)
							−0.847939	+	0.530093i	\(0.822157\pi\)
\(37\)	−7.73114	+	2.81391i	−1.27099	+	0.462603i	−0.887443	−	0.460917i	\(-0.847520\pi\)
							−0.383549	+	0.923521i	\(0.625298\pi\)
\(41\)	−9.76678	−	2.31477i	−1.52532	−	0.361506i	−0.619438	−	0.785046i	\(-0.712639\pi\)
							−0.905878	+	0.423540i	\(0.860787\pi\)
\(43\)	2.11947	+	4.91348i	0.323216	+	0.749299i	0.999915	+	0.0130390i	\(0.00415056\pi\)
							−0.676699	+	0.736260i	\(0.736590\pi\)
\(47\)	−0.160028	−	2.74758i	−0.0233425	−	0.400775i	−0.989661	−	0.143426i	\(-0.954188\pi\)
							0.966318	−	0.257349i	\(-0.0828490\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.2.g.b.157.3	yes	90
3.2	odd	2		486.2.g.b.361.4		90
81.16	even	27	inner	162.2.g.b.97.3	✓	90
81.65	odd	54		486.2.g.b.451.4		90

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.2.g.b.97.3	✓	90	81.16	even	27	inner
162.2.g.b.157.3	yes	90	1.1	even	1	trivial
486.2.g.b.361.4		90	3.2	odd	2
486.2.g.b.451.4		90	81.65	odd	54