Embedded newform 162.2.g.b.157.4

• Label: 162.2.g.b.157.4
• 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			157.4
Character	\(\chi\)	\(=\)	162.157
Dual form			162.2.g.b.97.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.973045 - 0.230616i) q^{2} +(0.538938 + 1.64607i) q^{3} +(0.893633 - 0.448799i) q^{4} +(-0.200298 - 0.269047i) q^{5} +(0.904020 + 1.47741i) q^{6} +(1.19293 - 0.784601i) q^{7} +(0.766044 - 0.642788i) q^{8} +(-2.41909 + 1.77426i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 9 q^{6}+O(q^{10}) \)

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
							\(3\)	0.538938	+	1.64607i	0.311156	+	0.950359i
\(5\)	−0.200298	−	0.269047i	−0.0895759	−	0.120321i								0.755076	−	0.655637i	\(-0.227600\pi\)
							−0.844652	+	0.535316i	\(0.820193\pi\)
\(7\)	1.19293	−	0.784601i	0.450884	−	0.296551i	−0.303684	−	0.952773i	\(-0.598217\pi\)
							0.754568	+	0.656222i	\(0.227846\pi\)
\(11\)	−1.81727	−	0.212408i	−0.547927	−	0.0640435i	−0.162374	−	0.986729i	\(-0.551915\pi\)
							−0.385553	+	0.922686i	\(0.625989\pi\)
\(13\)	0.451373	+	1.50769i	0.125188	+	0.418158i	0.997490	−	0.0708104i	\(-0.0225585\pi\)
							−0.872302	+	0.488968i	\(0.837373\pi\)
\(17\)	−0.187292	+	1.06218i	−0.0454249	+	0.257618i	−0.999060	−	0.0433477i	\(-0.986198\pi\)
							0.953635	+	0.300965i	\(0.0973088\pi\)
\(19\)	−0.689063	−	3.90787i	−0.158082	−	0.896527i	−0.955914	−	0.293647i	\(-0.905131\pi\)
							0.797832	−	0.602880i	\(-0.205980\pi\)
\(23\)	−7.14744	−	4.70095i	−1.49034	−	0.980215i	−0.993677	−	0.112274i	\(-0.964187\pi\)
							−0.496667	−	0.867941i	\(-0.665443\pi\)
\(29\)	0.506714	+	0.537085i	0.0940944	+	0.0997342i	0.772709	−	0.634760i	\(-0.218901\pi\)
							−0.678615	+	0.734494i	\(0.737420\pi\)
\(31\)	−0.386815	+	6.64136i	−0.0694741	+	1.19282i	0.765833	+	0.643040i	\(0.222327\pi\)
							−0.835307	+	0.549784i	\(0.814710\pi\)
\(37\)	8.37507	−	3.04828i	1.37685	−	0.501133i	0.455630	−	0.890169i	\(-0.349414\pi\)
							0.921223	+	0.389036i	\(0.127192\pi\)
\(41\)	−3.61087	−	0.855793i	−0.563924	−	0.133652i	−0.0612405	−	0.998123i	\(-0.519506\pi\)
							−0.502683	+	0.864471i	\(0.667654\pi\)
\(43\)	0.214504	+	0.497277i	0.0327116	+	0.0758340i	0.933789	−	0.357825i	\(-0.116482\pi\)
							−0.901077	+	0.433659i	\(0.857222\pi\)
\(47\)	0.159487	+	2.73829i	0.0232636	+	0.399421i	0.989760	+	0.142741i	\(0.0455916\pi\)
							−0.966496	+	0.256680i	\(0.917371\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.2.g.b.157.4	yes	90
3.2	odd	2		486.2.g.b.361.3		90
81.16	even	27	inner	162.2.g.b.97.4	✓	90
81.65	odd	54		486.2.g.b.451.3		90

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.2.g.b.97.4	✓	90	81.16	even	27	inner
162.2.g.b.157.4	yes	90	1.1	even	1	trivial
486.2.g.b.361.3		90	3.2	odd	2
486.2.g.b.451.3		90	81.65	odd	54