Embedded newform 162.4.g.b.13.1

• Label: 162.4.g.b.13.1
• Level: $162$
• Weight: $4$
• Character: 162.13
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(9.55830942093\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(14\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			13.1
Character	\(\chi\)	\(=\)	162.13
Dual form			162.4.g.b.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78727 - 0.897598i) q^{2} +(-5.19032 + 0.246219i) q^{3} +(2.38863 + 3.20849i) q^{4} +(4.68865 + 15.6612i) q^{5} +(9.49748 + 4.21876i) q^{6} +(-0.254938 - 0.591012i) q^{7} +(-1.38919 - 7.87846i) q^{8} +(26.8788 - 2.55591i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 36 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{4}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78727	−	0.897598i	−0.631894	−	0.317349i
							\(3\)	−5.19032	+	0.246219i	−0.998877	+	0.0473848i
\(5\)	4.68865	+	15.6612i	0.419366	+	1.40078i								0.863287	+	0.504713i	\(0.168402\pi\)
							−0.443921	+	0.896066i	\(0.646413\pi\)
\(7\)	−0.254938	−	0.591012i	−0.0137653	−	0.0319116i	0.911195	−	0.411976i	\(-0.135161\pi\)
							−0.924960	+	0.380064i	\(0.875902\pi\)
\(11\)	−15.3584	+	3.64000i	−0.420974	+	0.0997728i	−0.435643	−	0.900120i	\(-0.643479\pi\)
							0.0146683	+	0.999892i	\(0.495331\pi\)
\(13\)	14.7540	+	9.70384i	0.314770	+	0.207028i	0.697056	−	0.717016i	\(-0.254493\pi\)
							−0.382286	+	0.924044i	\(0.624863\pi\)
\(17\)	12.1756	−	4.43156i	0.173707	−	0.0632242i	−0.253702	−	0.967282i	\(-0.581648\pi\)
							0.427410	+	0.904058i	\(0.359426\pi\)
\(19\)	−90.3244	−	32.8754i	−1.09062	−	0.396954i	−0.266771	−	0.963760i	\(-0.585957\pi\)
							−0.823852	+	0.566805i	\(0.808179\pi\)
\(23\)	−39.7495	+	92.1496i	−0.360362	+	0.835414i	0.637524	+	0.770431i	\(0.279959\pi\)
							−0.997886	+	0.0649835i	\(0.979301\pi\)
\(29\)	11.3414	+	194.724i	0.0726219	+	1.24687i	0.816021	+	0.578022i	\(0.196175\pi\)
							−0.743399	+	0.668848i	\(0.766788\pi\)
\(31\)	−169.608	+	19.8243i	−0.982660	+	0.114857i	−0.592224	−	0.805773i	\(-0.701750\pi\)
							−0.390436	+	0.920630i	\(0.627676\pi\)
\(37\)	−328.206	−	275.398i	−1.45829	−	1.22365i	−0.926237	−	0.376942i	\(-0.876975\pi\)
							−0.532055	−	0.846710i	\(-0.678580\pi\)
\(41\)	−97.8896	+	49.1620i	−0.372873	+	0.187264i	−0.625360	−	0.780336i	\(-0.715048\pi\)
							0.252487	+	0.967600i	\(0.418752\pi\)
\(43\)	−245.621	−	260.343i	−0.871089	−	0.923300i	0.126551	−	0.991960i	\(-0.459609\pi\)
							−0.997640	+	0.0686600i	\(0.978128\pi\)
\(47\)	173.580	+	20.2886i	0.538707	+	0.0629659i	0.381096	−	0.924535i	\(-0.375547\pi\)
							0.157611	+	0.987501i	\(0.449621\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.b.13.1	✓	252
81.25	even	27	inner	162.4.g.b.25.1	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.b.13.1	✓	252	1.1	even	1	trivial
162.4.g.b.25.1	yes	252	81.25	even	27	inner