Embedded newform 162.4.g.b.13.3

• Label: 162.4.g.b.13.3
• 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			13.3
Character	\(\chi\)	\(=\)	162.13
Dual form			162.4.g.b.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78727 - 0.897598i) q^{2} +(-3.98094 - 3.33948i) q^{3} +(2.38863 + 3.20849i) q^{4} +(-3.33686 - 11.1459i) q^{5} +(4.11748 + 9.54182i) q^{6} +(11.7915 + 27.3357i) q^{7} +(-1.38919 - 7.87846i) q^{8} +(4.69572 + 26.5885i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 36 q^{6}+O(q^{10}) \)

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
							\(3\)	−3.98094	−	3.33948i	−0.766132	−	0.642684i
\(5\)	−3.33686	−	11.1459i	−0.298458	−	0.996919i								−0.967326	−	0.253538i	\(-0.918406\pi\)
							0.668867	−	0.743382i	\(-0.266779\pi\)
\(7\)	11.7915	+	27.3357i	0.636679	+	1.47599i	0.864678	+	0.502326i	\(0.167522\pi\)
							−0.228000	+	0.973661i	\(0.573219\pi\)
\(11\)	−51.7294	+	12.2601i	−1.41791	+	0.336050i	−0.866914	−	0.498458i	\(-0.833900\pi\)
							−0.550995	+	0.834509i	\(0.685752\pi\)
\(13\)	−5.97335	−	3.92874i	−0.127439	−	0.0838181i	0.484189	−	0.874963i	\(-0.339115\pi\)
							−0.611628	+	0.791145i	\(0.709485\pi\)
\(17\)	111.661	−	40.6412i	1.59304	−	0.579819i	0.615053	−	0.788486i	\(-0.289134\pi\)
							0.977987	+	0.208666i	\(0.0669122\pi\)
\(19\)	71.5412	+	26.0389i	0.863826	+	0.314407i	0.735664	−	0.677347i	\(-0.236870\pi\)
							0.128161	+	0.991753i	\(0.459092\pi\)
\(23\)	−15.9830	+	37.0528i	−0.144900	+	0.335915i	−0.975174	−	0.221440i	\(-0.928924\pi\)
							0.830274	+	0.557355i	\(0.188184\pi\)
\(29\)	9.57900	+	164.465i	0.0613371	+	1.05312i	0.878817	+	0.477160i	\(0.158334\pi\)
							−0.817479	+	0.575958i	\(0.804629\pi\)
\(31\)	122.056	−	14.2662i	0.707156	−	0.0826546i	0.245090	−	0.969500i	\(-0.421183\pi\)
							0.462066	+	0.886846i	\(0.347108\pi\)
\(37\)	−133.875	−	112.334i	−0.594833	−	0.499125i	0.294947	−	0.955514i	\(-0.404698\pi\)
							−0.889780	+	0.456389i	\(0.849143\pi\)
\(41\)	374.846	−	188.255i	1.42783	−	0.717084i	0.443974	−	0.896040i	\(-0.353568\pi\)
							0.983857	+	0.178956i	\(0.0572719\pi\)
\(43\)	97.2384	+	103.067i	0.344854	+	0.365524i	0.876466	−	0.481464i	\(-0.159895\pi\)
							−0.531612	+	0.846988i	\(0.678413\pi\)
\(47\)	241.663	+	28.2463i	0.750003	+	0.0876628i	0.482502	−	0.875895i	\(-0.339728\pi\)
							0.267501	+	0.963558i	\(0.413802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.b.13.3	✓	252
81.25	even	27	inner	162.4.g.b.25.3	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.b.13.3	✓	252	1.1	even	1	trivial
162.4.g.b.25.3	yes	252	81.25	even	27	inner