Embedded newform 162.4.g.b.13.4

• Label: 162.4.g.b.13.4
• 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			13.4
Character	\(\chi\)	\(=\)	162.13
Dual form			162.4.g.b.25.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78727 - 0.897598i) q^{2} +(-3.27564 + 4.03363i) q^{3} +(2.38863 + 3.20849i) q^{4} +(-0.224791 - 0.750853i) q^{5} +(9.47502 - 4.26895i) q^{6} +(-4.80994 - 11.1507i) q^{7} +(-1.38919 - 7.87846i) q^{8} +(-5.54032 - 26.4255i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 36 q^{6}+O(q^{10}) \)

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
							\(3\)	−3.27564	+	4.03363i	−0.630398	+	0.776272i
\(5\)	−0.224791	−	0.750853i	−0.0201059	−	0.0671583i								0.947348	−	0.320206i	\(-0.103752\pi\)
							−0.967454	+	0.253048i	\(0.918567\pi\)
\(7\)	−4.80994	−	11.1507i	−0.259713	−	0.602081i	0.737609	−	0.675229i	\(-0.235955\pi\)
							−0.997321	+	0.0731472i	\(0.976696\pi\)
\(11\)	−17.0651	+	4.04449i	−0.467755	+	0.110860i	−0.457733	−	0.889090i	\(-0.651338\pi\)
							−0.0100224	+	0.999950i	\(0.503190\pi\)
\(13\)	18.0677	+	11.8833i	0.385467	+	0.253526i	0.727422	−	0.686191i	\(-0.240718\pi\)
							−0.341954	+	0.939717i	\(0.611089\pi\)
\(17\)	24.8346	−	9.03907i	0.354311	−	0.128959i	−0.158732	−	0.987322i	\(-0.550740\pi\)
							0.513043	+	0.858363i	\(0.328518\pi\)
\(19\)	106.295	+	38.6881i	1.28346	+	0.467140i	0.891574	−	0.452875i	\(-0.149602\pi\)
							0.391883	+	0.920015i	\(0.371824\pi\)
\(23\)	−11.4645	+	26.5777i	−0.103935	+	0.240949i	−0.962246	−	0.272181i	\(-0.912255\pi\)
							0.858311	+	0.513130i	\(0.171514\pi\)
\(29\)	−3.82003	−	65.5874i	−0.0244607	−	0.419975i	−0.988204	−	0.153143i	\(-0.951060\pi\)
							0.963743	−	0.266831i	\(-0.0859767\pi\)
\(31\)	171.116	−	20.0006i	0.991397	−	0.115878i	0.395097	−	0.918639i	\(-0.370711\pi\)
							0.596300	+	0.802762i	\(0.296637\pi\)
\(37\)	207.137	+	173.808i	0.920352	+	0.772267i	0.974060	−	0.226289i	\(-0.0726595\pi\)
							−0.0537078	+	0.998557i	\(0.517104\pi\)
\(41\)	87.9498	−	44.1700i	0.335011	−	0.168249i	−0.273339	−	0.961918i	\(-0.588128\pi\)
							0.608350	+	0.793669i	\(0.291832\pi\)
\(43\)	84.7045	+	89.7815i	0.300403	+	0.318408i	0.859938	−	0.510398i	\(-0.170502\pi\)
							−0.559536	+	0.828806i	\(0.689020\pi\)
\(47\)	−162.126	−	18.9498i	−0.503159	−	0.0588109i	−0.139272	−	0.990254i	\(-0.544476\pi\)
							−0.363887	+	0.931443i	\(0.618550\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.b.13.4	✓	252
81.25	even	27	inner	162.4.g.b.25.4	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.b.13.4	✓	252	1.1	even	1	trivial
162.4.g.b.25.4	yes	252	81.25	even	27	inner