Embedded newform 162.4.g.a.13.3

• Label: 162.4.g.a.13.3
• Level: $162$
• Weight: $4$
• Character: 162.13
• Analytic conductor: $9.558$
• Analytic rank: $0$
• Dimension: $234$
• 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:	\(234\)
Relative dimension:	\(13\) 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.a.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.78727 + 0.897598i) q^{2} +(-4.27995 - 2.94653i) q^{3} +(2.38863 + 3.20849i) q^{4} +(-2.62306 - 8.76162i) q^{5} +(-5.00461 - 9.10790i) q^{6} +(0.612588 + 1.42014i) q^{7} +(1.38919 + 7.87846i) q^{8} +(9.63595 + 25.2220i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 234 q + 36 q^{6}+O(q^{10}) \)

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
							\(3\)	−4.27995	−	2.94653i	−0.823677	−	0.567059i
\(5\)	−2.62306	−	8.76162i	−0.234613	−	0.783663i								−0.991499	−	0.130116i	\(-0.958465\pi\)
							0.756885	−	0.653548i	\(-0.226720\pi\)
\(7\)	0.612588	+	1.42014i	0.0330767	+	0.0766803i	0.933954	−	0.357394i	\(-0.116335\pi\)
							−0.900877	+	0.434075i	\(0.857075\pi\)
\(11\)	−27.1910	+	6.44440i	−0.745310	+	0.176642i	−0.585688	−	0.810537i	\(-0.699176\pi\)
							−0.159622	+	0.987178i	\(0.551027\pi\)
\(13\)	−54.9885	−	36.1665i	−1.17316	−	0.771599i	−0.195276	−	0.980748i	\(-0.562560\pi\)
							−0.977884	+	0.209149i	\(0.932931\pi\)
\(17\)	−69.5692	+	25.3211i	−0.992530	+	0.361251i	−0.786699	−	0.617336i	\(-0.788212\pi\)
							−0.205830	+	0.978588i	\(0.565989\pi\)
\(19\)	−103.049	−	37.5067i	−1.24427	−	0.452876i	−0.365805	−	0.930691i	\(-0.619206\pi\)
							−0.878460	+	0.477816i	\(0.841429\pi\)
\(23\)	−31.9476	+	74.0629i	−0.289632	+	0.671443i	−0.999457	−	0.0329350i	\(-0.989515\pi\)
							0.709825	+	0.704378i	\(0.248774\pi\)
\(29\)	−5.04677	−	86.6497i	−0.0323159	−	0.554843i	−0.975145	−	0.221567i	\(-0.928883\pi\)
							0.942829	−	0.333276i	\(-0.108154\pi\)
\(31\)	45.8156	−	5.35508i	0.265443	−	0.0310258i	0.0176697	−	0.999844i	\(-0.494375\pi\)
							0.247773	+	0.968818i	\(0.420301\pi\)
\(37\)	−27.5655	−	23.1302i	−0.122479	−	0.102772i	0.579491	−	0.814979i	\(-0.303252\pi\)
							−0.701970	+	0.712206i	\(0.747696\pi\)
\(41\)	−289.024	+	145.153i	−1.10093	+	0.552906i	−0.904009	−	0.427514i	\(-0.859389\pi\)
							−0.196917	+	0.980420i	\(0.563093\pi\)
\(43\)	243.423	+	258.014i	0.863296	+	0.915040i	0.997110	−	0.0759665i	\(-0.0242042\pi\)
							−0.133815	+	0.991006i	\(0.542723\pi\)
\(47\)	−17.4284	−	2.03709i	−0.0540894	−	0.00632214i	0.0890052	−	0.996031i	\(-0.471631\pi\)
							−0.143095	+	0.989709i	\(0.545705\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.a.13.3	✓	234
81.25	even	27	inner	162.4.g.a.25.3	yes	234

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.a.13.3	✓	234	1.1	even	1	trivial
162.4.g.a.25.3	yes	234	81.25	even	27	inner