Embedded newform 162.4.g.a.7.2

• Label: 162.4.g.a.7.2
• Level: $162$
• Weight: $4$
• Character: 162.7
• 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			7.2
Character	\(\chi\)	\(=\)	162.7
Dual form			162.4.g.a.139.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19432 + 1.60425i) q^{2} +(-5.05924 + 1.18494i) q^{3} +(-1.14721 + 3.83196i) q^{4} +(-13.6695 + 8.99057i) q^{5} +(-7.94328 - 6.70107i) q^{6} +(-2.17193 + 2.30211i) q^{7} +(-7.51754 + 2.73616i) q^{8} +(24.1918 - 11.9898i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 234 q + 36 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)
\(\chi(n)\)	\(e\left(\frac{8}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19432	+	1.60425i	0.422255	+	0.567187i
							\(3\)	−5.05924	+	1.18494i	−0.973651	+	0.228043i
\(5\)	−13.6695	+	8.99057i	−1.22264	+	0.804141i								−0.985884	−	0.167429i	\(-0.946454\pi\)
							−0.236753	+	0.971570i	\(0.576083\pi\)
\(7\)	−2.17193	+	2.30211i	−0.117273	+	0.124302i	−0.783350	−	0.621581i	\(-0.786491\pi\)
							0.666077	+	0.745883i	\(0.267972\pi\)
\(11\)	19.8503	−	9.96918i	0.544098	−	0.273256i	−0.155448	−	0.987844i	\(-0.549682\pi\)
							0.699546	+	0.714588i	\(0.253386\pi\)
\(13\)	−10.9083	−	25.2883i	−0.232725	−	0.539517i	0.761334	−	0.648360i	\(-0.224545\pi\)
							−0.994059	+	0.108843i	\(0.965286\pi\)
\(17\)	40.7289	−	34.1756i	0.581071	−	0.487577i	−0.304227	−	0.952599i	\(-0.598398\pi\)
							0.885299	+	0.465023i	\(0.153954\pi\)
\(19\)	−87.3325	−	73.2806i	−1.05450	−	0.884828i	−0.0609375	−	0.998142i	\(-0.519409\pi\)
							−0.993559	+	0.113314i	\(0.963853\pi\)
\(23\)	30.1430	+	31.9498i	0.273272	+	0.289652i	0.849446	−	0.527676i	\(-0.176936\pi\)
							−0.576174	+	0.817327i	\(0.695455\pi\)
\(29\)	−80.6350	+	9.42488i	−0.516329	+	0.0603502i	−0.370267	−	0.928925i	\(-0.620734\pi\)
							−0.146062	+	0.989275i	\(0.546660\pi\)
\(31\)	−93.7072	+	22.2090i	−0.542913	+	0.128673i	−0.492917	−	0.870077i	\(-0.664069\pi\)
							−0.0499968	+	0.998749i	\(0.515921\pi\)
\(37\)	−19.9791	−	113.307i	−0.0887712	−	0.503447i	−0.996479	−	0.0838436i	\(-0.973280\pi\)
							0.907708	−	0.419603i	\(-0.137831\pi\)
\(41\)	−97.1674	+	130.519i	−0.370122	+	0.497160i	−0.947694	−	0.319179i	\(-0.896593\pi\)
							0.577572	+	0.816340i	\(0.304000\pi\)
\(43\)	29.5349	−	507.095i	0.104745	−	1.79840i	−0.381282	−	0.924459i	\(-0.624517\pi\)
							0.486027	−	0.873944i	\(-0.338446\pi\)
\(47\)	80.1650	+	18.9994i	0.248793	+	0.0589650i	0.353120	−	0.935578i	\(-0.385121\pi\)
							−0.104327	+	0.994543i	\(0.533269\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.a.7.2	✓	234
81.58	even	27	inner	162.4.g.a.139.2	yes	234

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.a.7.2	✓	234	1.1	even	1	trivial
162.4.g.a.139.2	yes	234	81.58	even	27	inner