Embedded newform 162.4.g.a.7.9

• Label: 162.4.g.a.7.9
• 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			7.9
Character	\(\chi\)	\(=\)	162.7
Dual form			162.4.g.a.139.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19432 + 1.60425i) q^{2} +(1.72596 - 4.90113i) q^{3} +(-1.14721 + 3.83196i) q^{4} +(-5.18053 + 3.40729i) q^{5} +(9.92396 - 3.08463i) q^{6} +(-19.8455 + 21.0350i) q^{7} +(-7.51754 + 2.73616i) q^{8} +(-21.0421 - 16.9183i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 234 q + 36 q^{6}+O(q^{10}) \)

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
							\(3\)	1.72596	−	4.90113i	0.332162	−	0.943222i
\(5\)	−5.18053	+	3.40729i	−0.463361	+	0.304757i								−0.759640	−	0.650344i	\(-0.774625\pi\)
							0.296279	+	0.955102i	\(0.404254\pi\)
\(7\)	−19.8455	+	21.0350i	−1.07156	+	1.13578i	−0.0812757	+	0.996692i	\(0.525899\pi\)
							−0.990280	+	0.139091i	\(0.955582\pi\)
\(11\)	−15.1202	+	7.59366i	−0.414447	+	0.208143i	−0.643789	−	0.765203i	\(-0.722639\pi\)
							0.229342	+	0.973346i	\(0.426342\pi\)
\(13\)	11.3068	+	26.2120i	0.241226	+	0.559224i	0.995220	−	0.0976567i	\(-0.0311347\pi\)
							−0.753994	+	0.656881i	\(0.771875\pi\)
\(17\)	−32.6007	+	27.3552i	−0.465107	+	0.390271i	−0.845006	−	0.534757i	\(-0.820403\pi\)
							0.379899	+	0.925028i	\(0.375959\pi\)
\(19\)	−28.4170	−	23.8447i	−0.343122	−	0.287914i	0.454899	−	0.890543i	\(-0.349675\pi\)
							−0.798021	+	0.602629i	\(0.794120\pi\)
\(23\)	63.0687	+	66.8489i	0.571771	+	0.606042i	0.947135	−	0.320836i	\(-0.103964\pi\)
							−0.375364	+	0.926878i	\(0.622482\pi\)
\(29\)	79.9302	−	9.34250i	0.511816	−	0.0598227i	0.143735	−	0.989616i	\(-0.454089\pi\)
							0.368082	+	0.929794i	\(0.380015\pi\)
\(31\)	−71.2888	+	16.8958i	−0.413027	+	0.0978893i	−0.431876	−	0.901933i	\(-0.642148\pi\)
							0.0188485	+	0.999822i	\(0.494000\pi\)
\(37\)	−68.7456	−	389.876i	−0.305452	−	1.73230i	−0.621371	−	0.783516i	\(-0.713424\pi\)
							0.315920	−	0.948786i	\(-0.397687\pi\)
\(41\)	−68.3966	+	91.8726i	−0.260531	+	0.349954i	−0.913071	−	0.407800i	\(-0.866296\pi\)
							0.652540	+	0.757754i	\(0.273703\pi\)
\(43\)	1.81620	−	31.1829i	0.00644111	−	0.110590i	−0.993555	−	0.113350i	\(-0.963842\pi\)
							0.999996	+	0.00276076i	\(0.000878778\pi\)
\(47\)	139.394	+	33.0369i	0.432609	+	0.102530i	0.441150	−	0.897433i	\(-0.354571\pi\)
							−0.00854095	+	0.999964i	\(0.502719\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.a.7.9	✓	234
81.58	even	27	inner	162.4.g.a.139.9	yes	234

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.a.7.9	✓	234	1.1	even	1	trivial
162.4.g.a.139.9	yes	234	81.58	even	27	inner