Embedded newform 162.4.g.b.7.6

• Label: 162.4.g.b.7.6
• Level: $162$
• Weight: $4$
• Character: 162.7
• 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			7.6
Character	\(\chi\)	\(=\)	162.7
Dual form			162.4.g.b.139.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19432 - 1.60425i) q^{2} +(-0.821241 - 5.13084i) q^{3} +(-1.14721 + 3.83196i) q^{4} +(1.88361 - 1.23887i) q^{5} +(-7.25032 + 7.44533i) q^{6} +(-2.53643 + 2.68846i) q^{7} +(7.51754 - 2.73616i) q^{8} +(-25.6511 + 8.42732i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 36 q^{6}+O(q^{10}) \)

Character values

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

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

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
							\(3\)	−0.821241	−	5.13084i	−0.158048	−	0.987431i
\(5\)	1.88361	−	1.23887i	0.168475	−	0.110808i								−0.462471	−	0.886634i	\(-0.653037\pi\)
							0.630947	+	0.775826i	\(0.282667\pi\)
\(7\)	−2.53643	+	2.68846i	−0.136954	+	0.145163i	−0.792220	−	0.610235i	\(-0.791075\pi\)
							0.655266	+	0.755398i	\(0.272556\pi\)
\(11\)	−44.7042	+	22.4513i	−1.22535	+	0.615393i	−0.939391	−	0.342849i	\(-0.888608\pi\)
							−0.285958	+	0.958242i	\(0.592312\pi\)
\(13\)	−13.6311	−	31.6005i	−0.290815	−	0.674185i	0.708694	−	0.705516i	\(-0.249285\pi\)
							−0.999509	+	0.0313308i	\(0.990025\pi\)
\(17\)	−62.4735	+	52.4215i	−0.891297	+	0.747887i	−0.968470	−	0.249131i	\(-0.919855\pi\)
							0.0771728	+	0.997018i	\(0.475411\pi\)
\(19\)	113.342	+	95.1052i	1.36855	+	1.14835i	0.973237	+	0.229802i	\(0.0738078\pi\)
							0.395312	+	0.918547i	\(0.370637\pi\)
\(23\)	−99.0330	−	104.969i	−0.897817	−	0.951631i	0.101248	−	0.994861i	\(-0.467717\pi\)
							−0.999065	+	0.0432303i	\(0.986235\pi\)
\(29\)	260.288	−	30.4233i	1.66670	−	0.194809i	0.770184	−	0.637822i	\(-0.220164\pi\)
							0.896515	+	0.443013i	\(0.146090\pi\)
\(31\)	−68.7151	+	16.2858i	−0.398116	+	0.0943552i	−0.424797	−	0.905289i	\(-0.639654\pi\)
							0.0266807	+	0.999644i	\(0.491506\pi\)
\(37\)	8.30159	+	47.0807i	0.0368858	+	0.209190i	0.997680	−	0.0680712i	\(-0.0216845\pi\)
							−0.960795	+	0.277261i	\(0.910573\pi\)
\(41\)	−127.439	+	171.180i	−0.485429	+	0.652044i	−0.975593	−	0.219586i	\(-0.929529\pi\)
							0.490164	+	0.871630i	\(0.336937\pi\)
\(43\)	−2.30007	+	39.4907i	−0.00815715	+	0.140053i	0.991771	+	0.128028i	\(0.0408647\pi\)
							−0.999928	+	0.0120248i	\(0.996172\pi\)
\(47\)	−566.405	−	134.240i	−1.75784	−	0.416616i	−0.779970	−	0.625817i	\(-0.784766\pi\)
							−0.977873	+	0.209200i	\(0.932914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.4.g.b.7.6	✓	252
81.58	even	27	inner	162.4.g.b.139.6	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.4.g.b.7.6	✓	252	1.1	even	1	trivial
162.4.g.b.139.6	yes	252	81.58	even	27	inner