Embedded newform 81.2.g.a.13.7

• Label: 81.2.g.a.13.7
• Level: $81$
• Weight: $2$
• Character: 81.13
• Analytic conductor: $0.647$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	81.g (of order \(27\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.646788256372\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(8\) over \(\Q(\zeta_{27})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label			13.7
Character	\(\chi\)	\(=\)	81.13
Dual form			81.2.g.a.25.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38037 + 0.693250i) q^{2} +(-0.395594 - 1.68627i) q^{3} +(0.230520 + 0.309643i) q^{4} +(0.145961 + 0.487543i) q^{5} +(0.622938 - 2.60193i) q^{6} +(1.57975 + 3.66226i) q^{7} +(-0.432916 - 2.45519i) q^{8} +(-2.68701 + 1.33416i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38037	+	0.693250i	0.976072	+	0.490202i	0.863996	−	0.503498i	\(-0.167954\pi\)
							0.112076	+	0.993700i	\(0.464250\pi\)
\(3\)	−0.395594	−	1.68627i	−0.228396	−	0.973568i
							\(5\)	0.145961	+	0.487543i	0.0652757	+	0.218036i	0.984404	−	0.175922i	\(-0.0562907\pi\)
−0.919128	+	0.393958i	\(0.871106\pi\)
\(7\)	1.57975	+	3.66226i	0.597088	+	1.38420i	0.901963	+	0.431813i	\(0.142126\pi\)
							−0.304876	+	0.952392i	\(0.598615\pi\)
\(11\)	−3.72537	+	0.882929i	−1.12324	+	0.266213i	−0.749946	−	0.661499i	\(-0.769921\pi\)
							−0.373296	+	0.927712i	\(0.621772\pi\)
\(13\)	−2.60109	−	1.71076i	−0.721412	−	0.474480i	0.134952	−	0.990852i	\(-0.456912\pi\)
							−0.856365	+	0.516372i	\(0.827282\pi\)
\(17\)	−1.41129	+	0.513668i	−0.342288	+	0.124583i	−0.507444	−	0.861685i	\(-0.669410\pi\)
							0.165156	+	0.986268i	\(0.447187\pi\)
\(19\)	6.30242	+	2.29389i	1.44587	+	0.526255i	0.941436	−	0.337192i	\(-0.109477\pi\)
							0.504439	+	0.863447i	\(0.331699\pi\)
\(23\)	−0.469931	+	1.08942i	−0.0979874	+	0.227160i	−0.960158	−	0.279458i	\(-0.909845\pi\)
							0.862171	+	0.506618i	\(0.169105\pi\)
\(29\)	−0.402098	−	6.90375i	−0.0746677	−	1.28199i	−0.802671	−	0.596422i	\(-0.796589\pi\)
							0.728004	−	0.685573i	\(-0.240448\pi\)
\(31\)	0.0460395	−	0.00538124i	0.00826893	−	0.000966499i	−0.111957	−	0.993713i	\(-0.535712\pi\)
							0.120226	+	0.992747i	\(0.461638\pi\)
\(37\)	−2.33905	−	1.96269i	−0.384537	−	0.322665i	0.429943	−	0.902856i	\(-0.358533\pi\)
							−0.814481	+	0.580191i	\(0.802978\pi\)
\(41\)	−5.30934	+	2.66645i	−0.829180	+	0.416430i	−0.812167	−	0.583425i	\(-0.801712\pi\)
							−0.0170132	+	0.999855i	\(0.505416\pi\)
\(43\)	−0.163456	−	0.173253i	−0.0249268	−	0.0264208i	0.714790	−	0.699339i	\(-0.246522\pi\)
							−0.739717	+	0.672918i	\(0.765041\pi\)
\(47\)	4.70556	+	0.550002i	0.686377	+	0.0802260i	0.452131	−	0.891951i	\(-0.350664\pi\)
							0.234246	+	0.972177i	\(0.424738\pi\)