Embedded newform 81.2.g.a.43.8

• Label: 81.2.g.a.43.8
• Level: $81$
• Weight: $2$
• Character: 81.43
• 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			43.8
Character	\(\chi\)	\(=\)	81.43
Dual form			81.2.g.a.49.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.761699 + 2.54425i) q^{2} +(1.50768 - 0.852593i) q^{3} +(-4.22205 + 2.77689i) q^{4} +(-1.12810 - 2.61524i) q^{5} +(3.31761 + 3.18649i) q^{6} +(-0.111224 + 1.90965i) q^{7} +(-6.21207 - 5.21255i) q^{8} +(1.54617 - 2.57087i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.761699	+	2.54425i	0.538602	+	1.79906i	0.598835	+	0.800873i	\(0.295631\pi\)
							−0.0602322	+	0.998184i	\(0.519184\pi\)
\(3\)	1.50768	−	0.852593i	0.870457	−	0.492245i
							\(5\)	−1.12810	−	2.61524i	−0.504504	−	1.16957i	−0.960169	−	0.279420i	\(-0.909858\pi\)
0.455666	−	0.890151i	\(-0.349401\pi\)
\(7\)	−0.111224	+	1.90965i	−0.0420389	+	0.721779i	0.909775	+	0.415102i	\(0.136254\pi\)
							−0.951814	+	0.306677i	\(0.900783\pi\)
\(11\)	−1.10623	−	1.48593i	−0.333541	−	0.448024i	0.603437	−	0.797411i	\(-0.293797\pi\)
							−0.936979	+	0.349387i	\(0.886390\pi\)
\(13\)	0.0925649	+	0.0981130i	0.0256729	+	0.0272117i	0.740083	−	0.672516i	\(-0.234786\pi\)
							−0.714410	+	0.699727i	\(0.753305\pi\)
\(17\)	0.193799	+	1.09909i	0.0470031	+	0.266568i	0.999248	−	0.0387665i	\(-0.0123428\pi\)
							−0.952245	+	0.305334i	\(0.901232\pi\)
\(19\)	−1.05411	+	5.97816i	−0.241830	+	1.37148i	0.585912	+	0.810375i	\(0.300737\pi\)
							−0.827741	+	0.561110i	\(0.810375\pi\)
\(23\)	−0.0271279	−	0.465768i	−0.00565655	−	0.0971192i	0.994298	−	0.106636i	\(-0.0340078\pi\)
							−0.999955	+	0.00951638i	\(0.996971\pi\)
\(29\)	0.426645	+	0.101117i	0.0792259	+	0.0187769i	0.270038	−	0.962850i	\(-0.412964\pi\)
							−0.190812	+	0.981627i	\(0.561112\pi\)
\(31\)	2.52937	+	1.27030i	0.454289	+	0.228152i	0.661204	−	0.750207i	\(-0.270046\pi\)
							−0.206915	+	0.978359i	\(0.566342\pi\)
\(37\)	−7.77306	−	2.82916i	−1.27788	−	0.465112i	−0.388152	−	0.921595i	\(-0.626886\pi\)
							−0.889732	+	0.456484i	\(0.849109\pi\)
\(41\)	−1.59594	+	5.33081i	−0.249244	+	0.832533i	0.738202	+	0.674579i	\(0.235675\pi\)
							−0.987446	+	0.157954i	\(0.949510\pi\)
\(43\)	7.45009	−	0.870791i	1.13613	−	0.132794i	0.472833	−	0.881152i	\(-0.343231\pi\)
							0.663295	+	0.748358i	\(0.269157\pi\)
\(47\)	−7.62373	+	3.82878i	−1.11204	+	0.558485i	−0.907364	−	0.420345i	\(-0.861909\pi\)
							−0.204671	+	0.978831i	\(0.565613\pi\)