Embedded newform 81.2.g.a.4.3

• Label: 81.2.g.a.4.3
• Level: $81$
• Weight: $2$
• Character: 81.4
• 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			4.3
Character	\(\chi\)	\(=\)	81.4
Dual form			81.2.g.a.61.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.907715 - 0.106097i) q^{2} +(-0.350962 + 1.69612i) q^{3} +(-1.13340 - 0.268621i) q^{4} +(-3.40320 + 1.70915i) q^{5} +(0.498526 - 1.50236i) q^{6} +(0.208293 - 0.695746i) q^{7} +(2.71786 + 0.989221i) q^{8} +(-2.75365 - 1.19055i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.907715	−	0.106097i	−0.641851	−	0.0750217i	−0.211060	−	0.977473i	\(-0.567691\pi\)
							−0.430791	+	0.902452i	\(0.641766\pi\)
\(3\)	−0.350962	+	1.69612i	−0.202628	+	0.979256i
							\(5\)	−3.40320	+	1.70915i	−1.52196	+	0.764356i	−0.996267	−	0.0863205i	\(-0.972489\pi\)
−0.525690	+	0.850676i	\(0.676193\pi\)
\(7\)	0.208293	−	0.695746i	0.0787273	−	0.262967i	−0.909560	−	0.415573i	\(-0.863581\pi\)
							0.988287	+	0.152605i	\(0.0487663\pi\)
\(11\)	0.156550	+	2.68786i	0.0472015	+	0.810419i	0.935974	+	0.352070i	\(0.114522\pi\)
							−0.888772	+	0.458349i	\(0.848441\pi\)
\(13\)	2.75139	+	3.69576i	0.763099	+	1.02502i	0.998587	+	0.0531430i	\(0.0169239\pi\)
							−0.235488	+	0.971877i	\(0.575669\pi\)
\(17\)	−0.692702	−	0.581246i	−0.168005	−	0.140973i	0.554909	−	0.831911i	\(-0.312753\pi\)
							−0.722914	+	0.690938i	\(0.757198\pi\)
\(19\)	−1.12072	+	0.940396i	−0.257111	+	0.215742i	−0.762227	−	0.647310i	\(-0.775894\pi\)
							0.505116	+	0.863051i	\(0.331450\pi\)
\(23\)	−1.11889	−	3.73735i	−0.233304	−	0.779291i	−0.991821	−	0.127637i	\(-0.959261\pi\)
							0.758517	−	0.651654i	\(-0.225924\pi\)
\(29\)	2.03616	+	4.72034i	0.378105	+	0.876545i	0.995922	+	0.0902230i	\(0.0287580\pi\)
							−0.617817	+	0.786322i	\(0.711983\pi\)
\(31\)	−6.85388	+	7.26468i	−1.23099	+	1.30478i	−0.293840	+	0.955855i	\(0.594933\pi\)
							−0.937152	+	0.348921i	\(0.886548\pi\)
\(37\)	−0.348559	+	1.97678i	−0.0573028	+	0.324980i	−0.999961	−	0.00877810i	\(-0.997206\pi\)
							0.942659	+	0.333758i	\(0.108317\pi\)
\(41\)	6.51467	−	0.761456i	1.01742	−	0.118919i	0.409000	−	0.912534i	\(-0.365877\pi\)
							0.608421	+	0.793615i	\(0.291803\pi\)
\(43\)	−6.08851	+	4.00448i	−0.928489	+	0.610677i	−0.921113	−	0.389296i	\(-0.872718\pi\)
							−0.00737620	+	0.999973i	\(0.502348\pi\)
\(47\)	3.06916	+	3.25312i	0.447683	+	0.474516i	0.911432	−	0.411450i	\(-0.134978\pi\)
							−0.463749	+	0.885967i	\(0.653496\pi\)