Embedded newform 81.2.g.a.4.4

• Label: 81.2.g.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	81.4
Dual form			81.2.g.a.61.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.702679 - 0.0821314i) q^{2} +(1.28612 + 1.16012i) q^{3} +(-1.45908 - 0.345808i) q^{4} +(3.06981 - 1.54172i) q^{5} +(-0.808449 - 0.920826i) q^{6} +(-0.775884 + 2.59163i) q^{7} +(2.32646 + 0.846761i) q^{8} +(0.308226 + 2.98412i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.702679	−	0.0821314i	−0.496869	−	0.0580757i	−0.136032	−	0.990704i	\(-0.543435\pi\)
							−0.360837	+	0.932629i	\(0.617509\pi\)
\(3\)	1.28612	+	1.16012i	0.742544	+	0.669798i
							\(5\)	3.06981	−	1.54172i	1.37286	−	0.689477i	0.399070	−	0.916920i	\(-0.369333\pi\)
0.973791	+	0.227443i	\(0.0730367\pi\)
\(7\)	−0.775884	+	2.59163i	−0.293257	+	0.979545i	0.676698	+	0.736260i	\(0.263410\pi\)
							−0.969955	+	0.243285i	\(0.921775\pi\)
\(11\)	−0.0859074	−	1.47497i	−0.0259021	−	0.444721i	−0.986180	−	0.165677i	\(-0.947019\pi\)
							0.960278	−	0.279045i	\(-0.0900178\pi\)
\(13\)	−3.31531	−	4.45324i	−0.919502	−	1.23511i	−0.971640	−	0.236466i	\(-0.924011\pi\)
							0.0521373	−	0.998640i	\(-0.483397\pi\)
\(17\)	−3.86594	−	3.24391i	−0.937627	−	0.786763i	0.0395437	−	0.999218i	\(-0.487410\pi\)
							−0.977171	+	0.212455i	\(0.931854\pi\)
\(19\)	−3.94119	+	3.30705i	−0.904171	+	0.758690i	−0.971001	−	0.239075i	\(-0.923156\pi\)
							0.0668301	+	0.997764i	\(0.478711\pi\)
\(23\)	−0.497394	−	1.66141i	−0.103714	−	0.346428i	0.890343	−	0.455290i	\(-0.150464\pi\)
							−0.994057	+	0.108862i	\(0.965279\pi\)
\(29\)	0.851040	+	1.97293i	0.158034	+	0.366364i	0.978775	−	0.204936i	\(-0.0656986\pi\)
							−0.820741	+	0.571300i	\(0.806439\pi\)
\(31\)	2.61206	−	2.76862i	0.469139	−	0.497259i	−0.448991	−	0.893536i	\(-0.648217\pi\)
							0.918131	+	0.396277i	\(0.129698\pi\)
\(37\)	−0.891723	+	5.05721i	−0.146598	+	0.831401i	0.819472	+	0.573120i	\(0.194267\pi\)
							−0.966070	+	0.258281i	\(0.916844\pi\)
\(41\)	4.48158	−	0.523822i	0.699906	−	0.0818072i	0.241304	−	0.970449i	\(-0.422425\pi\)
							0.458601	+	0.888642i	\(0.348351\pi\)
\(43\)	−0.688837	+	0.453055i	−0.105047	+	0.0690902i	−0.600945	−	0.799290i	\(-0.705209\pi\)
							0.495899	+	0.868380i	\(0.334839\pi\)
\(47\)	4.60630	+	4.88239i	0.671898	+	0.712170i	0.970201	−	0.242301i	\(-0.0779022\pi\)
							−0.298303	+	0.954471i	\(0.596421\pi\)