Embedded newform 81.2.g.a.40.6

• Label: 81.2.g.a.40.6
• Level: $81$
• Weight: $2$
• Character: 81.40
• 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			40.6
Character	\(\chi\)	\(=\)	81.40
Dual form			81.2.g.a.79.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0620019 + 1.06453i) q^{2} +(-0.676807 + 1.59434i) q^{3} +(0.857095 - 0.100180i) q^{4} +(-0.830696 - 0.196879i) q^{5} +(-1.73919 - 0.621630i) q^{6} +(-0.171051 + 0.229762i) q^{7} +(0.530120 + 3.00646i) q^{8} +(-2.08386 - 2.15813i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 18 q^{5} - 18 q^{6} - 18 q^{7} - 18 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0620019	+	1.06453i	0.0438419	+	0.752737i	0.946565	+	0.322514i	\(0.104528\pi\)
							−0.902723	+	0.430223i	\(0.858435\pi\)
\(3\)	−0.676807	+	1.59434i	−0.390755	+	0.920495i
							\(5\)	−0.830696	−	0.196879i	−0.371499	−	0.0880468i	0.0406256	−	0.999174i	\(-0.487065\pi\)
−0.412124	+	0.911128i	\(0.635213\pi\)
\(7\)	−0.171051	+	0.229762i	−0.0646514	+	0.0868419i	−0.833273	−	0.552862i	\(-0.813536\pi\)
							0.768621	+	0.639704i	\(0.220943\pi\)
\(11\)	−1.24185	−	1.31629i	−0.374433	−	0.396875i	0.512566	−	0.858648i	\(-0.328695\pi\)
							−0.886998	+	0.461773i	\(0.847214\pi\)
\(13\)	4.99610	−	2.50914i	1.38567	−	0.695910i	0.409378	−	0.912365i	\(-0.365746\pi\)
							0.976292	+	0.216455i	\(0.0694496\pi\)
\(17\)	4.42908	−	1.61205i	1.07421	−	0.390980i	0.256460	−	0.966555i	\(-0.417444\pi\)
							0.817749	+	0.575574i	\(0.195222\pi\)
\(19\)	−1.75427	−	0.638503i	−0.402458	−	0.146483i	0.132857	−	0.991135i	\(-0.457585\pi\)
							−0.535315	+	0.844652i	\(0.679807\pi\)
\(23\)	−0.809320	−	1.08711i	−0.168755	−	0.226677i	0.709698	−	0.704506i	\(-0.248831\pi\)
							−0.878453	+	0.477829i	\(0.841424\pi\)
\(29\)	−5.00267	+	3.29031i	−0.928973	+	0.610995i	−0.921248	−	0.388975i	\(-0.872829\pi\)
							−0.00772467	+	0.999970i	\(0.502459\pi\)
\(31\)	−1.68608	+	3.90878i	−0.302830	+	0.702038i	−0.999888	−	0.0149767i	\(-0.995233\pi\)
							0.697058	+	0.717014i	\(0.254492\pi\)
\(37\)	−5.46107	−	4.58238i	−0.897794	−	0.753339i	0.0719637	−	0.997407i	\(-0.477073\pi\)
							−0.969758	+	0.244068i	\(0.921518\pi\)
\(41\)	0.667300	−	11.4571i	0.104215	−	1.78930i	−0.390862	−	0.920449i	\(-0.627823\pi\)
							0.495077	−	0.868849i	\(-0.335140\pi\)
\(43\)	−3.64800	+	12.1852i	−0.556315	+	1.85822i	−0.0447098	+	0.999000i	\(0.514236\pi\)
							−0.511605	+	0.859221i	\(0.670949\pi\)
\(47\)	−0.811830	−	1.88203i	−0.118418	−	0.274523i	0.848690	−	0.528891i	\(-0.177392\pi\)
							−0.967108	+	0.254368i	\(0.918133\pi\)