Embedded newform 27.2.e.a.4.1

• Label: 27.2.e.a.4.1
• Level: $27$
• Weight: $2$
• Character: 27.4
• Analytic conductor: $0.216$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.215596085457\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	12.0.1952986685049.1
Defining polynomial:	\( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} - 258 x^{3} + 108 x^{2} - 27 x + 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.1
Root			\(0.500000 - 1.27297i\) of defining polynomial
Character	\(\chi\)	\(=\)	27.4
Dual form			27.2.e.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57954 - 0.574906i) q^{2} +(1.45446 - 0.940501i) q^{3} +(0.632343 + 0.530599i) q^{4} +(-0.196143 + 1.11238i) q^{5} +(-2.83808 + 0.649381i) q^{6} +(-2.99441 + 2.51261i) q^{7} +(0.987144 + 1.70978i) q^{8} +(1.23092 - 2.73584i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.57954	−	0.574906i	−1.11690	−	0.406520i	−0.283383	−	0.959007i	\(-0.591457\pi\)
							−0.833521	+	0.552487i	\(0.813679\pi\)
\(3\)	1.45446	−	0.940501i	0.839734	−	0.542999i
							\(5\)	−0.196143	+	1.11238i	−0.0877179	+	0.497473i	0.909019	+	0.416755i	\(0.136833\pi\)
−0.996737	+	0.0807185i	\(0.974279\pi\)
\(7\)	−2.99441	+	2.51261i	−1.13178	+	0.949676i	−0.999139	−	0.0414879i	\(-0.986790\pi\)
							−0.132641	+	0.991164i	\(0.542346\pi\)
\(11\)	−0.324801	−	1.84204i	−0.0979313	−	0.555396i	−0.993810	−	0.111096i	\(-0.964564\pi\)
							0.895878	−	0.444299i	\(-0.146547\pi\)
\(13\)	0.688417	−	0.250563i	0.190932	−	0.0694937i	−0.244784	−	0.969578i	\(-0.578717\pi\)
							0.435717	+	0.900084i	\(0.356495\pi\)
\(17\)	−0.944822	+	1.63648i	−0.229153	+	0.396905i	−0.957557	−	0.288243i	\(-0.906929\pi\)
							0.728404	+	0.685147i	\(0.240262\pi\)
\(19\)	−1.37143	−	2.37538i	−0.314627	−	0.544950i	0.664731	−	0.747083i	\(-0.268546\pi\)
							−0.979358	+	0.202133i	\(0.935213\pi\)
\(23\)	−4.46428	−	3.74597i	−0.930866	−	0.781089i	0.0451066	−	0.998982i	\(-0.485637\pi\)
							−0.975973	+	0.217893i	\(0.930082\pi\)
\(29\)	4.99910	+	1.81953i	0.928310	+	0.337877i	0.761540	−	0.648118i	\(-0.224444\pi\)
							0.166771	+	0.985996i	\(0.446666\pi\)
\(31\)	1.02696	+	0.861722i	0.184447	+	0.154770i	0.730336	−	0.683088i	\(-0.239363\pi\)
							−0.545889	+	0.837858i	\(0.683808\pi\)
\(37\)	−1.69806	+	2.94112i	−0.279159	+	0.483517i	−0.971176	−	0.238364i	\(-0.923389\pi\)
							0.692017	+	0.721881i	\(0.256722\pi\)
\(41\)	−1.68800	+	0.614382i	−0.263621	+	0.0959503i	−0.470449	−	0.882427i	\(-0.655908\pi\)
							0.206828	+	0.978377i	\(0.433686\pi\)
\(43\)	0.873477	+	4.95373i	0.133204	+	0.755437i	0.976094	+	0.217351i	\(0.0697415\pi\)
							−0.842890	+	0.538087i	\(0.819147\pi\)
\(47\)	1.30892	−	1.09832i	0.190926	−	0.160206i	−0.542314	−	0.840176i	\(-0.682452\pi\)
							0.733240	+	0.679970i	\(0.238007\pi\)