Embedded newform 27.2.e.a.16.1

• Label: 27.2.e.a.16.1
• Level: $27$
• Weight: $2$
• Character: 27.16
• 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			16.1
Root			\(0.500000 + 0.258654i\) of defining polynomial
Character	\(\chi\)	\(=\)	27.16
Dual form			27.2.e.a.22.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62143 - 1.36054i) q^{2} +(-0.986166 - 1.42389i) q^{3} +(0.430663 + 2.44241i) q^{4} +(2.52129 + 0.917674i) q^{5} +(-0.338267 + 3.65046i) q^{6} +(0.168844 - 0.957561i) q^{7} +(0.508086 - 0.880031i) q^{8} +(-1.05495 + 2.80839i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 6 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62143	−	1.36054i	−1.14652	−	0.962046i	−0.146889	−	0.989153i	\(-0.546926\pi\)
							−0.999633	+	0.0271067i	\(0.991371\pi\)
\(3\)	−0.986166	−	1.42389i	−0.569363	−	0.822086i
							\(5\)	2.52129	+	0.917674i	1.12755	+	0.410396i	0.837404	−	0.546585i	\(-0.184072\pi\)
0.290150	+	0.956981i	\(0.406295\pi\)
\(7\)	0.168844	−	0.957561i	0.0638170	−	0.361924i	−0.936130	−	0.351653i	\(-0.885620\pi\)
							0.999947	−	0.0102706i	\(-0.00326930\pi\)
\(11\)	0.297791	−	0.108387i	0.0897872	−	0.0326799i	−0.296736	−	0.954960i	\(-0.595898\pi\)
							0.386523	+	0.922280i	\(0.373676\pi\)
\(13\)	−1.15981	+	0.973200i	−0.321675	+	0.269917i	−0.789297	−	0.614011i	\(-0.789555\pi\)
							0.467623	+	0.883928i	\(0.345111\pi\)
\(17\)	−0.587342	−	1.01731i	−0.142451	−	0.246733i	0.785968	−	0.618267i	\(-0.212165\pi\)
							−0.928419	+	0.371534i	\(0.878832\pi\)
\(19\)	−3.11040	+	5.38737i	−0.713575	+	1.23595i	0.249931	+	0.968264i	\(0.419592\pi\)
							−0.963507	+	0.267685i	\(0.913741\pi\)
\(23\)	0.375556	+	2.12988i	0.0783089	+	0.444112i	0.998601	+	0.0528796i	\(0.0168399\pi\)
							−0.920292	+	0.391232i	\(0.872049\pi\)
\(29\)	−3.37436	−	2.83142i	−0.626602	−	0.525782i	0.273269	−	0.961938i	\(-0.411895\pi\)
							−0.899871	+	0.436156i	\(0.856340\pi\)
\(31\)	−1.50609	−	8.54146i	−0.270502	−	1.53409i	−0.752897	−	0.658138i	\(-0.771344\pi\)
							0.482395	−	0.875954i	\(-0.339767\pi\)
\(37\)	2.23332	+	3.86823i	0.367156	+	0.635933i	0.989120	−	0.147113i	\(-0.0469982\pi\)
							−0.621964	+	0.783046i	\(0.713665\pi\)
\(41\)	4.47767	−	3.75721i	0.699295	−	0.586778i	−0.222278	−	0.974983i	\(-0.571349\pi\)
							0.921573	+	0.388205i	\(0.126905\pi\)
\(43\)	−5.25381	+	1.91223i	−0.801199	+	0.291613i	−0.709983	−	0.704219i	\(-0.751298\pi\)
							−0.0912158	+	0.995831i	\(0.529075\pi\)
\(47\)	−0.429965	+	2.43845i	−0.0627168	+	0.355685i	0.937258	+	0.348636i	\(0.113355\pi\)
							−0.999975	+	0.00704911i	\(0.997756\pi\)