Embedded newform 27.5.f.a.2.8

• Label: 27.5.f.a.2.8
• Level: $27$
• Weight: $5$
• Character: 27.2
• Analytic conductor: $2.791$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79098900326\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			2.8
Character	\(\chi\)	\(=\)	27.2
Dual form			27.5.f.a.14.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.89555 + 0.510564i) q^{2} +(-3.30987 + 8.36927i) q^{3} +(-6.91152 - 2.51559i) q^{4} +(31.8237 + 37.9260i) q^{5} +(-13.8570 + 22.5438i) q^{6} +(28.2667 - 10.2882i) q^{7} +(-59.4692 - 34.3346i) q^{8} +(-59.0895 - 55.4025i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.89555	+	0.510564i	0.723889	+	0.127641i	0.523439	−	0.852063i	\(-0.324649\pi\)
							0.200449	+	0.979704i	\(0.435760\pi\)
\(3\)	−3.30987	+	8.36927i	−0.367764	+	0.929919i
							\(5\)	31.8237	+	37.9260i	1.27295	+	1.51704i	0.743573	+	0.668655i	\(0.233130\pi\)
0.529376	+	0.848387i	\(0.322426\pi\)
\(7\)	28.2667	−	10.2882i	0.576871	−	0.209964i	−0.0370743	−	0.999313i	\(-0.511804\pi\)
							0.613945	+	0.789349i	\(0.289582\pi\)
\(11\)	75.7716	−	90.3011i	0.626211	−	0.746290i	−0.355914	−	0.934519i	\(-0.615830\pi\)
							0.982125	+	0.188229i	\(0.0602747\pi\)
\(13\)	3.56465	+	20.2161i	0.0210926	+	0.119622i	0.993536	−	0.113517i	\(-0.0362115\pi\)
							−0.972443	+	0.233139i	\(0.925100\pi\)
\(17\)	203.984	−	117.770i	0.705827	−	0.407509i	−0.103687	−	0.994610i	\(-0.533064\pi\)
							0.809514	+	0.587101i	\(0.199731\pi\)
\(19\)	47.7809	−	82.7590i	0.132357	−	0.229249i	−0.792228	−	0.610226i	\(-0.791079\pi\)
							0.924585	+	0.380976i	\(0.124412\pi\)
\(23\)	85.2078	−	234.107i	0.161073	−	0.442545i	−0.832732	−	0.553676i	\(-0.813225\pi\)
							0.993806	+	0.111130i	\(0.0354470\pi\)
\(29\)	−1150.78	−	202.914i	−1.36835	−	0.241277i	−0.559274	−	0.828983i	\(-0.688920\pi\)
							−0.809074	+	0.587707i	\(0.800031\pi\)
\(31\)	562.326	+	204.670i	0.585147	+	0.212976i	0.617593	−	0.786497i	\(-0.288108\pi\)
							−0.0324468	+	0.999473i	\(0.510330\pi\)
\(37\)	−123.627	−	214.128i	−0.0903047	−	0.156412i	0.817335	−	0.576163i	\(-0.195451\pi\)
							−0.907639	+	0.419751i	\(0.862117\pi\)
\(41\)	−508.880	+	89.7293i	−0.302725	+	0.0533785i	−0.322947	−	0.946417i	\(-0.604674\pi\)
							0.0202226	+	0.999796i	\(0.493562\pi\)
\(43\)	−1990.16	−	1669.95i	−1.07635	−	0.903161i	−0.0807334	−	0.996736i	\(-0.525726\pi\)
							−0.995612	+	0.0935745i	\(0.970171\pi\)
\(47\)	−176.156	−	483.984i	−0.0797446	−	0.219097i	0.893413	−	0.449236i	\(-0.148304\pi\)
							−0.973158	+	0.230140i	\(0.926082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.f.a.2.8	✓	66
3.2	odd	2		81.5.f.a.8.4		66
27.13	even	9		81.5.f.a.71.4		66
27.14	odd	18	inner	27.5.f.a.14.8	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.2.8	✓	66	1.1	even	1	trivial
27.5.f.a.14.8	yes	66	27.14	odd	18	inner
81.5.f.a.8.4		66	3.2	odd	2
81.5.f.a.71.4		66	27.13	even	9