Embedded newform 27.5.f.a.2.3

• Label: 27.5.f.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	27.2
Dual form			27.5.f.a.14.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.35935 - 0.768672i) q^{2} +(-5.98497 - 6.72162i) q^{3} +(3.37802 + 1.22950i) q^{4} +(27.2417 + 32.4654i) q^{5} +(20.9239 + 33.9024i) q^{6} +(-49.1081 + 17.8739i) q^{7} +(47.5559 + 27.4564i) q^{8} +(-9.36028 + 80.4573i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.35935	−	0.768672i	−1.08984	−	0.192168i	−0.400276	−	0.916395i	\(-0.631086\pi\)
							−0.689562	+	0.724227i	\(0.742197\pi\)
\(3\)	−5.98497	−	6.72162i	−0.664997	−	0.746846i
							\(5\)	27.2417	+	32.4654i	1.08967	+	1.29862i	0.951311	+	0.308233i	\(0.0997375\pi\)
0.138357	+	0.990382i	\(0.455818\pi\)
\(7\)	−49.1081	+	17.8739i	−1.00221	+	0.364773i	−0.790435	−	0.612546i	\(-0.790145\pi\)
							−0.211771	+	0.977319i	\(0.567923\pi\)
\(11\)	11.4420	−	13.6360i	0.0945619	−	0.112694i	−0.716688	−	0.697394i	\(-0.754343\pi\)
							0.811250	+	0.584699i	\(0.198787\pi\)
\(13\)	21.2681	+	120.617i	0.125847	+	0.713713i	0.980801	+	0.195010i	\(0.0624739\pi\)
							−0.854954	+	0.518703i	\(0.826415\pi\)
\(17\)	−377.223	+	217.790i	−1.30527	+	0.753597i	−0.981303	−	0.192471i	\(-0.938350\pi\)
							−0.323966	+	0.946069i	\(0.605016\pi\)
\(19\)	65.4306	−	113.329i	0.181248	−	0.313931i	−0.761058	−	0.648684i	\(-0.775320\pi\)
							0.942306	+	0.334753i	\(0.108653\pi\)
\(23\)	−240.426	+	660.564i	−0.454491	+	1.24870i	0.475041	+	0.879963i	\(0.342433\pi\)
							−0.929532	+	0.368740i	\(0.879789\pi\)
\(29\)	809.117	+	142.669i	0.962090	+	0.169642i	0.632567	−	0.774505i	\(-0.282001\pi\)
							0.329522	+	0.944148i	\(0.393112\pi\)
\(31\)	−367.484	−	133.753i	−0.382397	−	0.139181i	0.143667	−	0.989626i	\(-0.454111\pi\)
							−0.526064	+	0.850445i	\(0.676333\pi\)
\(37\)	−905.030	−	1567.56i	−0.661088	−	1.14504i	−0.980330	−	0.197365i	\(-0.936762\pi\)
							0.319242	−	0.947673i	\(-0.396572\pi\)
\(41\)	1779.62	−	313.795i	1.05867	−	0.186672i	0.382903	−	0.923789i	\(-0.374924\pi\)
							0.675766	+	0.737117i	\(0.263813\pi\)
\(43\)	201.781	+	169.314i	0.109130	+	0.0915706i	0.695720	−	0.718313i	\(-0.255086\pi\)
							−0.586590	+	0.809884i	\(0.699530\pi\)
\(47\)	31.4023	+	86.2770i	0.0142156	+	0.0390570i	0.946597	−	0.322418i	\(-0.104496\pi\)
							−0.932382	+	0.361475i	\(0.882273\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.f.a.2.3	✓	66
3.2	odd	2		81.5.f.a.8.9		66
27.13	even	9		81.5.f.a.71.9		66
27.14	odd	18	inner	27.5.f.a.14.3	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.2.3	✓	66	1.1	even	1	trivial
27.5.f.a.14.3	yes	66	27.14	odd	18	inner
81.5.f.a.8.9		66	3.2	odd	2
81.5.f.a.71.9		66	27.13	even	9