Embedded newform 27.5.f.a.2.5

• Label: 27.5.f.a.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	27.2
Dual form			27.5.f.a.14.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.38785 - 0.421042i) q^{2} +(1.69434 + 8.83907i) q^{3} +(-9.51055 - 3.46156i) q^{4} +(-12.2061 - 14.5467i) q^{5} +(-0.324205 - 21.8197i) q^{6} +(-77.2633 + 28.1215i) q^{7} +(54.8497 + 31.6675i) q^{8} +(-75.2584 + 29.9528i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.38785	−	0.421042i	−0.596962	−	0.105260i	−0.133000	−	0.991116i	\(-0.542461\pi\)
							−0.463961	+	0.885856i	\(0.653572\pi\)
\(3\)	1.69434	+	8.83907i	0.188260	+	0.982119i
							\(5\)	−12.2061	−	14.5467i	−0.488245	−	0.581868i	0.464525	−	0.885560i	\(-0.346225\pi\)
−0.952770	+	0.303692i	\(0.901781\pi\)
\(7\)	−77.2633	+	28.1215i	−1.57680	+	0.573909i	−0.974505	−	0.224364i	\(-0.927970\pi\)
							−0.602296	+	0.798273i	\(0.705747\pi\)
\(11\)	12.1711	−	14.5050i	0.100588	−	0.119876i	−0.713402	−	0.700755i	\(-0.752847\pi\)
							0.813990	+	0.580879i	\(0.197291\pi\)
\(13\)	3.97248	+	22.5290i	0.0235058	+	0.133308i	0.994302	−	0.106596i	\(-0.0339952\pi\)
							−0.970797	+	0.239904i	\(0.922884\pi\)
\(17\)	324.171	−	187.160i	1.12170	−	0.647612i	0.179864	−	0.983692i	\(-0.442434\pi\)
							0.941834	+	0.336079i	\(0.109101\pi\)
\(19\)	−252.115	+	436.676i	−0.698380	+	1.20963i	0.270648	+	0.962678i	\(0.412762\pi\)
							−0.969028	+	0.246951i	\(0.920571\pi\)
\(23\)	−244.422	+	671.544i	−0.462045	+	1.26946i	0.461899	+	0.886933i	\(0.347168\pi\)
							−0.923944	+	0.382527i	\(0.875054\pi\)
\(29\)	−361.455	−	63.7342i	−0.429791	−	0.0757838i	−0.0454321	−	0.998967i	\(-0.514466\pi\)
							−0.384359	+	0.923184i	\(0.625578\pi\)
\(31\)	−812.225	−	295.626i	−0.845187	−	0.307623i	−0.117111	−	0.993119i	\(-0.537363\pi\)
							−0.728077	+	0.685496i	\(0.759585\pi\)
\(37\)	747.404	+	1294.54i	0.545949	+	0.945612i	0.998547	+	0.0538968i	\(0.0171642\pi\)
							−0.452597	+	0.891715i	\(0.649502\pi\)
\(41\)	−2056.90	+	362.687i	−1.22362	+	0.215757i	−0.747882	−	0.663832i	\(-0.768929\pi\)
							−0.475735	+	0.879589i	\(0.657818\pi\)
\(43\)	−208.251	−	174.743i	−0.112629	−	0.0945070i	0.584734	−	0.811225i	\(-0.301199\pi\)
							−0.697363	+	0.716718i	\(0.745643\pi\)
\(47\)	−646.583	−	1776.47i	−0.292704	−	0.804197i	−0.995669	−	0.0929724i	\(-0.970363\pi\)
							0.702965	−	0.711224i	\(-0.251859\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.f.a.2.5	✓	66
3.2	odd	2		81.5.f.a.8.7		66
27.13	even	9		81.5.f.a.71.7		66
27.14	odd	18	inner	27.5.f.a.14.5	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.2.5	✓	66	1.1	even	1	trivial
27.5.f.a.14.5	yes	66	27.14	odd	18	inner
81.5.f.a.8.7		66	3.2	odd	2
81.5.f.a.71.7		66	27.13	even	9