Embedded newform 27.4.e.a.13.2

• Label: 27.4.e.a.13.2
• Level: $27$
• Weight: $4$
• Character: 27.13
• Analytic conductor: $1.593$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.59305157015\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			13.2
Character	\(\chi\)	\(=\)	27.13
Dual form			27.4.e.a.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.592608 - 3.36085i) q^{2} +(-5.15968 - 0.614572i) q^{3} +(-3.42657 + 1.24717i) q^{4} +(-8.41296 - 7.05931i) q^{5} +(0.992184 + 17.7051i) q^{6} +(4.14024 + 1.50692i) q^{7} +(-7.42861 - 12.8667i) q^{8} +(26.2446 + 6.34199i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.592608	−	3.36085i	−0.209519	−	1.18824i	−0.890169	−	0.455631i	\(-0.849414\pi\)
							0.680650	−	0.732608i	\(-0.261697\pi\)
\(3\)	−5.15968	−	0.614572i	−0.992981	−	0.118275i
							\(5\)	−8.41296	−	7.05931i	−0.752478	−	0.631404i	0.183679	−	0.982986i	\(-0.441199\pi\)
−0.936157	+	0.351582i	\(0.885644\pi\)
\(7\)	4.14024	+	1.50692i	0.223552	+	0.0813663i	0.451368	−	0.892338i	\(-0.350936\pi\)
							−0.227816	+	0.973704i	\(0.573158\pi\)
\(11\)	44.1840	−	37.0748i	1.21109	−	1.01622i	0.211846	−	0.977303i	\(-0.432052\pi\)
							0.999242	−	0.0389212i	\(-0.0123921\pi\)
\(13\)	6.80702	−	38.6045i	0.145225	−	0.823613i	−0.821961	−	0.569544i	\(-0.807120\pi\)
							0.967186	−	0.254069i	\(-0.0817690\pi\)
\(17\)	−44.7356	+	77.4844i	−0.638235	+	1.10545i	0.347585	+	0.937648i	\(0.387002\pi\)
							−0.985820	+	0.167806i	\(0.946332\pi\)
\(19\)	29.6770	+	51.4022i	0.358336	+	0.620656i	0.987683	−	0.156468i	\(-0.0500109\pi\)
							−0.629347	+	0.777124i	\(0.716678\pi\)
\(23\)	21.4850	−	7.81991i	0.194780	−	0.0708941i	−0.242788	−	0.970079i	\(-0.578062\pi\)
							0.437568	+	0.899185i	\(0.355840\pi\)
\(29\)	−10.2292	−	58.0124i	−0.0655002	−	0.371470i	−0.999884	−	0.0152033i	\(-0.995160\pi\)
							0.934384	−	0.356267i	\(-0.115951\pi\)
\(31\)	313.847	−	114.231i	1.81834	−	0.661823i	0.822711	−	0.568460i	\(-0.192461\pi\)
							0.995632	−	0.0933628i	\(-0.0297616\pi\)
\(37\)	−47.4944	+	82.2627i	−0.211028	+	0.365511i	−0.952036	−	0.305985i	\(-0.901014\pi\)
							0.741009	+	0.671495i	\(0.234348\pi\)
\(41\)	−62.8116	+	356.222i	−0.239257	+	1.35689i	0.594204	+	0.804314i	\(0.297467\pi\)
							−0.833461	+	0.552578i	\(0.813644\pi\)
\(43\)	361.279	−	303.149i	1.28127	−	1.07511i	0.288198	−	0.957571i	\(-0.406944\pi\)
							0.993068	−	0.117540i	\(-0.0375007\pi\)
\(47\)	−222.580	−	81.0123i	−0.690778	−	0.251423i	−0.0273097	−	0.999627i	\(-0.508694\pi\)
							−0.663468	+	0.748204i	\(0.730916\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.13.2	✓	48
3.2	odd	2		81.4.e.a.10.7		48
9.2	odd	6		243.4.e.b.109.7		48
9.4	even	3		243.4.e.d.190.7		48
9.5	odd	6		243.4.e.a.190.2		48
9.7	even	3		243.4.e.c.109.2		48
27.2	odd	18		81.4.e.a.73.7		48
27.5	odd	18		729.4.a.c.1.19		24
27.7	even	9		243.4.e.c.136.2		48
27.11	odd	18		243.4.e.a.55.2		48
27.16	even	9		243.4.e.d.55.7		48
27.20	odd	18		243.4.e.b.136.7		48
27.22	even	9		729.4.a.d.1.6		24
27.25	even	9	inner	27.4.e.a.25.2	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.13.2	✓	48	1.1	even	1	trivial
27.4.e.a.25.2	yes	48	27.25	even	9	inner
81.4.e.a.10.7		48	3.2	odd	2
81.4.e.a.73.7		48	27.2	odd	18
243.4.e.a.55.2		48	27.11	odd	18
243.4.e.a.190.2		48	9.5	odd	6
243.4.e.b.109.7		48	9.2	odd	6
243.4.e.b.136.7		48	27.20	odd	18
243.4.e.c.109.2		48	9.7	even	3
243.4.e.c.136.2		48	27.7	even	9
243.4.e.d.55.7		48	27.16	even	9
243.4.e.d.190.7		48	9.4	even	3
729.4.a.c.1.19		24	27.5	odd	18
729.4.a.d.1.6		24	27.22	even	9