Embedded newform 27.8.e.a.4.16

• Label: 27.8.e.a.4.16
• Level: $27$
• Weight: $8$
• Character: 27.4
• Analytic conductor: $8.434$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.43439568807\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(20\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.16
Character	\(\chi\)	\(=\)	27.4
Dual form			27.8.e.a.7.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(11.7328 + 4.27037i) q^{2} +(-44.8117 + 13.3756i) q^{3} +(21.3678 + 17.9297i) q^{4} +(58.4278 - 331.360i) q^{5} +(-582.884 - 34.4297i) q^{6} +(-23.9260 + 20.0763i) q^{7} +(-624.951 - 1082.45i) q^{8} +(1829.18 - 1198.77i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	11.7328	+	4.27037i	1.03704	+	0.377451i	0.803757	−	0.594958i	\(-0.202831\pi\)
							0.233282	+	0.972409i	\(0.425053\pi\)
\(3\)	−44.8117	+	13.3756i	−0.958225	+	0.286016i
							\(5\)	58.4278	−	331.360i	0.209038	−	1.18551i	−0.681921	−	0.731426i	\(-0.738855\pi\)
0.890958	−	0.454085i	\(-0.150034\pi\)
\(7\)	−23.9260	+	20.0763i	−0.0263649	+	0.0221228i	−0.655875	−	0.754870i	\(-0.727700\pi\)
							0.629510	+	0.776992i	\(0.283256\pi\)
\(11\)	−823.923	−	4672.70i	−0.186643	−	1.05851i	−0.923826	−	0.382813i	\(-0.874955\pi\)
							0.737183	−	0.675693i	\(-0.236156\pi\)
\(13\)	−1318.00	+	479.713i	−0.166385	+	0.0605592i	−0.423870	−	0.905723i	\(-0.639329\pi\)
							0.257485	+	0.966282i	\(0.417106\pi\)
\(17\)	−8979.65	+	15553.2i	−0.443290	+	0.767801i	−0.997931	−	0.0642886i	\(-0.979522\pi\)
							0.554641	+	0.832090i	\(0.312856\pi\)
\(19\)	−291.761	−	505.346i	−0.00975866	−	0.0169025i	0.861105	−	0.508427i	\(-0.169773\pi\)
							−0.870863	+	0.491525i	\(0.836440\pi\)
\(23\)	−70143.6	−	58857.5i	−1.20210	−	1.00868i	−0.999568	−	0.0294069i	\(-0.990638\pi\)
							−0.202533	−	0.979275i	\(-0.564917\pi\)
\(29\)	165286.	+	60159.4i	1.25847	+	0.458047i	0.883257	−	0.468890i	\(-0.155346\pi\)
							0.375218	+	0.926937i	\(0.377568\pi\)
\(31\)	26748.2	+	22444.4i	0.161261	+	0.135314i	0.719847	−	0.694132i	\(-0.244212\pi\)
							−0.558587	+	0.829446i	\(0.688656\pi\)
\(37\)	−147008.	+	254625.i	−0.477127	+	0.826408i	−0.999656	−	0.0262129i	\(-0.991655\pi\)
							0.522529	+	0.852621i	\(0.324989\pi\)
\(41\)	765563.	−	278642.i	1.73475	−	0.631398i	0.735801	−	0.677198i	\(-0.236806\pi\)
							0.998951	+	0.0458001i	\(0.0145837\pi\)
\(43\)	−117769.	−	667900.i	−0.225887	−	1.28107i	−0.860983	−	0.508633i	\(-0.830151\pi\)
							0.635097	−	0.772433i	\(-0.280960\pi\)
\(47\)	717373.	−	601947.i	1.00786	−	0.845699i	0.0198102	−	0.999804i	\(-0.493694\pi\)
							0.988055	+	0.154105i	\(0.0492494\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.4.16	✓	120
3.2	odd	2		81.8.e.a.64.5		120
27.7	even	9	inner	27.8.e.a.7.16	yes	120
27.20	odd	18		81.8.e.a.19.5		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.16	✓	120	1.1	even	1	trivial
27.8.e.a.7.16	yes	120	27.7	even	9	inner
81.8.e.a.19.5		120	27.20	odd	18
81.8.e.a.64.5		120	3.2	odd	2