Embedded newform 27.8.e.a.7.16

• Label: 27.8.e.a.7.16
• Level: $27$
• Weight: $8$
• Character: 27.7
• 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			7.16
Character	\(\chi\)	\(=\)	27.7
Dual form			27.8.e.a.4.16

$q$-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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.4.16	✓	120	27.4	even	9	inner
27.8.e.a.7.16	yes	120	1.1	even	1	trivial
81.8.e.a.19.5		120	3.2	odd	2
81.8.e.a.64.5		120	27.23	odd	18