Embedded newform 27.8.e.a.13.15

• Label: 27.8.e.a.13.15
• Level: $27$
• Weight: $8$
• Character: 27.13
• 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			13.15
Character	\(\chi\)	\(=\)	27.13
Dual form			27.8.e.a.25.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.28109 + 12.9367i) q^{2} +(-25.7580 - 39.0324i) q^{3} +(-41.8741 + 15.2409i) q^{4} +(-86.4596 - 72.5482i) q^{5} +(446.194 - 422.260i) q^{6} +(1303.70 + 474.509i) q^{7} +(548.035 + 949.224i) q^{8} +(-860.050 + 2010.79i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.28109	+	12.9367i	0.201622	+	1.14345i	0.902668	+	0.430338i	\(0.141606\pi\)
							−0.701046	+	0.713116i	\(0.747283\pi\)
\(3\)	−25.7580	−	39.0324i	−0.550792	−	0.834642i
							\(5\)	−86.4596	−	72.5482i	−0.309327	−	0.259556i	0.474887	−	0.880047i	\(-0.342489\pi\)
−0.784214	+	0.620491i	\(0.786934\pi\)
\(7\)	1303.70	+	474.509i	1.43660	+	0.522879i	0.938815	−	0.344423i	\(-0.111925\pi\)
							0.497783	+	0.867301i	\(0.334147\pi\)
\(11\)	5485.58	−	4602.95i	1.24265	−	1.04271i	0.245335	−	0.969438i	\(-0.421102\pi\)
							0.997312	−	0.0732670i	\(-0.0233425\pi\)
\(13\)	−1148.28	+	6512.23i	−0.144960	+	0.822107i	0.822440	+	0.568852i	\(0.192612\pi\)
							−0.967400	+	0.253255i	\(0.918499\pi\)
\(17\)	11545.5	−	19997.4i	0.569956	−	0.987192i	−0.426614	−	0.904434i	\(-0.640294\pi\)
							0.996570	−	0.0827582i	\(-0.0263729\pi\)
\(19\)	22708.1	+	39331.6i	0.759528	+	1.31554i	0.943092	+	0.332533i	\(0.107903\pi\)
							−0.183564	+	0.983008i	\(0.558763\pi\)
\(23\)	37985.6	−	13825.6i	0.650986	−	0.236940i	0.00464640	−	0.999989i	\(-0.498521\pi\)
							0.646340	+	0.763050i	\(0.276299\pi\)
\(29\)	−10381.3	−	58875.1i	−0.0790419	−	0.448269i	−0.998484	−	0.0550435i	\(-0.982470\pi\)
							0.919442	−	0.393226i	\(-0.128641\pi\)
\(31\)	−122119.	+	44447.8i	−0.736239	+	0.267969i	−0.682803	−	0.730602i	\(-0.739239\pi\)
							−0.0534354	+	0.998571i	\(0.517017\pi\)
\(37\)	−81118.1	+	140501.i	−0.263276	+	0.456008i	−0.967111	−	0.254356i	\(-0.918136\pi\)
							0.703834	+	0.710364i	\(0.251470\pi\)
\(41\)	−78584.0	+	445672.i	−0.178070	+	1.00989i	0.756470	+	0.654028i	\(0.226922\pi\)
							−0.934540	+	0.355857i	\(0.884189\pi\)
\(43\)	409120.	−	343292.i	0.784714	−	0.658453i	−0.159717	−	0.987163i	\(-0.551058\pi\)
							0.944431	+	0.328710i	\(0.106614\pi\)
\(47\)	−796884.	−	290042.i	−1.11957	−	0.407491i	−0.285077	−	0.958505i	\(-0.592019\pi\)
							−0.834496	+	0.551013i	\(0.814241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.13.15	✓	120
3.2	odd	2		81.8.e.a.10.6		120
27.2	odd	18		81.8.e.a.73.6		120
27.25	even	9	inner	27.8.e.a.25.15	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.13.15	✓	120	1.1	even	1	trivial
27.8.e.a.25.15	yes	120	27.25	even	9	inner
81.8.e.a.10.6		120	3.2	odd	2
81.8.e.a.73.6		120	27.2	odd	18