Embedded newform 27.8.e.a.16.11

• Label: 27.8.e.a.16.11
• Level: $27$
• Weight: $8$
• Character: 27.16
• 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			16.11
Character	\(\chi\)	\(=\)	27.16
Dual form			27.8.e.a.22.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41760 + 1.18951i) q^{2} +(28.0806 + 37.3963i) q^{3} +(-21.6323 - 122.683i) q^{4} +(-280.299 - 102.020i) q^{5} +(-4.67608 + 86.4153i) q^{6} +(248.630 - 1410.05i) q^{7} +(233.702 - 404.783i) q^{8} +(-609.959 + 2100.22i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41760	+	1.18951i	0.125300	+	0.105139i	0.703284	−	0.710909i	\(-0.251716\pi\)
							−0.577985	+	0.816048i	\(0.696161\pi\)
\(3\)	28.0806	+	37.3963i	0.600457	+	0.799657i
							\(5\)	−280.299	−	102.020i	−1.00283	−	0.364999i	−0.212153	−	0.977236i	\(-0.568048\pi\)
−0.790674	+	0.612237i	\(0.790270\pi\)
\(7\)	248.630	−	1410.05i	0.273974	−	1.55378i	−0.468227	−	0.883608i	\(-0.655107\pi\)
							0.742201	−	0.670177i	\(-0.233782\pi\)
\(11\)	494.740	−	180.071i	0.112073	−	0.0407914i	−0.285375	−	0.958416i	\(-0.592118\pi\)
							0.397448	+	0.917625i	\(0.369896\pi\)
\(13\)	7010.04	−	5882.12i	0.884950	−	0.742561i	−0.0822410	−	0.996612i	\(-0.526208\pi\)
							0.967191	+	0.254051i	\(0.0817633\pi\)
\(17\)	−14947.1	−	25889.1i	−0.737879	−	1.27804i	−0.953449	−	0.301555i	\(-0.902494\pi\)
							0.215570	−	0.976488i	\(-0.430839\pi\)
\(19\)	2381.18	−	4124.32i	0.0796442	−	0.137948i	−0.823452	−	0.567386i	\(-0.807955\pi\)
							0.903096	+	0.429438i	\(0.141288\pi\)
\(23\)	8062.32	+	45723.7i	0.138170	+	0.783599i	0.972600	+	0.232485i	\(0.0746858\pi\)
							−0.834430	+	0.551114i	\(0.814203\pi\)
\(29\)	146259.	+	122726.i	1.11360	+	0.934421i	0.998264	−	0.0589055i	\(-0.0187611\pi\)
							0.115336	+	0.993327i	\(0.463206\pi\)
\(31\)	14595.3	+	82774.3i	0.0879931	+	0.499033i	0.996671	+	0.0815346i	\(0.0259821\pi\)
							−0.908677	+	0.417499i	\(0.862907\pi\)
\(37\)	−238662.	−	413375.i	−0.774600	−	1.34165i	−0.935019	−	0.354597i	\(-0.884618\pi\)
							0.160419	−	0.987049i	\(-0.448715\pi\)
\(41\)	−105498.	+	88523.6i	−0.239057	+	0.200593i	−0.754443	−	0.656365i	\(-0.772093\pi\)
							0.515386	+	0.856958i	\(0.327649\pi\)
\(43\)	404771.	−	147324.i	0.776371	−	0.282576i	0.0767124	−	0.997053i	\(-0.475558\pi\)
							0.699659	+	0.714477i	\(0.253335\pi\)
\(47\)	−69838.1	+	396071.i	−0.0981182	+	0.556456i	0.895629	+	0.444802i	\(0.146726\pi\)
							−0.993747	+	0.111654i	\(0.964385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.16.11	✓	120
3.2	odd	2		81.8.e.a.46.10		120
27.5	odd	18		81.8.e.a.37.10		120
27.22	even	9	inner	27.8.e.a.22.11	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.11	✓	120	1.1	even	1	trivial
27.8.e.a.22.11	yes	120	27.22	even	9	inner
81.8.e.a.37.10		120	27.5	odd	18
81.8.e.a.46.10		120	3.2	odd	2