Embedded newform 27.5.f.a.5.3

• Label: 27.5.f.a.5.3
• Level: $27$
• Weight: $5$
• Character: 27.5
• Analytic conductor: $2.791$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	27.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79098900326\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	27.5
Dual form			27.5.f.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.66286 - 3.17348i) q^{2} +(-8.41268 + 3.19794i) q^{3} +(-0.201747 + 1.14416i) q^{4} +(12.7013 + 34.8965i) q^{5} +(32.5504 + 18.1818i) q^{6} +(4.71194 + 26.7227i) q^{7} +(-53.2345 + 30.7349i) q^{8} +(60.5463 - 53.8065i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 3 q^{5} + 90 q^{6} - 6 q^{7} - 9 q^{8} - 108 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.66286	−	3.17348i	−0.665716	−	0.793370i	0.322478	−	0.946577i	\(-0.395484\pi\)
							−0.988194	+	0.153207i	\(0.951040\pi\)
\(3\)	−8.41268	+	3.19794i	−0.934742	+	0.355327i
							\(5\)	12.7013	+	34.8965i	0.508052	+	1.39586i	0.883244	+	0.468915i	\(0.155355\pi\)
−0.375191	+	0.926947i	\(0.622423\pi\)
\(7\)	4.71194	+	26.7227i	0.0961621	+	0.545362i	0.994385	+	0.105822i	\(0.0337474\pi\)
							−0.898223	+	0.439540i	\(0.855141\pi\)
\(11\)	−72.2767	+	198.578i	−0.597328	+	1.64114i	0.159248	+	0.987239i	\(0.449093\pi\)
							−0.756576	+	0.653906i	\(0.773129\pi\)
\(13\)	−129.366	−	108.551i	−0.765478	−	0.642312i	0.174069	−	0.984734i	\(-0.444309\pi\)
							−0.939547	+	0.342421i	\(0.888753\pi\)
\(17\)	−230.823	−	133.266i	−0.798695	−	0.461127i	0.0443197	−	0.999017i	\(-0.485888\pi\)
							−0.843015	+	0.537891i	\(0.819221\pi\)
\(19\)	168.179	+	291.294i	0.465869	+	0.806908i	0.999240	−	0.0389728i	\(-0.0124086\pi\)
							−0.533372	+	0.845881i	\(0.679075\pi\)
\(23\)	432.126	+	76.1954i	0.816873	+	0.144037i	0.566446	−	0.824099i	\(-0.308318\pi\)
							0.250427	+	0.968136i	\(0.419429\pi\)
\(29\)	6.73428	+	8.02560i	0.00800747	+	0.00954293i	0.770033	−	0.638004i	\(-0.220240\pi\)
							−0.762026	+	0.647547i	\(0.775795\pi\)
\(31\)	54.6110	−	309.714i	0.0568272	−	0.322283i	−0.943121	−	0.332449i	\(-0.892125\pi\)
							0.999948	+	0.0101662i	\(0.00323605\pi\)
\(37\)	−60.2948	+	104.434i	−0.0440430	+	0.0762847i	−0.887207	−	0.461372i	\(-0.847357\pi\)
							0.843164	+	0.537657i	\(0.180691\pi\)
\(41\)	1335.50	−	1591.59i	0.794470	−	0.946813i	−0.205019	−	0.978758i	\(-0.565726\pi\)
							0.999490	+	0.0319451i	\(0.0101702\pi\)
\(43\)	−441.200	−	160.584i	−0.238615	−	0.0868489i	0.219945	−	0.975512i	\(-0.429412\pi\)
							−0.458560	+	0.888663i	\(0.651635\pi\)
\(47\)	−2757.71	+	486.259i	−1.24840	+	0.220126i	−0.758511	−	0.651660i	\(-0.774073\pi\)
							−0.489887	+	0.871786i	\(0.662962\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.5.f.a.5.3	✓	66
3.2	odd	2		81.5.f.a.44.9		66
27.11	odd	18	inner	27.5.f.a.11.3	yes	66
27.16	even	9		81.5.f.a.35.9		66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.5.f.a.5.3	✓	66	1.1	even	1	trivial
27.5.f.a.11.3	yes	66	27.11	odd	18	inner
81.5.f.a.35.9		66	27.16	even	9
81.5.f.a.44.9		66	3.2	odd	2