Embedded newform 27.8.e.a.16.18

• Label: 27.8.e.a.16.18
• 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			16.18
Character	\(\chi\)	\(=\)	27.16
Dual form			27.8.e.a.22.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.6088 + 12.2582i) q^{2} +(-8.64267 + 45.9598i) q^{3} +(40.9257 + 232.101i) q^{4} +(357.226 + 130.020i) q^{5} +(-689.646 + 565.474i) q^{6} +(177.800 - 1008.35i) q^{7} +(-1026.77 + 1778.42i) q^{8} +(-2037.61 - 794.431i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 219 q^{5} + 1386 q^{6} - 6 q^{7} - 4611 q^{8} - 1728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.6088	+	12.2582i	1.29125	+	1.08349i	0.991587	+	0.129440i	\(0.0413181\pi\)
							0.299661	+	0.954046i	\(0.403126\pi\)
\(3\)	−8.64267	+	45.9598i	−0.184809	+	0.982774i
							\(5\)	357.226	+	130.020i	1.27805	+	0.465173i	0.889788	−	0.456375i	\(-0.150852\pi\)
0.388265	+	0.921548i	\(0.373075\pi\)
\(7\)	177.800	−	1008.35i	0.195924	−	1.11114i	−0.715173	−	0.698948i	\(-0.753652\pi\)
							0.911097	−	0.412192i	\(-0.135237\pi\)
\(11\)	−5223.41	+	1901.17i	−1.18326	+	0.430671i	−0.857352	−	0.514730i	\(-0.827892\pi\)
							−0.325908	+	0.945402i	\(0.605670\pi\)
\(13\)	7027.32	−	5896.62i	0.887132	−	0.744392i	−0.0805010	−	0.996755i	\(-0.525652\pi\)
							0.967633	+	0.252363i	\(0.0812076\pi\)
\(17\)	801.156	+	1387.64i	0.0395500	+	0.0685025i	0.885123	−	0.465358i	\(-0.154074\pi\)
							−0.845573	+	0.533860i	\(0.820741\pi\)
\(19\)	15337.1	−	26564.6i	0.512986	−	0.888518i	−0.486901	−	0.873457i	\(-0.661873\pi\)
							0.999887	−	0.0150602i	\(-0.00479400\pi\)
\(23\)	18756.1	+	106371.i	0.321437	+	1.82296i	0.533612	+	0.845730i	\(0.320834\pi\)
							−0.212174	+	0.977232i	\(0.568054\pi\)
\(29\)	−122778.	−	103023.i	−0.934820	−	0.784407i	0.0418564	−	0.999124i	\(-0.486673\pi\)
							−0.976676	+	0.214716i	\(0.931117\pi\)
\(31\)	1239.36	+	7028.76i	0.00747191	+	0.0423753i	0.988316	−	0.152420i	\(-0.0487067\pi\)
							−0.980844	+	0.194796i	\(0.937596\pi\)
\(37\)	15976.7	+	27672.4i	0.0518537	+	0.0898133i	0.890787	−	0.454421i	\(-0.150154\pi\)
							−0.838933	+	0.544234i	\(0.816820\pi\)
\(41\)	2334.49	−	1958.87i	0.00528992	−	0.00443877i	−0.640139	−	0.768259i	\(-0.721123\pi\)
							0.645429	+	0.763820i	\(0.276679\pi\)
\(43\)	−378626.	+	137809.i	−0.726225	+	0.264324i	−0.678566	−	0.734540i	\(-0.737398\pi\)
							−0.0476587	+	0.998864i	\(0.515176\pi\)
\(47\)	−44172.8	+	250516.i	−0.0620601	+	0.351960i	0.937927	+	0.346833i	\(0.112743\pi\)
							−0.999987	+	0.00512692i	\(0.998368\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.8.e.a.16.18	✓	120
3.2	odd	2		81.8.e.a.46.3		120
27.5	odd	18		81.8.e.a.37.3		120
27.22	even	9	inner	27.8.e.a.22.18	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.8.e.a.16.18	✓	120	1.1	even	1	trivial
27.8.e.a.22.18	yes	120	27.22	even	9	inner
81.8.e.a.37.3		120	27.5	odd	18
81.8.e.a.46.3		120	3.2	odd	2