Embedded newform 27.4.e.a.25.8

• Label: 27.4.e.a.25.8
• Level: $27$
• Weight: $4$
• Character: 27.25
• Analytic conductor: $1.593$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.59305157015\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			25.8
Character	\(\chi\)	\(=\)	27.25
Dual form			27.4.e.a.13.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.904728 - 5.13097i) q^{2} +(-4.65940 + 2.30000i) q^{3} +(-17.9907 - 6.54810i) q^{4} +(10.7601 - 9.02881i) q^{5} +(7.58576 + 25.9881i) q^{6} +(9.67890 - 3.52283i) q^{7} +(-29.0343 + 50.2889i) q^{8} +(16.4200 - 21.4333i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.904728	−	5.13097i	0.319870	−	1.81407i	−0.223645	−	0.974671i	\(-0.571796\pi\)
							0.543514	−	0.839400i	\(-0.317093\pi\)
\(3\)	−4.65940	+	2.30000i	−0.896701	+	0.442636i
							\(5\)	10.7601	−	9.02881i	0.962414	−	0.807562i	−0.0189297	−	0.999821i	\(-0.506026\pi\)
0.981344	+	0.192259i	\(0.0615814\pi\)
\(7\)	9.67890	−	3.52283i	0.522612	−	0.190215i	−0.0672248	−	0.997738i	\(-0.521414\pi\)
							0.589836	+	0.807523i	\(0.299192\pi\)
\(11\)	21.5580	+	18.0893i	0.590908	+	0.495831i	0.888509	−	0.458860i	\(-0.151742\pi\)
							−0.297601	+	0.954690i	\(0.596186\pi\)
\(13\)	−1.35031	−	7.65800i	−0.0288084	−	0.163381i	0.967010	−	0.254740i	\(-0.0819899\pi\)
							−0.995818	+	0.0913595i	\(0.970879\pi\)
\(17\)	4.59856	+	7.96494i	0.0656068	+	0.113634i	0.896963	−	0.442106i	\(-0.145768\pi\)
							−0.831356	+	0.555740i	\(0.812435\pi\)
\(19\)	−46.7770	+	81.0201i	−0.564809	+	0.978278i	0.432258	+	0.901750i	\(0.357717\pi\)
							−0.997067	+	0.0765282i	\(0.975616\pi\)
\(23\)	144.583	+	52.6238i	1.31077	+	0.477080i	0.900488	−	0.434881i	\(-0.143210\pi\)
							0.410278	+	0.911961i	\(0.365432\pi\)
\(29\)	30.3142	−	171.920i	0.194110	−	1.10085i	−0.719569	−	0.694420i	\(-0.755661\pi\)
							0.913680	−	0.406435i	\(-0.133228\pi\)
\(31\)	−199.014	−	72.4350i	−1.15303	−	0.419668i	−0.306427	−	0.951894i	\(-0.599133\pi\)
							−0.846602	+	0.532226i	\(0.821356\pi\)
\(37\)	51.0621	+	88.4422i	0.226880	+	0.392968i	0.956882	−	0.290478i	\(-0.0938142\pi\)
							−0.730002	+	0.683445i	\(0.760481\pi\)
\(41\)	7.73239	+	43.8526i	0.0294536	+	0.167040i	0.995987	−	0.0895028i	\(-0.0285278\pi\)
							−0.966533	+	0.256542i	\(0.917417\pi\)
\(43\)	−10.2238	−	8.57879i	−0.0362585	−	0.0304245i	0.624478	−	0.781042i	\(-0.285312\pi\)
							−0.660737	+	0.750618i	\(0.729756\pi\)
\(47\)	329.686	−	119.996i	1.02318	−	0.372409i	0.224703	−	0.974427i	\(-0.427859\pi\)
							0.798482	+	0.602019i	\(0.205637\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.25.8	yes	48
3.2	odd	2		81.4.e.a.73.1		48
9.2	odd	6		243.4.e.a.55.8		48
9.4	even	3		243.4.e.c.136.8		48
9.5	odd	6		243.4.e.b.136.1		48
9.7	even	3		243.4.e.d.55.1		48
27.4	even	9		243.4.e.c.109.8		48
27.5	odd	18		243.4.e.a.190.8		48
27.11	odd	18		729.4.a.c.1.3		24
27.13	even	9	inner	27.4.e.a.13.8	✓	48
27.14	odd	18		81.4.e.a.10.1		48
27.16	even	9		729.4.a.d.1.22		24
27.22	even	9		243.4.e.d.190.1		48
27.23	odd	18		243.4.e.b.109.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.13.8	✓	48	27.13	even	9	inner
27.4.e.a.25.8	yes	48	1.1	even	1	trivial
81.4.e.a.10.1		48	27.14	odd	18
81.4.e.a.73.1		48	3.2	odd	2
243.4.e.a.55.8		48	9.2	odd	6
243.4.e.a.190.8		48	27.5	odd	18
243.4.e.b.109.1		48	27.23	odd	18
243.4.e.b.136.1		48	9.5	odd	6
243.4.e.c.109.8		48	27.4	even	9
243.4.e.c.136.8		48	9.4	even	3
243.4.e.d.55.1		48	9.7	even	3
243.4.e.d.190.1		48	27.22	even	9
729.4.a.c.1.3		24	27.11	odd	18
729.4.a.d.1.22		24	27.16	even	9