Embedded newform 27.4.e.a.22.6

• Label: 27.4.e.a.22.6
• Level: $27$
• Weight: $4$
• Character: 27.22
• 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			22.6
Character	\(\chi\)	\(=\)	27.22
Dual form			27.4.e.a.16.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.26133 - 1.89748i) q^{2} +(5.05876 - 1.18697i) q^{3} +(0.123997 - 0.703222i) q^{4} +(-16.0156 + 5.82920i) q^{5} +(9.18729 - 12.2831i) q^{6} +(-2.22794 - 12.6353i) q^{7} +(10.7539 + 18.6263i) q^{8} +(24.1822 - 12.0092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.26133	−	1.89748i	0.799502	−	0.670862i	−0.148576	−	0.988901i	\(-0.547469\pi\)
							0.948078	+	0.318039i	\(0.103024\pi\)
\(3\)	5.05876	−	1.18697i	0.973560	−	0.228433i
							\(5\)	−16.0156	+	5.82920i	−1.43248	+	0.521379i	−0.937640	−	0.347607i	\(-0.886994\pi\)
−0.494837	+	0.868986i	\(0.664772\pi\)
\(7\)	−2.22794	−	12.6353i	−0.120298	−	0.682242i	−0.983990	−	0.178223i	\(-0.942965\pi\)
							0.863693	−	0.504019i	\(-0.168146\pi\)
\(11\)	−48.2711	−	17.5693i	−1.32312	−	0.481576i	−0.418662	−	0.908142i	\(-0.637501\pi\)
							−0.904456	+	0.426567i	\(0.859723\pi\)
\(13\)	42.4812	+	35.6460i	0.906320	+	0.760493i	0.971415	−	0.237386i	\(-0.0762906\pi\)
							−0.0650951	+	0.997879i	\(0.520735\pi\)
\(17\)	18.6211	−	32.2527i	0.265663	−	0.460142i	−0.702074	−	0.712104i	\(-0.747742\pi\)
							0.967737	+	0.251962i	\(0.0810757\pi\)
\(19\)	−3.43235	−	5.94500i	−0.0414439	−	0.0717830i	0.844559	−	0.535462i	\(-0.179862\pi\)
							−0.886003	+	0.463679i	\(0.846529\pi\)
\(23\)	5.91382	−	33.5389i	0.0536138	−	0.304059i	−0.946195	−	0.323596i	\(-0.895108\pi\)
							0.999809	+	0.0195372i	\(0.00621927\pi\)
\(29\)	−78.5252	+	65.8905i	−0.502819	+	0.421916i	−0.858594	−	0.512656i	\(-0.828662\pi\)
							0.355775	+	0.934572i	\(0.384217\pi\)
\(31\)	38.1788	−	216.523i	0.221197	−	1.25447i	−0.648625	−	0.761108i	\(-0.724656\pi\)
							0.869822	−	0.493365i	\(-0.164233\pi\)
\(37\)	−159.206	+	275.752i	−0.707385	+	1.22523i	0.258438	+	0.966028i	\(0.416792\pi\)
							−0.965824	+	0.259200i	\(0.916541\pi\)
\(41\)	−77.7716	−	65.2581i	−0.296241	−	0.248576i	0.482537	−	0.875876i	\(-0.339716\pi\)
							−0.778778	+	0.627300i	\(0.784160\pi\)
\(43\)	−97.2504	−	35.3962i	−0.344896	−	0.125532i	0.163763	−	0.986500i	\(-0.447637\pi\)
							−0.508660	+	0.860968i	\(0.669859\pi\)
\(47\)	85.0649	+	482.427i	0.264000	+	1.49722i	0.771868	+	0.635783i	\(0.219323\pi\)
							−0.507868	+	0.861435i	\(0.669566\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.22.6	yes	48
3.2	odd	2		81.4.e.a.37.3		48
9.2	odd	6		243.4.e.b.190.6		48
9.4	even	3		243.4.e.d.28.6		48
9.5	odd	6		243.4.e.a.28.3		48
9.7	even	3		243.4.e.c.190.3		48
27.2	odd	18		243.4.e.b.55.6		48
27.4	even	9		729.4.a.d.1.17		24
27.7	even	9		243.4.e.d.217.6		48
27.11	odd	18		81.4.e.a.46.3		48
27.16	even	9	inner	27.4.e.a.16.6	✓	48
27.20	odd	18		243.4.e.a.217.3		48
27.23	odd	18		729.4.a.c.1.8		24
27.25	even	9		243.4.e.c.55.3		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.16.6	✓	48	27.16	even	9	inner
27.4.e.a.22.6	yes	48	1.1	even	1	trivial
81.4.e.a.37.3		48	3.2	odd	2
81.4.e.a.46.3		48	27.11	odd	18
243.4.e.a.28.3		48	9.5	odd	6
243.4.e.a.217.3		48	27.20	odd	18
243.4.e.b.55.6		48	27.2	odd	18
243.4.e.b.190.6		48	9.2	odd	6
243.4.e.c.55.3		48	27.25	even	9
243.4.e.c.190.3		48	9.7	even	3
243.4.e.d.28.6		48	9.4	even	3
243.4.e.d.217.6		48	27.7	even	9
729.4.a.c.1.8		24	27.23	odd	18
729.4.a.d.1.17		24	27.4	even	9