Embedded newform 27.4.e.a.22.3

• Label: 27.4.e.a.22.3
• 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			22.3
Character	\(\chi\)	\(=\)	27.22
Dual form			27.4.e.a.16.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39552 + 1.17098i) q^{2} +(4.34503 - 2.84969i) q^{3} +(-0.812901 + 4.61019i) q^{4} +(12.7121 - 4.62683i) q^{5} +(-2.72666 + 9.06477i) q^{6} +(3.96931 + 22.5111i) q^{7} +(-11.5509 - 20.0068i) q^{8} +(10.7586 - 24.7639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39552	+	1.17098i	−0.493392	+	0.414005i	−0.855240	−	0.518232i	\(-0.826590\pi\)
							0.361848	+	0.932237i	\(0.382146\pi\)
\(3\)	4.34503	−	2.84969i	0.836202	−	0.548422i
							\(5\)	12.7121	−	4.62683i	1.13700	−	0.413836i	0.296174	−	0.955134i	\(-0.404289\pi\)
0.840831	+	0.541298i	\(0.182067\pi\)
\(7\)	3.96931	+	22.5111i	0.214322	+	1.21548i	0.882078	+	0.471103i	\(0.156144\pi\)
							−0.667756	+	0.744380i	\(0.732745\pi\)
\(11\)	−31.0845	−	11.3138i	−0.852030	−	0.310114i	−0.121162	−	0.992633i	\(-0.538662\pi\)
							−0.730868	+	0.682519i	\(0.760884\pi\)
\(13\)	−52.2664	−	43.8567i	−1.11508	−	0.935666i	−0.116738	−	0.993163i	\(-0.537244\pi\)
							−0.998346	+	0.0574967i	\(0.981688\pi\)
\(17\)	−18.6324	+	32.2722i	−0.265825	+	0.460422i	−0.967779	−	0.251800i	\(-0.918978\pi\)
							0.701955	+	0.712222i	\(0.252311\pi\)
\(19\)	47.5146	+	82.2977i	0.573715	+	0.993704i	0.996180	+	0.0873247i	\(0.0278317\pi\)
							−0.422465	+	0.906379i	\(0.638835\pi\)
\(23\)	16.6296	−	94.3109i	0.150761	−	0.855008i	−0.811798	−	0.583938i	\(-0.801511\pi\)
							0.962559	−	0.271071i	\(-0.0873777\pi\)
\(29\)	−109.835	+	92.1623i	−0.703303	+	0.590142i	−0.922711	−	0.385492i	\(-0.874032\pi\)
							0.219408	+	0.975633i	\(0.429587\pi\)
\(31\)	13.9405	−	79.0604i	0.0807673	−	0.458054i	−0.917423	−	0.397914i	\(-0.869734\pi\)
							0.998190	−	0.0601399i	\(-0.0191547\pi\)
\(37\)	32.8251	−	56.8548i	0.145849	−	0.252618i	−0.783840	−	0.620962i	\(-0.786742\pi\)
							0.929689	+	0.368344i	\(0.120075\pi\)
\(41\)	−75.7866	−	63.5925i	−0.288680	−	0.242231i	0.486934	−	0.873439i	\(-0.338115\pi\)
							−0.775614	+	0.631207i	\(0.782560\pi\)
\(43\)	413.325	+	150.438i	1.46585	+	0.533525i	0.946970	−	0.321322i	\(-0.104127\pi\)
							0.518879	+	0.854848i	\(0.326350\pi\)
\(47\)	46.4461	+	263.409i	0.144146	+	0.817493i	0.968049	+	0.250762i	\(0.0806810\pi\)
							−0.823903	+	0.566731i	\(0.808208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.22.3	yes	48
3.2	odd	2		81.4.e.a.37.6		48
9.2	odd	6		243.4.e.b.190.3		48
9.4	even	3		243.4.e.d.28.3		48
9.5	odd	6		243.4.e.a.28.6		48
9.7	even	3		243.4.e.c.190.6		48
27.2	odd	18		243.4.e.b.55.3		48
27.4	even	9		729.4.a.d.1.9		24
27.7	even	9		243.4.e.d.217.3		48
27.11	odd	18		81.4.e.a.46.6		48
27.16	even	9	inner	27.4.e.a.16.3	✓	48
27.20	odd	18		243.4.e.a.217.6		48
27.23	odd	18		729.4.a.c.1.16		24
27.25	even	9		243.4.e.c.55.6		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.16.3	✓	48	27.16	even	9	inner
27.4.e.a.22.3	yes	48	1.1	even	1	trivial
81.4.e.a.37.6		48	3.2	odd	2
81.4.e.a.46.6		48	27.11	odd	18
243.4.e.a.28.6		48	9.5	odd	6
243.4.e.a.217.6		48	27.20	odd	18
243.4.e.b.55.3		48	27.2	odd	18
243.4.e.b.190.3		48	9.2	odd	6
243.4.e.c.55.6		48	27.25	even	9
243.4.e.c.190.6		48	9.7	even	3
243.4.e.d.28.3		48	9.4	even	3
243.4.e.d.217.3		48	27.7	even	9
729.4.a.c.1.16		24	27.23	odd	18
729.4.a.d.1.9		24	27.4	even	9