Embedded newform 27.4.e.a.22.2

• Label: 27.4.e.a.22.2
• 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			22.2
Character	\(\chi\)	\(=\)	27.22
Dual form			27.4.e.a.16.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.18975 + 2.67652i) q^{2} +(3.68036 + 3.66810i) q^{3} +(1.62157 - 9.19635i) q^{4} +(-14.0025 + 5.09651i) q^{5} +(-21.5571 - 1.84978i) q^{6} +(0.758337 + 4.30074i) q^{7} +(2.78612 + 4.82571i) q^{8} +(0.0900554 + 26.9998i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.18975	+	2.67652i	−1.12775	+	0.946291i	−0.998970	−	0.0453825i	\(-0.985549\pi\)
							−0.128776	+	0.991674i	\(0.541105\pi\)
\(3\)	3.68036	+	3.66810i	0.708285	+	0.705927i
							\(5\)	−14.0025	+	5.09651i	−1.25243	+	0.455845i	−0.881220	−	0.472706i	\(-0.843277\pi\)
−0.371205	+	0.928551i	\(0.621055\pi\)
\(7\)	0.758337	+	4.30074i	0.0409463	+	0.232218i	0.998412	−	0.0563279i	\(-0.0179392\pi\)
							−0.957466	+	0.288546i	\(0.906828\pi\)
\(11\)	60.8046	+	22.1311i	1.66666	+	0.606615i	0.991389	−	0.130953i	\(-0.0418037\pi\)
							0.675273	+	0.737568i	\(0.264026\pi\)
\(13\)	−2.75952	−	2.31552i	−0.0588734	−	0.0494007i	0.612876	−	0.790179i	\(-0.290012\pi\)
							−0.671750	+	0.740778i	\(0.734457\pi\)
\(17\)	−18.0500	+	31.2635i	−0.257516	+	0.446030i	−0.965576	−	0.260122i	\(-0.916237\pi\)
							0.708060	+	0.706152i	\(0.249571\pi\)
\(19\)	37.4017	+	64.7817i	0.451608	+	0.782207i	0.998486	−	0.0550047i	\(-0.0175174\pi\)
							−0.546878	+	0.837212i	\(0.684184\pi\)
\(23\)	36.4054	−	206.466i	0.330046	−	1.87178i	−0.141493	−	0.989939i	\(-0.545190\pi\)
							0.471539	−	0.881845i	\(-0.343699\pi\)
\(29\)	68.7272	−	57.6690i	0.440080	−	0.369271i	−0.395659	−	0.918397i	\(-0.629484\pi\)
							0.835739	+	0.549126i	\(0.185039\pi\)
\(31\)	−4.29738	+	24.3717i	−0.0248978	+	0.141203i	−0.994723	−	0.102600i	\(-0.967284\pi\)
							0.969825	+	0.243803i	\(0.0783949\pi\)
\(37\)	38.9926	−	67.5371i	0.173252	−	0.300082i	−0.766303	−	0.642480i	\(-0.777906\pi\)
							0.939555	+	0.342398i	\(0.111239\pi\)
\(41\)	−91.8433	−	77.0657i	−0.349842	−	0.293552i	0.450885	−	0.892582i	\(-0.351109\pi\)
							−0.800727	+	0.599030i	\(0.795553\pi\)
\(43\)	−294.934	−	107.347i	−1.04598	−	0.380705i	−0.238835	−	0.971060i	\(-0.576765\pi\)
							−0.807143	+	0.590355i	\(0.798988\pi\)
\(47\)	50.7235	+	287.667i	0.157421	+	0.892779i	0.956539	+	0.291604i	\(0.0941891\pi\)
							−0.799118	+	0.601174i	\(0.794700\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.22.2	yes	48
3.2	odd	2		81.4.e.a.37.7		48
9.2	odd	6		243.4.e.b.190.2		48
9.4	even	3		243.4.e.d.28.2		48
9.5	odd	6		243.4.e.a.28.7		48
9.7	even	3		243.4.e.c.190.7		48
27.2	odd	18		243.4.e.b.55.2		48
27.4	even	9		729.4.a.d.1.4		24
27.7	even	9		243.4.e.d.217.2		48
27.11	odd	18		81.4.e.a.46.7		48
27.16	even	9	inner	27.4.e.a.16.2	✓	48
27.20	odd	18		243.4.e.a.217.7		48
27.23	odd	18		729.4.a.c.1.21		24
27.25	even	9		243.4.e.c.55.7		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.16.2	✓	48	27.16	even	9	inner
27.4.e.a.22.2	yes	48	1.1	even	1	trivial
81.4.e.a.37.7		48	3.2	odd	2
81.4.e.a.46.7		48	27.11	odd	18
243.4.e.a.28.7		48	9.5	odd	6
243.4.e.a.217.7		48	27.20	odd	18
243.4.e.b.55.2		48	27.2	odd	18
243.4.e.b.190.2		48	9.2	odd	6
243.4.e.c.55.7		48	27.25	even	9
243.4.e.c.190.7		48	9.7	even	3
243.4.e.d.28.2		48	9.4	even	3
243.4.e.d.217.2		48	27.7	even	9
729.4.a.c.1.21		24	27.23	odd	18
729.4.a.d.1.4		24	27.4	even	9