Embedded newform 27.4.e.a.22.1

• Label: 27.4.e.a.22.1
• 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			22.1
Character	\(\chi\)	\(=\)	27.22
Dual form			27.4.e.a.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.58593 + 3.00895i) q^{2} +(-2.57206 - 4.51492i) q^{3} +(2.41590 - 13.7013i) q^{4} +(2.94522 - 1.07197i) q^{5} +(22.8084 + 8.45097i) q^{6} +(-5.29680 - 30.0397i) q^{7} +(13.8388 + 23.9695i) q^{8} +(-13.7690 + 23.2253i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.58593	+	3.00895i	−1.26782	+	1.06382i	−0.273013	+	0.962010i	\(0.588020\pi\)
							−0.994804	+	0.101813i	\(0.967536\pi\)
\(3\)	−2.57206	−	4.51492i	−0.494993	−	0.868897i
							\(5\)	2.94522	−	1.07197i	0.263429	−	0.0958802i	−0.206929	−	0.978356i	\(-0.566347\pi\)
0.470358	+	0.882476i	\(0.344125\pi\)
\(7\)	−5.29680	−	30.0397i	−0.286000	−	1.62199i	−0.701690	−	0.712482i	\(-0.747571\pi\)
							0.415690	−	0.909507i	\(-0.363540\pi\)
\(11\)	−49.0358	−	17.8476i	−1.34408	−	0.489204i	−0.432982	−	0.901402i	\(-0.642539\pi\)
							−0.911094	+	0.412199i	\(0.864761\pi\)
\(13\)	11.6707	+	9.79286i	0.248990	+	0.208927i	0.758737	−	0.651397i	\(-0.225817\pi\)
							−0.509748	+	0.860324i	\(0.670261\pi\)
\(17\)	35.3229	−	61.1810i	0.503945	−	0.872858i	−0.496045	−	0.868297i	\(-0.665215\pi\)
							0.999990	−	0.00456086i	\(-0.00145177\pi\)
\(19\)	32.0937	+	55.5879i	0.387515	+	0.671196i	0.992115	−	0.125333i	\(-0.0400001\pi\)
							−0.604599	+	0.796530i	\(0.706667\pi\)
\(23\)	9.07464	−	51.4649i	0.0822693	−	0.466572i	−0.915643	−	0.401992i	\(-0.868318\pi\)
							0.997913	−	0.0645805i	\(-0.0205709\pi\)
\(29\)	20.2760	−	17.0136i	0.129833	−	0.108943i	−0.575559	−	0.817760i	\(-0.695215\pi\)
							0.705392	+	0.708817i	\(0.250771\pi\)
\(31\)	18.8621	−	106.972i	0.109282	−	0.619766i	−0.880142	−	0.474711i	\(-0.842553\pi\)
							0.989423	−	0.145056i	\(-0.0463362\pi\)
\(37\)	175.269	−	303.574i	0.778756	−	1.34885i	−0.153903	−	0.988086i	\(-0.549184\pi\)
							0.932659	−	0.360760i	\(-0.117482\pi\)
\(41\)	147.723	+	123.955i	0.562695	+	0.472157i	0.879213	−	0.476429i	\(-0.158069\pi\)
							−0.316518	+	0.948587i	\(0.602514\pi\)
\(43\)	132.385	+	48.1843i	0.469501	+	0.170885i	0.565926	−	0.824456i	\(-0.308519\pi\)
							−0.0964251	+	0.995340i	\(0.530741\pi\)
\(47\)	−20.0049	−	113.454i	−0.0620855	−	0.352104i	−0.999986	−	0.00520123i	\(-0.998344\pi\)
							0.937901	−	0.346903i	\(-0.112767\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.22.1	yes	48
3.2	odd	2		81.4.e.a.37.8		48
9.2	odd	6		243.4.e.b.190.1		48
9.4	even	3		243.4.e.d.28.1		48
9.5	odd	6		243.4.e.a.28.8		48
9.7	even	3		243.4.e.c.190.8		48
27.2	odd	18		243.4.e.b.55.1		48
27.4	even	9		729.4.a.d.1.2		24
27.7	even	9		243.4.e.d.217.1		48
27.11	odd	18		81.4.e.a.46.8		48
27.16	even	9	inner	27.4.e.a.16.1	✓	48
27.20	odd	18		243.4.e.a.217.8		48
27.23	odd	18		729.4.a.c.1.23		24
27.25	even	9		243.4.e.c.55.8		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.16.1	✓	48	27.16	even	9	inner
27.4.e.a.22.1	yes	48	1.1	even	1	trivial
81.4.e.a.37.8		48	3.2	odd	2
81.4.e.a.46.8		48	27.11	odd	18
243.4.e.a.28.8		48	9.5	odd	6
243.4.e.a.217.8		48	27.20	odd	18
243.4.e.b.55.1		48	27.2	odd	18
243.4.e.b.190.1		48	9.2	odd	6
243.4.e.c.55.8		48	27.25	even	9
243.4.e.c.190.8		48	9.7	even	3
243.4.e.d.28.1		48	9.4	even	3
243.4.e.d.217.1		48	27.7	even	9
729.4.a.c.1.23		24	27.23	odd	18
729.4.a.d.1.2		24	27.4	even	9