Embedded newform 27.4.e.a.22.8

• Label: 27.4.e.a.22.8
• 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			22.8
Character	\(\chi\)	\(=\)	27.22
Dual form			27.4.e.a.16.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.06576 - 3.41158i) q^{2} +(-1.93603 + 4.82201i) q^{3} +(3.50237 - 19.8629i) q^{4} +(-5.18200 + 1.88609i) q^{5} +(8.57922 + 26.2101i) q^{6} +(4.27574 + 24.2490i) q^{7} +(-32.2942 - 55.9352i) q^{8} +(-19.5035 - 18.6711i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.06576	−	3.41158i	1.43746	−	1.20618i	0.496335	−	0.868131i	\(-0.334679\pi\)
							0.941130	−	0.338045i	\(-0.109765\pi\)
\(3\)	−1.93603	+	4.82201i	−0.372590	+	0.927996i
							\(5\)	−5.18200	+	1.88609i	−0.463492	+	0.168697i	−0.563202	−	0.826319i	\(-0.690431\pi\)
0.0997101	+	0.995017i	\(0.468208\pi\)
\(7\)	4.27574	+	24.2490i	0.230868	+	1.30932i	0.851142	+	0.524935i	\(0.175910\pi\)
							−0.620274	+	0.784385i	\(0.712979\pi\)
\(11\)	0.554625	+	0.201867i	0.0152023	+	0.00553320i	0.349610	−	0.936895i	\(-0.386314\pi\)
							−0.334408	+	0.942428i	\(0.608536\pi\)
\(13\)	−1.25967	−	1.05699i	−0.0268745	−	0.0225504i	0.629251	−	0.777202i	\(-0.283362\pi\)
							−0.656126	+	0.754651i	\(0.727806\pi\)
\(17\)	33.3077	−	57.6906i	0.475194	−	0.823060i	−0.524403	−	0.851470i	\(-0.675711\pi\)
							0.999596	+	0.0284107i	\(0.00904462\pi\)
\(19\)	6.62771	+	11.4795i	0.0800264	+	0.138610i	0.903261	−	0.429091i	\(-0.141166\pi\)
							−0.823235	+	0.567701i	\(0.807833\pi\)
\(23\)	28.1084	−	159.411i	0.254827	−	1.44519i	−0.541691	−	0.840578i	\(-0.682216\pi\)
							0.796517	−	0.604616i	\(-0.206673\pi\)
\(29\)	−54.0249	+	45.3323i	−0.345937	+	0.290276i	−0.799156	−	0.601124i	\(-0.794720\pi\)
							0.453219	+	0.891399i	\(0.350276\pi\)
\(31\)	−24.1904	+	137.191i	−0.140152	+	0.794844i	0.830980	+	0.556303i	\(0.187780\pi\)
							−0.971132	+	0.238542i	\(0.923331\pi\)
\(37\)	−40.2843	+	69.7744i	−0.178992	+	0.310023i	−0.941535	−	0.336914i	\(-0.890617\pi\)
							0.762544	+	0.646937i	\(0.223950\pi\)
\(41\)	130.925	+	109.859i	0.498710	+	0.418467i	0.857135	−	0.515091i	\(-0.172242\pi\)
							−0.358426	+	0.933558i	\(0.616686\pi\)
\(43\)	456.650	+	166.207i	1.61950	+	0.589449i	0.983287	−	0.182064i	\(-0.0582779\pi\)
							0.636213	+	0.771514i	\(0.280500\pi\)
\(47\)	17.5170	+	99.3440i	0.0543643	+	0.308315i	0.999849	−	0.0173499i	\(-0.00552292\pi\)
							−0.945485	+	0.325665i	\(0.894412\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.22.8	yes	48
3.2	odd	2		81.4.e.a.37.1		48
9.2	odd	6		243.4.e.b.190.8		48
9.4	even	3		243.4.e.d.28.8		48
9.5	odd	6		243.4.e.a.28.1		48
9.7	even	3		243.4.e.c.190.1		48
27.2	odd	18		243.4.e.b.55.8		48
27.4	even	9		729.4.a.d.1.24		24
27.7	even	9		243.4.e.d.217.8		48
27.11	odd	18		81.4.e.a.46.1		48
27.16	even	9	inner	27.4.e.a.16.8	✓	48
27.20	odd	18		243.4.e.a.217.1		48
27.23	odd	18		729.4.a.c.1.1		24
27.25	even	9		243.4.e.c.55.1		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.16.8	✓	48	27.16	even	9	inner
27.4.e.a.22.8	yes	48	1.1	even	1	trivial
81.4.e.a.37.1		48	3.2	odd	2
81.4.e.a.46.1		48	27.11	odd	18
243.4.e.a.28.1		48	9.5	odd	6
243.4.e.a.217.1		48	27.20	odd	18
243.4.e.b.55.8		48	27.2	odd	18
243.4.e.b.190.8		48	9.2	odd	6
243.4.e.c.55.1		48	27.25	even	9
243.4.e.c.190.1		48	9.7	even	3
243.4.e.d.28.8		48	9.4	even	3
243.4.e.d.217.8		48	27.7	even	9
729.4.a.c.1.1		24	27.23	odd	18
729.4.a.d.1.24		24	27.4	even	9