Embedded newform 27.4.e.a.25.7

• Label: 27.4.e.a.25.7
• Level: $27$
• Weight: $4$
• Character: 27.25
• 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			25.7
Character	\(\chi\)	\(=\)	27.25
Dual form			27.4.e.a.13.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.640527 - 3.63261i) q^{2} +(5.12390 + 0.863506i) q^{3} +(-5.26804 - 1.91741i) q^{4} +(-1.23763 + 1.03850i) q^{5} +(6.41878 - 18.0600i) q^{6} +(-31.9169 + 11.6168i) q^{7} +(4.41508 - 7.64714i) q^{8} +(25.5087 + 8.84904i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.640527	−	3.63261i	0.226461	−	1.28432i	−0.633413	−	0.773814i	\(-0.718346\pi\)
							0.859873	−	0.510507i	\(-0.170542\pi\)
\(3\)	5.12390	+	0.863506i	0.986095	+	0.166182i
							\(5\)	−1.23763	+	1.03850i	−0.110697	+	0.0928860i	−0.696456	−	0.717599i	\(-0.745241\pi\)
0.585759	+	0.810485i	\(0.300797\pi\)
\(7\)	−31.9169	+	11.6168i	−1.72335	+	0.627248i	−0.998122	−	0.0612602i	\(-0.980488\pi\)
							−0.725228	+	0.688508i	\(0.758266\pi\)
\(11\)	18.8740	+	15.8372i	0.517339	+	0.434099i	0.863703	−	0.504001i	\(-0.168139\pi\)
							−0.346364	+	0.938100i	\(0.612584\pi\)
\(13\)	5.35351	+	30.3613i	0.114215	+	0.647746i	0.987136	+	0.159884i	\(0.0511121\pi\)
							−0.872921	+	0.487862i	\(0.837777\pi\)
\(17\)	−18.2417	−	31.5955i	−0.260250	−	0.450766i	0.706058	−	0.708154i	\(-0.250472\pi\)
							−0.966308	+	0.257387i	\(0.917138\pi\)
\(19\)	36.5170	−	63.2494i	0.440925	−	0.763705i	−0.556833	−	0.830625i	\(-0.687984\pi\)
							0.997758	+	0.0669192i	\(0.0213170\pi\)
\(23\)	−62.9523	−	22.9128i	−0.570716	−	0.207724i	0.0405108	−	0.999179i	\(-0.487101\pi\)
							−0.611227	+	0.791455i	\(0.709324\pi\)
\(29\)	41.0688	−	232.913i	0.262975	−	1.49141i	−0.511764	−	0.859126i	\(-0.671008\pi\)
							0.774739	−	0.632281i	\(-0.217881\pi\)
\(31\)	−152.414	−	55.4740i	−0.883042	−	0.321401i	−0.139605	−	0.990207i	\(-0.544583\pi\)
							−0.743437	+	0.668806i	\(0.766806\pi\)
\(37\)	93.1046	+	161.262i	0.413684	+	0.716521i	0.995289	−	0.0969496i	\(-0.0309086\pi\)
							−0.581605	+	0.813471i	\(0.697575\pi\)
\(41\)	−37.9265	−	215.092i	−0.144466	−	0.819309i	−0.967794	−	0.251743i	\(-0.918996\pi\)
							0.823328	−	0.567566i	\(-0.192115\pi\)
\(43\)	156.989	+	131.730i	0.556759	+	0.467176i	0.877222	−	0.480085i	\(-0.159394\pi\)
							−0.320463	+	0.947261i	\(0.603839\pi\)
\(47\)	−450.185	+	163.854i	−1.39715	+	0.508522i	−0.927332	−	0.374239i	\(-0.877904\pi\)
							−0.469821	+	0.882762i	\(0.655682\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.25.7	yes	48
3.2	odd	2		81.4.e.a.73.2		48
9.2	odd	6		243.4.e.a.55.7		48
9.4	even	3		243.4.e.c.136.7		48
9.5	odd	6		243.4.e.b.136.2		48
9.7	even	3		243.4.e.d.55.2		48
27.4	even	9		243.4.e.c.109.7		48
27.5	odd	18		243.4.e.a.190.7		48
27.11	odd	18		729.4.a.c.1.5		24
27.13	even	9	inner	27.4.e.a.13.7	✓	48
27.14	odd	18		81.4.e.a.10.2		48
27.16	even	9		729.4.a.d.1.20		24
27.22	even	9		243.4.e.d.190.2		48
27.23	odd	18		243.4.e.b.109.2		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.13.7	✓	48	27.13	even	9	inner
27.4.e.a.25.7	yes	48	1.1	even	1	trivial
81.4.e.a.10.2		48	27.14	odd	18
81.4.e.a.73.2		48	3.2	odd	2
243.4.e.a.55.7		48	9.2	odd	6
243.4.e.a.190.7		48	27.5	odd	18
243.4.e.b.109.2		48	27.23	odd	18
243.4.e.b.136.2		48	9.5	odd	6
243.4.e.c.109.7		48	27.4	even	9
243.4.e.c.136.7		48	9.4	even	3
243.4.e.d.55.2		48	9.7	even	3
243.4.e.d.190.2		48	27.22	even	9
729.4.a.c.1.5		24	27.11	odd	18
729.4.a.d.1.20		24	27.16	even	9