Embedded newform 27.4.e.a.13.3

• Label: 27.4.e.a.13.3
• Level: $27$
• Weight: $4$
• Character: 27.13
• 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			13.3
Character	\(\chi\)	\(=\)	27.13
Dual form			27.4.e.a.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.404735 - 2.29536i) q^{2} +(4.52897 + 2.54724i) q^{3} +(2.41266 - 0.878135i) q^{4} +(-4.78113 - 4.01185i) q^{5} +(4.01380 - 11.4266i) q^{6} +(2.53990 + 0.924448i) q^{7} +(-12.3152 - 21.3306i) q^{8} +(14.0232 + 23.0727i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.404735	−	2.29536i	−0.143095	−	0.811534i	−0.968877	−	0.247544i	\(-0.920377\pi\)
							0.825781	−	0.563990i	\(-0.190734\pi\)
\(3\)	4.52897	+	2.54724i	0.871601	+	0.490216i
							\(5\)	−4.78113	−	4.01185i	−0.427637	−	0.358830i	0.403422	−	0.915014i	\(-0.367821\pi\)
−0.831059	+	0.556184i	\(0.812265\pi\)
\(7\)	2.53990	+	0.924448i	0.137142	+	0.0499155i	0.409679	−	0.912230i	\(-0.365641\pi\)
							−0.272537	+	0.962145i	\(0.587863\pi\)
\(11\)	−30.2808	+	25.4086i	−0.830000	+	0.696453i	−0.955291	−	0.295667i	\(-0.904458\pi\)
							0.125291	+	0.992120i	\(0.460014\pi\)
\(13\)	−12.8853	+	73.0761i	−0.274903	+	1.55905i	0.464367	+	0.885643i	\(0.346282\pi\)
							−0.739270	+	0.673409i	\(0.764829\pi\)
\(17\)	29.1160	−	50.4303i	0.415392	−	0.719479i	−0.580078	−	0.814561i	\(-0.696978\pi\)
							0.995469	+	0.0950816i	\(0.0303112\pi\)
\(19\)	−41.2204	−	71.3959i	−0.497717	−	0.862070i	0.502280	−	0.864705i	\(-0.332495\pi\)
							−0.999997	+	0.00263470i	\(0.999161\pi\)
\(23\)	−25.8401	+	9.40502i	−0.234262	+	0.0852644i	−0.456484	−	0.889732i	\(-0.650891\pi\)
							0.222222	+	0.974996i	\(0.428669\pi\)
\(29\)	30.1245	+	170.844i	0.192896	+	1.09397i	0.915384	+	0.402582i	\(0.131887\pi\)
							−0.722488	+	0.691383i	\(0.757002\pi\)
\(31\)	149.460	−	54.3991i	0.865931	−	0.315173i	0.129413	−	0.991591i	\(-0.458691\pi\)
							0.736518	+	0.676418i	\(0.236469\pi\)
\(37\)	220.701	−	382.265i	0.980621	−	1.69849i	0.320644	−	0.947200i	\(-0.396101\pi\)
							0.659977	−	0.751286i	\(-0.270566\pi\)
\(41\)	14.7471	−	83.6352i	0.0561736	−	0.318576i	−0.943754	−	0.330650i	\(-0.892732\pi\)
							0.999927	+	0.0120735i	\(0.00384321\pi\)
\(43\)	−150.412	+	126.210i	−0.533431	+	0.447602i	−0.869284	−	0.494312i	\(-0.835420\pi\)
							0.335853	+	0.941914i	\(0.390975\pi\)
\(47\)	402.616	+	146.540i	1.24952	+	0.454789i	0.880240	−	0.474528i	\(-0.157381\pi\)
							0.369283	+	0.929317i	\(0.379603\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.13.3	✓	48
3.2	odd	2		81.4.e.a.10.6		48
9.2	odd	6		243.4.e.b.109.6		48
9.4	even	3		243.4.e.d.190.6		48
9.5	odd	6		243.4.e.a.190.3		48
9.7	even	3		243.4.e.c.109.3		48
27.2	odd	18		81.4.e.a.73.6		48
27.5	odd	18		729.4.a.c.1.18		24
27.7	even	9		243.4.e.c.136.3		48
27.11	odd	18		243.4.e.a.55.3		48
27.16	even	9		243.4.e.d.55.6		48
27.20	odd	18		243.4.e.b.136.6		48
27.22	even	9		729.4.a.d.1.7		24
27.25	even	9	inner	27.4.e.a.25.3	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.13.3	✓	48	1.1	even	1	trivial
27.4.e.a.25.3	yes	48	27.25	even	9	inner
81.4.e.a.10.6		48	3.2	odd	2
81.4.e.a.73.6		48	27.2	odd	18
243.4.e.a.55.3		48	27.11	odd	18
243.4.e.a.190.3		48	9.5	odd	6
243.4.e.b.109.6		48	9.2	odd	6
243.4.e.b.136.6		48	27.20	odd	18
243.4.e.c.109.3		48	9.7	even	3
243.4.e.c.136.3		48	27.7	even	9
243.4.e.d.55.6		48	27.16	even	9
243.4.e.d.190.6		48	9.4	even	3
729.4.a.c.1.18		24	27.5	odd	18
729.4.a.d.1.7		24	27.22	even	9