Embedded newform 27.4.e.a.7.7

• Label: 27.4.e.a.7.7
• Level: $27$
• Weight: $4$
• Character: 27.7
• 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			7.7
Character	\(\chi\)	\(=\)	27.7
Dual form			27.4.e.a.4.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.53135 - 1.28531i) q^{2} +(-2.46998 - 4.57157i) q^{3} +(4.69008 - 3.93544i) q^{4} +(1.06026 + 6.01302i) q^{5} +(-14.5982 - 12.9691i) q^{6} +(13.2194 + 11.0924i) q^{7} +(-3.52788 + 6.11047i) q^{8} +(-14.7984 + 22.5833i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 6 q^{3} - 6 q^{4} + 6 q^{5} - 18 q^{6} - 6 q^{7} - 75 q^{8} - 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.53135	−	1.28531i	1.24852	−	0.454425i	0.368620	−	0.929580i	\(-0.379830\pi\)
							0.879902	+	0.475156i	\(0.157608\pi\)
\(3\)	−2.46998	−	4.57157i	−0.475347	−	0.879798i
							\(5\)	1.06026	+	6.01302i	0.0948323	+	0.537821i	0.994799	+	0.101861i	\(0.0324798\pi\)
−0.899966	+	0.435959i	\(0.856409\pi\)
\(7\)	13.2194	+	11.0924i	0.713781	+	0.598934i	0.925657	−	0.378363i	\(-0.123513\pi\)
							−0.211876	+	0.977297i	\(0.567957\pi\)
\(11\)	6.89885	−	39.1253i	0.189098	−	1.07243i	−0.731478	−	0.681865i	\(-0.761169\pi\)
							0.920576	−	0.390564i	\(-0.127720\pi\)
\(13\)	−25.7091	−	9.35736i	−0.548495	−	0.199636i	0.0528828	−	0.998601i	\(-0.483159\pi\)
							−0.601378	+	0.798965i	\(0.705381\pi\)
\(17\)	−54.8631	−	95.0258i	−0.782722	−	1.35571i	−0.930351	−	0.366671i	\(-0.880498\pi\)
							0.147629	−	0.989043i	\(-0.452836\pi\)
\(19\)	−61.0814	+	105.796i	−0.737528	+	1.27744i	0.216077	+	0.976376i	\(0.430674\pi\)
							−0.953605	+	0.301060i	\(0.902660\pi\)
\(23\)	96.2392	−	80.7543i	0.872490	−	0.732106i	−0.0921311	−	0.995747i	\(-0.529368\pi\)
							0.964621	+	0.263641i	\(0.0849234\pi\)
\(29\)	−104.238	+	37.9395i	−0.667465	+	0.242937i	−0.653456	−	0.756964i	\(-0.726682\pi\)
							−0.0140090	+	0.999902i	\(0.504459\pi\)
\(31\)	148.234	−	124.383i	0.858826	−	0.720641i	−0.102889	−	0.994693i	\(-0.532809\pi\)
							0.961715	+	0.274052i	\(0.0883641\pi\)
\(37\)	46.0186	+	79.7065i	0.204471	+	0.354153i	0.949964	−	0.312360i	\(-0.101119\pi\)
							−0.745493	+	0.666513i	\(0.767786\pi\)
\(41\)	−103.221	−	37.5694i	−0.393181	−	0.143106i	0.137861	−	0.990452i	\(-0.455977\pi\)
							−0.531042	+	0.847345i	\(0.678199\pi\)
\(43\)	−74.3465	+	421.640i	−0.263668	+	1.49534i	0.509132	+	0.860688i	\(0.329966\pi\)
							−0.772801	+	0.634649i	\(0.781145\pi\)
\(47\)	−77.3688	−	64.9201i	−0.240115	−	0.201480i	0.514787	−	0.857318i	\(-0.327871\pi\)
							−0.754902	+	0.655838i	\(0.772315\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.4.e.a.7.7	yes	48
3.2	odd	2		81.4.e.a.19.2		48
9.2	odd	6		243.4.e.a.136.7		48
9.4	even	3		243.4.e.c.217.2		48
9.5	odd	6		243.4.e.b.217.7		48
9.7	even	3		243.4.e.d.136.2		48
27.2	odd	18		729.4.a.c.1.20		24
27.4	even	9	inner	27.4.e.a.4.7	✓	48
27.5	odd	18		243.4.e.b.28.7		48
27.13	even	9		243.4.e.d.109.2		48
27.14	odd	18		243.4.e.a.109.7		48
27.22	even	9		243.4.e.c.28.2		48
27.23	odd	18		81.4.e.a.64.2		48
27.25	even	9		729.4.a.d.1.5		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.e.a.4.7	✓	48	27.4	even	9	inner
27.4.e.a.7.7	yes	48	1.1	even	1	trivial
81.4.e.a.19.2		48	3.2	odd	2
81.4.e.a.64.2		48	27.23	odd	18
243.4.e.a.109.7		48	27.14	odd	18
243.4.e.a.136.7		48	9.2	odd	6
243.4.e.b.28.7		48	27.5	odd	18
243.4.e.b.217.7		48	9.5	odd	6
243.4.e.c.28.2		48	27.22	even	9
243.4.e.c.217.2		48	9.4	even	3
243.4.e.d.109.2		48	27.13	even	9
243.4.e.d.136.2		48	9.7	even	3
729.4.a.c.1.20		24	27.2	odd	18
729.4.a.d.1.5		24	27.25	even	9