Embedded newform 27.6.e.a.4.1

• Label: 27.6.e.a.4.1
• Level: $27$
• Weight: $6$
• Character: 27.4
• Analytic conductor: $4.330$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.33036313495\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	27.4
Dual form			27.6.e.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.69504 - 3.52870i) q^{2} +(-5.72592 + 14.4988i) q^{3} +(57.0285 + 47.8526i) q^{4} +(2.39953 - 13.6084i) q^{5} +(106.675 - 120.361i) q^{6} +(-83.8831 + 70.3863i) q^{7} +(-218.960 - 379.250i) q^{8} +(-177.428 - 166.037i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 93 q^{5} - 126 q^{6} - 6 q^{7} + 573 q^{8} + 324 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.69504	−	3.52870i	−1.71386	−	0.623793i	−0.716577	−	0.697508i	\(-0.754292\pi\)
							−0.997279	+	0.0737152i	\(0.976514\pi\)
\(3\)	−5.72592	+	14.4988i	−0.367318	+	0.930096i
							\(5\)	2.39953	−	13.6084i	0.0429242	−	0.243435i	−0.955795	−	0.294034i	\(-0.905002\pi\)
0.998719	+	0.0505992i	\(0.0161131\pi\)
\(7\)	−83.8831	+	70.3863i	−0.647038	+	0.542929i	−0.906170	−	0.422913i	\(-0.861008\pi\)
							0.259133	+	0.965842i	\(0.416563\pi\)
\(11\)	−96.8959	−	549.524i	−0.241448	−	1.36932i	−0.828599	−	0.559842i	\(-0.810862\pi\)
							0.587151	−	0.809477i	\(-0.300249\pi\)
\(13\)	857.851	−	312.232i	1.40784	−	0.512412i	0.477345	−	0.878716i	\(-0.341599\pi\)
							0.930495	+	0.366304i	\(0.119377\pi\)
\(17\)	−249.360	+	431.905i	−0.209269	+	0.362465i	−0.951485	−	0.307697i	\(-0.900442\pi\)
							0.742215	+	0.670161i	\(0.233775\pi\)
\(19\)	−1431.28	−	2479.06i	−0.909582	−	1.57544i	−0.814645	−	0.579960i	\(-0.803068\pi\)
							−0.0949374	−	0.995483i	\(-0.530265\pi\)
\(23\)	−296.790	−	249.036i	−0.116985	−	0.0981618i	0.582419	−	0.812889i	\(-0.302106\pi\)
							−0.699403	+	0.714727i	\(0.746551\pi\)
\(29\)	−612.595	−	222.966i	−0.135263	−	0.0492317i	0.273502	−	0.961872i	\(-0.411818\pi\)
							−0.408765	+	0.912640i	\(0.634040\pi\)
\(31\)	−4646.87	−	3899.19i	−0.868473	−	0.728736i	0.0953028	−	0.995448i	\(-0.469618\pi\)
							−0.963776	+	0.266713i	\(0.914063\pi\)
\(37\)	325.245	−	563.342i	0.0390577	−	0.0676500i	−0.845836	−	0.533443i	\(-0.820898\pi\)
							0.884893	+	0.465794i	\(0.154231\pi\)
\(41\)	3373.52	−	1227.86i	0.313418	−	0.114075i	−0.180522	−	0.983571i	\(-0.557779\pi\)
							0.493940	+	0.869496i	\(0.335556\pi\)
\(43\)	−2554.34	−	14486.4i	−0.210672	−	1.19478i	−0.888260	−	0.459341i	\(-0.848086\pi\)
							0.677588	−	0.735442i	\(-0.263025\pi\)
\(47\)	−11740.4	+	9851.33i	−0.775241	+	0.650504i	−0.942045	−	0.335486i	\(-0.891100\pi\)
							0.166804	+	0.985990i	\(0.446655\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.6.e.a.4.1	✓	84
3.2	odd	2		81.6.e.a.64.14		84
27.7	even	9	inner	27.6.e.a.7.1	yes	84
27.13	even	9		729.6.a.c.1.41		42
27.14	odd	18		729.6.a.e.1.2		42
27.20	odd	18		81.6.e.a.19.14		84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.6.e.a.4.1	✓	84	1.1	even	1	trivial
27.6.e.a.7.1	yes	84	27.7	even	9	inner
81.6.e.a.19.14		84	27.20	odd	18
81.6.e.a.64.14		84	3.2	odd	2
729.6.a.c.1.41		42	27.13	even	9
729.6.a.e.1.2		42	27.14	odd	18