Embedded newform 539.2.r.a.144.14

• Label: 539.2.r.a.144.14
• Level: $539$
• Weight: $2$
• Character: 539.144
• Analytic conductor: $4.304$
• Analytic rank: $0$
• Dimension: $276$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.r (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.30393666895\)
Analytic rank:	\(0\)
Dimension:	\(276\)
Relative dimension:	\(23\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			144.14
Character	\(\chi\)	\(=\)	539.144
Dual form			539.2.r.a.408.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.210907 - 0.537382i) q^{2} +(0.0614017 - 0.819348i) q^{3} +(1.22181 + 1.13367i) q^{4} +(-2.44131 - 1.66446i) q^{5} +(-0.427353 - 0.205802i) q^{6} +(2.38379 + 1.14785i) q^{7} +(1.90714 - 0.918429i) q^{8} +(2.29893 + 0.346508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 276 q - q^{3} + 24 q^{4} - 2 q^{5} - 12 q^{6} - 2 q^{7} + 6 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/539\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(442\)
\(\chi(n)\)	\(e\left(\frac{11}{21}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.210907	−	0.537382i	0.149134	−	0.379986i	−0.836712	−	0.547644i	\(-0.815525\pi\)
							0.985845	+	0.167657i	\(0.0536202\pi\)
\(3\)	0.0614017	−	0.819348i	0.0354503	−	0.473051i	−0.951022	−	0.309123i	\(-0.899964\pi\)
							0.986472	−	0.163928i	\(-0.0524165\pi\)
\(5\)	−2.44131	−	1.66446i	−1.09179	−	0.744369i	−0.123037	−	0.992402i	\(-0.539263\pi\)
							−0.968751	+	0.248033i	\(0.920216\pi\)
\(7\)	2.38379	+	1.14785i	0.900988	+	0.433845i
							\(11\)	0.988831	−	0.149042i	0.298144	−	0.0449379i
\(13\)	−0.820882	−	1.02935i	−0.227672	−	0.285491i								0.654854	−	0.755755i	\(-0.272730\pi\)
							−0.882526	+	0.470264i	\(0.844159\pi\)
\(17\)	−0.731643	−	0.225682i	−0.177449	−	0.0547359i	0.204757	−	0.978813i	\(-0.434360\pi\)
							−0.382207	+	0.924077i	\(0.624836\pi\)
\(19\)	0.598287	+	1.03626i	0.137256	+	0.237735i	0.926457	−	0.376400i	\(-0.122838\pi\)
							−0.789201	+	0.614135i	\(0.789505\pi\)
\(23\)	1.98359	−	0.611856i	0.413606	−	0.127581i	−0.0809665	−	0.996717i	\(-0.525801\pi\)
							0.494573	+	0.869136i	\(0.335324\pi\)
\(29\)	−1.43187	−	6.27341i	−0.265891	−	1.16494i	−0.914745	−	0.404031i	\(-0.867609\pi\)
							0.648854	−	0.760913i	\(-0.275248\pi\)
\(31\)	5.26586	−	9.12074i	0.945776	−	1.63813i	0.191587	−	0.981476i	\(-0.438636\pi\)
							0.754189	−	0.656657i	\(-0.228030\pi\)
\(37\)	−3.78897	+	3.51565i	−0.622903	+	0.577970i	−0.927181	−	0.374614i	\(-0.877775\pi\)
							0.304278	+	0.952583i	\(0.401585\pi\)
\(41\)	0.820575	−	0.395168i	0.128152	−	0.0617149i	−0.368709	−	0.929545i	\(-0.620200\pi\)
							0.496861	+	0.867830i	\(0.334486\pi\)
\(43\)	3.36228	+	1.61919i	0.512743	+	0.246924i	0.672319	−	0.740262i	\(-0.265298\pi\)
							−0.159576	+	0.987186i	\(0.551013\pi\)
\(47\)	−3.29610	+	8.39831i	−0.480785	+	1.22502i	0.460576	+	0.887620i	\(0.347643\pi\)
							−0.941361	+	0.337400i	\(0.890452\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.r.a.144.14	✓	276
49.16	even	21	inner	539.2.r.a.408.14	yes	276

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.r.a.144.14	✓	276	1.1	even	1	trivial
539.2.r.a.408.14	yes	276	49.16	even	21	inner