Embedded newform 531.2.p.a.128.42

• Label: 531.2.p.a.128.42
• Level: $531$
• Weight: $2$
• Character: 531.128
• Analytic conductor: $4.240$
• Analytic rank: $0$
• Dimension: $3248$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.p (of order \(174\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24005634733\)
Analytic rank:	\(0\)
Dimension:	\(3248\)
Relative dimension:	\(58\) over \(\Q(\zeta_{174})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{174}]$

Embedding invariants

Embedding label			128.42
Character	\(\chi\)	\(=\)	531.128
Dual form			531.2.p.a.419.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.789160 + 1.00012i) q^{2} +(-0.304068 + 1.70515i) q^{3} +(0.0876771 - 0.366659i) q^{4} +(-0.419645 + 2.29893i) q^{5} +(-1.94531 + 1.04153i) q^{6} +(-2.07688 + 1.14960i) q^{7} +(2.74834 - 1.27152i) q^{8} +(-2.81509 - 1.03696i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 87 q^{2} - 54 q^{3} + 29 q^{4} - 87 q^{5} - 58 q^{6} - 27 q^{7} - 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/531\mathbb{Z}\right)^\times\).

\(n\)	\(119\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{58}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.789160	+	1.00012i	0.558020	+	0.707188i	0.979906	−	0.199461i	\(-0.0639190\pi\)
							−0.421886	+	0.906649i	\(0.638632\pi\)
\(3\)	−0.304068	+	1.70515i	−0.175553	+	0.984470i
							\(5\)	−0.419645	+	2.29893i	−0.187671	+	1.02811i	0.744913	+	0.667161i	\(0.232491\pi\)
−0.932584	+	0.360952i	\(0.882452\pi\)
\(7\)	−2.07688	+	1.14960i	−0.784989	+	0.434509i	−0.825311	−	0.564678i	\(-0.809000\pi\)
							0.0403228	+	0.999187i	\(0.487161\pi\)
\(11\)	−4.36321	−	2.84472i	−1.31556	−	0.857715i	−0.319602	−	0.947552i	\(-0.603549\pi\)
							−0.995956	+	0.0898372i	\(0.971365\pi\)
\(13\)	−3.29184	+	3.00732i	−0.912993	+	0.834080i	−0.986592	−	0.163208i	\(-0.947816\pi\)
							0.0735989	+	0.997288i	\(0.476552\pi\)
\(17\)	3.51156	+	5.83626i	0.851679	+	1.41550i	0.908843	+	0.417137i	\(0.136967\pi\)
							−0.0571641	+	0.998365i	\(0.518206\pi\)
\(19\)	−0.479898	+	1.72844i	−0.110096	+	0.396530i	−0.997979	−	0.0635385i	\(-0.979761\pi\)
							0.887883	+	0.460069i	\(0.152175\pi\)
\(23\)	−0.0469146	+	0.0967799i	−0.00978238	+	0.0201800i	−0.904681	−	0.426089i	\(-0.859891\pi\)
							0.894899	+	0.446269i	\(0.147247\pi\)
\(29\)	0.687947	−	4.72965i	0.127748	−	0.878274i	−0.822940	−	0.568128i	\(-0.807668\pi\)
							0.950689	−	0.310146i	\(-0.100378\pi\)
\(31\)	4.44534	+	4.36580i	0.798407	+	0.784121i	0.979946	−	0.199263i	\(-0.0638549\pi\)
							−0.181539	+	0.983384i	\(0.558108\pi\)
\(37\)	−1.02626	+	2.21822i	−0.168716	+	0.364673i	−0.973428	−	0.228991i	\(-0.926457\pi\)
							0.804713	+	0.593664i	\(0.202319\pi\)
\(41\)	0.513212	−	7.09382i	0.0801502	−	1.10787i	−0.789353	−	0.613940i	\(-0.789584\pi\)
							0.869503	−	0.493928i	\(-0.164439\pi\)
\(43\)	−4.54813	+	8.96620i	−0.693584	+	1.36733i	0.225879	+	0.974155i	\(0.427475\pi\)
							−0.919463	+	0.393177i	\(0.871376\pi\)
\(47\)	−1.86979	+	0.341310i	−0.272737	+	0.0497851i	−0.314207	−	0.949355i	\(-0.601738\pi\)
							0.0414699	+	0.999140i	\(0.486796\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.p.a.128.42	yes	3248
9.5	odd	6	inner	531.2.p.a.482.42	yes	3248
59.6	odd	58	inner	531.2.p.a.65.42	✓	3248
531.419	even	174	inner	531.2.p.a.419.42	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.p.a.65.42	✓	3248	59.6	odd	58	inner
531.2.p.a.128.42	yes	3248	1.1	even	1	trivial
531.2.p.a.419.42	yes	3248	531.419	even	174	inner
531.2.p.a.482.42	yes	3248	9.5	odd	6	inner