Embedded newform 531.2.m.a.130.51

• Label: 531.2.m.a.130.51
• Level: $531$
• Weight: $2$
• Character: 531.130
• 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.m (of order \(87\), degree \(56\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.24005634733\)
Analytic rank:	\(0\)
Dimension:	\(3248\)
Relative dimension:	\(58\) over \(\Q(\zeta_{87})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{87}]$

Embedding invariants

Embedding label			130.51
Character	\(\chi\)	\(=\)	531.130
Dual form			531.2.m.a.241.51

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.69431 + 1.49265i) q^{2} +(-0.860227 - 1.50333i) q^{3} +(0.390591 + 3.07399i) q^{4} +(2.14482 + 0.311972i) q^{5} +(0.786458 - 3.83114i) q^{6} +(-0.283332 - 3.12999i) q^{7} +(-1.39226 + 2.05343i) q^{8} +(-1.52002 + 2.58641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 29 q^{2} - 58 q^{3} + 29 q^{4} - 25 q^{5} - 66 q^{6} - 27 q^{7} - 116 q^{8} - 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(119\)	\(415\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{26}{29}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.69431	+	1.49265i	1.19806	+	1.05546i	0.997118	+	0.0758611i	\(0.0241706\pi\)
							0.200943	+	0.979603i	\(0.435600\pi\)
\(3\)	−0.860227	−	1.50333i	−0.496652	−	0.867950i
							\(5\)	2.14482	+	0.311972i	0.959192	+	0.139518i	0.605501	−	0.795844i	\(-0.292973\pi\)
0.353691	+	0.935362i	\(0.384927\pi\)
\(7\)	−0.283332	−	3.12999i	−0.107089	−	1.18303i	−0.854784	−	0.518983i	\(-0.826311\pi\)
							0.747695	−	0.664042i	\(-0.231161\pi\)
\(11\)	1.92920	−	1.96435i	0.581676	−	0.592275i	−0.358261	−	0.933621i	\(-0.616630\pi\)
							0.939937	+	0.341347i	\(0.110883\pi\)
\(13\)	3.28357	+	0.118622i	0.910697	+	0.0328998i	0.487486	−	0.873131i	\(-0.337914\pi\)
							0.423212	+	0.906031i	\(0.360903\pi\)
\(17\)	−4.21453	+	1.94985i	−1.02217	+	0.472907i	−0.858068	−	0.513536i	\(-0.828335\pi\)
							−0.164104	+	0.986443i	\(0.552473\pi\)
\(19\)	1.47592	+	0.324874i	0.338599	+	0.0745313i	0.381015	−	0.924569i	\(-0.375575\pi\)
							−0.0424163	+	0.999100i	\(0.513506\pi\)
\(23\)	−1.53865	−	0.581089i	−0.320830	−	0.121165i	0.187658	−	0.982234i	\(-0.439910\pi\)
							−0.508488	+	0.861069i	\(0.669795\pi\)
\(29\)	1.68868	+	8.39054i	0.313579	+	1.55808i	0.751907	+	0.659269i	\(0.229134\pi\)
							−0.438327	+	0.898815i	\(0.644429\pi\)
\(31\)	0.396337	−	1.25046i	0.0711842	−	0.224589i	−0.911034	−	0.412331i	\(-0.864715\pi\)
							0.982218	+	0.187742i	\(0.0601170\pi\)
\(37\)	2.04040	+	3.00937i	0.335440	+	0.494737i	0.957579	−	0.288169i	\(-0.0930466\pi\)
							−0.622139	+	0.782907i	\(0.713736\pi\)
\(41\)	−5.28287	−	4.32557i	−0.825045	−	0.675540i	0.123444	−	0.992352i	\(-0.460606\pi\)
							−0.948489	+	0.316811i	\(0.897388\pi\)
\(43\)	−1.04666	+	0.270349i	−0.159615	+	0.0412279i	−0.329045	−	0.944314i	\(-0.606727\pi\)
							0.169430	+	0.985542i	\(0.445807\pi\)
\(47\)	−8.85925	+	1.28861i	−1.29225	+	0.187964i	−0.755150	−	0.655552i	\(-0.772436\pi\)
							−0.537105	+	0.843516i	\(0.680482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.m.a.130.51	✓	3248
9.7	even	3	inner	531.2.m.a.484.51	yes	3248
59.5	even	29	inner	531.2.m.a.418.51	yes	3248
531.241	even	87	inner	531.2.m.a.241.51	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.m.a.130.51	✓	3248	1.1	even	1	trivial
531.2.m.a.241.51	yes	3248	531.241	even	87	inner
531.2.m.a.418.51	yes	3248	59.5	even	29	inner
531.2.m.a.484.51	yes	3248	9.7	even	3	inner