Embedded newform 531.2.p.a.128.36

• Label: 531.2.p.a.128.36
• 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			128.36
Character	\(\chi\)	\(=\)	531.128
Dual form			531.2.p.a.419.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.341948 + 0.433356i) q^{2} +(0.504199 - 1.65704i) q^{3} +(0.394266 - 1.64879i) q^{4} +(0.0144711 - 0.0792767i) q^{5} +(0.890498 - 0.348124i) q^{6} +(-4.36255 + 2.41477i) q^{7} +(1.85133 - 0.856517i) q^{8} +(-2.49157 - 1.67096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 87 q^{2} - 54 q^{3} + 29 q^{4} - 87 q^{5} - 58 q^{6} - 27 q^{7} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.341948	+	0.433356i	0.241793	+	0.306429i	0.890838	−	0.454322i	\(-0.150118\pi\)
							−0.649044	+	0.760751i	\(0.724831\pi\)
\(3\)	0.504199	−	1.65704i	0.291099	−	0.956693i
							\(5\)	0.0144711	−	0.0792767i	0.00647168	−	0.0354536i	−0.980349	−	0.197270i	\(-0.936793\pi\)
0.986821	+	0.161816i	\(0.0517351\pi\)
\(7\)	−4.36255	+	2.41477i	−1.64889	+	0.912697i	−0.662345	+	0.749199i	\(0.730439\pi\)
							−0.986544	+	0.163498i	\(0.947722\pi\)
\(11\)	−4.51086	−	2.94098i	−1.36008	−	0.886739i	−0.361125	−	0.932517i	\(-0.617607\pi\)
							−0.998952	+	0.0457780i	\(0.985423\pi\)
\(13\)	2.43460	−	2.22417i	0.675236	−	0.616874i	−0.262207	−	0.965012i	\(-0.584450\pi\)
							0.937443	+	0.348138i	\(0.113186\pi\)
\(17\)	0.996637	+	1.65642i	0.241720	+	0.401742i	0.953840	−	0.300316i	\(-0.0970922\pi\)
							−0.712120	+	0.702058i	\(0.752265\pi\)
\(19\)	0.988139	−	3.55896i	0.226695	−	0.816480i	−0.759446	−	0.650570i	\(-0.774530\pi\)
							0.986141	−	0.165910i	\(-0.0530562\pi\)
\(23\)	0.781872	−	1.61292i	0.163032	−	0.336317i	−0.802479	−	0.596680i	\(-0.796486\pi\)
							0.965511	+	0.260363i	\(0.0838424\pi\)
\(29\)	−0.436762	+	3.00275i	−0.0811047	+	0.557597i	0.908949	+	0.416907i	\(0.136886\pi\)
							−0.990054	+	0.140690i	\(0.955068\pi\)
\(31\)	−6.19640	−	6.08553i	−1.11291	−	1.09299i	−0.994991	−	0.0999692i	\(-0.968126\pi\)
							−0.117917	−	0.993024i	\(-0.537622\pi\)
\(37\)	3.49781	−	7.56040i	0.575037	−	1.24292i	−0.373904	−	0.927467i	\(-0.621981\pi\)
							0.948941	−	0.315454i	\(-0.102157\pi\)
\(41\)	−0.841773	+	11.6353i	−0.131463	+	1.81713i	0.345732	+	0.938333i	\(0.387631\pi\)
							−0.477194	+	0.878798i	\(0.658346\pi\)
\(43\)	−1.03079	+	2.03210i	−0.157194	+	0.309893i	−0.956858	−	0.290555i	\(-0.906160\pi\)
							0.799664	+	0.600447i	\(0.205011\pi\)
\(47\)	8.36728	−	1.52736i	1.22049	−	0.222788i	0.469401	−	0.882985i	\(-0.344470\pi\)
							0.751092	+	0.660197i	\(0.229527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.p.a.128.36	yes	3248
9.5	odd	6	inner	531.2.p.a.482.36	yes	3248
59.6	odd	58	inner	531.2.p.a.65.36	✓	3248
531.419	even	174	inner	531.2.p.a.419.36	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.p.a.65.36	✓	3248	59.6	odd	58	inner
531.2.p.a.128.36	yes	3248	1.1	even	1	trivial
531.2.p.a.419.36	yes	3248	531.419	even	174	inner
531.2.p.a.482.36	yes	3248	9.5	odd	6	inner