Embedded newform 531.2.p.a.128.37

• Label: 531.2.p.a.128.37
• 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			128.37
Character	\(\chi\)	\(=\)	531.128
Dual form			531.2.p.a.419.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.502433 + 0.636742i) q^{2} +(1.35950 - 1.07321i) q^{3} +(0.312133 - 1.30532i) q^{4} +(0.0705256 - 0.386359i) q^{5} +(1.36641 + 0.326435i) q^{6} +(0.590174 - 0.326675i) q^{7} +(2.46024 - 1.13823i) q^{8} +(0.696461 - 2.91804i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 87 q^{2} - 54 q^{3} + 29 q^{4} - 87 q^{5} - 58 q^{6} - 27 q^{7} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.502433	+	0.636742i	0.355274	+	0.450245i	0.929703	−	0.368309i	\(-0.120063\pi\)
							−0.574429	+	0.818554i	\(0.694776\pi\)
\(3\)	1.35950	−	1.07321i	0.784906	−	0.619615i
							\(5\)	0.0705256	−	0.386359i	0.0315400	−	0.172785i	−0.964174	−	0.265271i	\(-0.914539\pi\)
0.995714	+	0.0924857i	\(0.0294813\pi\)
\(7\)	0.590174	−	0.326675i	0.223065	−	0.123471i	−0.368800	−	0.929509i	\(-0.620231\pi\)
							0.591865	+	0.806037i	\(0.298392\pi\)
\(11\)	−0.0272708	−	0.0177800i	−0.00822246	−	0.00536086i	0.542035	−	0.840356i	\(-0.317654\pi\)
							−0.550257	+	0.834995i	\(0.685470\pi\)
\(13\)	−2.29104	+	2.09302i	−0.635421	+	0.580499i	−0.926544	−	0.376186i	\(-0.877235\pi\)
							0.291123	+	0.956685i	\(0.405971\pi\)
\(17\)	−2.69937	−	4.48639i	−0.654694	−	1.08811i	−0.989320	−	0.145759i	\(-0.953438\pi\)
							0.334626	−	0.942351i	\(-0.391390\pi\)
\(19\)	−1.41883	+	5.11016i	−0.325502	+	1.17235i	0.601989	+	0.798504i	\(0.294375\pi\)
							−0.927491	+	0.373846i	\(0.878039\pi\)
\(23\)	−1.41275	+	2.91435i	−0.294578	+	0.607683i	−0.994286	−	0.106746i	\(-0.965957\pi\)
							0.699708	+	0.714429i	\(0.253313\pi\)
\(29\)	0.0206719	−	0.142120i	0.00383868	−	0.0263910i	−0.987579	−	0.157122i	\(-0.949778\pi\)
							0.991418	+	0.130731i	\(0.0417324\pi\)
\(31\)	7.65474	+	7.51777i	1.37483	+	1.35023i	0.875012	+	0.484101i	\(0.160853\pi\)
							0.499820	+	0.866129i	\(0.333400\pi\)
\(37\)	2.99777	−	6.47957i	0.492830	−	1.06523i	−0.488440	−	0.872598i	\(-0.662434\pi\)
							0.981270	−	0.192637i	\(-0.0617041\pi\)
\(41\)	−0.275054	+	3.80191i	−0.0429563	+	0.593759i	0.930707	+	0.365767i	\(0.119193\pi\)
							−0.973663	+	0.227992i	\(0.926784\pi\)
\(43\)	1.31581	−	2.59399i	0.200659	−	0.395580i	−0.769285	−	0.638905i	\(-0.779388\pi\)
							0.969945	+	0.243325i	\(0.0782382\pi\)
\(47\)	9.87539	−	1.80265i	1.44047	−	0.262943i	0.597920	−	0.801556i	\(-0.295994\pi\)
							0.842553	+	0.538613i	\(0.181051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.p.a.128.37	yes	3248
9.5	odd	6	inner	531.2.p.a.482.37	yes	3248
59.6	odd	58	inner	531.2.p.a.65.37	✓	3248
531.419	even	174	inner	531.2.p.a.419.37	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.p.a.65.37	✓	3248	59.6	odd	58	inner
531.2.p.a.128.37	yes	3248	1.1	even	1	trivial
531.2.p.a.419.37	yes	3248	531.419	even	174	inner
531.2.p.a.482.37	yes	3248	9.5	odd	6	inner