Embedded newform 531.2.m.a.130.52

• Label: 531.2.m.a.130.52
• 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			130.52
Character	\(\chi\)	\(=\)	531.130
Dual form			531.2.m.a.241.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.75731 + 1.54815i) q^{2} +(-1.66806 - 0.466452i) q^{3} +(0.439268 + 3.45708i) q^{4} +(-3.69570 - 0.537555i) q^{5} +(-2.20915 - 3.40210i) q^{6} +(-0.274434 - 3.03169i) q^{7} +(-1.95156 + 2.87834i) q^{8} +(2.56484 + 1.55614i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 29 q^{2} - 58 q^{3} + 29 q^{4} - 25 q^{5} - 66 q^{6} - 27 q^{7} - 116 q^{8} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.75731	+	1.54815i	1.24260	+	1.09470i	0.991950	+	0.126627i	\(0.0404153\pi\)
							0.250653	+	0.968077i	\(0.419355\pi\)
\(3\)	−1.66806	−	0.466452i	−0.963055	−	0.269306i
							\(5\)	−3.69570	−	0.537555i	−1.65277	−	0.240402i	−0.747199	−	0.664600i	\(-0.768602\pi\)
−0.905569	+	0.424198i	\(0.860556\pi\)
\(7\)	−0.274434	−	3.03169i	−0.103726	−	1.14587i	−0.866490	−	0.499195i	\(-0.833629\pi\)
							0.762763	−	0.646678i	\(-0.223842\pi\)
\(11\)	3.06385	−	3.11968i	0.923786	−	0.940618i	−0.0746109	−	0.997213i	\(-0.523771\pi\)
							0.998397	+	0.0565949i	\(0.0180243\pi\)
\(13\)	−3.11047	−	0.112369i	−0.862688	−	0.0311654i	−0.398778	−	0.917048i	\(-0.630566\pi\)
							−0.463910	+	0.885882i	\(0.653554\pi\)
\(17\)	4.72512	−	2.18607i	1.14601	−	0.530201i	0.247385	−	0.968917i	\(-0.420429\pi\)
							0.898626	+	0.438716i	\(0.144567\pi\)
\(19\)	−6.56146	−	1.44429i	−1.50530	−	0.331342i	−0.615664	−	0.788009i	\(-0.711112\pi\)
							−0.889638	+	0.456667i	\(0.849043\pi\)
\(23\)	−6.38785	−	2.41245i	−1.33196	−	0.503030i	−0.417856	−	0.908513i	\(-0.637218\pi\)
							−0.914103	+	0.405483i	\(0.867103\pi\)
\(29\)	−1.18927	−	5.90915i	−0.220842	−	1.09730i	−0.922858	−	0.385140i	\(-0.874153\pi\)
							0.702016	−	0.712161i	\(-0.252283\pi\)
\(31\)	−0.603867	+	1.90522i	−0.108458	+	0.342188i	−0.992012	−	0.126140i	\(-0.959741\pi\)
							0.883555	+	0.468328i	\(0.155143\pi\)
\(37\)	0.744993	+	1.09878i	0.122476	+	0.180639i	0.883984	−	0.467517i	\(-0.154851\pi\)
							−0.761508	+	0.648155i	\(0.775541\pi\)
\(41\)	−3.11711	−	2.55227i	−0.486811	−	0.398597i	0.357950	−	0.933741i	\(-0.383476\pi\)
							−0.844761	+	0.535144i	\(0.820257\pi\)
\(43\)	−11.1504	+	2.88012i	−1.70043	+	0.439215i	−0.969867	−	0.243634i	\(-0.921661\pi\)
							−0.730560	+	0.682848i	\(0.760741\pi\)
\(47\)	3.62535	−	0.527322i	0.528812	−	0.0769178i	0.125700	−	0.992068i	\(-0.459882\pi\)
							0.403112	+	0.915151i	\(0.367928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.m.a.130.52	✓	3248
9.7	even	3	inner	531.2.m.a.484.52	yes	3248
59.5	even	29	inner	531.2.m.a.418.52	yes	3248
531.241	even	87	inner	531.2.m.a.241.52	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.m.a.130.52	✓	3248	1.1	even	1	trivial
531.2.m.a.241.52	yes	3248	531.241	even	87	inner
531.2.m.a.418.52	yes	3248	59.5	even	29	inner
531.2.m.a.484.52	yes	3248	9.7	even	3	inner