Embedded newform 531.2.m.a.130.58

• Label: 531.2.m.a.130.58
• 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			130.58
Character	\(\chi\)	\(=\)	531.130
Dual form			531.2.m.a.241.58

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.02760 + 1.78627i) q^{2} +(-0.305910 + 1.70482i) q^{3} +(0.668307 + 5.25964i) q^{4} +(-0.249388 - 0.0362744i) q^{5} +(-3.66554 + 2.91026i) q^{6} +(-0.0230235 - 0.254342i) q^{7} +(-5.00720 + 7.38507i) q^{8} +(-2.81284 - 1.04304i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3248 q - 29 q^{2} - 58 q^{3} + 29 q^{4} - 25 q^{5} - 66 q^{6} - 27 q^{7} - 116 q^{8} - 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.02760	+	1.78627i	1.43373	+	1.26308i	0.912134	+	0.409892i	\(0.134434\pi\)
							0.521597	+	0.853192i	\(0.325336\pi\)
\(3\)	−0.305910	+	1.70482i	−0.176617	+	0.984280i
							\(5\)	−0.249388	−	0.0362744i	−0.111530	−	0.0162224i	0.0879458	−	0.996125i	\(-0.471970\pi\)
−0.199475	+	0.979903i	\(0.563924\pi\)
\(7\)	−0.0230235	−	0.254342i	−0.00870205	−	0.0961322i	0.990416	−	0.138114i	\(-0.0441041\pi\)
							−0.999118	+	0.0419819i	\(0.986633\pi\)
\(11\)	2.39159	−	2.43516i	0.721091	−	0.734229i	−0.251218	−	0.967931i	\(-0.580831\pi\)
							0.972308	+	0.233702i	\(0.0750839\pi\)
\(13\)	4.36407	+	0.157656i	1.21037	+	0.0437259i	0.633917	−	0.773401i	\(-0.281446\pi\)
							0.576458	+	0.817127i	\(0.304435\pi\)
\(17\)	3.26650	−	1.51125i	0.792243	−	0.366531i	0.0183379	−	0.999832i	\(-0.494163\pi\)
							0.773906	+	0.633301i	\(0.218300\pi\)
\(19\)	−3.45314	−	0.760093i	−0.792204	−	0.174377i	−0.199607	−	0.979876i	\(-0.563967\pi\)
							−0.592598	+	0.805499i	\(0.701898\pi\)
\(23\)	0.916255	+	0.346035i	0.191052	+	0.0721532i	0.446985	−	0.894541i	\(-0.352498\pi\)
							−0.255933	+	0.966695i	\(0.582383\pi\)
\(29\)	0.634250	+	3.15140i	0.117777	+	0.585201i	0.994547	+	0.104290i	\(0.0332571\pi\)
							−0.876770	+	0.480911i	\(0.840306\pi\)
\(31\)	−1.15008	+	3.62853i	−0.206560	+	0.651703i	0.792562	+	0.609792i	\(0.208747\pi\)
							−0.999121	+	0.0419113i	\(0.986655\pi\)
\(37\)	−2.27229	−	3.35138i	−0.373562	−	0.550963i	0.593718	−	0.804673i	\(-0.297659\pi\)
							−0.967280	+	0.253710i	\(0.918349\pi\)
\(41\)	5.09479	+	4.17158i	0.795673	+	0.651491i	0.941195	−	0.337864i	\(-0.109704\pi\)
							−0.145522	+	0.989355i	\(0.546486\pi\)
\(43\)	9.77455	−	2.52473i	1.49061	−	0.385018i	0.585664	−	0.810554i	\(-0.300834\pi\)
							0.904942	+	0.425536i	\(0.139914\pi\)
\(47\)	−5.55446	+	0.807918i	−0.810201	+	0.117847i	−0.536427	−	0.843947i	\(-0.680226\pi\)
							−0.273774	+	0.961794i	\(0.588272\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.m.a.130.58	✓	3248
9.7	even	3	inner	531.2.m.a.484.58	yes	3248
59.5	even	29	inner	531.2.m.a.418.58	yes	3248
531.241	even	87	inner	531.2.m.a.241.58	yes	3248

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.2.m.a.130.58	✓	3248	1.1	even	1	trivial
531.2.m.a.241.58	yes	3248	531.241	even	87	inner
531.2.m.a.418.58	yes	3248	59.5	even	29	inner
531.2.m.a.484.58	yes	3248	9.7	even	3	inner