Embedded newform 572.2.j.a.463.9

• Label: 572.2.j.a.463.9
• Level: $572$
• Weight: $2$
• Character: 572.463
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(70\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			463.9
Character	\(\chi\)	\(=\)	572.463
Dual form			572.2.j.a.551.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31145 + 0.529251i) q^{2} +2.04664i q^{3} +(1.43979 - 1.38817i) q^{4} +(-1.50803 - 1.50803i) q^{5} +(-1.08318 - 2.68406i) q^{6} +(0.828927 + 0.828927i) q^{7} +(-1.15351 + 2.58252i) q^{8} -1.18872 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q + 4 q^{5} - 12 q^{6} + 12 q^{8} - 140 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31145	+	0.529251i	−0.927333	+	0.374237i
							\(3\)			2.04664i			1.18163i	0.806808	+	0.590813i	\(0.201193\pi\)
−0.806808	+	0.590813i	\(0.798807\pi\)
\(5\)	−1.50803	−	1.50803i	−0.674411	−	0.674411i	0.284319	−	0.958730i	\(-0.408233\pi\)
							−0.958730	+	0.284319i	\(0.908233\pi\)
\(7\)	0.828927	+	0.828927i	0.313305	+	0.313305i	0.846189	−	0.532884i	\(-0.178892\pi\)
							−0.532884	+	0.846189i	\(0.678892\pi\)
\(11\)	−0.707107	−	0.707107i	−0.213201	−	0.213201i
							\(13\)	−2.54308	+	2.55592i	−0.705324	+	0.708885i
\(17\)		−	0.715993i		−	0.173654i								−0.996223	−	0.0868269i	\(-0.972327\pi\)
							0.996223	−	0.0868269i	\(-0.0276727\pi\)
\(19\)	−4.98401	+	4.98401i	−1.14341	+	1.14341i	−0.155588	+	0.987822i	\(0.549727\pi\)
							−0.987822	+	0.155588i	\(0.950273\pi\)
\(23\)	−4.09441			−0.853743			−0.426872	−	0.904312i	\(-0.640384\pi\)
							−0.426872	+	0.904312i	\(0.640384\pi\)
\(29\)	−4.13281			−0.767444			−0.383722	−	0.923449i	\(-0.625358\pi\)
							−0.383722	+	0.923449i	\(0.625358\pi\)
\(31\)	−1.32626	+	1.32626i	−0.238203	+	0.238203i	−0.816106	−	0.577903i	\(-0.803871\pi\)
							0.577903	+	0.816106i	\(0.303871\pi\)
\(37\)	−5.64413	+	5.64413i	−0.927889	+	0.927889i	−0.997569	−	0.0696807i	\(-0.977802\pi\)
							0.0696807	+	0.997569i	\(0.477802\pi\)
\(41\)	−4.49792	−	4.49792i	−0.702458	−	0.702458i	0.262480	−	0.964937i	\(-0.415460\pi\)
							−0.964937	+	0.262480i	\(0.915460\pi\)
\(43\)	12.8102			1.95354			0.976771	−	0.214286i	\(-0.0687423\pi\)
							0.976771	+	0.214286i	\(0.0687423\pi\)
\(47\)	−5.25220	−	5.25220i	−0.766112	−	0.766112i	0.211307	−	0.977420i	\(-0.432228\pi\)
							−0.977420	+	0.211307i	\(0.932228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.j.a.463.9	✓	140
4.3	odd	2	inner	572.2.j.a.463.45	yes	140
13.5	odd	4	inner	572.2.j.a.551.45	yes	140
52.31	even	4	inner	572.2.j.a.551.9	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.j.a.463.9	✓	140	1.1	even	1	trivial
572.2.j.a.463.45	yes	140	4.3	odd	2	inner
572.2.j.a.551.9	yes	140	52.31	even	4	inner
572.2.j.a.551.45	yes	140	13.5	odd	4	inner