Embedded newform 572.2.bk.a.31.4

• Label: 572.2.bk.a.31.4
• Level: $572$
• Weight: $2$
• Character: 572.31
• Analytic conductor: $4.567$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.bk (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.56744299562\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(80\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			31.4
Character	\(\chi\)	\(=\)	572.31
Dual form			572.2.bk.a.203.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39464 + 0.234483i) q^{2} +(0.111085 + 0.152895i) q^{3} +(1.89004 - 0.654039i) q^{4} +(-2.01367 - 1.02602i) q^{5} +(-0.190775 - 0.187186i) q^{6} +(3.86875 - 0.612750i) q^{7} +(-2.48256 + 1.35533i) q^{8} +(0.916014 - 2.81920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 6 q^{2} - 12 q^{5} - 14 q^{6} - 6 q^{8} + 120 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(287\)	\(353\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39464	+	0.234483i	−0.986159	+	0.165805i
							\(3\)	0.111085	+	0.152895i	0.0641349	+	0.0882741i	0.839880	−	0.542773i	\(-0.182626\pi\)
−0.775745	+	0.631047i	\(0.782626\pi\)
\(5\)	−2.01367	−	1.02602i	−0.900541	−	0.458849i	−0.0585156	−	0.998286i	\(-0.518637\pi\)
							−0.842025	+	0.539438i	\(0.818637\pi\)
\(7\)	3.86875	−	0.612750i	1.46225	−	0.231598i	0.625947	−	0.779865i	\(-0.284712\pi\)
							0.836303	+	0.548268i	\(0.184712\pi\)
\(11\)	−1.55914	−	2.92730i	−0.470100	−	0.882613i
							\(13\)	−2.30038	+	2.77637i	−0.638011	+	0.770027i
\(17\)	−1.36729	+	0.444258i	−0.331615	+	0.107748i								−0.470092	−	0.882617i	\(-0.655779\pi\)
							0.138477	+	0.990366i	\(0.455779\pi\)
\(19\)	−2.78938	−	0.441794i	−0.639927	−	0.101354i	−0.171965	−	0.985103i	\(-0.555012\pi\)
							−0.467962	+	0.883749i	\(0.655012\pi\)
\(23\)	3.14251			0.655259			0.327630	−	0.944806i	\(-0.393750\pi\)
							0.327630	+	0.944806i	\(0.393750\pi\)
\(29\)	−5.14287	−	3.73651i	−0.955006	−	0.693853i	−0.00302053	−	0.999995i	\(-0.500961\pi\)
							−0.951986	+	0.306143i	\(0.900961\pi\)
\(31\)	−2.78028	+	1.41663i	−0.499354	+	0.254433i	−0.685479	−	0.728093i	\(-0.740407\pi\)
							0.186125	+	0.982526i	\(0.440407\pi\)
\(37\)	−1.36105	−	8.59332i	−0.223755	−	1.41273i	−0.802221	−	0.597027i	\(-0.796348\pi\)
							0.578466	−	0.815707i	\(-0.303652\pi\)
\(41\)	0.942323	−	5.94959i	0.147166	−	0.929170i	−0.798018	−	0.602633i	\(-0.794118\pi\)
							0.945184	−	0.326537i	\(-0.105882\pi\)
\(43\)	9.94526			1.51664			0.758319	−	0.651884i	\(-0.226021\pi\)
							0.758319	+	0.651884i	\(0.226021\pi\)
\(47\)	−1.22238	+	7.71782i	−0.178303	+	1.12576i	0.722448	+	0.691425i	\(0.243017\pi\)
							−0.900751	+	0.434335i	\(0.856983\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.bk.a.31.4	✓	640
4.3	odd	2	inner	572.2.bk.a.31.60	yes	640
11.5	even	5	inner	572.2.bk.a.291.60	yes	640
13.8	odd	4	inner	572.2.bk.a.515.37	yes	640
44.27	odd	10	inner	572.2.bk.a.291.37	yes	640
52.47	even	4	inner	572.2.bk.a.515.60	yes	640
143.60	odd	20	inner	572.2.bk.a.203.60	yes	640
572.203	even	20	inner	572.2.bk.a.203.4	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.bk.a.31.4	✓	640	1.1	even	1	trivial
572.2.bk.a.31.60	yes	640	4.3	odd	2	inner
572.2.bk.a.203.4	yes	640	572.203	even	20	inner
572.2.bk.a.203.60	yes	640	143.60	odd	20	inner
572.2.bk.a.291.37	yes	640	44.27	odd	10	inner
572.2.bk.a.291.60	yes	640	11.5	even	5	inner
572.2.bk.a.515.37	yes	640	13.8	odd	4	inner
572.2.bk.a.515.60	yes	640	52.47	even	4	inner