Embedded newform 560.2.cp.a.221.20

• Label: 560.2.cp.a.221.20
• Level: $560$
• Weight: $2$
• Character: 560.221
• Analytic conductor: $4.472$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.cp (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47162251319\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			221.20
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.828573 + 1.14607i) q^{2} +(-1.27872 + 0.342633i) q^{3} +(-0.626934 - 1.89920i) q^{4} +(-0.965926 - 0.258819i) q^{5} +(0.666836 - 1.74940i) q^{6} +(0.500742 + 2.59793i) q^{7} +(2.69607 + 0.855117i) q^{8} +(-1.08034 + 0.623734i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(351\)	\(421\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.828573	+	1.14607i	−0.585890	+	0.810391i
							\(3\)	−1.27872	+	0.342633i	−0.738272	+	0.197819i	−0.608310	−	0.793700i	\(-0.708152\pi\)
−0.129962	+	0.991519i	\(0.541485\pi\)
\(5\)	−0.965926	−	0.258819i	−0.431975	−	0.115747i
							\(7\)	0.500742	+	2.59793i	0.189263	+	0.981927i
\(11\)	−0.579080	−	2.16116i	−0.174599	−	0.651613i								−0.996620	−	0.0821552i	\(-0.973820\pi\)
							0.822020	−	0.569458i	\(-0.192847\pi\)
\(13\)	−2.04779	−	2.04779i	−0.567954	−	0.567954i	0.363601	−	0.931555i	\(-0.381547\pi\)
							−0.931555	+	0.363601i	\(0.881547\pi\)
\(17\)	−1.49556	+	2.59039i	−0.362726	+	0.628261i	−0.988409	−	0.151817i	\(-0.951487\pi\)
							0.625682	+	0.780078i	\(0.284821\pi\)
\(19\)	0.905203	−	3.37826i	0.207668	−	0.775027i	−0.780952	−	0.624591i	\(-0.785266\pi\)
							0.988620	−	0.150436i	\(-0.0480676\pi\)
\(23\)	1.48852	−	0.859397i	0.310378	−	0.179197i	−0.336718	−	0.941606i	\(-0.609317\pi\)
							0.647095	+	0.762409i	\(0.275984\pi\)
\(29\)	−2.48453	−	2.48453i	−0.461365	−	0.461365i	0.437738	−	0.899103i	\(-0.355780\pi\)
							−0.899103	+	0.437738i	\(0.855780\pi\)
\(31\)	2.55475	−	4.42496i	0.458847	−	0.794747i	−0.540053	−	0.841631i	\(-0.681596\pi\)
							0.998900	+	0.0468840i	\(0.0149291\pi\)
\(37\)	5.59326	+	1.49871i	0.919527	+	0.246386i	0.687383	−	0.726295i	\(-0.258759\pi\)
							0.232144	+	0.972682i	\(0.425426\pi\)
\(41\)		−	9.35961i		−	1.46173i	−0.682525	−	0.730863i	\(-0.739118\pi\)
							0.682525	−	0.730863i	\(-0.260882\pi\)
\(43\)	7.05383	−	7.05383i	1.07570	−	1.07570i	0.0788097	−	0.996890i	\(-0.474888\pi\)
							0.996890	−	0.0788097i	\(-0.0251119\pi\)
\(47\)	−3.27427	−	5.67120i	−0.477601	−	0.827229i	0.522070	−	0.852903i	\(-0.325160\pi\)
							−0.999670	+	0.0256741i	\(0.991827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.20	✓	256
7.2	even	3	inner	560.2.cp.a.541.62	yes	256
16.5	even	4	inner	560.2.cp.a.501.62	yes	256
112.37	even	12	inner	560.2.cp.a.261.20	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.20	✓	256	1.1	even	1	trivial
560.2.cp.a.261.20	yes	256	112.37	even	12	inner
560.2.cp.a.501.62	yes	256	16.5	even	4	inner
560.2.cp.a.541.62	yes	256	7.2	even	3	inner