Embedded newform 560.2.cp.a.221.29

• Label: 560.2.cp.a.221.29
• 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			221.29
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.293008 + 1.38353i) q^{2} +(2.13619 - 0.572390i) q^{3} +(-1.82829 - 0.810770i) q^{4} +(-0.965926 - 0.258819i) q^{5} +(0.165995 + 3.12319i) q^{6} +(1.39832 + 2.24604i) q^{7} +(1.65743 - 2.29193i) q^{8} +(1.63759 - 0.945465i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.293008	+	1.38353i	−0.207188	+	0.978301i
							\(3\)	2.13619	−	0.572390i	1.23333	−	0.330469i	0.417454	−	0.908698i	\(-0.362923\pi\)
0.815875	+	0.578229i	\(0.196256\pi\)
\(5\)	−0.965926	−	0.258819i	−0.431975	−	0.115747i
							\(7\)	1.39832	+	2.24604i	0.528515	+	0.848924i
\(11\)	0.592210	+	2.21016i	0.178558	+	0.666388i								0.995918	+	0.0902607i	\(0.0287700\pi\)
							−0.817360	+	0.576127i	\(0.804563\pi\)
\(13\)	2.68950	+	2.68950i	0.745934	+	0.745934i	0.973713	−	0.227779i	\(-0.0731464\pi\)
							−0.227779	+	0.973713i	\(0.573146\pi\)
\(17\)	2.19357	−	3.79937i	0.532018	−	0.921482i	−0.467283	−	0.884108i	\(-0.654767\pi\)
							0.999301	−	0.0373747i	\(-0.0118995\pi\)
\(19\)	0.250597	−	0.935240i	0.0574908	−	0.214559i	−0.931205	−	0.364497i	\(-0.881241\pi\)
							0.988695	+	0.149938i	\(0.0479076\pi\)
\(23\)	1.49204	−	0.861427i	0.311111	−	0.179620i	−0.336313	−	0.941750i	\(-0.609180\pi\)
							0.647423	+	0.762130i	\(0.275847\pi\)
\(29\)	3.50076	+	3.50076i	0.650075	+	0.650075i	0.953011	−	0.302936i	\(-0.0979668\pi\)
							−0.302936	+	0.953011i	\(0.597967\pi\)
\(31\)	−3.18289	+	5.51292i	−0.571663	+	0.990149i	0.424732	+	0.905319i	\(0.360368\pi\)
							−0.996395	+	0.0848305i	\(0.972965\pi\)
\(37\)	0.0696912	+	0.0186737i	0.0114572	+	0.00306994i	0.264543	−	0.964374i	\(-0.414779\pi\)
							−0.253086	+	0.967444i	\(0.581446\pi\)
\(41\)		−	1.82694i		−	0.285321i	−0.989772	−	0.142660i	\(-0.954434\pi\)
							0.989772	−	0.142660i	\(-0.0455657\pi\)
\(43\)	5.09952	−	5.09952i	0.777669	−	0.777669i	−0.201765	−	0.979434i	\(-0.564668\pi\)
							0.979434	+	0.201765i	\(0.0646676\pi\)
\(47\)	2.76625	+	4.79129i	0.403499	+	0.698881i	0.994146	−	0.108049i	\(-0.0344605\pi\)
							−0.590646	+	0.806931i	\(0.701127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.29	✓	256
7.2	even	3	inner	560.2.cp.a.541.59	yes	256
16.5	even	4	inner	560.2.cp.a.501.59	yes	256
112.37	even	12	inner	560.2.cp.a.261.29	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.29	✓	256	1.1	even	1	trivial
560.2.cp.a.261.29	yes	256	112.37	even	12	inner
560.2.cp.a.501.59	yes	256	16.5	even	4	inner
560.2.cp.a.541.59	yes	256	7.2	even	3	inner