Embedded newform 560.2.cp.a.221.13

• Label: 560.2.cp.a.221.13
• 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			221.13
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13864 + 0.838751i) q^{2} +(-2.01239 + 0.539219i) q^{3} +(0.592992 - 1.91007i) q^{4} +(0.965926 + 0.258819i) q^{5} +(1.83912 - 2.30187i) q^{6} +(0.308773 - 2.62767i) q^{7} +(0.926869 + 2.67225i) q^{8} +(1.16089 - 0.670242i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13864	+	0.838751i	−0.805139	+	0.593087i
							\(3\)	−2.01239	+	0.539219i	−1.16186	+	0.311318i	−0.787707	−	0.616050i	\(-0.788732\pi\)
−0.374149	+	0.927369i	\(0.622065\pi\)
\(5\)	0.965926	+	0.258819i	0.431975	+	0.115747i
							\(7\)	0.308773	−	2.62767i	0.116705	−	0.993167i
\(11\)	−0.473055	−	1.76546i	−0.142631	−	0.532308i								−0.999849	−	0.0173540i	\(-0.994476\pi\)
							0.857218	−	0.514954i	\(-0.172191\pi\)
\(13\)	0.888531	+	0.888531i	0.246434	+	0.246434i	0.819505	−	0.573071i	\(-0.194248\pi\)
							−0.573071	+	0.819505i	\(0.694248\pi\)
\(17\)	−3.54942	+	6.14777i	−0.860860	+	1.49105i	0.0102401	+	0.999948i	\(0.496740\pi\)
							−0.871100	+	0.491106i	\(0.836593\pi\)
\(19\)	0.611028	−	2.28039i	0.140180	−	0.523157i	−0.859743	−	0.510727i	\(-0.829376\pi\)
							0.999923	−	0.0124306i	\(-0.00395687\pi\)
\(23\)	−6.40255	+	3.69652i	−1.33502	+	0.770777i	−0.986065	−	0.166361i	\(-0.946798\pi\)
							−0.348960	+	0.937138i	\(0.613465\pi\)
\(29\)	−4.60896	−	4.60896i	−0.855863	−	0.855863i	0.134985	−	0.990848i	\(-0.456901\pi\)
							−0.990848	+	0.134985i	\(0.956901\pi\)
\(31\)	0.980894	−	1.69896i	0.176174	−	0.305142i	−0.764393	−	0.644751i	\(-0.776961\pi\)
							0.940567	+	0.339609i	\(0.110295\pi\)
\(37\)	−5.76056	−	1.54354i	−0.947030	−	0.253756i	−0.247928	−	0.968778i	\(-0.579750\pi\)
							−0.699101	+	0.715022i	\(0.746416\pi\)
\(41\)			4.87909i			0.761986i	0.924578	+	0.380993i	\(0.124418\pi\)
							−0.924578	+	0.380993i	\(0.875582\pi\)
\(43\)	−0.178550	+	0.178550i	−0.0272287	+	0.0272287i	−0.720590	−	0.693361i	\(-0.756129\pi\)
							0.693361	+	0.720590i	\(0.256129\pi\)
\(47\)	−4.96385	−	8.59764i	−0.724052	−	1.25410i	−0.959363	−	0.282175i	\(-0.908944\pi\)
							0.235311	−	0.971920i	\(-0.424389\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.13	✓	256
7.2	even	3	inner	560.2.cp.a.541.56	yes	256
16.5	even	4	inner	560.2.cp.a.501.56	yes	256
112.37	even	12	inner	560.2.cp.a.261.13	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.13	✓	256	1.1	even	1	trivial
560.2.cp.a.261.13	yes	256	112.37	even	12	inner
560.2.cp.a.501.56	yes	256	16.5	even	4	inner
560.2.cp.a.541.56	yes	256	7.2	even	3	inner