Embedded newform 560.2.cp.a.221.2

• Label: 560.2.cp.a.221.2
• 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			221.2
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41379 + 0.0345737i) q^{2} +(1.56132 - 0.418355i) q^{3} +(1.99761 - 0.0977599i) q^{4} +(-0.965926 - 0.258819i) q^{5} +(-2.19292 + 0.645447i) q^{6} +(-0.806804 + 2.51974i) q^{7} +(-2.82082 + 0.207277i) q^{8} +(-0.335371 + 0.193626i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41379	+	0.0345737i	−0.999701	+	0.0244473i
							\(3\)	1.56132	−	0.418355i	0.901430	−	0.241537i	0.221800	−	0.975092i	\(-0.428807\pi\)
0.679630	+	0.733555i	\(0.262140\pi\)
\(5\)	−0.965926	−	0.258819i	−0.431975	−	0.115747i
							\(7\)	−0.806804	+	2.51974i	−0.304943	+	0.952371i
\(11\)	−0.0736025	−	0.274688i	−0.0221920	−	0.0828216i								0.953942	−	0.299992i	\(-0.0969840\pi\)
							−0.976134	+	0.217170i	\(0.930317\pi\)
\(13\)	1.80794	+	1.80794i	0.501431	+	0.501431i	0.911883	−	0.410451i	\(-0.134629\pi\)
							−0.410451	+	0.911883i	\(0.634629\pi\)
\(17\)	−3.45105	+	5.97740i	−0.837003	+	1.44973i	0.0553860	+	0.998465i	\(0.482361\pi\)
							−0.892389	+	0.451267i	\(0.850972\pi\)
\(19\)	−1.90964	+	7.12687i	−0.438101	+	1.63502i	0.295434	+	0.955363i	\(0.404536\pi\)
							−0.733535	+	0.679652i	\(0.762131\pi\)
\(23\)	3.59282	−	2.07432i	0.749155	−	0.432525i	−0.0762336	−	0.997090i	\(-0.524289\pi\)
							0.825388	+	0.564565i	\(0.190956\pi\)
\(29\)	−0.292960	−	0.292960i	−0.0544013	−	0.0544013i	0.679383	−	0.733784i	\(-0.262248\pi\)
							−0.733784	+	0.679383i	\(0.762248\pi\)
\(31\)	4.94023	−	8.55672i	0.887291	−	1.53683i	0.0442256	−	0.999022i	\(-0.485918\pi\)
							0.843065	−	0.537811i	\(-0.180749\pi\)
\(37\)	8.46812	+	2.26903i	1.39215	+	0.373025i	0.875519	−	0.483183i	\(-0.160519\pi\)
							0.516630	+	0.856208i	\(0.327186\pi\)
\(41\)		−	3.42567i		−	0.535000i	−0.963558	−	0.267500i	\(-0.913803\pi\)
							0.963558	−	0.267500i	\(-0.0861975\pi\)
\(43\)	−5.01855	+	5.01855i	−0.765321	+	0.765321i	−0.977279	−	0.211958i	\(-0.932016\pi\)
							0.211958	+	0.977279i	\(0.432016\pi\)
\(47\)	3.34555	+	5.79466i	0.487998	+	0.845238i	0.999905	−	0.0138033i	\(-0.00439387\pi\)
							−0.511906	+	0.859041i	\(0.671061\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.2	✓	256
7.2	even	3	inner	560.2.cp.a.541.44	yes	256
16.5	even	4	inner	560.2.cp.a.501.44	yes	256
112.37	even	12	inner	560.2.cp.a.261.2	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.2	✓	256	1.1	even	1	trivial
560.2.cp.a.261.2	yes	256	112.37	even	12	inner
560.2.cp.a.501.44	yes	256	16.5	even	4	inner
560.2.cp.a.541.44	yes	256	7.2	even	3	inner