Embedded newform 560.2.cp.a.221.12

• Label: 560.2.cp.a.221.12
• 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			221.12
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15092 - 0.821810i) q^{2} +(-1.44743 + 0.387838i) q^{3} +(0.649255 + 1.89168i) q^{4} +(-0.965926 - 0.258819i) q^{5} +(1.98462 + 0.743143i) q^{6} +(-1.97064 + 1.76539i) q^{7} +(0.807362 - 2.71075i) q^{8} +(-0.653433 + 0.377260i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15092	−	0.821810i	−0.813827	−	0.581108i
							\(3\)	−1.44743	+	0.387838i	−0.835676	+	0.223919i	−0.651188	−	0.758916i	\(-0.725729\pi\)
−0.184487	+	0.982835i	\(0.559063\pi\)
\(5\)	−0.965926	−	0.258819i	−0.431975	−	0.115747i
							\(7\)	−1.97064	+	1.76539i	−0.744831	+	0.667253i
\(11\)	0.475636	+	1.77510i	0.143410	+	0.535213i								0.999821	+	0.0189183i	\(0.00602225\pi\)
							−0.856411	+	0.516294i	\(0.827311\pi\)
\(13\)	−1.44604	−	1.44604i	−0.401060	−	0.401060i	0.477547	−	0.878606i	\(-0.341526\pi\)
							−0.878606	+	0.477547i	\(0.841526\pi\)
\(17\)	2.91011	−	5.04045i	0.705804	−	1.22249i	−0.260596	−	0.965448i	\(-0.583919\pi\)
							0.966400	−	0.257041i	\(-0.0827476\pi\)
\(19\)	−1.15698	+	4.31791i	−0.265430	+	0.990597i	0.696557	+	0.717501i	\(0.254714\pi\)
							−0.961987	+	0.273096i	\(0.911952\pi\)
\(23\)	1.05616	−	0.609773i	0.220224	−	0.127147i	−0.385830	−	0.922570i	\(-0.626085\pi\)
							0.606054	+	0.795423i	\(0.292751\pi\)
\(29\)	0.796743	+	0.796743i	0.147951	+	0.147951i	0.777202	−	0.629251i	\(-0.216638\pi\)
							−0.629251	+	0.777202i	\(0.716638\pi\)
\(31\)	4.29268	−	7.43514i	0.770989	−	1.33539i	−0.166033	−	0.986120i	\(-0.553096\pi\)
							0.937022	−	0.349271i	\(-0.113571\pi\)
\(37\)	−8.09813	−	2.16989i	−1.33133	−	0.356727i	−0.478116	−	0.878297i	\(-0.658680\pi\)
							−0.853209	+	0.521569i	\(0.825347\pi\)
\(41\)		−	3.78756i		−	0.591518i	−0.955263	−	0.295759i	\(-0.904427\pi\)
							0.955263	−	0.295759i	\(-0.0955725\pi\)
\(43\)	4.19220	−	4.19220i	0.639305	−	0.639305i	−0.311079	−	0.950384i	\(-0.600690\pi\)
							0.950384	+	0.311079i	\(0.100690\pi\)
\(47\)	−0.445237	−	0.771174i	−0.0649445	−	0.112487i	0.831725	−	0.555188i	\(-0.187354\pi\)
							−0.896669	+	0.442701i	\(0.854020\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.12	✓	256
7.2	even	3	inner	560.2.cp.a.541.31	yes	256
16.5	even	4	inner	560.2.cp.a.501.31	yes	256
112.37	even	12	inner	560.2.cp.a.261.12	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.12	✓	256	1.1	even	1	trivial
560.2.cp.a.261.12	yes	256	112.37	even	12	inner
560.2.cp.a.501.31	yes	256	16.5	even	4	inner
560.2.cp.a.541.31	yes	256	7.2	even	3	inner