Embedded newform 560.2.cp.a.221.28

• Label: 560.2.cp.a.221.28
• 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			221.28
Character	\(\chi\)	\(=\)	560.221
Dual form			560.2.cp.a.261.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.381380 - 1.36182i) q^{2} +(-3.13300 + 0.839485i) q^{3} +(-1.70910 + 1.03874i) q^{4} +(-0.965926 - 0.258819i) q^{5} +(2.33809 + 3.94642i) q^{6} +(1.24650 + 2.33372i) q^{7} +(2.06639 + 1.93133i) q^{8} +(6.51289 - 3.76022i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 4 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.381380	−	1.36182i	−0.269676	−	0.962951i
							\(3\)	−3.13300	+	0.839485i	−1.80884	+	0.484677i	−0.995301	−	0.0968330i	\(-0.969129\pi\)
−0.813539	+	0.581510i	\(0.802462\pi\)
\(5\)	−0.965926	−	0.258819i	−0.431975	−	0.115747i
							\(7\)	1.24650	+	2.33372i	0.471131	+	0.882063i
\(11\)	0.622973	+	2.32497i	0.187834	+	0.701004i								0.994006	+	0.109324i	\(0.0348686\pi\)
							−0.806173	+	0.591680i	\(0.798465\pi\)
\(13\)	−2.48767	−	2.48767i	−0.689956	−	0.689956i	0.272266	−	0.962222i	\(-0.412227\pi\)
							−0.962222	+	0.272266i	\(0.912227\pi\)
\(17\)	−2.14369	+	3.71298i	−0.519921	+	0.900530i	0.479811	+	0.877372i	\(0.340705\pi\)
							−0.999732	+	0.0231578i	\(0.992628\pi\)
\(19\)	0.438624	−	1.63697i	0.100627	−	0.375546i	−0.897185	−	0.441655i	\(-0.854392\pi\)
							0.997812	+	0.0661086i	\(0.0210584\pi\)
\(23\)	−0.0444779	+	0.0256793i	−0.00927428	+	0.00535451i	−0.504630	−	0.863336i	\(-0.668371\pi\)
							0.495356	+	0.868690i	\(0.335038\pi\)
\(29\)	−1.11663	−	1.11663i	−0.207353	−	0.207353i	0.595789	−	0.803141i	\(-0.296840\pi\)
							−0.803141	+	0.595789i	\(0.796840\pi\)
\(31\)	−1.93189	+	3.34613i	−0.346978	+	0.600983i	−0.985711	−	0.168445i	\(-0.946125\pi\)
							0.638733	+	0.769428i	\(0.279459\pi\)
\(37\)	−1.90956	−	0.511665i	−0.313930	−	0.0841173i	0.0984139	−	0.995146i	\(-0.468623\pi\)
							−0.412344	+	0.911028i	\(0.635290\pi\)
\(41\)		−	11.5810i		−	1.80864i	−0.426851	−	0.904322i	\(-0.640377\pi\)
							0.426851	−	0.904322i	\(-0.359623\pi\)
\(43\)	−7.23946	+	7.23946i	−1.10401	+	1.10401i	−0.110085	+	0.993922i	\(0.535112\pi\)
							−0.993922	+	0.110085i	\(0.964888\pi\)
\(47\)	−2.31414	−	4.00821i	−0.337552	−	0.584658i	0.646419	−	0.762982i	\(-0.276266\pi\)
							−0.983972	+	0.178324i	\(0.942932\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.cp.a.221.28	✓	256
7.2	even	3	inner	560.2.cp.a.541.16	yes	256
16.5	even	4	inner	560.2.cp.a.501.16	yes	256
112.37	even	12	inner	560.2.cp.a.261.28	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.cp.a.221.28	✓	256	1.1	even	1	trivial
560.2.cp.a.261.28	yes	256	112.37	even	12	inner
560.2.cp.a.501.16	yes	256	16.5	even	4	inner
560.2.cp.a.541.16	yes	256	7.2	even	3	inner