Embedded newform 560.2.r.a.237.20

• Label: 560.2.r.a.237.20
• Level: $560$
• Weight: $2$
• Character: 560.237
• Analytic conductor: $4.472$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.47162251319\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			237.20
Character	\(\chi\)	\(=\)	560.237
Dual form			560.2.r.a.293.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.13458 - 0.844232i) q^{2} +3.14518i q^{3} +(0.574546 + 1.91570i) q^{4} +(1.64124 - 1.51866i) q^{5} +(2.65526 - 3.56847i) q^{6} +(-2.03745 + 1.68783i) q^{7} +(0.965423 - 2.65856i) q^{8} -6.89218 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\)	\(241\)	\(337\)	\(351\)	\(421\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.13458	−	0.844232i	−0.802270	−	0.596962i
							\(3\)			3.14518i			1.81587i	0.419108	+	0.907937i	\(0.362343\pi\)
−0.419108	+	0.907937i	\(0.637657\pi\)
\(5\)	1.64124	−	1.51866i	0.733987	−	0.679164i
							\(7\)	−2.03745	+	1.68783i	−0.770085	+	0.637941i
\(11\)	−1.12382	+	1.12382i	−0.338843	+	0.338843i								−0.855932	−	0.517089i	\(-0.827016\pi\)
							0.517089	+	0.855932i	\(0.327016\pi\)
\(13\)		−	1.04039i		−	0.288551i	−0.989538	−	0.144276i	\(-0.953915\pi\)
							0.989538	−	0.144276i	\(-0.0460852\pi\)
\(17\)	−5.19127	+	5.19127i	−1.25907	+	1.25907i	−0.307529	+	0.951539i	\(0.599502\pi\)
							−0.951539	+	0.307529i	\(0.900498\pi\)
\(19\)	−2.08587	−	2.08587i	−0.478532	−	0.478532i	0.426130	−	0.904662i	\(-0.359877\pi\)
							−0.904662	+	0.426130i	\(0.859877\pi\)
\(23\)	4.81118	+	4.81118i	1.00320	+	1.00320i	0.999995	+	0.00320642i	\(0.00102064\pi\)
							0.00320642	+	0.999995i	\(0.498979\pi\)
\(29\)	−2.43496	+	2.43496i	−0.452160	+	0.452160i	−0.896071	−	0.443911i	\(-0.853591\pi\)
							0.443911	+	0.896071i	\(0.353591\pi\)
\(31\)		−	6.72312i		−	1.20751i	−0.797171	−	0.603754i	\(-0.793671\pi\)
							0.797171	−	0.603754i	\(-0.206329\pi\)
\(37\)	−5.10902			−0.839917			−0.419959	−	0.907543i	\(-0.637955\pi\)
							−0.419959	+	0.907543i	\(0.637955\pi\)
\(41\)	−5.70956			−0.891683			−0.445842	−	0.895112i	\(-0.647096\pi\)
							−0.445842	+	0.895112i	\(0.647096\pi\)
\(43\)			10.3376i			1.57647i	0.615376	+	0.788234i	\(0.289004\pi\)
							−0.615376	+	0.788234i	\(0.710996\pi\)
\(47\)	−1.63249	+	1.63249i	−0.238124	+	0.238124i	−0.816073	−	0.577949i	\(-0.803853\pi\)
							0.577949	+	0.816073i	\(0.303853\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.r.a.237.20	yes	184
5.3	odd	4		560.2.bn.a.13.29	yes	184
7.6	odd	2	inner	560.2.r.a.237.19	✓	184
16.5	even	4		560.2.bn.a.517.30	yes	184
35.13	even	4		560.2.bn.a.13.30	yes	184
80.53	odd	4	inner	560.2.r.a.293.20	yes	184
112.69	odd	4		560.2.bn.a.517.29	yes	184
560.293	even	4	inner	560.2.r.a.293.19	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.r.a.237.19	✓	184	7.6	odd	2	inner
560.2.r.a.237.20	yes	184	1.1	even	1	trivial
560.2.r.a.293.19	yes	184	560.293	even	4	inner
560.2.r.a.293.20	yes	184	80.53	odd	4	inner
560.2.bn.a.13.29	yes	184	5.3	odd	4
560.2.bn.a.13.30	yes	184	35.13	even	4
560.2.bn.a.517.29	yes	184	112.69	odd	4
560.2.bn.a.517.30	yes	184	16.5	even	4