Embedded newform 560.2.r.a.237.5

• Label: 560.2.r.a.237.5
• 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			237.5
Character	\(\chi\)	\(=\)	560.237
Dual form			560.2.r.a.293.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39265 - 0.246023i) q^{2} -2.36202i q^{3} +(1.87894 + 0.685249i) q^{4} +(2.19592 + 0.421835i) q^{5} +(-0.581111 + 3.28946i) q^{6} +(1.13552 + 2.38968i) q^{7} +(-2.44812 - 1.41658i) q^{8} -2.57912 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39265	−	0.246023i	−0.984752	−	0.173965i
							\(3\)		−	2.36202i		−	1.36371i	−0.731487	−	0.681855i	\(-0.761173\pi\)
0.731487	−	0.681855i	\(-0.238827\pi\)
\(5\)	2.19592	+	0.421835i	0.982044	+	0.188651i
							\(7\)	1.13552	+	2.38968i	0.429186	+	0.903216i
\(11\)	3.21839	−	3.21839i	0.970380	−	0.970380i								−0.0291941	−	0.999574i	\(-0.509294\pi\)
							0.999574	+	0.0291941i	\(0.00929409\pi\)
\(13\)		−	1.27152i		−	0.352655i	−0.984332	−	0.176328i	\(-0.943578\pi\)
							0.984332	−	0.176328i	\(-0.0564218\pi\)
\(17\)	−4.03395	+	4.03395i	−0.978377	+	0.978377i	−0.999771	−	0.0213937i	\(-0.993190\pi\)
							0.0213937	+	0.999771i	\(0.493190\pi\)
\(19\)	5.63108	+	5.63108i	1.29186	+	1.29186i	0.933637	+	0.358222i	\(0.116617\pi\)
							0.358222	+	0.933637i	\(0.383383\pi\)
\(23\)	3.43187	+	3.43187i	0.715595	+	0.715595i	0.967700	−	0.252105i	\(-0.0811228\pi\)
							−0.252105	+	0.967700i	\(0.581123\pi\)
\(29\)	4.18267	−	4.18267i	0.776702	−	0.776702i	−0.202566	−	0.979269i	\(-0.564928\pi\)
							0.979269	+	0.202566i	\(0.0649282\pi\)
\(31\)		−	4.23986i		−	0.761501i	−0.924678	−	0.380751i	\(-0.875666\pi\)
							0.924678	−	0.380751i	\(-0.124334\pi\)
\(37\)	−6.41925			−1.05532			−0.527659	−	0.849456i	\(-0.676930\pi\)
							−0.527659	+	0.849456i	\(0.676930\pi\)
\(41\)	−6.81751			−1.06472			−0.532358	−	0.846519i	\(-0.678694\pi\)
							−0.532358	+	0.846519i	\(0.678694\pi\)
\(43\)			0.836025i			0.127493i	0.997966	+	0.0637463i	\(0.0203048\pi\)
							−0.997966	+	0.0637463i	\(0.979695\pi\)
\(47\)	3.39314	−	3.39314i	0.494940	−	0.494940i	−0.414919	−	0.909859i	\(-0.636190\pi\)
							0.909859	+	0.414919i	\(0.136190\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.r.a.237.5	✓	184
5.3	odd	4		560.2.bn.a.13.42	yes	184
7.6	odd	2	inner	560.2.r.a.237.6	yes	184
16.5	even	4		560.2.bn.a.517.41	yes	184
35.13	even	4		560.2.bn.a.13.41	yes	184
80.53	odd	4	inner	560.2.r.a.293.5	yes	184
112.69	odd	4		560.2.bn.a.517.42	yes	184
560.293	even	4	inner	560.2.r.a.293.6	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.r.a.237.5	✓	184	1.1	even	1	trivial
560.2.r.a.237.6	yes	184	7.6	odd	2	inner
560.2.r.a.293.5	yes	184	80.53	odd	4	inner
560.2.r.a.293.6	yes	184	560.293	even	4	inner
560.2.bn.a.13.41	yes	184	35.13	even	4
560.2.bn.a.13.42	yes	184	5.3	odd	4
560.2.bn.a.517.41	yes	184	16.5	even	4
560.2.bn.a.517.42	yes	184	112.69	odd	4