Embedded newform 560.2.r.a.237.17

• Label: 560.2.r.a.237.17
• 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			237.17
Character	\(\chi\)	\(=\)	560.237
Dual form			560.2.r.a.293.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18156 + 0.777113i) q^{2} -2.96802i q^{3} +(0.792190 - 1.83642i) q^{4} +(-0.835108 + 2.07427i) q^{5} +(2.30649 + 3.50691i) q^{6} +(2.64524 - 0.0522070i) q^{7} +(0.491081 + 2.78547i) q^{8} -5.80916 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18156	+	0.777113i	−0.835492	+	0.549502i
							\(3\)		−	2.96802i		−	1.71359i	−0.515658	−	0.856794i	\(-0.672453\pi\)
0.515658	−	0.856794i	\(-0.327547\pi\)
\(5\)	−0.835108	+	2.07427i	−0.373471	+	0.927642i
							\(7\)	2.64524	−	0.0522070i	0.999805	−	0.0197324i
\(11\)	−2.45276	+	2.45276i	−0.739534	+	0.739534i								−0.972488	−	0.232953i	\(-0.925161\pi\)
							0.232953	+	0.972488i	\(0.425161\pi\)
\(13\)			6.32066i			1.75303i	0.481370	+	0.876517i	\(0.340139\pi\)
							−0.481370	+	0.876517i	\(0.659861\pi\)
\(17\)	−4.86440	+	4.86440i	−1.17979	+	1.17979i	−0.199994	+	0.979797i	\(0.564092\pi\)
							−0.979797	+	0.199994i	\(0.935908\pi\)
\(19\)	2.74894	+	2.74894i	0.630650	+	0.630650i	0.948231	−	0.317581i	\(-0.102870\pi\)
							−0.317581	+	0.948231i	\(0.602870\pi\)
\(23\)	−2.15413	−	2.15413i	−0.449166	−	0.449166i	0.445911	−	0.895077i	\(-0.352880\pi\)
							−0.895077	+	0.445911i	\(0.852880\pi\)
\(29\)	1.22661	−	1.22661i	0.227775	−	0.227775i	−0.583988	−	0.811763i	\(-0.698508\pi\)
							0.811763	+	0.583988i	\(0.198508\pi\)
\(31\)		−	0.368577i		−	0.0661983i	−0.999452	−	0.0330992i	\(-0.989462\pi\)
							0.999452	−	0.0330992i	\(-0.0105377\pi\)
\(37\)	1.24127			0.204064			0.102032	−	0.994781i	\(-0.467466\pi\)
							0.102032	+	0.994781i	\(0.467466\pi\)
\(41\)	−1.62340			−0.253533			−0.126766	−	0.991933i	\(-0.540460\pi\)
							−0.126766	+	0.991933i	\(0.540460\pi\)
\(43\)			5.47490i			0.834914i	0.908697	+	0.417457i	\(0.137079\pi\)
							−0.908697	+	0.417457i	\(0.862921\pi\)
\(47\)	−4.87855	+	4.87855i	−0.711609	+	0.711609i	−0.966872	−	0.255263i	\(-0.917838\pi\)
							0.255263	+	0.966872i	\(0.417838\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.r.a.237.17	✓	184
5.3	odd	4		560.2.bn.a.13.64	yes	184
7.6	odd	2	inner	560.2.r.a.237.18	yes	184
16.5	even	4		560.2.bn.a.517.63	yes	184
35.13	even	4		560.2.bn.a.13.63	yes	184
80.53	odd	4	inner	560.2.r.a.293.17	yes	184
112.69	odd	4		560.2.bn.a.517.64	yes	184
560.293	even	4	inner	560.2.r.a.293.18	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.r.a.237.17	✓	184	1.1	even	1	trivial
560.2.r.a.237.18	yes	184	7.6	odd	2	inner
560.2.r.a.293.17	yes	184	80.53	odd	4	inner
560.2.r.a.293.18	yes	184	560.293	even	4	inner
560.2.bn.a.13.63	yes	184	35.13	even	4
560.2.bn.a.13.64	yes	184	5.3	odd	4
560.2.bn.a.517.63	yes	184	16.5	even	4
560.2.bn.a.517.64	yes	184	112.69	odd	4