Embedded newform 560.2.r.a.237.12

• Label: 560.2.r.a.237.12
• 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			237.12
Character	\(\chi\)	\(=\)	560.237
Dual form			560.2.r.a.293.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33522 + 0.466046i) q^{2} +0.958223i q^{3} +(1.56560 - 1.24454i) q^{4} +(-0.666131 + 2.13454i) q^{5} +(-0.446576 - 1.27943i) q^{6} +(2.54325 - 0.729292i) q^{7} +(-1.51040 + 2.39138i) q^{8} +2.08181 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33522	+	0.466046i	−0.944140	+	0.329544i
							\(3\)			0.958223i			0.553230i	0.960981	+	0.276615i	\(0.0892127\pi\)
−0.960981	+	0.276615i	\(0.910787\pi\)
\(5\)	−0.666131	+	2.13454i	−0.297903	+	0.954596i
							\(7\)	2.54325	−	0.729292i	0.961259	−	0.275646i
\(11\)	3.64406	−	3.64406i	1.09873	−	1.09873i								0.104165	−	0.994560i	\(-0.466783\pi\)
							0.994560	−	0.104165i	\(-0.0332170\pi\)
\(13\)		−	5.12086i		−	1.42027i	−0.704065	−	0.710135i	\(-0.748634\pi\)
							0.704065	−	0.710135i	\(-0.251366\pi\)
\(17\)	0.135731	−	0.135731i	0.0329196	−	0.0329196i	−0.690455	−	0.723375i	\(-0.742590\pi\)
							0.723375	+	0.690455i	\(0.242590\pi\)
\(19\)	0.763055	+	0.763055i	0.175057	+	0.175057i	0.789197	−	0.614140i	\(-0.210497\pi\)
							−0.614140	+	0.789197i	\(0.710497\pi\)
\(23\)	1.64227	+	1.64227i	0.342436	+	0.342436i	0.857283	−	0.514846i	\(-0.172151\pi\)
							−0.514846	+	0.857283i	\(0.672151\pi\)
\(29\)	1.76859	−	1.76859i	0.328419	−	0.328419i	−0.523566	−	0.851985i	\(-0.675399\pi\)
							0.851985	+	0.523566i	\(0.175399\pi\)
\(31\)		−	6.27749i		−	1.12747i	−0.825955	−	0.563735i	\(-0.809364\pi\)
							0.825955	−	0.563735i	\(-0.190636\pi\)
\(37\)	−3.32890			−0.547267			−0.273634	−	0.961834i	\(-0.588226\pi\)
							−0.273634	+	0.961834i	\(0.588226\pi\)
\(41\)	8.79916			1.37420			0.687099	−	0.726564i	\(-0.258884\pi\)
							0.687099	+	0.726564i	\(0.258884\pi\)
\(43\)			6.58815i			1.00468i	0.864669	+	0.502342i	\(0.167528\pi\)
							−0.864669	+	0.502342i	\(0.832472\pi\)
\(47\)	−3.60682	+	3.60682i	−0.526109	+	0.526109i	−0.919410	−	0.393301i	\(-0.871333\pi\)
							0.393301	+	0.919410i	\(0.371333\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.r.a.237.12	yes	184
5.3	odd	4		560.2.bn.a.13.57	yes	184
7.6	odd	2	inner	560.2.r.a.237.11	✓	184
16.5	even	4		560.2.bn.a.517.58	yes	184
35.13	even	4		560.2.bn.a.13.58	yes	184
80.53	odd	4	inner	560.2.r.a.293.12	yes	184
112.69	odd	4		560.2.bn.a.517.57	yes	184
560.293	even	4	inner	560.2.r.a.293.11	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.r.a.237.11	✓	184	7.6	odd	2	inner
560.2.r.a.237.12	yes	184	1.1	even	1	trivial
560.2.r.a.293.11	yes	184	560.293	even	4	inner
560.2.r.a.293.12	yes	184	80.53	odd	4	inner
560.2.bn.a.13.57	yes	184	5.3	odd	4
560.2.bn.a.13.58	yes	184	35.13	even	4
560.2.bn.a.517.57	yes	184	112.69	odd	4
560.2.bn.a.517.58	yes	184	16.5	even	4