Embedded newform 560.2.r.a.237.9

• Label: 560.2.r.a.237.9
• 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			237.9
Character	\(\chi\)	\(=\)	560.237
Dual form			560.2.r.a.293.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33659 + 0.462100i) q^{2} -0.445450i q^{3} +(1.57293 - 1.23527i) q^{4} +(-2.22590 - 0.213020i) q^{5} +(0.205843 + 0.595383i) q^{6} +(-1.93702 - 1.80221i) q^{7} +(-1.53153 + 2.37790i) q^{8} +2.80157 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33659	+	0.462100i	−0.945109	+	0.326754i
							\(3\)		−	0.445450i		−	0.257181i	−0.991698	−	0.128590i	\(-0.958955\pi\)
0.991698	−	0.128590i	\(-0.0410452\pi\)
\(5\)	−2.22590	−	0.213020i	−0.995452	−	0.0952656i
							\(7\)	−1.93702	−	1.80221i	−0.732125	−	0.681171i
\(11\)	−3.77867	+	3.77867i	−1.13931	+	1.13931i								−0.150738	+	0.988574i	\(0.548165\pi\)
							−0.988574	+	0.150738i	\(0.951835\pi\)
\(13\)		−	1.83398i		−	0.508655i	−0.967118	−	0.254327i	\(-0.918146\pi\)
							0.967118	−	0.254327i	\(-0.0818541\pi\)
\(17\)	−1.51562	+	1.51562i	−0.367593	+	0.367593i	−0.866599	−	0.499006i	\(-0.833699\pi\)
							0.499006	+	0.866599i	\(0.333699\pi\)
\(19\)	4.13008	+	4.13008i	0.947506	+	0.947506i	0.998689	−	0.0511832i	\(-0.0162993\pi\)
							−0.0511832	+	0.998689i	\(0.516299\pi\)
\(23\)	3.56829	+	3.56829i	0.744040	+	0.744040i	0.973353	−	0.229313i	\(-0.0736479\pi\)
							−0.229313	+	0.973353i	\(0.573648\pi\)
\(29\)	−0.276311	+	0.276311i	−0.0513097	+	0.0513097i	−0.732296	−	0.680986i	\(-0.761551\pi\)
							0.680986	+	0.732296i	\(0.261551\pi\)
\(31\)			5.44952i			0.978762i	0.872070	+	0.489381i	\(0.162777\pi\)
							−0.872070	+	0.489381i	\(0.837223\pi\)
\(37\)	−1.20113			−0.197464			−0.0987319	−	0.995114i	\(-0.531479\pi\)
							−0.0987319	+	0.995114i	\(0.531479\pi\)
\(41\)	6.73436			1.05173			0.525865	−	0.850568i	\(-0.323742\pi\)
							0.525865	+	0.850568i	\(0.323742\pi\)
\(43\)			1.62354i			0.247587i	0.992308	+	0.123793i	\(0.0395060\pi\)
							−0.992308	+	0.123793i	\(0.960494\pi\)
\(47\)	3.18427	−	3.18427i	0.464474	−	0.464474i	−0.435645	−	0.900119i	\(-0.643480\pi\)
							0.900119	+	0.435645i	\(0.143480\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.r.a.237.9	✓	184
5.3	odd	4		560.2.bn.a.13.56	yes	184
7.6	odd	2	inner	560.2.r.a.237.10	yes	184
16.5	even	4		560.2.bn.a.517.55	yes	184
35.13	even	4		560.2.bn.a.13.55	yes	184
80.53	odd	4	inner	560.2.r.a.293.9	yes	184
112.69	odd	4		560.2.bn.a.517.56	yes	184
560.293	even	4	inner	560.2.r.a.293.10	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.r.a.237.9	✓	184	1.1	even	1	trivial
560.2.r.a.237.10	yes	184	7.6	odd	2	inner
560.2.r.a.293.9	yes	184	80.53	odd	4	inner
560.2.r.a.293.10	yes	184	560.293	even	4	inner
560.2.bn.a.13.55	yes	184	35.13	even	4
560.2.bn.a.13.56	yes	184	5.3	odd	4
560.2.bn.a.517.55	yes	184	16.5	even	4
560.2.bn.a.517.56	yes	184	112.69	odd	4