Embedded newform 560.2.bn.a.13.38

• Label: 560.2.bn.a.13.38
• Level: $560$
• Weight: $2$
• Character: 560.13
• 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.bn (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			13.38
Character	\(\chi\)	\(=\)	560.13
Dual form			560.2.bn.a.517.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.429109 + 1.34754i) q^{2} +2.77875 q^{3} +(-1.63173 - 1.15648i) q^{4} +(-1.63611 + 1.52419i) q^{5} +(-1.19239 + 3.74447i) q^{6} +(-2.48617 + 0.904951i) q^{7} +(2.25860 - 1.70257i) q^{8} +4.72144 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} + 8 q^{8} + 168 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.429109	+	1.34754i	−0.303426	+	0.952855i
							\(3\)	2.77875			1.60431			0.802155	−	0.597115i	\(-0.203687\pi\)
0.802155	+	0.597115i	\(0.203687\pi\)
\(5\)	−1.63611	+	1.52419i	−0.731689	+	0.681639i
							\(7\)	−2.48617	+	0.904951i	−0.939686	+	0.342039i
\(11\)	−4.66171	+	4.66171i	−1.40556	+	1.40556i								−0.624673	+	0.780886i	\(0.714768\pi\)
							−0.780886	+	0.624673i	\(0.785232\pi\)
\(13\)	−0.783504			−0.217305			−0.108652	−	0.994080i	\(-0.534654\pi\)
							−0.108652	+	0.994080i	\(0.534654\pi\)
\(17\)	1.44597	+	1.44597i	0.350698	+	0.350698i	0.860369	−	0.509671i	\(-0.170233\pi\)
							−0.509671	+	0.860369i	\(0.670233\pi\)
\(19\)	2.51677	+	2.51677i	0.577386	+	0.577386i	0.934182	−	0.356797i	\(-0.116131\pi\)
							−0.356797	+	0.934182i	\(0.616131\pi\)
\(23\)	−0.0180574	+	0.0180574i	−0.00376522	+	0.00376522i	−0.708987	−	0.705222i	\(-0.750847\pi\)
							0.705222	+	0.708987i	\(0.250847\pi\)
\(29\)	−3.87232	+	3.87232i	−0.719072	+	0.719072i	−0.968415	−	0.249343i	\(-0.919785\pi\)
							0.249343	+	0.968415i	\(0.419785\pi\)
\(31\)		−	4.17072i		−	0.749083i	−0.927210	−	0.374541i	\(-0.877800\pi\)
							0.927210	−	0.374541i	\(-0.122200\pi\)
\(37\)			3.09358i			0.508582i	0.967128	+	0.254291i	\(0.0818421\pi\)
							−0.967128	+	0.254291i	\(0.918158\pi\)
\(41\)	7.88509			1.23144			0.615722	−	0.787963i	\(-0.288864\pi\)
							0.615722	+	0.787963i	\(0.288864\pi\)
\(43\)	6.47404			0.987282			0.493641	−	0.869666i	\(-0.335666\pi\)
							0.493641	+	0.869666i	\(0.335666\pi\)
\(47\)	6.82602	+	6.82602i	0.995678	+	0.995678i	0.999991	−	0.00431268i	\(-0.00137277\pi\)
							−0.00431268	+	0.999991i	\(0.501373\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.bn.a.13.38	yes	184
5.2	odd	4		560.2.r.a.237.7	✓	184
7.6	odd	2	inner	560.2.bn.a.13.37	yes	184
16.5	even	4		560.2.r.a.293.7	yes	184
35.27	even	4		560.2.r.a.237.8	yes	184
80.37	odd	4	inner	560.2.bn.a.517.37	yes	184
112.69	odd	4		560.2.r.a.293.8	yes	184
560.517	even	4	inner	560.2.bn.a.517.38	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.r.a.237.7	✓	184	5.2	odd	4
560.2.r.a.237.8	yes	184	35.27	even	4
560.2.r.a.293.7	yes	184	16.5	even	4
560.2.r.a.293.8	yes	184	112.69	odd	4
560.2.bn.a.13.37	yes	184	7.6	odd	2	inner
560.2.bn.a.13.38	yes	184	1.1	even	1	trivial
560.2.bn.a.517.37	yes	184	80.37	odd	4	inner
560.2.bn.a.517.38	yes	184	560.517	even	4	inner