Embedded newform 560.2.r.a.237.14

• Label: 560.2.r.a.237.14
• 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			237.14
Character	\(\chi\)	\(=\)	560.237
Dual form			560.2.r.a.293.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.31458 + 0.521428i) q^{2} +2.31341i q^{3} +(1.45623 - 1.37091i) q^{4} +(-1.44475 - 1.70666i) q^{5} +(-1.20628 - 3.04115i) q^{6} +(0.698510 + 2.55188i) q^{7} +(-1.19949 + 2.56149i) q^{8} -2.35185 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.31458	+	0.521428i	−0.929546	+	0.368705i
							\(3\)			2.31341i			1.33565i	0.744320	+	0.667823i	\(0.232774\pi\)
−0.744320	+	0.667823i	\(0.767226\pi\)
\(5\)	−1.44475	−	1.70666i	−0.646113	−	0.763242i
							\(7\)	0.698510	+	2.55188i	0.264012	+	0.964519i
\(11\)	0.741925	−	0.741925i	0.223699	−	0.223699i								−0.586355	−	0.810054i	\(-0.699438\pi\)
							0.810054	+	0.586355i	\(0.199438\pi\)
\(13\)			1.38126i			0.383091i	0.981484	+	0.191546i	\(0.0613500\pi\)
							−0.981484	+	0.191546i	\(0.938650\pi\)
\(17\)	−5.56949	+	5.56949i	−1.35080	+	1.35080i	−0.466030	+	0.884769i	\(0.654316\pi\)
							−0.884769	+	0.466030i	\(0.845684\pi\)
\(19\)	0.508972	+	0.508972i	0.116766	+	0.116766i	0.763076	−	0.646309i	\(-0.223688\pi\)
							−0.646309	+	0.763076i	\(0.723688\pi\)
\(23\)	−4.53089	−	4.53089i	−0.944756	−	0.944756i	0.0537963	−	0.998552i	\(-0.482868\pi\)
							−0.998552	+	0.0537963i	\(0.982868\pi\)
\(29\)	−0.661827	+	0.661827i	−0.122898	+	0.122898i	−0.765881	−	0.642983i	\(-0.777697\pi\)
							0.642983	+	0.765881i	\(0.277697\pi\)
\(31\)			2.75434i			0.494695i	0.968927	+	0.247347i	\(0.0795589\pi\)
							−0.968927	+	0.247347i	\(0.920441\pi\)
\(37\)	−5.94245			−0.976932			−0.488466	−	0.872583i	\(-0.662443\pi\)
							−0.488466	+	0.872583i	\(0.662443\pi\)
\(41\)	−2.71787			−0.424460			−0.212230	−	0.977220i	\(-0.568073\pi\)
							−0.212230	+	0.977220i	\(0.568073\pi\)
\(43\)		−	7.19715i		−	1.09755i	−0.835969	−	0.548777i	\(-0.815093\pi\)
							0.835969	−	0.548777i	\(-0.184907\pi\)
\(47\)	3.68458	−	3.68458i	0.537451	−	0.537451i	−0.385328	−	0.922780i	\(-0.625912\pi\)
							0.922780	+	0.385328i	\(0.125912\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.r.a.237.14	yes	184
5.3	odd	4		560.2.bn.a.13.59	yes	184
7.6	odd	2	inner	560.2.r.a.237.13	✓	184
16.5	even	4		560.2.bn.a.517.60	yes	184
35.13	even	4		560.2.bn.a.13.60	yes	184
80.53	odd	4	inner	560.2.r.a.293.14	yes	184
112.69	odd	4		560.2.bn.a.517.59	yes	184
560.293	even	4	inner	560.2.r.a.293.13	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.r.a.237.13	✓	184	7.6	odd	2	inner
560.2.r.a.237.14	yes	184	1.1	even	1	trivial
560.2.r.a.293.13	yes	184	560.293	even	4	inner
560.2.r.a.293.14	yes	184	80.53	odd	4	inner
560.2.bn.a.13.59	yes	184	5.3	odd	4
560.2.bn.a.13.60	yes	184	35.13	even	4
560.2.bn.a.517.59	yes	184	112.69	odd	4
560.2.bn.a.517.60	yes	184	16.5	even	4