Embedded newform 560.2.r.a.237.2

• Label: 560.2.r.a.237.2
• 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			237.2
Character	\(\chi\)	\(=\)	560.237
Dual form			560.2.r.a.293.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41385 + 0.0320021i) q^{2} +2.37744i q^{3} +(1.99795 - 0.0904924i) q^{4} +(1.88118 + 1.20879i) q^{5} +(-0.0760829 - 3.36134i) q^{6} +(-0.753095 - 2.53631i) q^{7} +(-2.82191 + 0.191881i) q^{8} -2.65221 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 4 q^{2} + 8 q^{8} - 168 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41385	+	0.0320021i	−0.999744	+	0.0226289i
							\(3\)			2.37744i			1.37261i	0.727312	+	0.686307i	\(0.240769\pi\)
−0.727312	+	0.686307i	\(0.759231\pi\)
\(5\)	1.88118	+	1.20879i	0.841289	+	0.540586i
							\(7\)	−0.753095	−	2.53631i	−0.284643	−	0.958634i
\(11\)	0.0135673	−	0.0135673i	0.00409070	−	0.00409070i								−0.705058	−	0.709149i	\(-0.749079\pi\)
							0.709149	+	0.705058i	\(0.249079\pi\)
\(13\)			4.78314i			1.32660i	0.748352	+	0.663302i	\(0.230846\pi\)
							−0.748352	+	0.663302i	\(0.769154\pi\)
\(17\)	−2.51605	+	2.51605i	−0.610233	+	0.610233i	−0.943007	−	0.332774i	\(-0.892015\pi\)
							0.332774	+	0.943007i	\(0.392015\pi\)
\(19\)	3.95431	+	3.95431i	0.907180	+	0.907180i	0.996044	−	0.0888637i	\(-0.0283236\pi\)
							−0.0888637	+	0.996044i	\(0.528324\pi\)
\(23\)	−2.87838	−	2.87838i	−0.600184	−	0.600184i	0.340177	−	0.940361i	\(-0.389513\pi\)
							−0.940361	+	0.340177i	\(0.889513\pi\)
\(29\)	−4.88550	+	4.88550i	−0.907214	+	0.907214i	−0.996047	−	0.0888321i	\(-0.971687\pi\)
							0.0888321	+	0.996047i	\(0.471687\pi\)
\(31\)		−	4.34601i		−	0.780567i	−0.920695	−	0.390284i	\(-0.872377\pi\)
							0.920695	−	0.390284i	\(-0.127623\pi\)
\(37\)	1.76042			0.289411			0.144706	−	0.989475i	\(-0.453776\pi\)
							0.144706	+	0.989475i	\(0.453776\pi\)
\(41\)	3.90061			0.609173			0.304587	−	0.952485i	\(-0.401482\pi\)
							0.304587	+	0.952485i	\(0.401482\pi\)
\(43\)			2.66169i			0.405904i	0.979189	+	0.202952i	\(0.0650535\pi\)
							−0.979189	+	0.202952i	\(0.934946\pi\)
\(47\)	0.843962	−	0.843962i	0.123104	−	0.123104i	−0.642870	−	0.765975i	\(-0.722257\pi\)
							0.765975	+	0.642870i	\(0.222257\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.r.a.237.2	yes	184
5.3	odd	4		560.2.bn.a.13.47	yes	184
7.6	odd	2	inner	560.2.r.a.237.1	✓	184
16.5	even	4		560.2.bn.a.517.48	yes	184
35.13	even	4		560.2.bn.a.13.48	yes	184
80.53	odd	4	inner	560.2.r.a.293.2	yes	184
112.69	odd	4		560.2.bn.a.517.47	yes	184
560.293	even	4	inner	560.2.r.a.293.1	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.r.a.237.1	✓	184	7.6	odd	2	inner
560.2.r.a.237.2	yes	184	1.1	even	1	trivial
560.2.r.a.293.1	yes	184	560.293	even	4	inner
560.2.r.a.293.2	yes	184	80.53	odd	4	inner
560.2.bn.a.13.47	yes	184	5.3	odd	4
560.2.bn.a.13.48	yes	184	35.13	even	4
560.2.bn.a.517.47	yes	184	112.69	odd	4
560.2.bn.a.517.48	yes	184	16.5	even	4