Embedded newform 560.2.be.a.139.4

• Label: 560.2.be.a.139.4
• Level: $560$
• Weight: $2$
• Character: 560.139
• Analytic conductor: $4.472$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.be (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			139.4
Character	\(\chi\)	\(=\)	560.139
Dual form			560.2.be.a.419.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40363 + 0.172701i) q^{2} +(0.710544 + 0.710544i) q^{3} +(1.94035 - 0.484816i) q^{4} +(-1.39309 - 1.74909i) q^{5} +(-1.12005 - 0.874629i) q^{6} +(2.62345 + 0.342784i) q^{7} +(-2.63980 + 1.01560i) q^{8} -1.99025i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 8 q^{4}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40363	+	0.172701i	−0.992516	+	0.122118i
							\(3\)	0.710544	+	0.710544i	0.410233	+	0.410233i	0.881820	−	0.471587i	\(-0.156319\pi\)
−0.471587	+	0.881820i	\(0.656319\pi\)
\(5\)	−1.39309	−	1.74909i	−0.623009	−	0.782215i
							\(7\)	2.62345	+	0.342784i	0.991572	+	0.129560i
\(11\)	1.25363	−	1.25363i	0.377985	−	0.377985i								−0.492390	−	0.870375i	\(-0.663877\pi\)
							0.870375	+	0.492390i	\(0.163877\pi\)
\(13\)	−1.04503	+	1.04503i	−0.289840	+	0.289840i	−0.837017	−	0.547177i	\(-0.815702\pi\)
							0.547177	+	0.837017i	\(0.315702\pi\)
\(17\)	−4.22346			−1.02434			−0.512170	−	0.858884i	\(-0.671158\pi\)
							−0.512170	+	0.858884i	\(0.671158\pi\)
\(19\)	2.54075	−	2.54075i	0.582889	−	0.582889i	−0.352807	−	0.935696i	\(-0.614773\pi\)
							0.935696	+	0.352807i	\(0.114773\pi\)
\(23\)		−	7.07967i		−	1.47621i	−0.674684	−	0.738106i	\(-0.735720\pi\)
							0.674684	−	0.738106i	\(-0.264280\pi\)
\(29\)	2.76891	−	2.76891i	0.514174	−	0.514174i	−0.401629	−	0.915803i	\(-0.631556\pi\)
							0.915803	+	0.401629i	\(0.131556\pi\)
\(31\)	−7.99208			−1.43542			−0.717710	−	0.696342i	\(-0.754810\pi\)
							−0.717710	+	0.696342i	\(0.754810\pi\)
\(37\)	4.40780	−	4.40780i	0.724638	−	0.724638i	−0.244908	−	0.969546i	\(-0.578758\pi\)
							0.969546	+	0.244908i	\(0.0787578\pi\)
\(41\)	0.571658			0.0892779			0.0446390	−	0.999003i	\(-0.485786\pi\)
							0.0446390	+	0.999003i	\(0.485786\pi\)
\(43\)	4.18907	+	4.18907i	0.638827	+	0.638827i	0.950266	−	0.311439i	\(-0.100811\pi\)
							−0.311439	+	0.950266i	\(0.600811\pi\)
\(47\)			4.81785i			0.702755i	0.936234	+	0.351377i	\(0.114287\pi\)
							−0.936234	+	0.351377i	\(0.885713\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.be.a.139.4	yes	184
5.4	even	2	inner	560.2.be.a.139.89	yes	184
7.6	odd	2	inner	560.2.be.a.139.3	✓	184
16.3	odd	4	inner	560.2.be.a.419.90	yes	184
35.34	odd	2	inner	560.2.be.a.139.90	yes	184
80.19	odd	4	inner	560.2.be.a.419.3	yes	184
112.83	even	4	inner	560.2.be.a.419.89	yes	184
560.419	even	4	inner	560.2.be.a.419.4	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.be.a.139.3	✓	184	7.6	odd	2	inner
560.2.be.a.139.4	yes	184	1.1	even	1	trivial
560.2.be.a.139.89	yes	184	5.4	even	2	inner
560.2.be.a.139.90	yes	184	35.34	odd	2	inner
560.2.be.a.419.3	yes	184	80.19	odd	4	inner
560.2.be.a.419.4	yes	184	560.419	even	4	inner
560.2.be.a.419.89	yes	184	112.83	even	4	inner
560.2.be.a.419.90	yes	184	16.3	odd	4	inner