Embedded newform 560.2.be.a.139.14

• Label: 560.2.be.a.139.14
• 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			139.14
Character	\(\chi\)	\(=\)	560.139
Dual form			560.2.be.a.419.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27264 + 0.616746i) q^{2} +(0.0442516 + 0.0442516i) q^{3} +(1.23925 - 1.56980i) q^{4} +(0.0460683 - 2.23559i) q^{5} +(-0.0836085 - 0.0290246i) q^{6} +(-2.16475 + 1.52114i) q^{7} +(-0.608959 + 2.76210i) q^{8} -2.99608i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27264	+	0.616746i	−0.899896	+	0.436105i
							\(3\)	0.0442516	+	0.0442516i	0.0255487	+	0.0255487i	0.719766	−	0.694217i	\(-0.244249\pi\)
−0.694217	+	0.719766i	\(0.744249\pi\)
\(5\)	0.0460683	−	2.23559i	0.0206023	−	0.999788i
							\(7\)	−2.16475	+	1.52114i	−0.818197	+	0.574938i
\(11\)	−3.11043	+	3.11043i	−0.937831	+	0.937831i								−0.998177	−	0.0603463i	\(-0.980779\pi\)
							0.0603463	+	0.998177i	\(0.480779\pi\)
\(13\)	3.90340	−	3.90340i	1.08261	−	1.08261i	0.0863421	−	0.996266i	\(-0.472482\pi\)
							0.996266	−	0.0863421i	\(-0.0275178\pi\)
\(17\)	−4.09495			−0.993172			−0.496586	−	0.867987i	\(-0.665413\pi\)
							−0.496586	+	0.867987i	\(0.665413\pi\)
\(19\)	−2.25827	+	2.25827i	−0.518082	+	0.518082i	−0.916991	−	0.398909i	\(-0.869389\pi\)
							0.398909	+	0.916991i	\(0.369389\pi\)
\(23\)			1.06026i			0.221080i	0.993872	+	0.110540i	\(0.0352580\pi\)
							−0.993872	+	0.110540i	\(0.964742\pi\)
\(29\)	−4.11635	+	4.11635i	−0.764387	+	0.764387i	−0.977112	−	0.212725i	\(-0.931766\pi\)
							0.212725	+	0.977112i	\(0.431766\pi\)
\(31\)	−3.75761			−0.674886			−0.337443	−	0.941346i	\(-0.609562\pi\)
							−0.337443	+	0.941346i	\(0.609562\pi\)
\(37\)	−3.75605	+	3.75605i	−0.617490	+	0.617490i	−0.944887	−	0.327397i	\(-0.893829\pi\)
							0.327397	+	0.944887i	\(0.393829\pi\)
\(41\)	2.26283			0.353394			0.176697	−	0.984265i	\(-0.443459\pi\)
							0.176697	+	0.984265i	\(0.443459\pi\)
\(43\)	−6.22500	−	6.22500i	−0.949304	−	0.949304i	0.0494712	−	0.998776i	\(-0.484246\pi\)
							−0.998776	+	0.0494712i	\(0.984246\pi\)
\(47\)		−	7.17348i		−	1.04636i	−0.852222	−	0.523180i	\(-0.824746\pi\)
							0.852222	−	0.523180i	\(-0.175254\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.be.a.139.14	yes	184
5.4	even	2	inner	560.2.be.a.139.79	yes	184
7.6	odd	2	inner	560.2.be.a.139.13	✓	184
16.3	odd	4	inner	560.2.be.a.419.80	yes	184
35.34	odd	2	inner	560.2.be.a.139.80	yes	184
80.19	odd	4	inner	560.2.be.a.419.13	yes	184
112.83	even	4	inner	560.2.be.a.419.79	yes	184
560.419	even	4	inner	560.2.be.a.419.14	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.be.a.139.13	✓	184	7.6	odd	2	inner
560.2.be.a.139.14	yes	184	1.1	even	1	trivial
560.2.be.a.139.79	yes	184	5.4	even	2	inner
560.2.be.a.139.80	yes	184	35.34	odd	2	inner
560.2.be.a.419.13	yes	184	80.19	odd	4	inner
560.2.be.a.419.14	yes	184	560.419	even	4	inner
560.2.be.a.419.79	yes	184	112.83	even	4	inner
560.2.be.a.419.80	yes	184	16.3	odd	4	inner