Embedded newform 560.2.be.a.139.6

• Label: 560.2.be.a.139.6
• 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			139.6
Character	\(\chi\)	\(=\)	560.139
Dual form			560.2.be.a.419.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40049 - 0.196510i) q^{2} +(1.64523 + 1.64523i) q^{3} +(1.92277 + 0.550422i) q^{4} +(1.65460 - 1.50409i) q^{5} +(-1.98083 - 2.62744i) q^{6} +(0.240324 + 2.63481i) q^{7} +(-2.58466 - 1.14871i) q^{8} +2.41357i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40049	−	0.196510i	−0.990299	−	0.138953i
							\(3\)	1.64523	+	1.64523i	0.949875	+	0.949875i	0.998802	−	0.0489277i	\(-0.0155804\pi\)
−0.0489277	+	0.998802i	\(0.515580\pi\)
\(5\)	1.65460	−	1.50409i	0.739960	−	0.672650i
							\(7\)	0.240324	+	2.63481i	0.0908340	+	0.995866i
\(11\)	−2.59553	+	2.59553i	−0.782581	+	0.782581i								−0.980266	−	0.197684i	\(-0.936658\pi\)
							0.197684	+	0.980266i	\(0.436658\pi\)
\(13\)	−3.28686	+	3.28686i	−0.911612	+	0.911612i	−0.996399	−	0.0847871i	\(-0.972979\pi\)
							0.0847871	+	0.996399i	\(0.472979\pi\)
\(17\)	3.04384			0.738239			0.369120	−	0.929382i	\(-0.379659\pi\)
							0.369120	+	0.929382i	\(0.379659\pi\)
\(19\)	−4.60008	+	4.60008i	−1.05533	+	1.05533i	−0.0569541	+	0.998377i	\(0.518139\pi\)
							−0.998377	+	0.0569541i	\(0.981861\pi\)
\(23\)			4.00330i			0.834746i	0.908735	+	0.417373i	\(0.137049\pi\)
							−0.908735	+	0.417373i	\(0.862951\pi\)
\(29\)	6.65601	−	6.65601i	1.23599	−	1.23599i	0.274364	−	0.961626i	\(-0.411533\pi\)
							0.961626	−	0.274364i	\(-0.0884673\pi\)
\(31\)	7.05897			1.26783			0.633914	−	0.773403i	\(-0.281447\pi\)
							0.633914	+	0.773403i	\(0.281447\pi\)
\(37\)	0.842371	−	0.842371i	0.138485	−	0.138485i	−0.634466	−	0.772951i	\(-0.718780\pi\)
							0.772951	+	0.634466i	\(0.218780\pi\)
\(41\)	−0.244028			−0.0381108			−0.0190554	−	0.999818i	\(-0.506066\pi\)
							−0.0190554	+	0.999818i	\(0.506066\pi\)
\(43\)	−4.19864	−	4.19864i	−0.640286	−	0.640286i	0.310340	−	0.950626i	\(-0.399557\pi\)
							−0.950626	+	0.310340i	\(0.899557\pi\)
\(47\)			3.65883i			0.533696i	0.963739	+	0.266848i	\(0.0859821\pi\)
							−0.963739	+	0.266848i	\(0.914018\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.be.a.139.6	yes	184
5.4	even	2	inner	560.2.be.a.139.87	yes	184
7.6	odd	2	inner	560.2.be.a.139.5	✓	184
16.3	odd	4	inner	560.2.be.a.419.88	yes	184
35.34	odd	2	inner	560.2.be.a.139.88	yes	184
80.19	odd	4	inner	560.2.be.a.419.5	yes	184
112.83	even	4	inner	560.2.be.a.419.87	yes	184
560.419	even	4	inner	560.2.be.a.419.6	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.be.a.139.5	✓	184	7.6	odd	2	inner
560.2.be.a.139.6	yes	184	1.1	even	1	trivial
560.2.be.a.139.87	yes	184	5.4	even	2	inner
560.2.be.a.139.88	yes	184	35.34	odd	2	inner
560.2.be.a.419.5	yes	184	80.19	odd	4	inner
560.2.be.a.419.6	yes	184	560.419	even	4	inner
560.2.be.a.419.87	yes	184	112.83	even	4	inner
560.2.be.a.419.88	yes	184	16.3	odd	4	inner