Embedded newform 560.2.be.a.139.8

• Label: 560.2.be.a.139.8
• 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			139.8
Character	\(\chi\)	\(=\)	560.139
Dual form			560.2.be.a.419.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33517 - 0.466181i) q^{2} +(1.96435 + 1.96435i) q^{3} +(1.56535 + 1.24486i) q^{4} +(-0.776425 - 2.09694i) q^{5} +(-1.70699 - 3.53848i) q^{6} +(-1.99873 - 1.73352i) q^{7} +(-1.50968 - 2.39184i) q^{8} +4.71733i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33517	−	0.466181i	−0.944107	−	0.329640i
							\(3\)	1.96435	+	1.96435i	1.13412	+	1.13412i	0.989486	+	0.144631i	\(0.0461996\pi\)
0.144631	+	0.989486i	\(0.453800\pi\)
\(5\)	−0.776425	−	2.09694i	−0.347228	−	0.937781i
							\(7\)	−1.99873	−	1.73352i	−0.755448	−	0.655209i
\(11\)	0.903406	−	0.903406i	0.272387	−	0.272387i								−0.557673	−	0.830061i	\(-0.688306\pi\)
							0.830061	+	0.557673i	\(0.188306\pi\)
\(13\)	4.57272	−	4.57272i	1.26824	−	1.26824i	0.321250	−	0.946994i	\(-0.395897\pi\)
							0.946994	−	0.321250i	\(-0.104103\pi\)
\(17\)	0.186789			0.0453030			0.0226515	−	0.999743i	\(-0.492789\pi\)
							0.0226515	+	0.999743i	\(0.492789\pi\)
\(19\)	3.99327	−	3.99327i	0.916118	−	0.916118i	−0.0806259	−	0.996744i	\(-0.525692\pi\)
							0.996744	+	0.0806259i	\(0.0256919\pi\)
\(23\)			1.80338i			0.376030i	0.982166	+	0.188015i	\(0.0602053\pi\)
							−0.982166	+	0.188015i	\(0.939795\pi\)
\(29\)	2.73482	−	2.73482i	0.507843	−	0.507843i	−0.406021	−	0.913864i	\(-0.633084\pi\)
							0.913864	+	0.406021i	\(0.133084\pi\)
\(31\)	7.33459			1.31733			0.658666	−	0.752436i	\(-0.271121\pi\)
							0.658666	+	0.752436i	\(0.271121\pi\)
\(37\)	−4.27368	+	4.27368i	−0.702589	+	0.702589i	−0.964966	−	0.262377i	\(-0.915494\pi\)
							0.262377	+	0.964966i	\(0.415494\pi\)
\(41\)	−10.0686			−1.57245			−0.786225	−	0.617940i	\(-0.787967\pi\)
							−0.786225	+	0.617940i	\(0.787967\pi\)
\(43\)	8.51075	+	8.51075i	1.29788	+	1.29788i	0.929792	+	0.368085i	\(0.119986\pi\)
							0.368085	+	0.929792i	\(0.380014\pi\)
\(47\)		−	2.51534i		−	0.366900i	−0.983029	−	0.183450i	\(-0.941274\pi\)
							0.983029	−	0.183450i	\(-0.0587265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.be.a.139.8	yes	184
5.4	even	2	inner	560.2.be.a.139.85	yes	184
7.6	odd	2	inner	560.2.be.a.139.7	✓	184
16.3	odd	4	inner	560.2.be.a.419.86	yes	184
35.34	odd	2	inner	560.2.be.a.139.86	yes	184
80.19	odd	4	inner	560.2.be.a.419.7	yes	184
112.83	even	4	inner	560.2.be.a.419.85	yes	184
560.419	even	4	inner	560.2.be.a.419.8	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.be.a.139.7	✓	184	7.6	odd	2	inner
560.2.be.a.139.8	yes	184	1.1	even	1	trivial
560.2.be.a.139.85	yes	184	5.4	even	2	inner
560.2.be.a.139.86	yes	184	35.34	odd	2	inner
560.2.be.a.419.7	yes	184	80.19	odd	4	inner
560.2.be.a.419.8	yes	184	560.419	even	4	inner
560.2.be.a.419.85	yes	184	112.83	even	4	inner
560.2.be.a.419.86	yes	184	16.3	odd	4	inner