Embedded newform 560.2.be.a.139.17

• Label: 560.2.be.a.139.17
• 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			139.17
Character	\(\chi\)	\(=\)	560.139
Dual form			560.2.be.a.419.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16263 + 0.805161i) q^{2} +(-2.03615 - 2.03615i) q^{3} +(0.703430 - 1.87221i) q^{4} +(-1.56280 - 1.59927i) q^{5} +(4.00673 + 0.727866i) q^{6} +(0.889133 + 2.49188i) q^{7} +(0.689603 + 2.74307i) q^{8} +5.29182i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16263	+	0.805161i	−0.822106	+	0.569335i
							\(3\)	−2.03615	−	2.03615i	−1.17557	−	1.17557i	−0.980860	−	0.194712i	\(-0.937623\pi\)
−0.194712	−	0.980860i	\(-0.562377\pi\)
\(5\)	−1.56280	−	1.59927i	−0.698906	−	0.715213i
							\(7\)	0.889133	+	2.49188i	0.336061	+	0.941840i
\(11\)	0.0647842	−	0.0647842i	0.0195332	−	0.0195332i								−0.697273	−	0.716806i	\(-0.745603\pi\)
							0.716806	+	0.697273i	\(0.245603\pi\)
\(13\)	−2.20336	+	2.20336i	−0.611102	+	0.611102i	−0.943233	−	0.332131i	\(-0.892232\pi\)
							0.332131	+	0.943233i	\(0.392232\pi\)
\(17\)	6.40701			1.55393			0.776965	−	0.629544i	\(-0.216758\pi\)
							0.776965	+	0.629544i	\(0.216758\pi\)
\(19\)	−2.92231	+	2.92231i	−0.670425	+	0.670425i	−0.957814	−	0.287389i	\(-0.907213\pi\)
							0.287389	+	0.957814i	\(0.407213\pi\)
\(23\)		−	1.32521i		−	0.276326i	−0.990410	−	0.138163i	\(-0.955880\pi\)
							0.990410	−	0.138163i	\(-0.0441198\pi\)
\(29\)	3.22310	−	3.22310i	0.598515	−	0.598515i	−0.341402	−	0.939917i	\(-0.610902\pi\)
							0.939917	+	0.341402i	\(0.110902\pi\)
\(31\)	−2.73582			−0.491368			−0.245684	−	0.969350i	\(-0.579012\pi\)
							−0.245684	+	0.969350i	\(0.579012\pi\)
\(37\)	7.36052	−	7.36052i	1.21006	−	1.21006i	0.239057	−	0.971006i	\(-0.423162\pi\)
							0.971006	−	0.239057i	\(-0.0768382\pi\)
\(41\)	7.60012			1.18694			0.593469	−	0.804857i	\(-0.297758\pi\)
							0.593469	+	0.804857i	\(0.297758\pi\)
\(43\)	3.46698	+	3.46698i	0.528709	+	0.528709i	0.920187	−	0.391478i	\(-0.128036\pi\)
							−0.391478	+	0.920187i	\(0.628036\pi\)
\(47\)		−	3.15943i		−	0.460851i	−0.973090	−	0.230425i	\(-0.925988\pi\)
							0.973090	−	0.230425i	\(-0.0740118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.be.a.139.17	✓	184
5.4	even	2	inner	560.2.be.a.139.76	yes	184
7.6	odd	2	inner	560.2.be.a.139.18	yes	184
16.3	odd	4	inner	560.2.be.a.419.75	yes	184
35.34	odd	2	inner	560.2.be.a.139.75	yes	184
80.19	odd	4	inner	560.2.be.a.419.18	yes	184
112.83	even	4	inner	560.2.be.a.419.76	yes	184
560.419	even	4	inner	560.2.be.a.419.17	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.be.a.139.17	✓	184	1.1	even	1	trivial
560.2.be.a.139.18	yes	184	7.6	odd	2	inner
560.2.be.a.139.75	yes	184	35.34	odd	2	inner
560.2.be.a.139.76	yes	184	5.4	even	2	inner
560.2.be.a.419.17	yes	184	560.419	even	4	inner
560.2.be.a.419.18	yes	184	80.19	odd	4	inner
560.2.be.a.419.75	yes	184	16.3	odd	4	inner
560.2.be.a.419.76	yes	184	112.83	even	4	inner