Embedded newform 560.2.be.a.139.12

• Label: 560.2.be.a.139.12
• 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			139.12
Character	\(\chi\)	\(=\)	560.139
Dual form			560.2.be.a.419.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30750 - 0.538918i) q^{2} +(0.289321 + 0.289321i) q^{3} +(1.41913 + 1.40928i) q^{4} +(2.02485 + 0.948664i) q^{5} +(-0.222368 - 0.534210i) q^{6} +(1.73658 + 1.99607i) q^{7} +(-1.09604 - 2.60743i) q^{8} -2.83259i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30750	−	0.538918i	−0.924545	−	0.381073i
							\(3\)	0.289321	+	0.289321i	0.167040	+	0.167040i	0.785677	−	0.618637i	\(-0.212315\pi\)
−0.618637	+	0.785677i	\(0.712315\pi\)
\(5\)	2.02485	+	0.948664i	0.905543	+	0.424255i
							\(7\)	1.73658	+	1.99607i	0.656366	+	0.754443i
\(11\)	2.25797	−	2.25797i	0.680802	−	0.680802i								−0.279379	−	0.960181i	\(-0.590128\pi\)
							0.960181	+	0.279379i	\(0.0901285\pi\)
\(13\)	0.811208	−	0.811208i	0.224988	−	0.224988i	−0.585607	−	0.810595i	\(-0.699144\pi\)
							0.810595	+	0.585607i	\(0.199144\pi\)
\(17\)	−3.28948			−0.797815			−0.398908	−	0.916991i	\(-0.630611\pi\)
							−0.398908	+	0.916991i	\(0.630611\pi\)
\(19\)	4.99749	−	4.99749i	1.14650	−	1.14650i	0.159268	−	0.987235i	\(-0.449086\pi\)
							0.987235	−	0.159268i	\(-0.0509135\pi\)
\(23\)			2.40806i			0.502114i	0.967972	+	0.251057i	\(0.0807782\pi\)
							−0.967972	+	0.251057i	\(0.919222\pi\)
\(29\)	−6.60291	+	6.60291i	−1.22613	+	1.22613i	−0.260712	+	0.965417i	\(0.583957\pi\)
							−0.965417	+	0.260712i	\(0.916043\pi\)
\(31\)	0.610944			0.109729			0.0548644	−	0.998494i	\(-0.482527\pi\)
							0.0548644	+	0.998494i	\(0.482527\pi\)
\(37\)	−1.76325	+	1.76325i	−0.289877	+	0.289877i	−0.837032	−	0.547155i	\(-0.815711\pi\)
							0.547155	+	0.837032i	\(0.315711\pi\)
\(41\)	5.43309			0.848506			0.424253	−	0.905544i	\(-0.360537\pi\)
							0.424253	+	0.905544i	\(0.360537\pi\)
\(43\)	2.72027	+	2.72027i	0.414837	+	0.414837i	0.883420	−	0.468583i	\(-0.155235\pi\)
							−0.468583	+	0.883420i	\(0.655235\pi\)
\(47\)		−	9.86589i		−	1.43909i	−0.694447	−	0.719544i	\(-0.744351\pi\)
							0.694447	−	0.719544i	\(-0.255649\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.be.a.139.12	yes	184
5.4	even	2	inner	560.2.be.a.139.81	yes	184
7.6	odd	2	inner	560.2.be.a.139.11	✓	184
16.3	odd	4	inner	560.2.be.a.419.82	yes	184
35.34	odd	2	inner	560.2.be.a.139.82	yes	184
80.19	odd	4	inner	560.2.be.a.419.11	yes	184
112.83	even	4	inner	560.2.be.a.419.81	yes	184
560.419	even	4	inner	560.2.be.a.419.12	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.be.a.139.11	✓	184	7.6	odd	2	inner
560.2.be.a.139.12	yes	184	1.1	even	1	trivial
560.2.be.a.139.81	yes	184	5.4	even	2	inner
560.2.be.a.139.82	yes	184	35.34	odd	2	inner
560.2.be.a.419.11	yes	184	80.19	odd	4	inner
560.2.be.a.419.12	yes	184	560.419	even	4	inner
560.2.be.a.419.81	yes	184	112.83	even	4	inner
560.2.be.a.419.82	yes	184	16.3	odd	4	inner