Embedded newform 560.2.be.a.139.2

• Label: 560.2.be.a.139.2
• 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			139.2
Character	\(\chi\)	\(=\)	560.139
Dual form			560.2.be.a.419.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41383 - 0.0328882i) q^{2} +(1.31817 + 1.31817i) q^{3} +(1.99784 + 0.0929966i) q^{4} +(-1.05420 + 1.97197i) q^{5} +(-1.82031 - 1.90702i) q^{6} +(-2.24452 - 1.40075i) q^{7} +(-2.82155 - 0.197187i) q^{8} +0.475122i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41383	−	0.0328882i	−0.999730	−	0.0232554i
							\(3\)	1.31817	+	1.31817i	0.761043	+	0.761043i	0.976511	−	0.215468i	\(-0.0691276\pi\)
−0.215468	+	0.976511i	\(0.569128\pi\)
\(5\)	−1.05420	+	1.97197i	−0.471454	+	0.881891i
							\(7\)	−2.24452	−	1.40075i	−0.848351	−	0.529435i
\(11\)	0.521295	−	0.521295i	0.157176	−	0.157176i								−0.624138	−	0.781314i	\(-0.714550\pi\)
							0.781314	+	0.624138i	\(0.214550\pi\)
\(13\)	−2.71344	+	2.71344i	−0.752574	+	0.752574i	−0.974959	−	0.222385i	\(-0.928616\pi\)
							0.222385	+	0.974959i	\(0.428616\pi\)
\(17\)	−7.22655			−1.75270			−0.876348	−	0.481678i	\(-0.840027\pi\)
							−0.876348	+	0.481678i	\(0.840027\pi\)
\(19\)	−0.996080	+	0.996080i	−0.228516	+	0.228516i	−0.812073	−	0.583556i	\(-0.801661\pi\)
							0.583556	+	0.812073i	\(0.301661\pi\)
\(23\)			8.64009i			1.80158i	0.434252	+	0.900792i	\(0.357013\pi\)
							−0.434252	+	0.900792i	\(0.642987\pi\)
\(29\)	−3.30789	+	3.30789i	−0.614260	+	0.614260i	−0.944053	−	0.329793i	\(-0.893021\pi\)
							0.329793	+	0.944053i	\(0.393021\pi\)
\(31\)	−4.83062			−0.867604			−0.433802	−	0.901008i	\(-0.642828\pi\)
							−0.433802	+	0.901008i	\(0.642828\pi\)
\(37\)	6.42522	−	6.42522i	1.05630	−	1.05630i	0.0579821	−	0.998318i	\(-0.481533\pi\)
							0.998318	−	0.0579821i	\(-0.0184666\pi\)
\(41\)	6.00233			0.937406			0.468703	−	0.883356i	\(-0.344721\pi\)
							0.468703	+	0.883356i	\(0.344721\pi\)
\(43\)	−5.03525	−	5.03525i	−0.767868	−	0.767868i	0.209863	−	0.977731i	\(-0.432698\pi\)
							−0.977731	+	0.209863i	\(0.932698\pi\)
\(47\)			4.11795i			0.600664i	0.953835	+	0.300332i	\(0.0970975\pi\)
							−0.953835	+	0.300332i	\(0.902903\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.be.a.139.2	yes	184
5.4	even	2	inner	560.2.be.a.139.91	yes	184
7.6	odd	2	inner	560.2.be.a.139.1	✓	184
16.3	odd	4	inner	560.2.be.a.419.92	yes	184
35.34	odd	2	inner	560.2.be.a.139.92	yes	184
80.19	odd	4	inner	560.2.be.a.419.1	yes	184
112.83	even	4	inner	560.2.be.a.419.91	yes	184
560.419	even	4	inner	560.2.be.a.419.2	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.be.a.139.1	✓	184	7.6	odd	2	inner
560.2.be.a.139.2	yes	184	1.1	even	1	trivial
560.2.be.a.139.91	yes	184	5.4	even	2	inner
560.2.be.a.139.92	yes	184	35.34	odd	2	inner
560.2.be.a.419.1	yes	184	80.19	odd	4	inner
560.2.be.a.419.2	yes	184	560.419	even	4	inner
560.2.be.a.419.91	yes	184	112.83	even	4	inner
560.2.be.a.419.92	yes	184	16.3	odd	4	inner