Embedded newform 560.2.be.a.139.19

• Label: 560.2.be.a.139.19
• 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			139.19
Character	\(\chi\)	\(=\)	560.139
Dual form			560.2.be.a.419.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15591 + 0.814782i) q^{2} +(-2.22344 - 2.22344i) q^{3} +(0.672261 - 1.88363i) q^{4} +(2.21099 + 0.333957i) q^{5} +(4.38172 + 0.758481i) q^{6} +(-1.99535 - 1.73741i) q^{7} +(0.757674 + 2.72506i) q^{8} +6.88739i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 8 q^{4}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15591	+	0.814782i	−0.817353	+	0.576138i
							\(3\)	−2.22344	−	2.22344i	−1.28370	−	1.28370i	−0.938543	−	0.345162i	\(-0.887824\pi\)
−0.345162	−	0.938543i	\(-0.612176\pi\)
\(5\)	2.21099	+	0.333957i	0.988784	+	0.149350i
							\(7\)	−1.99535	−	1.73741i	−0.754171	−	0.656678i
\(11\)	−3.57424	+	3.57424i	−1.07767	+	1.07767i								−0.0809561	+	0.996718i	\(0.525797\pi\)
							−0.996718	+	0.0809561i	\(0.974203\pi\)
\(13\)	−0.234145	+	0.234145i	−0.0649401	+	0.0649401i	−0.738831	−	0.673891i	\(-0.764622\pi\)
							0.673891	+	0.738831i	\(0.264622\pi\)
\(17\)	−1.25034			−0.303252			−0.151626	−	0.988438i	\(-0.548451\pi\)
							−0.151626	+	0.988438i	\(0.548451\pi\)
\(19\)	3.59835	−	3.59835i	0.825519	−	0.825519i	−0.161374	−	0.986893i	\(-0.551593\pi\)
							0.986893	+	0.161374i	\(0.0515927\pi\)
\(23\)			8.60589i			1.79445i	0.441572	+	0.897226i	\(0.354421\pi\)
							−0.441572	+	0.897226i	\(0.645579\pi\)
\(29\)	−1.87245	+	1.87245i	−0.347704	+	0.347704i	−0.859254	−	0.511549i	\(-0.829072\pi\)
							0.511549	+	0.859254i	\(0.329072\pi\)
\(31\)	−1.97739			−0.355150			−0.177575	−	0.984107i	\(-0.556825\pi\)
							−0.177575	+	0.984107i	\(0.556825\pi\)
\(37\)	−1.08762	+	1.08762i	−0.178804	+	0.178804i	−0.790834	−	0.612030i	\(-0.790353\pi\)
							0.612030	+	0.790834i	\(0.290353\pi\)
\(41\)	−6.11578			−0.955124			−0.477562	−	0.878598i	\(-0.658479\pi\)
							−0.477562	+	0.878598i	\(0.658479\pi\)
\(43\)	3.31107	+	3.31107i	0.504933	+	0.504933i	0.912967	−	0.408034i	\(-0.133785\pi\)
							−0.408034	+	0.912967i	\(0.633785\pi\)
\(47\)			6.64089i			0.968674i	0.874882	+	0.484337i	\(0.160939\pi\)
							−0.874882	+	0.484337i	\(0.839061\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.be.a.139.19	✓	184
5.4	even	2	inner	560.2.be.a.139.74	yes	184
7.6	odd	2	inner	560.2.be.a.139.20	yes	184
16.3	odd	4	inner	560.2.be.a.419.73	yes	184
35.34	odd	2	inner	560.2.be.a.139.73	yes	184
80.19	odd	4	inner	560.2.be.a.419.20	yes	184
112.83	even	4	inner	560.2.be.a.419.74	yes	184
560.419	even	4	inner	560.2.be.a.419.19	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.be.a.139.19	✓	184	1.1	even	1	trivial
560.2.be.a.139.20	yes	184	7.6	odd	2	inner
560.2.be.a.139.73	yes	184	35.34	odd	2	inner
560.2.be.a.139.74	yes	184	5.4	even	2	inner
560.2.be.a.419.19	yes	184	560.419	even	4	inner
560.2.be.a.419.20	yes	184	80.19	odd	4	inner
560.2.be.a.419.73	yes	184	16.3	odd	4	inner
560.2.be.a.419.74	yes	184	112.83	even	4	inner