Embedded newform 568.2.bf.a.211.57

• Label: 568.2.bf.a.211.57
• Level: $568$
• Weight: $2$
• Character: 568.211
• Analytic conductor: $4.536$
• Analytic rank: $0$
• Dimension: $1680$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 568 = 2^{3} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	568.bf (of order \(70\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.53550283481\)
Analytic rank:	\(0\)
Dimension:	\(1680\)
Relative dimension:	\(70\) over \(\Q(\zeta_{70})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{70}]$

Embedding invariants

Embedding label			211.57
Character	\(\chi\)	\(=\)	568.211
Dual form			568.2.bf.a.35.57

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.10422 + 0.883570i) q^{2} +(-2.21915 - 1.93881i) q^{3} +(0.438608 + 1.95131i) q^{4} +(1.15895 - 0.376567i) q^{5} +(-0.737356 - 4.10165i) q^{6} +(-2.62613 - 0.724766i) q^{7} +(-1.23980 + 2.54222i) q^{8} +(0.762932 + 5.63219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1680 q - 22 q^{2} - 50 q^{3} - 22 q^{4} - 20 q^{6} - 7 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/568\mathbb{Z}\right)^\times\).

\(n\)	\(143\)	\(285\)	\(433\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{41}{70}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.10422	+	0.883570i	0.780802	+	0.624778i
							\(3\)	−2.21915	−	1.93881i	−1.28123	−	1.11937i	−0.986494	−	0.163795i	\(-0.947627\pi\)
−0.294733	−	0.955580i	\(-0.595231\pi\)
\(5\)	1.15895	−	0.376567i	0.518300	−	0.168406i	−0.0381736	−	0.999271i	\(-0.512154\pi\)
							0.556474	+	0.830865i	\(0.312154\pi\)
\(7\)	−2.62613	−	0.724766i	−0.992584	−	0.273936i	−0.268223	−	0.963357i	\(-0.586436\pi\)
							−0.724361	+	0.689421i	\(0.757865\pi\)
\(11\)	−4.89582	−	2.63455i	−1.47615	−	0.794348i	−0.479608	−	0.877483i	\(-0.659221\pi\)
							−0.996538	+	0.0831349i	\(0.973507\pi\)
\(13\)	0.256569	+	0.476785i	0.0711595	+	0.132236i	0.913690	−	0.406413i	\(-0.133221\pi\)
							−0.842530	+	0.538649i	\(0.818935\pi\)
\(17\)	−1.98897	+	2.73758i	−0.482395	+	0.663960i	−0.978963	−	0.204038i	\(-0.934593\pi\)
							0.496568	+	0.867998i	\(0.334593\pi\)
\(19\)	0.995047	+	2.32803i	0.228279	+	0.534086i	0.994037	−	0.109043i	\(-0.0347786\pi\)
							−0.765758	+	0.643129i	\(0.777636\pi\)
\(23\)	1.14508	+	1.43589i	0.238766	+	0.299403i	0.886749	−	0.462252i	\(-0.152958\pi\)
							−0.647982	+	0.761655i	\(0.724387\pi\)
\(29\)	−3.85757	−	6.45649i	−0.716333	−	1.19894i	−0.972634	−	0.232343i	\(-0.925361\pi\)
							0.256301	−	0.966597i	\(-0.417496\pi\)
\(31\)	−7.31305	+	1.32712i	−1.31346	+	0.238358i	−0.789693	−	0.613502i	\(-0.789760\pi\)
							−0.523769	+	0.851860i	\(0.675475\pi\)
\(37\)	−5.32870	−	4.24949i	−0.876032	−	0.698613i	0.0784377	−	0.996919i	\(-0.475007\pi\)
							−0.954470	+	0.298306i	\(0.903578\pi\)
\(41\)	2.25212	−	4.67658i	0.351723	−	0.730360i	−0.647782	−	0.761826i	\(-0.724303\pi\)
							0.999505	+	0.0314659i	\(0.0100176\pi\)
\(43\)	10.0040	+	3.75457i	1.52560	+	0.572567i	0.966530	−	0.256554i	\(-0.0825872\pi\)
							0.559069	+	0.829121i	\(0.311159\pi\)
\(47\)	−5.64399	+	4.93101i	−0.823261	+	0.719262i	−0.962583	−	0.270985i	\(-0.912650\pi\)
							0.139322	+	0.990247i	\(0.455508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	568.2.bf.a.211.57	yes	1680
8.3	odd	2	inner	568.2.bf.a.211.13	yes	1680
71.35	odd	70	inner	568.2.bf.a.35.13	✓	1680
568.35	even	70	inner	568.2.bf.a.35.57	yes	1680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
568.2.bf.a.35.13	✓	1680	71.35	odd	70	inner
568.2.bf.a.35.57	yes	1680	568.35	even	70	inner
568.2.bf.a.211.13	yes	1680	8.3	odd	2	inner
568.2.bf.a.211.57	yes	1680	1.1	even	1	trivial