Embedded newform 568.2.bf.a.211.13

• Label: 568.2.bf.a.211.13
• 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			211.13
Character	\(\chi\)	\(=\)	568.211
Dual form			568.2.bf.a.35.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17898 + 0.781031i) q^{2} +(-2.21915 - 1.93881i) q^{3} +(0.779980 - 1.84164i) q^{4} +(-1.15895 + 0.376567i) q^{5} +(4.13060 + 0.552593i) q^{6} +(2.62613 + 0.724766i) q^{7} +(0.518797 + 2.78044i) q^{8} +(0.762932 + 5.63219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1680 q - 22 q^{2} - 50 q^{3} - 22 q^{4} - 20 q^{6} - 7 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17898	+	0.781031i	−0.833664	+	0.552273i
							\(3\)	−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	−0.556474	−	0.830865i	\(-0.687846\pi\)
							0.0381736	+	0.999271i	\(0.487846\pi\)
\(7\)	2.62613	+	0.724766i	0.992584	+	0.273936i	0.724361	−	0.689421i	\(-0.242135\pi\)
							0.268223	+	0.963357i	\(0.413564\pi\)
\(11\)	−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	0.842530	−	0.538649i	\(-0.181065\pi\)
							−0.913690	+	0.406413i	\(0.866779\pi\)
\(17\)	−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	0.647982	−	0.761655i	\(-0.275613\pi\)
							−0.886749	+	0.462252i	\(0.847042\pi\)
\(29\)	3.85757	+	6.45649i	0.716333	+	1.19894i	0.972634	+	0.232343i	\(0.0746391\pi\)
							−0.256301	+	0.966597i	\(0.582504\pi\)
\(31\)	7.31305	−	1.32712i	1.31346	−	0.238358i	0.523769	−	0.851860i	\(-0.324525\pi\)
							0.789693	+	0.613502i	\(0.210240\pi\)
\(37\)	5.32870	+	4.24949i	0.876032	+	0.698613i	0.954470	−	0.298306i	\(-0.0964218\pi\)
							−0.0784377	+	0.996919i	\(0.524993\pi\)
\(41\)	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	−0.139322	−	0.990247i	\(-0.544492\pi\)
							0.962583	+	0.270985i	\(0.0873495\pi\)

(See \(a_n\) instead)

Twists

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

    

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