Embedded newform 700.2.bj.a.323.15

• Label: 700.2.bj.a.323.15
• Level: $700$
• Weight: $2$
• Character: 700.323
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $720$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.bj (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.58952814149\)
Analytic rank:	\(0\)
Dimension:	\(720\)
Relative dimension:	\(90\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			323.15
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.22102 + 0.713522i) q^{2} +(1.38699 + 2.72212i) q^{3} +(0.981773 - 1.74245i) q^{4} +(-0.521836 + 2.17432i) q^{5} +(-3.63583 - 2.33411i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(0.0445120 + 2.82808i) q^{8} +(-3.72285 + 5.12406i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{11}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.22102	+	0.713522i	−0.863390	+	0.504536i
							\(3\)	1.38699	+	2.72212i	0.800779	+	1.57162i	0.820386	+	0.571810i	\(0.193759\pi\)
−0.0196069	+	0.999808i	\(0.506241\pi\)
\(5\)	−0.521836	+	2.17432i	−0.233372	+	0.972388i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	3.29096	+	4.52962i	0.992262	+	1.36573i								0.929954	+	0.367675i	\(0.119846\pi\)
							0.0623076	+	0.998057i	\(0.480154\pi\)
\(13\)	−0.466368	−	2.94453i	−0.129347	−	0.816665i	−0.964003	−	0.265893i	\(-0.914333\pi\)
							0.834655	−	0.550772i	\(-0.185667\pi\)
\(17\)	−0.336967	−	0.171693i	−0.0817265	−	0.0416417i	0.412650	−	0.910890i	\(-0.364603\pi\)
							−0.494376	+	0.869248i	\(0.664603\pi\)
\(19\)	1.65135	+	5.08233i	0.378845	+	1.16597i	0.940848	+	0.338830i	\(0.110031\pi\)
							−0.562003	+	0.827135i	\(0.689969\pi\)
\(23\)	0.403083	−	2.54496i	0.0840486	−	0.530662i	−0.909358	−	0.416015i	\(-0.863426\pi\)
							0.993406	−	0.114647i	\(-0.0365736\pi\)
\(29\)	−4.85833	−	1.57857i	−0.902170	−	0.293133i	−0.179038	−	0.983842i	\(-0.557298\pi\)
							−0.723132	+	0.690709i	\(0.757298\pi\)
\(31\)	3.80180	−	1.23528i	0.682823	−	0.221863i	0.0529920	−	0.998595i	\(-0.483124\pi\)
							0.629831	+	0.776732i	\(0.283124\pi\)
\(37\)	−9.90352	+	1.56856i	−1.62813	+	0.257870i	−0.902651	−	0.430374i	\(-0.858382\pi\)
							−0.725479	+	0.688245i	\(0.758382\pi\)
\(41\)	7.26889	+	5.28115i	1.13521	+	0.824778i	0.986445	−	0.164095i	\(-0.0524703\pi\)
							0.148765	+	0.988873i	\(0.452470\pi\)
\(43\)	6.36022	−	6.36022i	0.969924	−	0.969924i	−0.0296368	−	0.999561i	\(-0.509435\pi\)
							0.999561	+	0.0296368i	\(0.00943506\pi\)
\(47\)	−2.63896	+	1.34462i	−0.384931	+	0.196132i	−0.635738	−	0.771905i	\(-0.719304\pi\)
							0.250807	+	0.968037i	\(0.419304\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.15	✓	720
4.3	odd	2	inner	700.2.bj.a.323.68	yes	720
25.12	odd	20	inner	700.2.bj.a.687.68	yes	720
100.87	even	20	inner	700.2.bj.a.687.15	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.15	✓	720	1.1	even	1	trivial
700.2.bj.a.323.68	yes	720	4.3	odd	2	inner
700.2.bj.a.687.15	yes	720	100.87	even	20	inner
700.2.bj.a.687.68	yes	720	25.12	odd	20	inner