Embedded newform 700.2.bj.a.323.42

• Label: 700.2.bj.a.323.42
• 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			323.42
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.200916 - 1.39987i) q^{2} +(1.06998 + 2.09996i) q^{3} +(-1.91927 + 0.562512i) q^{4} +(2.18229 - 0.487438i) q^{5} +(2.72469 - 1.91975i) q^{6} +(0.707107 + 0.707107i) q^{7} +(1.17305 + 2.57370i) q^{8} +(-1.50161 + 2.06679i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.200916	−	1.39987i	−0.142069	−	0.989857i
							\(3\)	1.06998	+	2.09996i	0.617755	+	1.21241i	0.961876	+	0.273487i	\(0.0881771\pi\)
−0.344121	+	0.938925i	\(0.611823\pi\)
\(5\)	2.18229	−	0.487438i	0.975951	−	0.217989i
							\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
\(11\)	1.50295	+	2.06863i	0.453156	+	0.623715i								0.973072	−	0.230502i	\(-0.0740368\pi\)
							−0.519916	+	0.854217i	\(0.674037\pi\)
\(13\)	0.340781	+	2.15161i	0.0945156	+	0.596748i	0.988800	+	0.149249i	\(0.0476855\pi\)
							−0.894284	+	0.447500i	\(0.852315\pi\)
\(17\)	−4.17197	−	2.12572i	−1.01185	−	0.515564i	−0.132220	−	0.991220i	\(-0.542210\pi\)
							−0.879631	+	0.475657i	\(0.842210\pi\)
\(19\)	0.492019	+	1.51428i	0.112877	+	0.347399i	0.991498	−	0.130120i	\(-0.0415362\pi\)
							−0.878621	+	0.477519i	\(0.841536\pi\)
\(23\)	0.0316412	−	0.199775i	0.00659765	−	0.0416559i	−0.984170	−	0.177228i	\(-0.943287\pi\)
							0.990768	+	0.135572i	\(0.0432871\pi\)
\(29\)	−1.29506	−	0.420791i	−0.240487	−	0.0781390i	0.186294	−	0.982494i	\(-0.440352\pi\)
							−0.426781	+	0.904355i	\(0.640352\pi\)
\(31\)	−6.26478	+	2.03555i	−1.12519	+	0.365595i	−0.811745	−	0.584012i	\(-0.801482\pi\)
							−0.313442	+	0.949607i	\(0.601482\pi\)
\(37\)	−2.44295	+	0.386926i	−0.401619	+	0.0636102i	−0.353977	−	0.935254i	\(-0.615171\pi\)
							−0.0476423	+	0.998864i	\(0.515171\pi\)
\(41\)	7.08302	+	5.14612i	1.10618	+	0.803689i	0.982058	−	0.188578i	\(-0.0603878\pi\)
							0.124124	+	0.992267i	\(0.460388\pi\)
\(43\)	−1.12096	+	1.12096i	−0.170944	+	0.170944i	−0.787394	−	0.616450i	\(-0.788570\pi\)
							0.616450	+	0.787394i	\(0.288570\pi\)
\(47\)	5.86807	−	2.98993i	0.855945	−	0.436126i	0.0297819	−	0.999556i	\(-0.490519\pi\)
							0.826163	+	0.563430i	\(0.190519\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.42	✓	720
4.3	odd	2	inner	700.2.bj.a.323.58	yes	720
25.12	odd	20	inner	700.2.bj.a.687.58	yes	720
100.87	even	20	inner	700.2.bj.a.687.42	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.42	✓	720	1.1	even	1	trivial
700.2.bj.a.323.58	yes	720	4.3	odd	2	inner
700.2.bj.a.687.42	yes	720	100.87	even	20	inner
700.2.bj.a.687.58	yes	720	25.12	odd	20	inner