Embedded newform 700.2.bj.a.323.33

• Label: 700.2.bj.a.323.33
• 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			323.33
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.483634 + 1.32895i) q^{2} +(-0.00732929 - 0.0143845i) q^{3} +(-1.53220 - 1.28545i) q^{4} +(-2.23382 + 0.100315i) q^{5} +(0.0226610 - 0.00278338i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(2.44931 - 1.41452i) q^{8} +(1.76320 - 2.42684i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.483634	+	1.32895i	−0.341981	+	0.939707i
							\(3\)	−0.00732929	−	0.0143845i	−0.00423157	−	0.00830492i	0.888881	−	0.458138i	\(-0.151483\pi\)
−0.893113	+	0.449833i	\(0.851483\pi\)
\(5\)	−2.23382	+	0.100315i	−0.998993	+	0.0448624i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	1.01643	+	1.39899i	0.306465	+	0.421812i								0.934275	−	0.356554i	\(-0.116049\pi\)
							−0.627810	+	0.778367i	\(0.716049\pi\)
\(13\)	0.349731	+	2.20812i	0.0969980	+	0.612421i	0.987522	+	0.157484i	\(0.0503383\pi\)
							−0.890524	+	0.454937i	\(0.849662\pi\)
\(17\)	−2.42695	−	1.23659i	−0.588621	−	0.299917i	0.134198	−	0.990954i	\(-0.457154\pi\)
							−0.722819	+	0.691037i	\(0.757154\pi\)
\(19\)	2.57329	+	7.91976i	0.590353	+	1.81692i	0.576620	+	0.817012i	\(0.304371\pi\)
							0.0137325	+	0.999906i	\(0.495629\pi\)
\(23\)	−0.352797	+	2.22748i	−0.0735634	+	0.464461i	0.923217	+	0.384279i	\(0.125550\pi\)
							−0.996780	+	0.0801815i	\(0.974450\pi\)
\(29\)	3.47298	+	1.12844i	0.644916	+	0.209546i	0.613171	−	0.789950i	\(-0.289894\pi\)
							0.0317446	+	0.999496i	\(0.489894\pi\)
\(31\)	−9.19869	+	2.98884i	−1.65213	+	0.536811i	−0.979201	−	0.202892i	\(-0.934966\pi\)
							−0.672933	+	0.739703i	\(0.734966\pi\)
\(37\)	−3.89988	+	0.617681i	−0.641137	+	0.101546i	−0.468534	−	0.883445i	\(-0.655218\pi\)
							−0.172603	+	0.984992i	\(0.555218\pi\)
\(41\)	6.44297	+	4.68109i	1.00622	+	0.731063i	0.963413	−	0.268020i	\(-0.0863693\pi\)
							0.0428089	+	0.999083i	\(0.486369\pi\)
\(43\)	−3.18099	+	3.18099i	−0.485097	+	0.485097i	−0.906755	−	0.421658i	\(-0.861448\pi\)
							0.421658	+	0.906755i	\(0.361448\pi\)
\(47\)	0.782664	−	0.398787i	0.114163	−	0.0581691i	−0.395976	−	0.918261i	\(-0.629594\pi\)
							0.510140	+	0.860092i	\(0.329594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.33	✓	720
4.3	odd	2	inner	700.2.bj.a.323.47	yes	720
25.12	odd	20	inner	700.2.bj.a.687.47	yes	720
100.87	even	20	inner	700.2.bj.a.687.33	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.33	✓	720	1.1	even	1	trivial
700.2.bj.a.323.47	yes	720	4.3	odd	2	inner
700.2.bj.a.687.33	yes	720	100.87	even	20	inner
700.2.bj.a.687.47	yes	720	25.12	odd	20	inner