Embedded newform 700.2.bj.a.323.65

• Label: 700.2.bj.a.323.65
• 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			323.65
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.65

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.817760 + 1.15381i) q^{2} +(1.44006 + 2.82629i) q^{3} +(-0.662538 + 1.88707i) q^{4} +(1.37788 - 1.76109i) q^{5} +(-2.08336 + 3.97278i) q^{6} +(0.707107 + 0.707107i) q^{7} +(-2.71911 + 0.778731i) q^{8} +(-4.15075 + 5.71302i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.817760	+	1.15381i	0.578243	+	0.815864i
							\(3\)	1.44006	+	2.82629i	0.831422	+	1.63176i	0.773810	+	0.633418i	\(0.218349\pi\)
0.0576121	+	0.998339i	\(0.481651\pi\)
\(5\)	1.37788	−	1.76109i	0.616208	−	0.787584i
							\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
\(11\)	2.81431	+	3.87356i	0.848545	+	1.16792i								0.984181	+	0.177164i	\(0.0566922\pi\)
							−0.135636	+	0.990759i	\(0.543308\pi\)
\(13\)	−1.01495	−	6.40812i	−0.281495	−	1.77729i	−0.571835	−	0.820368i	\(-0.693768\pi\)
							0.290340	−	0.956924i	\(-0.406232\pi\)
\(17\)	1.84432	+	0.939727i	0.447313	+	0.227917i	0.663112	−	0.748521i	\(-0.269236\pi\)
							−0.215799	+	0.976438i	\(0.569236\pi\)
\(19\)	−1.91643	−	5.89817i	−0.439660	−	1.35313i	−0.888235	−	0.459389i	\(-0.848069\pi\)
							0.448576	−	0.893745i	\(-0.351931\pi\)
\(23\)	−0.775307	+	4.89509i	−0.161663	+	1.02070i	0.764788	+	0.644282i	\(0.222844\pi\)
							−0.926451	+	0.376416i	\(0.877156\pi\)
\(29\)	1.53543	+	0.498892i	0.285123	+	0.0926419i	0.448087	−	0.893990i	\(-0.352106\pi\)
							−0.162964	+	0.986632i	\(0.552106\pi\)
\(31\)	3.93661	−	1.27908i	0.707036	−	0.229730i	0.0666425	−	0.997777i	\(-0.478771\pi\)
							0.640393	+	0.768047i	\(0.278771\pi\)
\(37\)	−0.987853	+	0.156460i	−0.162402	+	0.0257219i	−0.237106	−	0.971484i	\(-0.576199\pi\)
							0.0747039	+	0.997206i	\(0.476199\pi\)
\(41\)	1.40272	+	1.01914i	0.219069	+	0.159163i	0.691907	−	0.721986i	\(-0.256771\pi\)
							−0.472839	+	0.881149i	\(0.656771\pi\)
\(43\)	−1.72822	+	1.72822i	−0.263551	+	0.263551i	−0.826495	−	0.562944i	\(-0.809669\pi\)
							0.562944	+	0.826495i	\(0.309669\pi\)
\(47\)	4.01870	−	2.04763i	0.586188	−	0.298678i	−0.135631	−	0.990759i	\(-0.543306\pi\)
							0.721819	+	0.692082i	\(0.243306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.65	yes	720
4.3	odd	2	inner	700.2.bj.a.323.17	✓	720
25.12	odd	20	inner	700.2.bj.a.687.17	yes	720
100.87	even	20	inner	700.2.bj.a.687.65	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.17	✓	720	4.3	odd	2	inner
700.2.bj.a.323.65	yes	720	1.1	even	1	trivial
700.2.bj.a.687.17	yes	720	25.12	odd	20	inner
700.2.bj.a.687.65	yes	720	100.87	even	20	inner