Embedded newform 700.2.bj.a.687.65

• Label: 700.2.bj.a.687.65
• Level: $700$
• Weight: $2$
• Character: 700.687
• Analytic conductor: $5.590$
• Analytic rank: $0$
• Dimension: $720$
• 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			687.65
Character	\(\chi\)	\(=\)	700.687
Dual form			700.2.bj.a.323.65

$q$-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 \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(477\)
\(\chi(n)\)	\(1\)	\(-1\)	\(e\left(\frac{9}{20}\right)\)

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	0.0576121	−	0.998339i	\(-0.481651\pi\)
0.773810	−	0.633418i	\(-0.218349\pi\)
\(5\)	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								−0.135636	−	0.990759i	\(-0.543308\pi\)
							0.984181	−	0.177164i	\(-0.0566922\pi\)
\(13\)	−1.01495	+	6.40812i	−0.281495	+	1.77729i	0.290340	+	0.956924i	\(0.406232\pi\)
							−0.571835	+	0.820368i	\(0.693768\pi\)
\(17\)	1.84432	−	0.939727i	0.447313	−	0.227917i	−0.215799	−	0.976438i	\(-0.569236\pi\)
							0.663112	+	0.748521i	\(0.269236\pi\)
\(19\)	−1.91643	+	5.89817i	−0.439660	+	1.35313i	0.448576	+	0.893745i	\(0.351931\pi\)
							−0.888235	+	0.459389i	\(0.848069\pi\)
\(23\)	−0.775307	−	4.89509i	−0.161663	−	1.02070i	−0.926451	−	0.376416i	\(-0.877156\pi\)
							0.764788	−	0.644282i	\(-0.222844\pi\)
\(29\)	1.53543	−	0.498892i	0.285123	−	0.0926419i	−0.162964	−	0.986632i	\(-0.552106\pi\)
							0.448087	+	0.893990i	\(0.352106\pi\)
\(31\)	3.93661	+	1.27908i	0.707036	+	0.229730i	0.640393	−	0.768047i	\(-0.278771\pi\)
							0.0666425	+	0.997777i	\(0.478771\pi\)
\(37\)	−0.987853	−	0.156460i	−0.162402	−	0.0257219i	0.0747039	−	0.997206i	\(-0.476199\pi\)
							−0.237106	+	0.971484i	\(0.576199\pi\)
\(41\)	1.40272	−	1.01914i	0.219069	−	0.159163i	−0.472839	−	0.881149i	\(-0.656771\pi\)
							0.691907	+	0.721986i	\(0.256771\pi\)
\(43\)	−1.72822	−	1.72822i	−0.263551	−	0.263551i	0.562944	−	0.826495i	\(-0.309669\pi\)
							−0.826495	+	0.562944i	\(0.809669\pi\)
\(47\)	4.01870	+	2.04763i	0.586188	+	0.298678i	0.721819	−	0.692082i	\(-0.243306\pi\)
							−0.135631	+	0.990759i	\(0.543306\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.17	✓	720	25.23	odd	20	inner
700.2.bj.a.323.65	yes	720	100.23	even	20	inner
700.2.bj.a.687.17	yes	720	4.3	odd	2	inner
700.2.bj.a.687.65	yes	720	1.1	even	1	trivial