Embedded newform 700.2.bj.a.323.48

• Label: 700.2.bj.a.323.48
• 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			323.48
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0764030 - 1.41215i) q^{2} +(0.602372 + 1.18222i) q^{3} +(-1.98833 - 0.215785i) q^{4} +(-2.08430 - 0.809739i) q^{5} +(1.71550 - 0.760313i) q^{6} +(0.707107 + 0.707107i) q^{7} +(-0.456634 + 2.79132i) q^{8} +(0.728560 - 1.00278i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0764030	−	1.41215i	0.0540251	−	0.998540i
							\(3\)	0.602372	+	1.18222i	0.347780	+	0.682556i	0.996946	−	0.0780902i	\(-0.0248822\pi\)
−0.649167	+	0.760646i	\(0.724882\pi\)
\(5\)	−2.08430	−	0.809739i	−0.932129	−	0.362126i
							\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
\(11\)	2.41003	+	3.31711i	0.726650	+	1.00015i								0.999277	+	0.0380279i	\(0.0121076\pi\)
							−0.272627	+	0.962120i	\(0.587892\pi\)
\(13\)	−0.684643	−	4.32266i	−0.189886	−	1.19889i	−0.879923	−	0.475117i	\(-0.842406\pi\)
							0.690037	−	0.723774i	\(-0.257594\pi\)
\(17\)	4.81746	+	2.45462i	1.16841	+	0.595333i	0.926989	−	0.375089i	\(-0.122388\pi\)
							0.241417	+	0.970421i	\(0.422388\pi\)
\(19\)	−1.14827	−	3.53402i	−0.263432	−	0.810761i	−0.992050	−	0.125841i	\(-0.959837\pi\)
							0.728618	−	0.684920i	\(-0.240163\pi\)
\(23\)	1.18763	−	7.49840i	0.247638	−	1.56353i	−0.479816	−	0.877369i	\(-0.659297\pi\)
							0.727454	−	0.686156i	\(-0.240703\pi\)
\(29\)	−0.734772	−	0.238742i	−0.136444	−	0.0443333i	0.239999	−	0.970773i	\(-0.422853\pi\)
							−0.376443	+	0.926440i	\(0.622853\pi\)
\(31\)	7.18714	−	2.33524i	1.29085	−	0.419422i	0.418460	−	0.908235i	\(-0.362570\pi\)
							0.872389	+	0.488813i	\(0.162570\pi\)
\(37\)	0.865275	−	0.137046i	0.142250	−	0.0225302i	−0.0849031	−	0.996389i	\(-0.527058\pi\)
							0.227154	+	0.973859i	\(0.427058\pi\)
\(41\)	−6.17492	−	4.48634i	−0.964360	−	0.700649i	−0.0102011	−	0.999948i	\(-0.503247\pi\)
							−0.954159	+	0.299299i	\(0.903247\pi\)
\(43\)	−6.75136	+	6.75136i	−1.02957	+	1.02957i	−0.0300235	+	0.999549i	\(0.509558\pi\)
							−0.999549	+	0.0300235i	\(0.990442\pi\)
\(47\)	3.96724	−	2.02141i	0.578681	−	0.294853i	−0.140046	−	0.990145i	\(-0.544725\pi\)
							0.718727	+	0.695292i	\(0.244725\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.48	✓	720
4.3	odd	2	inner	700.2.bj.a.323.54	yes	720
25.12	odd	20	inner	700.2.bj.a.687.54	yes	720
100.87	even	20	inner	700.2.bj.a.687.48	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.48	✓	720	1.1	even	1	trivial
700.2.bj.a.323.54	yes	720	4.3	odd	2	inner
700.2.bj.a.687.48	yes	720	100.87	even	20	inner
700.2.bj.a.687.54	yes	720	25.12	odd	20	inner