Embedded newform 700.2.bj.a.323.23

• Label: 700.2.bj.a.323.23
• 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			323.23
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.01783 - 0.981847i) q^{2} +(-0.0725586 - 0.142404i) q^{3} +(0.0719512 + 1.99871i) q^{4} +(1.73352 + 1.41241i) q^{5} +(-0.0659670 + 0.216185i) q^{6} +(0.707107 + 0.707107i) q^{7} +(1.88919 - 2.10499i) q^{8} +(1.74834 - 2.40639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.01783	−	0.981847i	−0.719714	−	0.694271i
							\(3\)	−0.0725586	−	0.142404i	−0.0418917	−	0.0822172i	0.869112	−	0.494616i	\(-0.164691\pi\)
−0.911004	+	0.412398i	\(0.864691\pi\)
\(5\)	1.73352	+	1.41241i	0.775254	+	0.631650i
							\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
\(11\)	0.485209	+	0.667833i	0.146296	+	0.201359i								0.875876	−	0.482536i	\(-0.160284\pi\)
							−0.729580	+	0.683896i	\(0.760284\pi\)
\(13\)	0.338912	+	2.13981i	0.0939973	+	0.593475i	0.989058	+	0.147530i	\(0.0471324\pi\)
							−0.895060	+	0.445945i	\(0.852868\pi\)
\(17\)	−0.484253	−	0.246739i	−0.117449	−	0.0598431i	0.394279	−	0.918991i	\(-0.370994\pi\)
							−0.511728	+	0.859148i	\(0.670994\pi\)
\(19\)	−0.598424	−	1.84176i	−0.137288	−	0.422528i	0.858651	−	0.512561i	\(-0.171303\pi\)
							−0.995939	+	0.0900322i	\(0.971303\pi\)
\(23\)	−0.531250	+	3.35418i	−0.110773	+	0.699395i	0.868323	+	0.495999i	\(0.165198\pi\)
							−0.979097	+	0.203396i	\(0.934802\pi\)
\(29\)	5.91398	+	1.92157i	1.09820	+	0.356826i	0.801409	−	0.598117i	\(-0.204084\pi\)
							0.296789	+	0.954943i	\(0.404084\pi\)
\(31\)	4.81633	−	1.56492i	0.865039	−	0.281068i	0.157308	−	0.987550i	\(-0.449719\pi\)
							0.707732	+	0.706481i	\(0.249719\pi\)
\(37\)	−2.61814	+	0.414672i	−0.430419	+	0.0681717i	−0.367885	−	0.929871i	\(-0.619918\pi\)
							−0.0625337	+	0.998043i	\(0.519918\pi\)
\(41\)	3.28660	+	2.38785i	0.513280	+	0.372920i	0.814066	−	0.580772i	\(-0.197249\pi\)
							−0.300786	+	0.953692i	\(0.597249\pi\)
\(43\)	−4.18330	+	4.18330i	−0.637947	+	0.637947i	−0.950049	−	0.312102i	\(-0.898967\pi\)
							0.312102	+	0.950049i	\(0.398967\pi\)
\(47\)	7.12131	−	3.62849i	1.03875	−	0.529269i	0.150490	−	0.988612i	\(-0.451915\pi\)
							0.888259	+	0.459342i	\(0.151915\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.23	✓	720
4.3	odd	2	inner	700.2.bj.a.323.77	yes	720
25.12	odd	20	inner	700.2.bj.a.687.77	yes	720
100.87	even	20	inner	700.2.bj.a.687.23	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.23	✓	720	1.1	even	1	trivial
700.2.bj.a.323.77	yes	720	4.3	odd	2	inner
700.2.bj.a.687.23	yes	720	100.87	even	20	inner
700.2.bj.a.687.77	yes	720	25.12	odd	20	inner