Embedded newform 700.2.bj.a.323.10

• Label: 700.2.bj.a.323.10
• 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			323.10
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34215 + 0.445689i) q^{2} +(0.601550 + 1.18061i) q^{3} +(1.60272 - 1.19636i) q^{4} +(-2.23175 - 0.138830i) q^{5} +(-1.33355 - 1.31645i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(-1.61789 + 2.32001i) q^{8} +(0.731381 - 1.00666i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34215	+	0.445689i	−0.949042	+	0.315150i
							\(3\)	0.601550	+	1.18061i	0.347305	+	0.681625i	0.996902	−	0.0786541i	\(-0.0250623\pi\)
−0.649597	+	0.760279i	\(0.725062\pi\)
\(5\)	−2.23175	−	0.138830i	−0.998071	−	0.0620868i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	−1.48866	−	2.04896i	−0.448847	−	0.617785i								0.523302	−	0.852147i	\(-0.324700\pi\)
							−0.972149	+	0.234362i	\(0.924700\pi\)
\(13\)	0.469492	+	2.96425i	0.130214	+	0.822136i	0.963189	+	0.268826i	\(0.0866357\pi\)
							−0.832975	+	0.553310i	\(0.813364\pi\)
\(17\)	3.39902	+	1.73189i	0.824384	+	0.420045i	0.814682	−	0.579908i	\(-0.196912\pi\)
							0.00970227	+	0.999953i	\(0.496912\pi\)
\(19\)	−0.631085	−	1.94228i	−0.144781	−	0.445589i	0.852202	−	0.523213i	\(-0.175267\pi\)
							−0.996983	+	0.0776236i	\(0.975267\pi\)
\(23\)	0.787025	−	4.96908i	0.164106	−	1.03613i	−0.758863	−	0.651251i	\(-0.774245\pi\)
							0.922969	−	0.384875i	\(-0.125755\pi\)
\(29\)	7.44062	+	2.41760i	1.38169	+	0.448938i	0.903224	−	0.429169i	\(-0.141194\pi\)
							0.478464	+	0.878107i	\(0.341194\pi\)
\(31\)	2.98663	−	0.970414i	0.536414	−	0.174291i	−0.0282676	−	0.999600i	\(-0.508999\pi\)
							0.564681	+	0.825309i	\(0.308999\pi\)
\(37\)	6.27964	−	0.994597i	1.03237	−	0.163511i	0.382812	−	0.923826i	\(-0.374956\pi\)
							0.649554	+	0.760315i	\(0.274956\pi\)
\(41\)	−7.63890	−	5.54999i	−1.19300	−	0.866762i	−0.199418	−	0.979914i	\(-0.563905\pi\)
							−0.993578	+	0.113152i	\(0.963905\pi\)
\(43\)	8.51143	−	8.51143i	1.29798	−	1.29798i	0.368257	−	0.929724i	\(-0.379955\pi\)
							0.929724	−	0.368257i	\(-0.120045\pi\)
\(47\)	−11.4326	+	5.82519i	−1.66761	+	0.849692i	−0.673766	+	0.738945i	\(0.735324\pi\)
							−0.993849	+	0.110747i	\(0.964676\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.10	✓	720
4.3	odd	2	inner	700.2.bj.a.323.72	yes	720
25.12	odd	20	inner	700.2.bj.a.687.72	yes	720
100.87	even	20	inner	700.2.bj.a.687.10	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.10	✓	720	1.1	even	1	trivial
700.2.bj.a.323.72	yes	720	4.3	odd	2	inner
700.2.bj.a.687.10	yes	720	100.87	even	20	inner
700.2.bj.a.687.72	yes	720	25.12	odd	20	inner