Embedded newform 700.2.bj.a.323.40

• Label: 700.2.bj.a.323.40
• 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			323.40
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.306420 - 1.38062i) q^{2} +(-0.713396 - 1.40012i) q^{3} +(-1.81221 + 0.846097i) q^{4} +(-1.23504 + 1.86405i) q^{5} +(-1.71443 + 1.41395i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(1.72344 + 2.24271i) q^{8} +(0.311958 - 0.429374i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.306420	−	1.38062i	−0.216671	−	0.976245i
							\(3\)	−0.713396	−	1.40012i	−0.411879	−	0.808359i	0.588121	−	0.808773i	\(-0.299868\pi\)
−1.00000		0.000414566i	\(0.999868\pi\)
\(5\)	−1.23504	+	1.86405i	−0.552329	+	0.833626i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	−0.0883436	−	0.121595i	−0.0266366	−	0.0366621i								0.795491	−	0.605965i	\(-0.207213\pi\)
							−0.822128	+	0.569303i	\(0.807213\pi\)
\(13\)	0.899241	+	5.67759i	0.249405	+	1.57468i	0.721045	+	0.692888i	\(0.243662\pi\)
							−0.471640	+	0.881791i	\(0.656338\pi\)
\(17\)	4.70413	+	2.39688i	1.14092	+	0.581328i	0.919203	−	0.393783i	\(-0.128834\pi\)
							0.221717	+	0.975111i	\(0.428834\pi\)
\(19\)	−0.209420	−	0.644530i	−0.0480443	−	0.147865i	0.924156	−	0.382015i	\(-0.124770\pi\)
							−0.972201	+	0.234149i	\(0.924770\pi\)
\(23\)	0.420176	−	2.65289i	0.0876129	−	0.553166i	−0.904366	−	0.426759i	\(-0.859655\pi\)
							0.991978	−	0.126407i	\(-0.0403446\pi\)
\(29\)	0.763961	+	0.248226i	0.141864	+	0.0460944i	0.379088	−	0.925361i	\(-0.376238\pi\)
							−0.237224	+	0.971455i	\(0.576238\pi\)
\(31\)	6.71778	−	2.18274i	1.20655	−	0.392032i	0.364383	−	0.931249i	\(-0.381280\pi\)
							0.842167	+	0.539217i	\(0.181280\pi\)
\(37\)	3.42562	−	0.542565i	0.563169	−	0.0891972i	0.131642	−	0.991297i	\(-0.457975\pi\)
							0.431527	+	0.902100i	\(0.357975\pi\)
\(41\)	4.00323	+	2.90851i	0.625199	+	0.454233i	0.854734	−	0.519067i	\(-0.173720\pi\)
							−0.229535	+	0.973300i	\(0.573720\pi\)
\(43\)	−3.01693	+	3.01693i	−0.460078	+	0.460078i	−0.898681	−	0.438603i	\(-0.855473\pi\)
							0.438603	+	0.898681i	\(0.355473\pi\)
\(47\)	5.73075	−	2.91996i	0.835916	−	0.425920i	0.0170148	−	0.999855i	\(-0.494584\pi\)
							0.818901	+	0.573935i	\(0.194584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.40	✓	720
4.3	odd	2	inner	700.2.bj.a.323.60	yes	720
25.12	odd	20	inner	700.2.bj.a.687.60	yes	720
100.87	even	20	inner	700.2.bj.a.687.40	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.40	✓	720	1.1	even	1	trivial
700.2.bj.a.323.60	yes	720	4.3	odd	2	inner
700.2.bj.a.687.40	yes	720	100.87	even	20	inner
700.2.bj.a.687.60	yes	720	25.12	odd	20	inner