Embedded newform 700.2.bj.a.323.24

• Label: 700.2.bj.a.323.24
• 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			323.24
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.989390 - 1.01050i) q^{2} +(-0.658566 - 1.29251i) q^{3} +(-0.0422154 + 1.99955i) q^{4} +(-1.06784 - 1.96462i) q^{5} +(-0.654500 + 1.94428i) q^{6} +(0.707107 + 0.707107i) q^{7} +(2.06231 - 1.93568i) q^{8} +(0.526485 - 0.724644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.989390	−	1.01050i	−0.699604	−	0.714531i
							\(3\)	−0.658566	−	1.29251i	−0.380224	−	0.746231i	0.619010	−	0.785383i	\(-0.287534\pi\)
−0.999233	+	0.0391526i	\(0.987534\pi\)
\(5\)	−1.06784	−	1.96462i	−0.477552	−	0.878603i
							\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
\(11\)	3.77391	+	5.19434i	1.13788	+	1.56615i								0.772163	+	0.635425i	\(0.219175\pi\)
							0.365713	+	0.930728i	\(0.380825\pi\)
\(13\)	0.559523	+	3.53269i	0.155184	+	0.979792i	0.935223	+	0.354060i	\(0.115199\pi\)
							−0.780039	+	0.625731i	\(0.784801\pi\)
\(17\)	4.07228	+	2.07493i	0.987674	+	0.503245i	0.871717	−	0.490009i	\(-0.163007\pi\)
							0.115957	+	0.993254i	\(0.463007\pi\)
\(19\)	1.70534	+	5.24849i	0.391232	+	1.20409i	0.931858	+	0.362824i	\(0.118187\pi\)
							−0.540626	+	0.841263i	\(0.681813\pi\)
\(23\)	0.748491	−	4.72579i	0.156071	−	0.985395i	−0.777989	−	0.628278i	\(-0.783760\pi\)
							0.934060	−	0.357116i	\(-0.116240\pi\)
\(29\)	0.578283	+	0.187896i	0.107385	+	0.0348913i	0.362216	−	0.932094i	\(-0.382020\pi\)
							−0.254832	+	0.966985i	\(0.582020\pi\)
\(31\)	−4.60312	+	1.49565i	−0.826745	+	0.268626i	−0.691674	−	0.722210i	\(-0.743126\pi\)
							−0.135071	+	0.990836i	\(0.543126\pi\)
\(37\)	5.07091	−	0.803154i	0.833653	−	0.132038i	0.275005	−	0.961443i	\(-0.411320\pi\)
							0.558648	+	0.829405i	\(0.311320\pi\)
\(41\)	6.30602	+	4.58159i	0.984834	+	0.715524i	0.958784	−	0.284137i	\(-0.0917070\pi\)
							0.0260504	+	0.999661i	\(0.491707\pi\)
\(43\)	8.66104	−	8.66104i	1.32080	−	1.32080i	0.407663	−	0.913132i	\(-0.366344\pi\)
							0.913132	−	0.407663i	\(-0.133656\pi\)
\(47\)	−6.22899	+	3.17383i	−0.908592	+	0.462951i	−0.844842	−	0.535015i	\(-0.820306\pi\)
							−0.0637495	+	0.997966i	\(0.520306\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.24	✓	720
4.3	odd	2	inner	700.2.bj.a.323.76	yes	720
25.12	odd	20	inner	700.2.bj.a.687.76	yes	720
100.87	even	20	inner	700.2.bj.a.687.24	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.24	✓	720	1.1	even	1	trivial
700.2.bj.a.323.76	yes	720	4.3	odd	2	inner
700.2.bj.a.687.24	yes	720	100.87	even	20	inner
700.2.bj.a.687.76	yes	720	25.12	odd	20	inner