Embedded newform 700.2.bj.a.323.39

• Label: 700.2.bj.a.323.39
• 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			323.39
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.39

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.315016 + 1.37868i) q^{2} +(1.45856 + 2.86259i) q^{3} +(-1.80153 - 0.868615i) q^{4} +(1.35907 + 1.77565i) q^{5} +(-4.40607 + 1.10913i) q^{6} +(0.707107 + 0.707107i) q^{7} +(1.76506 - 2.21011i) q^{8} +(-4.30364 + 5.92345i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.315016	+	1.37868i	−0.222750	+	0.974876i
							\(3\)	1.45856	+	2.86259i	0.842100	+	1.65271i	0.754256	+	0.656580i	\(0.227998\pi\)
0.0878436	+	0.996134i	\(0.472002\pi\)
\(5\)	1.35907	+	1.77565i	0.607794	+	0.794095i
							\(7\)	0.707107	+	0.707107i	0.267261	+	0.267261i
\(11\)	−2.04399	−	2.81331i	−0.616285	−	0.848244i								0.380791	−	0.924661i	\(-0.375652\pi\)
							−0.997076	+	0.0764173i	\(0.975652\pi\)
\(13\)	0.0115670	+	0.0730309i	0.00320810	+	0.0202551i	0.989240	−	0.146299i	\(-0.0467361\pi\)
							−0.986032	+	0.166554i	\(0.946736\pi\)
\(17\)	−2.40861	−	1.22725i	−0.584173	−	0.297651i	0.136817	−	0.990596i	\(-0.456313\pi\)
							−0.720990	+	0.692945i	\(0.756313\pi\)
\(19\)	0.280546	+	0.863431i	0.0643616	+	0.198085i	0.978066	−	0.208295i	\(-0.0667912\pi\)
							−0.913705	+	0.406379i	\(0.866791\pi\)
\(23\)	0.183083	−	1.15594i	0.0381754	−	0.241030i	−0.961221	−	0.275781i	\(-0.911064\pi\)
							0.999396	+	0.0347507i	\(0.0110637\pi\)
\(29\)	7.67871	+	2.49497i	1.42590	+	0.463303i	0.917472	−	0.397800i	\(-0.130226\pi\)
							0.508429	+	0.861104i	\(0.330226\pi\)
\(31\)	−5.36384	+	1.74282i	−0.963375	+	0.313019i	−0.748138	−	0.663543i	\(-0.769052\pi\)
							−0.215236	+	0.976562i	\(0.569052\pi\)
\(37\)	3.47004	−	0.549601i	0.570472	−	0.0903538i	0.135467	−	0.990782i	\(-0.456747\pi\)
							0.435005	+	0.900428i	\(0.356747\pi\)
\(41\)	8.36936	+	6.08070i	1.30708	+	0.949646i	0.999998	−	0.00203903i	\(-0.000649043\pi\)
							0.307077	+	0.951685i	\(0.400649\pi\)
\(43\)	4.55361	−	4.55361i	0.694419	−	0.694419i	−0.268782	−	0.963201i	\(-0.586621\pi\)
							0.963201	+	0.268782i	\(0.0866212\pi\)
\(47\)	−11.8098	+	6.01737i	−1.72263	+	0.877724i	−0.745093	+	0.666961i	\(0.767595\pi\)
							−0.977537	+	0.210763i	\(0.932405\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.39	✓	720
4.3	odd	2	inner	700.2.bj.a.323.43	yes	720
25.12	odd	20	inner	700.2.bj.a.687.43	yes	720
100.87	even	20	inner	700.2.bj.a.687.39	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.39	✓	720	1.1	even	1	trivial
700.2.bj.a.323.43	yes	720	4.3	odd	2	inner
700.2.bj.a.687.39	yes	720	100.87	even	20	inner
700.2.bj.a.687.43	yes	720	25.12	odd	20	inner