Embedded newform 700.2.bj.a.323.28

• Label: 700.2.bj.a.323.28
• 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			323.28
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.899697 + 1.09112i) q^{2} +(0.799472 + 1.56905i) q^{3} +(-0.381091 - 1.96336i) q^{4} +(2.23385 - 0.0995329i) q^{5} +(-2.43131 - 0.539351i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(2.48513 + 1.35061i) q^{8} +(-0.0594155 + 0.0817784i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.899697	+	1.09112i	−0.636182	+	0.771539i
							\(3\)	0.799472	+	1.56905i	0.461576	+	0.905893i	0.998077	+	0.0619935i	\(0.0197458\pi\)
−0.536501	+	0.843900i	\(0.680254\pi\)
\(5\)	2.23385	−	0.0995329i	0.999009	−	0.0445125i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	−0.257725	−	0.354728i	−0.0777070	−	0.106955i								0.768394	−	0.639977i	\(-0.221056\pi\)
							−0.846101	+	0.533022i	\(0.821056\pi\)
\(13\)	−0.0479319	−	0.302630i	−0.0132939	−	0.0839344i	0.980149	−	0.198260i	\(-0.0635292\pi\)
							−0.993443	+	0.114326i	\(0.963529\pi\)
\(17\)	4.64793	+	2.36824i	1.12729	+	0.574382i	0.915253	−	0.402880i	\(-0.131991\pi\)
							0.212036	+	0.977262i	\(0.431991\pi\)
\(19\)	−0.262194	−	0.806950i	−0.0601514	−	0.185127i	0.916466	−	0.400113i	\(-0.131029\pi\)
							−0.976617	+	0.214986i	\(0.931029\pi\)
\(23\)	−0.747959	+	4.72242i	−0.155960	+	0.984694i	0.778246	+	0.627960i	\(0.216110\pi\)
							−0.934206	+	0.356734i	\(0.883890\pi\)
\(29\)	4.75264	+	1.54423i	0.882544	+	0.286756i	0.715013	−	0.699111i	\(-0.246421\pi\)
							0.167531	+	0.985867i	\(0.446421\pi\)
\(31\)	−3.63845	+	1.18220i	−0.653485	+	0.212330i	−0.616950	−	0.787002i	\(-0.711632\pi\)
							−0.0365347	+	0.999332i	\(0.511632\pi\)
\(37\)	4.47023	−	0.708014i	0.734901	−	0.116397i	0.222245	−	0.974991i	\(-0.428662\pi\)
							0.512656	+	0.858594i	\(0.328662\pi\)
\(41\)	2.57793	+	1.87298i	0.402605	+	0.292510i	0.770601	−	0.637318i	\(-0.219956\pi\)
							−0.367996	+	0.929827i	\(0.619956\pi\)
\(43\)	−0.555848	+	0.555848i	−0.0847660	+	0.0847660i	−0.748218	−	0.663452i	\(-0.769091\pi\)
							0.663452	+	0.748218i	\(0.269091\pi\)
\(47\)	4.58810	−	2.33775i	0.669242	−	0.340996i	−0.0861373	−	0.996283i	\(-0.527452\pi\)
							0.755380	+	0.655287i	\(0.227452\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.28	✓	720
4.3	odd	2	inner	700.2.bj.a.323.55	yes	720
25.12	odd	20	inner	700.2.bj.a.687.55	yes	720
100.87	even	20	inner	700.2.bj.a.687.28	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.28	✓	720	1.1	even	1	trivial
700.2.bj.a.323.55	yes	720	4.3	odd	2	inner
700.2.bj.a.687.28	yes	720	100.87	even	20	inner
700.2.bj.a.687.55	yes	720	25.12	odd	20	inner