Embedded newform 700.2.bj.a.323.18

• Label: 700.2.bj.a.323.18
• 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			323.18
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.12620 - 0.855386i) q^{2} +(0.416111 + 0.816663i) q^{3} +(0.536631 + 1.92666i) q^{4} +(-1.86169 + 1.23859i) q^{5} +(0.229940 - 1.27566i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(1.04369 - 2.62882i) q^{8} +(1.26957 - 1.74741i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.12620	−	0.855386i	−0.796340	−	0.604849i
							\(3\)	0.416111	+	0.816663i	0.240242	+	0.471501i	0.979373	−	0.202061i	\(-0.0647638\pi\)
−0.739131	+	0.673561i	\(0.764764\pi\)
\(5\)	−1.86169	+	1.23859i	−0.832574	+	0.553914i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	0.567397	+	0.780956i	0.171077	+	0.235467i								0.885943	−	0.463794i	\(-0.153512\pi\)
							−0.714866	+	0.699261i	\(0.753512\pi\)
\(13\)	−0.905020	−	5.71407i	−0.251007	−	1.58480i	−0.715107	−	0.699015i	\(-0.753622\pi\)
							0.464099	−	0.885783i	\(-0.346378\pi\)
\(17\)	3.01641	+	1.53694i	0.731587	+	0.372762i	0.779761	−	0.626077i	\(-0.215340\pi\)
							−0.0481745	+	0.998839i	\(0.515340\pi\)
\(19\)	0.700863	+	2.15703i	0.160789	+	0.494858i	0.998701	−	0.0509472i	\(-0.0162240\pi\)
							−0.837912	+	0.545805i	\(0.816224\pi\)
\(23\)	−0.00245706	+	0.0155133i	−0.000512333	+	0.00323474i	−0.987943	−	0.154817i	\(-0.950521\pi\)
							0.987431	+	0.158052i	\(0.0505212\pi\)
\(29\)	3.99424	+	1.29781i	0.741712	+	0.240997i	0.655411	−	0.755272i	\(-0.272495\pi\)
							0.0863009	+	0.996269i	\(0.472495\pi\)
\(31\)	1.33869	−	0.434966i	0.240435	−	0.0781222i	−0.186321	−	0.982489i	\(-0.559656\pi\)
							0.426756	+	0.904367i	\(0.359656\pi\)
\(37\)	3.05320	−	0.483580i	0.501944	−	0.0795001i	0.0996719	−	0.995020i	\(-0.468221\pi\)
							0.402272	+	0.915520i	\(0.368221\pi\)
\(41\)	7.08323	+	5.14627i	1.10621	+	0.803712i	0.982063	−	0.188552i	\(-0.0603793\pi\)
							0.124151	+	0.992263i	\(0.460379\pi\)
\(43\)	−0.460575	+	0.460575i	−0.0702370	+	0.0702370i	−0.741353	−	0.671116i	\(-0.765815\pi\)
							0.671116	+	0.741353i	\(0.265815\pi\)
\(47\)	8.21191	−	4.18418i	1.19783	−	0.610325i	0.262786	−	0.964854i	\(-0.415359\pi\)
							0.935045	+	0.354529i	\(0.115359\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.18	✓	720
4.3	odd	2	inner	700.2.bj.a.323.82	yes	720
25.12	odd	20	inner	700.2.bj.a.687.82	yes	720
100.87	even	20	inner	700.2.bj.a.687.18	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.18	✓	720	1.1	even	1	trivial
700.2.bj.a.323.82	yes	720	4.3	odd	2	inner
700.2.bj.a.687.18	yes	720	100.87	even	20	inner
700.2.bj.a.687.82	yes	720	25.12	odd	20	inner