Embedded newform 700.2.bj.a.323.13

• Label: 700.2.bj.a.323.13
• 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			323.13
Character	\(\chi\)	\(=\)	700.323
Dual form			700.2.bj.a.687.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29418 - 0.570165i) q^{2} +(0.886576 + 1.74000i) q^{3} +(1.34982 + 1.47580i) q^{4} +(1.52028 - 1.63974i) q^{5} +(-0.155304 - 2.75738i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(-0.905473 - 2.67957i) q^{8} +(-0.478237 + 0.658237i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 720 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29418	−	0.570165i	−0.915126	−	0.403167i
							\(3\)	0.886576	+	1.74000i	0.511865	+	1.00459i	0.991862	+	0.127321i	\(0.0406378\pi\)
−0.479997	+	0.877270i	\(0.659362\pi\)
\(5\)	1.52028	−	1.63974i	0.679888	−	0.733316i
							\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
\(11\)	2.71014	+	3.73019i	0.817138	+	1.12469i								0.990183	+	0.139780i	\(0.0446396\pi\)
							−0.173044	+	0.984914i	\(0.555360\pi\)
\(13\)	0.637663	+	4.02605i	0.176856	+	1.11662i	0.903177	+	0.429269i	\(0.141229\pi\)
							−0.726321	+	0.687356i	\(0.758771\pi\)
\(17\)	0.876257	+	0.446475i	0.212524	+	0.108286i	0.557012	−	0.830504i	\(-0.311948\pi\)
							−0.344489	+	0.938790i	\(0.611948\pi\)
\(19\)	−1.78607	−	5.49697i	−0.409753	−	1.26109i	−0.916860	−	0.399208i	\(-0.869285\pi\)
							0.507107	−	0.861883i	\(-0.330715\pi\)
\(23\)	−0.419549	+	2.64893i	−0.0874821	+	0.552340i	0.904551	+	0.426365i	\(0.140206\pi\)
							−0.992033	+	0.125975i	\(0.959794\pi\)
\(29\)	−0.643831	−	0.209193i	−0.119556	−	0.0388462i	0.248628	−	0.968599i	\(-0.420020\pi\)
							−0.368184	+	0.929753i	\(0.620020\pi\)
\(31\)	8.63031	−	2.80416i	1.55005	−	0.503642i	0.595922	−	0.803042i	\(-0.296787\pi\)
							0.954127	+	0.299401i	\(0.0967868\pi\)
\(37\)	−4.07581	+	0.645545i	−0.670059	+	0.106127i	−0.482189	−	0.876067i	\(-0.660158\pi\)
							−0.187870	+	0.982194i	\(0.560158\pi\)
\(41\)	8.62041	+	6.26310i	1.34628	+	0.978131i	0.999188	+	0.0403010i	\(0.0128317\pi\)
							0.347094	+	0.937830i	\(0.387168\pi\)
\(43\)	3.25263	−	3.25263i	0.496021	−	0.496021i	−0.414176	−	0.910197i	\(-0.635930\pi\)
							0.910197	+	0.414176i	\(0.135930\pi\)
\(47\)	−9.81968	+	5.00338i	−1.43235	+	0.729818i	−0.986264	−	0.165174i	\(-0.947182\pi\)
							−0.446083	+	0.894991i	\(0.647182\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.2.bj.a.323.13	✓	720
4.3	odd	2	inner	700.2.bj.a.323.86	yes	720
25.12	odd	20	inner	700.2.bj.a.687.86	yes	720
100.87	even	20	inner	700.2.bj.a.687.13	yes	720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.bj.a.323.13	✓	720	1.1	even	1	trivial
700.2.bj.a.323.86	yes	720	4.3	odd	2	inner
700.2.bj.a.687.13	yes	720	100.87	even	20	inner
700.2.bj.a.687.86	yes	720	25.12	odd	20	inner