Embedded newform 859.2.f.a.100.9

• Label: 859.2.f.a.100.9
• Level: $859$
• Weight: $2$
• Character: 859.100
• Analytic conductor: $6.859$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 859 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	859.f (of order \(13\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.85914953363\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(70\) over \(\Q(\zeta_{13})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{13}]$

Embedding invariants

Embedding label			100.9
Character	\(\chi\)	\(=\)	859.100
Dual form			859.2.f.a.524.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.04140 + 1.07141i) q^{2} +(-1.39826 + 2.02573i) q^{3} +(1.88326 - 2.72837i) q^{4} +(-0.257120 - 0.372503i) q^{5} +(0.684023 - 5.63344i) q^{6} +(0.629477 + 5.18421i) q^{7} +(-0.365492 + 3.01010i) q^{8} +(-1.08464 - 2.85997i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/859\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{13}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.04140	+	1.07141i	−1.44349	+	0.757600i	−0.990961	−	0.134152i	\(-0.957169\pi\)
							−0.452525	+	0.891752i	\(0.649477\pi\)
\(3\)	−1.39826	+	2.02573i	−0.807288	+	1.16956i	0.175438	+	0.984490i	\(0.443866\pi\)
							−0.982726	+	0.185068i	\(0.940750\pi\)
\(5\)	−0.257120	−	0.372503i	−0.114988	−	0.166589i	0.761264	−	0.648442i	\(-0.224579\pi\)
							−0.876252	+	0.481854i	\(0.839964\pi\)
\(7\)	0.629477	+	5.18421i	0.237920	+	1.95945i	0.279004	+	0.960290i	\(0.409996\pi\)
							−0.0410839	+	0.999156i	\(0.513081\pi\)
\(11\)	−0.439291	−	3.61789i	−0.132451	−	1.09083i	−0.895678	−	0.444703i	\(-0.853309\pi\)
							0.763227	−	0.646131i	\(-0.223614\pi\)
\(13\)	3.59136			0.996063			0.498031	−	0.867159i	\(-0.334056\pi\)
							0.498031	+	0.867159i	\(0.334056\pi\)
\(17\)	−2.95292	+	0.727828i	−0.716187	+	0.176524i	−0.580539	−	0.814233i	\(-0.697158\pi\)
							−0.135649	+	0.990757i	\(0.543312\pi\)
\(19\)	−4.52470			−1.03804			−0.519018	−	0.854763i	\(-0.673702\pi\)
							−0.519018	+	0.854763i	\(0.673702\pi\)
\(23\)	−1.72630	−	0.906034i	−0.359959	−	0.188921i	0.275042	−	0.961432i	\(-0.411308\pi\)
							−0.635001	+	0.772511i	\(0.719000\pi\)
\(29\)	1.17449	−	3.09688i	0.218098	−	0.575077i	−0.780657	−	0.624959i	\(-0.785116\pi\)
							0.998755	+	0.0498827i	\(0.0158848\pi\)
\(31\)	−5.93320	−	5.25635i	−1.06563	−	0.944069i	−0.0670754	−	0.997748i	\(-0.521367\pi\)
							−0.998558	+	0.0536789i	\(0.982905\pi\)
\(37\)	0.341644	+	0.900840i	0.0561659	+	0.148097i	0.960162	−	0.279442i	\(-0.0901496\pi\)
							−0.903997	+	0.427540i	\(0.859380\pi\)
\(41\)	5.63032	−	8.15693i	0.879308	−	1.27390i	−0.0814969	−	0.996674i	\(-0.525970\pi\)
							0.960805	−	0.277224i	\(-0.0894145\pi\)
\(43\)	−11.3493			−1.73075			−0.865376	−	0.501124i	\(-0.832920\pi\)
							−0.865376	+	0.501124i	\(0.832920\pi\)
\(47\)	3.35465	−	2.97196i	0.489325	−	0.433504i	−0.382035	−	0.924148i	\(-0.624777\pi\)
							0.871360	+	0.490643i	\(0.163238\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	859.2.f.a.100.9	✓	840
859.524	even	13	inner	859.2.f.a.524.9	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.9	✓	840	1.1	even	1	trivial
859.2.f.a.524.9	yes	840	859.524	even	13	inner