Embedded newform 859.2.f.a.524.9

• Label: 859.2.f.a.524.9
• Level: $859$
• Weight: $2$
• Character: 859.524
• Analytic conductor: $6.859$
• Analytic rank: $0$
• Dimension: $840$
• 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			524.9
Character	\(\chi\)	\(=\)	859.524
Dual form			859.2.f.a.100.9

$q$-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 \(\left(\mathbb{Z}/859\mathbb{Z}\right)^\times\).

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

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	−0.452525	−	0.891752i	\(-0.649477\pi\)
							−0.990961	+	0.134152i	\(0.957169\pi\)
\(3\)	−1.39826	−	2.02573i	−0.807288	−	1.16956i	−0.982726	−	0.185068i	\(-0.940750\pi\)
							0.175438	−	0.984490i	\(-0.443866\pi\)
\(5\)	−0.257120	+	0.372503i	−0.114988	+	0.166589i	−0.876252	−	0.481854i	\(-0.839964\pi\)
							0.761264	+	0.648442i	\(0.224579\pi\)
\(7\)	0.629477	−	5.18421i	0.237920	−	1.95945i	−0.0410839	−	0.999156i	\(-0.513081\pi\)
							0.279004	−	0.960290i	\(-0.409996\pi\)
\(11\)	−0.439291	+	3.61789i	−0.132451	+	1.09083i	0.763227	+	0.646131i	\(0.223614\pi\)
							−0.895678	+	0.444703i	\(0.853309\pi\)
\(13\)	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	−0.135649	−	0.990757i	\(-0.543312\pi\)
							−0.580539	+	0.814233i	\(0.697158\pi\)
\(19\)	−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	−0.635001	−	0.772511i	\(-0.719000\pi\)
							0.275042	+	0.961432i	\(0.411308\pi\)
\(29\)	1.17449	+	3.09688i	0.218098	+	0.575077i	0.998755	−	0.0498827i	\(-0.0158848\pi\)
							−0.780657	+	0.624959i	\(0.785116\pi\)
\(31\)	−5.93320	+	5.25635i	−1.06563	+	0.944069i	−0.998558	−	0.0536789i	\(-0.982905\pi\)
							−0.0670754	+	0.997748i	\(0.521367\pi\)
\(37\)	0.341644	−	0.900840i	0.0561659	−	0.148097i	−0.903997	−	0.427540i	\(-0.859380\pi\)
							0.960162	+	0.279442i	\(0.0901496\pi\)
\(41\)	5.63032	+	8.15693i	0.879308	+	1.27390i	0.960805	+	0.277224i	\(0.0894145\pi\)
							−0.0814969	+	0.996674i	\(0.525970\pi\)
\(43\)	−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	0.871360	−	0.490643i	\(-0.163238\pi\)
							−0.382035	+	0.924148i	\(0.624777\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.9	✓	840	859.100	even	13	inner
859.2.f.a.524.9	yes	840	1.1	even	1	trivial