Embedded newform 859.2.f.a.100.14

• Label: 859.2.f.a.100.14
• Level: $859$
• Weight: $2$
• Character: 859.100
• 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			100.14
Character	\(\chi\)	\(=\)	859.100
Dual form			859.2.f.a.524.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.70030 + 0.892384i) q^{2} +(1.83435 - 2.65752i) q^{3} +(0.958526 - 1.38866i) q^{4} +(-0.866259 - 1.25499i) q^{5} +(-0.747414 + 6.15551i) q^{6} +(-0.526271 - 4.33423i) q^{7} +(0.0723652 - 0.595981i) q^{8} +(-2.63374 - 6.94460i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.70030	+	0.892384i	−1.20229	+	0.631011i	−0.942355	−	0.334615i	\(-0.891394\pi\)
							−0.259936	+	0.965626i	\(0.583701\pi\)
\(3\)	1.83435	−	2.65752i	1.05906	−	1.53432i	0.231590	−	0.972813i	\(-0.425607\pi\)
							0.827473	−	0.561505i	\(-0.189777\pi\)
\(5\)	−0.866259	−	1.25499i	−0.387403	−	0.561250i	0.579960	−	0.814645i	\(-0.303068\pi\)
							−0.967363	+	0.253395i	\(0.918453\pi\)
\(7\)	−0.526271	−	4.33423i	−0.198912	−	1.63819i	−0.660310	−	0.750993i	\(-0.729575\pi\)
							0.461399	−	0.887193i	\(-0.347348\pi\)
\(11\)	−0.0122990	−	0.101291i	−0.00370828	−	0.0305405i	0.990737	−	0.135793i	\(-0.0433581\pi\)
							−0.994446	+	0.105252i	\(0.966435\pi\)
\(13\)	6.73620			1.86829			0.934143	−	0.356900i	\(-0.116166\pi\)
							0.934143	+	0.356900i	\(0.116166\pi\)
\(17\)	2.82794	−	0.697025i	0.685877	−	0.169053i	0.119047	−	0.992889i	\(-0.462016\pi\)
							0.566829	+	0.823835i	\(0.308170\pi\)
\(19\)	−1.46845			−0.336885			−0.168443	−	0.985711i	\(-0.553874\pi\)
							−0.168443	+	0.985711i	\(0.553874\pi\)
\(23\)	1.96821	+	1.03300i	0.410401	+	0.215395i	0.657272	−	0.753653i	\(-0.271710\pi\)
							−0.246871	+	0.969048i	\(0.579402\pi\)
\(29\)	0.570934	−	1.50543i	0.106020	−	0.279551i	−0.871500	−	0.490396i	\(-0.836852\pi\)
							0.977520	+	0.210845i	\(0.0676214\pi\)
\(31\)	0.940788	+	0.833466i	0.168971	+	0.149695i	0.743370	−	0.668880i	\(-0.233226\pi\)
							−0.574400	+	0.818575i	\(0.694764\pi\)
\(37\)	1.68839	+	4.45191i	0.277569	+	0.731889i	0.999215	+	0.0396265i	\(0.0126168\pi\)
							−0.721646	+	0.692263i	\(0.756614\pi\)
\(41\)	−4.66197	+	6.75403i	−0.728078	+	1.05480i	0.267744	+	0.963490i	\(0.413722\pi\)
							−0.995822	+	0.0913128i	\(0.970894\pi\)
\(43\)	12.0086			1.83129			0.915645	−	0.401987i	\(-0.131680\pi\)
							0.915645	+	0.401987i	\(0.131680\pi\)
\(47\)	−4.48331	+	3.97187i	−0.653958	+	0.579356i	−0.923484	−	0.383637i	\(-0.874672\pi\)
							0.269526	+	0.962993i	\(0.413133\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	859.2.f.a.100.14	✓	840
859.524	even	13	inner	859.2.f.a.524.14	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.14	✓	840	1.1	even	1	trivial
859.2.f.a.524.14	yes	840	859.524	even	13	inner