Embedded newform 859.2.f.a.524.13

• Label: 859.2.f.a.524.13
• 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			524.13
Character	\(\chi\)	\(=\)	859.524
Dual form			859.2.f.a.100.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.74103 - 0.913764i) q^{2} +(0.895484 + 1.29733i) q^{3} +(1.06010 + 1.53582i) q^{4} +(1.65604 - 2.39919i) q^{5} +(-0.373610 - 3.07696i) q^{6} +(0.380352 - 3.13248i) q^{7} +(0.0317215 + 0.261250i) q^{8} +(0.182634 - 0.481567i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.74103	−	0.913764i	−1.23110	−	0.646129i	−0.281486	−	0.959565i	\(-0.590827\pi\)
							−0.949609	+	0.313436i	\(0.898520\pi\)
\(3\)	0.895484	+	1.29733i	0.517008	+	0.749015i	0.991339	−	0.131331i	\(-0.0419252\pi\)
							−0.474331	+	0.880347i	\(0.657310\pi\)
\(5\)	1.65604	−	2.39919i	0.740605	−	1.07295i	−0.253794	−	0.967258i	\(-0.581679\pi\)
							0.994399	−	0.105693i	\(-0.0337060\pi\)
\(7\)	0.380352	−	3.13248i	0.143759	−	1.18397i	−0.725006	−	0.688742i	\(-0.758163\pi\)
							0.868766	−	0.495223i	\(-0.164914\pi\)
\(11\)	0.704727	−	5.80395i	0.212483	−	1.74996i	−0.362675	−	0.931916i	\(-0.618137\pi\)
							0.575159	−	0.818042i	\(-0.304940\pi\)
\(13\)	4.81075			1.33426			0.667131	−	0.744941i	\(-0.267522\pi\)
							0.667131	+	0.744941i	\(0.267522\pi\)
\(17\)	2.31149	+	0.569732i	0.560619	+	0.138180i	0.509435	−	0.860509i	\(-0.329854\pi\)
							0.0511839	+	0.998689i	\(0.483701\pi\)
\(19\)	−7.47027			−1.71380			−0.856899	−	0.515485i	\(-0.827612\pi\)
							−0.856899	+	0.515485i	\(0.827612\pi\)
\(23\)	−7.71644	+	4.04990i	−1.60899	+	0.844462i	−0.610359	+	0.792125i	\(0.708975\pi\)
							−0.998629	+	0.0523379i	\(0.983333\pi\)
\(29\)	−0.868595	−	2.29030i	−0.161294	−	0.425297i	0.829867	−	0.557961i	\(-0.188416\pi\)
							−0.991161	+	0.132664i	\(0.957647\pi\)
\(31\)	−4.83502	+	4.28345i	−0.868395	+	0.769331i	−0.974350	−	0.225040i	\(-0.927749\pi\)
							0.105954	+	0.994371i	\(0.466210\pi\)
\(37\)	−3.69093	+	9.73218i	−0.606785	+	1.59996i	0.180640	+	0.983549i	\(0.442183\pi\)
							−0.787425	+	0.616411i	\(0.788586\pi\)
\(41\)	0.0166531	+	0.0241262i	0.00260078	+	0.00376788i	0.824282	−	0.566179i	\(-0.191579\pi\)
							−0.821681	+	0.569947i	\(0.806964\pi\)
\(43\)	4.44260			0.677490			0.338745	−	0.940878i	\(-0.389998\pi\)
							0.338745	+	0.940878i	\(0.389998\pi\)
\(47\)	−2.05614	−	1.82158i	−0.299919	−	0.265705i	0.499719	−	0.866188i	\(-0.333437\pi\)
							−0.799638	+	0.600483i	\(0.794975\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	859.2.f.a.524.13	yes	840
859.100	even	13	inner	859.2.f.a.100.13	✓	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.13	✓	840	859.100	even	13	inner
859.2.f.a.524.13	yes	840	1.1	even	1	trivial