Embedded newform 859.2.f.a.100.19

• Label: 859.2.f.a.100.19
• 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			100.19
Character	\(\chi\)	\(=\)	859.100
Dual form			859.2.f.a.524.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40677 + 0.738332i) q^{2} +(-0.456049 + 0.660700i) q^{3} +(0.297748 - 0.431363i) q^{4} +(2.02811 + 2.93823i) q^{5} +(0.153741 - 1.26617i) q^{6} +(-0.0376523 - 0.310095i) q^{7} +(0.282631 - 2.32768i) q^{8} +(0.835270 + 2.20243i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40677	+	0.738332i	−0.994739	+	0.522079i	−0.881851	−	0.471528i	\(-0.843702\pi\)
							−0.112888	+	0.993608i	\(0.536010\pi\)
\(3\)	−0.456049	+	0.660700i	−0.263300	+	0.381456i	−0.932228	−	0.361871i	\(-0.882138\pi\)
							0.668928	+	0.743327i	\(0.266753\pi\)
\(5\)	2.02811	+	2.93823i	0.906999	+	1.31401i	0.949124	+	0.314902i	\(0.101972\pi\)
							−0.0421254	+	0.999112i	\(0.513413\pi\)
\(7\)	−0.0376523	−	0.310095i	−0.0142312	−	0.117205i	0.983862	−	0.178929i	\(-0.0572632\pi\)
							−0.998093	+	0.0617241i	\(0.980340\pi\)
\(11\)	−0.307228	−	2.53025i	−0.0926327	−	0.762899i	−0.962814	−	0.270166i	\(-0.912921\pi\)
							0.870181	−	0.492732i	\(-0.164002\pi\)
\(13\)	−5.98525			−1.66001			−0.830005	−	0.557755i	\(-0.811663\pi\)
							−0.830005	+	0.557755i	\(0.811663\pi\)
\(17\)	0.868637	−	0.214100i	0.210675	−	0.0519268i	−0.132565	−	0.991174i	\(-0.542321\pi\)
							0.343241	+	0.939247i	\(0.388475\pi\)
\(19\)	−2.02631			−0.464867			−0.232434	−	0.972612i	\(-0.574669\pi\)
							−0.232434	+	0.972612i	\(0.574669\pi\)
\(23\)	−3.67416	−	1.92835i	−0.766115	−	0.402088i	0.0359279	−	0.999354i	\(-0.488561\pi\)
							−0.802043	+	0.597266i	\(0.796254\pi\)
\(29\)	−2.51934	+	6.64295i	−0.467830	+	1.23357i	0.468819	+	0.883294i	\(0.344680\pi\)
							−0.936648	+	0.350271i	\(0.886089\pi\)
\(31\)	−4.92462	−	4.36283i	−0.884487	−	0.783587i	0.0927606	−	0.995688i	\(-0.470431\pi\)
							−0.977248	+	0.212101i	\(0.931969\pi\)
\(37\)	4.13528	+	10.9038i	0.679835	+	1.79258i	0.606228	+	0.795291i	\(0.292682\pi\)
							0.0736077	+	0.997287i	\(0.476549\pi\)
\(41\)	5.75493	−	8.33746i	0.898769	−	1.30209i	−0.0540454	−	0.998538i	\(-0.517212\pi\)
							0.952814	−	0.303553i	\(-0.0981730\pi\)
\(43\)	0.477566			0.0728282			0.0364141	−	0.999337i	\(-0.488406\pi\)
							0.0364141	+	0.999337i	\(0.488406\pi\)
\(47\)	−8.16332	+	7.23207i	−1.19074	+	1.05491i	−0.193470	+	0.981106i	\(0.561974\pi\)
							−0.997273	+	0.0738003i	\(0.976487\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	859.2.f.a.100.19	✓	840
859.524	even	13	inner	859.2.f.a.524.19	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.19	✓	840	1.1	even	1	trivial
859.2.f.a.524.19	yes	840	859.524	even	13	inner