Embedded newform 859.2.f.a.100.4

• Label: 859.2.f.a.100.4
• 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			100.4
Character	\(\chi\)	\(=\)	859.100
Dual form			859.2.f.a.524.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.20441 + 1.15696i) q^{2} +(1.29739 - 1.87959i) q^{3} +(2.38474 - 3.45489i) q^{4} +(-0.179952 - 0.260706i) q^{5} +(-0.685356 + 5.64442i) q^{6} +(0.00469214 + 0.0386433i) q^{7} +(-0.659588 + 5.43220i) q^{8} +(-0.785826 - 2.07205i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.20441	+	1.15696i	−1.55875	+	0.818098i	−0.999937	−	0.0111929i	\(-0.996437\pi\)
							−0.558818	+	0.829291i	\(0.688745\pi\)
\(3\)	1.29739	−	1.87959i	0.749046	−	1.08518i	−0.244265	−	0.969709i	\(-0.578547\pi\)
							0.993311	−	0.115472i	\(-0.0368381\pi\)
\(5\)	−0.179952	−	0.260706i	−0.0804770	−	0.116591i	0.780678	−	0.624933i	\(-0.214874\pi\)
							−0.861155	+	0.508342i	\(0.830258\pi\)
\(7\)	0.00469214	+	0.0386433i	0.00177346	+	0.0146058i	0.993569	−	0.113231i	\(-0.0361199\pi\)
							−0.991795	+	0.127836i	\(0.959197\pi\)
\(11\)	−0.248869	−	2.04962i	−0.0750369	−	0.617985i	−0.980914	−	0.194443i	\(-0.937710\pi\)
							0.905877	−	0.423541i	\(-0.139213\pi\)
\(13\)	−1.98457			−0.550420			−0.275210	−	0.961384i	\(-0.588747\pi\)
							−0.275210	+	0.961384i	\(0.588747\pi\)
\(17\)	−7.44131	+	1.83412i	−1.80478	+	0.444839i	−0.990718	−	0.135935i	\(-0.956596\pi\)
							−0.814065	+	0.580774i	\(0.802750\pi\)
\(19\)	−4.18117			−0.959226			−0.479613	−	0.877480i	\(-0.659223\pi\)
							−0.479613	+	0.877480i	\(0.659223\pi\)
\(23\)	−2.75053	−	1.44359i	−0.573526	−	0.301009i	0.152914	−	0.988239i	\(-0.451134\pi\)
							−0.726440	+	0.687230i	\(0.758826\pi\)
\(29\)	0.646273	−	1.70408i	0.120010	−	0.316440i	−0.861527	−	0.507711i	\(-0.830492\pi\)
							0.981537	+	0.191271i	\(0.0612610\pi\)
\(31\)	0.477445	+	0.422979i	0.0857517	+	0.0759693i	0.704910	−	0.709297i	\(-0.250988\pi\)
							−0.619158	+	0.785266i	\(0.712526\pi\)
\(37\)	−0.417728	−	1.10146i	−0.0686741	−	0.181079i	0.896285	−	0.443478i	\(-0.146256\pi\)
							−0.964959	+	0.262399i	\(0.915486\pi\)
\(41\)	−2.47468	+	3.58519i	−0.386479	+	0.559912i	−0.967143	−	0.254231i	\(-0.918178\pi\)
							0.580664	+	0.814143i	\(0.302793\pi\)
\(43\)	0.328996			0.0501714			0.0250857	−	0.999685i	\(-0.492014\pi\)
							0.0250857	+	0.999685i	\(0.492014\pi\)
\(47\)	−8.82036	+	7.81416i	−1.28658	+	1.13981i	−0.304265	+	0.952587i	\(0.598411\pi\)
							−0.982317	+	0.187225i	\(0.940051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	859.2.f.a.100.4	✓	840
859.524	even	13	inner	859.2.f.a.524.4	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.4	✓	840	1.1	even	1	trivial
859.2.f.a.524.4	yes	840	859.524	even	13	inner