Embedded newform 859.2.f.a.100.20

• Label: 859.2.f.a.100.20
• 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			100.20
Character	\(\chi\)	\(=\)	859.100
Dual form			859.2.f.a.524.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37011 + 0.719088i) q^{2} +(-0.554289 + 0.803027i) q^{3} +(0.223979 - 0.324490i) q^{4} +(0.478083 + 0.692622i) q^{5} +(0.181989 - 1.49882i) q^{6} +(0.117665 + 0.969059i) q^{7} +(0.299485 - 2.46648i) q^{8} +(0.726199 + 1.91483i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37011	+	0.719088i	−0.968813	+	0.508472i	−0.873420	−	0.486968i	\(-0.838103\pi\)
							−0.0953927	+	0.995440i	\(0.530411\pi\)
\(3\)	−0.554289	+	0.803027i	−0.320019	+	0.463628i	−0.949671	−	0.313250i	\(-0.898582\pi\)
							0.629652	+	0.776878i	\(0.283198\pi\)
\(5\)	0.478083	+	0.692622i	0.213805	+	0.309750i	0.915182	−	0.403040i	\(-0.132047\pi\)
							−0.701377	+	0.712790i	\(0.747431\pi\)
\(7\)	0.117665	+	0.969059i	0.0444732	+	0.366270i	0.997907	+	0.0646672i	\(0.0205986\pi\)
							−0.953434	+	0.301603i	\(0.902478\pi\)
\(11\)	−0.555220	−	4.57265i	−0.167405	−	1.37871i	−0.798030	−	0.602618i	\(-0.794124\pi\)
							0.630625	−	0.776088i	\(-0.282799\pi\)
\(13\)	2.58710			0.717533			0.358766	−	0.933427i	\(-0.383197\pi\)
							0.358766	+	0.933427i	\(0.383197\pi\)
\(17\)	7.12190	−	1.75539i	1.72731	−	0.425745i	0.754307	−	0.656521i	\(-0.227973\pi\)
							0.973007	+	0.230777i	\(0.0741267\pi\)
\(19\)	−0.113591			−0.0260595			−0.0130298	−	0.999915i	\(-0.504148\pi\)
							−0.0130298	+	0.999915i	\(0.504148\pi\)
\(23\)	1.35192	+	0.709543i	0.281895	+	0.147950i	0.599745	−	0.800191i	\(-0.295269\pi\)
							−0.317850	+	0.948141i	\(0.602961\pi\)
\(29\)	2.43340	−	6.41634i	0.451870	−	1.19148i	−0.494722	−	0.869051i	\(-0.664730\pi\)
							0.946592	−	0.322433i	\(-0.104501\pi\)
\(31\)	6.69609	+	5.93221i	1.20265	+	1.06546i	0.996208	+	0.0869988i	\(0.0277276\pi\)
							0.206444	+	0.978458i	\(0.433811\pi\)
\(37\)	−2.90106	−	7.64947i	−0.476932	−	1.25757i	−0.930483	−	0.366335i	\(-0.880612\pi\)
							0.453551	−	0.891230i	\(-0.350157\pi\)
\(41\)	−0.748818	+	1.08485i	−0.116946	+	0.169425i	−0.877085	−	0.480335i	\(-0.840515\pi\)
							0.760139	+	0.649760i	\(0.225131\pi\)
\(43\)	−3.79882			−0.579315			−0.289658	−	0.957130i	\(-0.593541\pi\)
							−0.289658	+	0.957130i	\(0.593541\pi\)
\(47\)	−5.64654	+	5.00240i	−0.823633	+	0.729675i	−0.965554	−	0.260203i	\(-0.916211\pi\)
							0.141921	+	0.989878i	\(0.454672\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	859.2.f.a.100.20	✓	840
859.524	even	13	inner	859.2.f.a.524.20	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.20	✓	840	1.1	even	1	trivial
859.2.f.a.524.20	yes	840	859.524	even	13	inner