Embedded newform 859.2.f.a.100.6

• Label: 859.2.f.a.100.6
• 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			100.6
Character	\(\chi\)	\(=\)	859.100
Dual form			859.2.f.a.524.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.15192 + 1.12941i) q^{2} +(-0.589027 + 0.853353i) q^{3} +(2.21904 - 3.21484i) q^{4} +(-0.851299 - 1.23332i) q^{5} +(0.303749 - 2.50160i) q^{6} +(-0.487978 - 4.01886i) q^{7} +(-0.558435 + 4.59913i) q^{8} +(0.682556 + 1.79975i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.15192	+	1.12941i	−1.52163	+	0.798615i	−0.998559	−	0.0536706i	\(-0.982908\pi\)
							−0.523076	+	0.852286i	\(0.675216\pi\)
\(3\)	−0.589027	+	0.853353i	−0.340075	+	0.492684i	−0.955288	−	0.295677i	\(-0.904455\pi\)
							0.615213	+	0.788361i	\(0.289070\pi\)
\(5\)	−0.851299	−	1.23332i	−0.380712	−	0.551557i	0.585046	−	0.811000i	\(-0.301076\pi\)
							−0.965758	+	0.259443i	\(0.916461\pi\)
\(7\)	−0.487978	−	4.01886i	−0.184438	−	1.51899i	−0.731470	−	0.681874i	\(-0.761165\pi\)
							0.547032	−	0.837112i	\(-0.315758\pi\)
\(11\)	−0.766492	−	6.31263i	−0.231106	−	1.90333i	−0.398052	−	0.917363i	\(-0.630314\pi\)
							0.166946	−	0.985966i	\(-0.446609\pi\)
\(13\)	−0.504906			−0.140036			−0.0700178	−	0.997546i	\(-0.522306\pi\)
							−0.0700178	+	0.997546i	\(0.522306\pi\)
\(17\)	1.27233	−	0.313601i	0.308586	−	0.0760595i	−0.0819819	−	0.996634i	\(-0.526125\pi\)
							0.390567	+	0.920574i	\(0.372279\pi\)
\(19\)	2.50671			0.575078			0.287539	−	0.957769i	\(-0.407163\pi\)
							0.287539	+	0.957769i	\(0.407163\pi\)
\(23\)	−6.17266	−	3.23966i	−1.28709	−	0.675517i	−0.324347	−	0.945938i	\(-0.605145\pi\)
							−0.962742	+	0.270422i	\(0.912837\pi\)
\(29\)	−2.88332	+	7.60269i	−0.535419	+	1.41178i	0.345456	+	0.938435i	\(0.387724\pi\)
							−0.880875	+	0.473349i	\(0.843045\pi\)
\(31\)	2.85954	+	2.53333i	0.513588	+	0.455000i	0.879671	−	0.475583i	\(-0.157763\pi\)
							−0.366083	+	0.930582i	\(0.619301\pi\)
\(37\)	0.959058	+	2.52883i	0.157668	+	0.415737i	0.990451	−	0.137862i	\(-0.0440232\pi\)
							−0.832783	+	0.553599i	\(0.813254\pi\)
\(41\)	−5.01498	+	7.26545i	−0.783208	+	1.13467i	0.204577	+	0.978850i	\(0.434418\pi\)
							−0.987785	+	0.155822i	\(0.950197\pi\)
\(43\)	−9.33681			−1.42385			−0.711925	−	0.702255i	\(-0.752177\pi\)
							−0.711925	+	0.702255i	\(0.752177\pi\)
\(47\)	4.56043	−	4.04019i	0.665207	−	0.589322i	−0.261456	−	0.965215i	\(-0.584203\pi\)
							0.926663	+	0.375893i	\(0.122664\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	859.2.f.a.100.6	✓	840
859.524	even	13	inner	859.2.f.a.524.6	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.6	✓	840	1.1	even	1	trivial
859.2.f.a.524.6	yes	840	859.524	even	13	inner