Embedded newform 859.2.f.a.100.1

• Label: 859.2.f.a.100.1
• 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			100.1
Character	\(\chi\)	\(=\)	859.100
Dual form			859.2.f.a.524.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.48316 + 1.30326i) q^{2} +(-0.107649 + 0.155957i) q^{3} +(3.33145 - 4.82644i) q^{4} +(0.315645 + 0.457291i) q^{5} +(0.0640574 - 0.527560i) q^{6} +(-0.162393 - 1.33743i) q^{7} +(-1.30634 + 10.7587i) q^{8} +(1.05108 + 2.77147i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 10 q^{2} - 5 q^{3} - 74 q^{4} - 20 q^{5} + 5 q^{6} - 3 q^{7} + 2 q^{8} - 69 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.48316	+	1.30326i	−1.75586	+	0.921545i	−0.817462	+	0.575983i	\(0.804619\pi\)
							−0.938396	+	0.345562i	\(0.887688\pi\)
\(3\)	−0.107649	+	0.155957i	−0.0621513	+	0.0900416i	−0.852827	−	0.522193i	\(-0.825114\pi\)
							0.790676	+	0.612235i	\(0.209729\pi\)
\(5\)	0.315645	+	0.457291i	0.141161	+	0.204507i	0.887187	−	0.461410i	\(-0.152656\pi\)
							−0.746026	+	0.665916i	\(0.768041\pi\)
\(7\)	−0.162393	−	1.33743i	−0.0613787	−	0.505499i	−0.990683	−	0.136185i	\(-0.956516\pi\)
							0.929305	−	0.369314i	\(-0.120407\pi\)
\(11\)	0.445108	+	3.66580i	0.134205	+	1.10528i	0.891760	+	0.452508i	\(0.149471\pi\)
							−0.757555	+	0.652771i	\(0.773606\pi\)
\(13\)	5.88536			1.63230			0.816152	−	0.577837i	\(-0.196103\pi\)
							0.816152	+	0.577837i	\(0.196103\pi\)
\(17\)	−0.137731	+	0.0339478i	−0.0334048	+	0.00823354i	−0.255983	−	0.966681i	\(-0.582399\pi\)
							0.222578	+	0.974915i	\(0.428553\pi\)
\(19\)	−2.45468			−0.563142			−0.281571	−	0.959540i	\(-0.590856\pi\)
							−0.281571	+	0.959540i	\(0.590856\pi\)
\(23\)	−0.885034	−	0.464502i	−0.184542	−	0.0968553i	0.369922	−	0.929063i	\(-0.379384\pi\)
							−0.554464	+	0.832208i	\(0.687077\pi\)
\(29\)	0.189875	−	0.500659i	0.0352589	−	0.0929700i	−0.916231	−	0.400651i	\(-0.868784\pi\)
							0.951489	+	0.307681i	\(0.0995531\pi\)
\(31\)	−1.07134	−	0.949122i	−0.192418	−	0.170467i	0.561413	−	0.827536i	\(-0.310258\pi\)
							−0.753831	+	0.657068i	\(0.771796\pi\)
\(37\)	4.13803	+	10.9111i	0.680289	+	1.79377i	0.604434	+	0.796655i	\(0.293399\pi\)
							0.0758545	+	0.997119i	\(0.475832\pi\)
\(41\)	5.05797	−	7.32774i	0.789923	−	1.14440i	−0.196553	−	0.980493i	\(-0.562975\pi\)
							0.986476	−	0.163907i	\(-0.0524098\pi\)
\(43\)	−5.45910			−0.832505			−0.416253	−	0.909249i	\(-0.636657\pi\)
							−0.416253	+	0.909249i	\(0.636657\pi\)
\(47\)	−7.41265	+	6.56704i	−1.08125	+	0.957901i	−0.999250	−	0.0387315i	\(-0.987668\pi\)
							−0.0819972	+	0.996633i	\(0.526130\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	859.2.f.a.100.1	✓	840
859.524	even	13	inner	859.2.f.a.524.1	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
859.2.f.a.100.1	✓	840	1.1	even	1	trivial
859.2.f.a.524.1	yes	840	859.524	even	13	inner