Properties

Label 8820.2.a.k
Level $8820$
Weight $2$
Character orbit 8820.a
Self dual yes
Analytic conductor $70.428$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.4280545828\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + O(q^{10}) \) \( q - q^{5} + 2 q^{11} + 6 q^{13} + 2 q^{17} + 9 q^{23} + q^{25} - 3 q^{29} - 2 q^{31} + 8 q^{37} + 5 q^{41} + q^{43} + 8 q^{47} - 4 q^{53} - 2 q^{55} - 8 q^{59} - 7 q^{61} - 6 q^{65} - 3 q^{67} - 8 q^{71} - 14 q^{73} + 4 q^{79} - q^{83} - 2 q^{85} + 13 q^{89} + 10 q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −1.00000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8820.2.a.k 1
3.b odd 2 1 980.2.a.i 1
7.b odd 2 1 8820.2.a.w 1
7.d odd 6 2 1260.2.s.b 2
12.b even 2 1 3920.2.a.d 1
15.d odd 2 1 4900.2.a.a 1
15.e even 4 2 4900.2.e.b 2
21.c even 2 1 980.2.a.a 1
21.g even 6 2 140.2.i.b 2
21.h odd 6 2 980.2.i.a 2
84.h odd 2 1 3920.2.a.bi 1
84.j odd 6 2 560.2.q.a 2
105.g even 2 1 4900.2.a.v 1
105.k odd 4 2 4900.2.e.c 2
105.p even 6 2 700.2.i.a 2
105.w odd 12 4 700.2.r.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.b 2 21.g even 6 2
560.2.q.a 2 84.j odd 6 2
700.2.i.a 2 105.p even 6 2
700.2.r.c 4 105.w odd 12 4
980.2.a.a 1 21.c even 2 1
980.2.a.i 1 3.b odd 2 1
980.2.i.a 2 21.h odd 6 2
1260.2.s.b 2 7.d odd 6 2
3920.2.a.d 1 12.b even 2 1
3920.2.a.bi 1 84.h odd 2 1
4900.2.a.a 1 15.d odd 2 1
4900.2.a.v 1 105.g even 2 1
4900.2.e.b 2 15.e even 4 2
4900.2.e.c 2 105.k odd 4 2
8820.2.a.k 1 1.a even 1 1 trivial
8820.2.a.w 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11} - 2 \)
\( T_{13} - 6 \)
\( T_{17} - 2 \)
\( T_{31} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( -6 + T \)
$17$ \( -2 + T \)
$19$ \( T \)
$23$ \( -9 + T \)
$29$ \( 3 + T \)
$31$ \( 2 + T \)
$37$ \( -8 + T \)
$41$ \( -5 + T \)
$43$ \( -1 + T \)
$47$ \( -8 + T \)
$53$ \( 4 + T \)
$59$ \( 8 + T \)
$61$ \( 7 + T \)
$67$ \( 3 + T \)
$71$ \( 8 + T \)
$73$ \( 14 + T \)
$79$ \( -4 + T \)
$83$ \( 1 + T \)
$89$ \( -13 + T \)
$97$ \( -10 + T \)
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