Properties

Label 8820.2.a.j
Level $8820$
Weight $2$
Character orbit 8820.a
Self dual yes
Analytic conductor $70.428$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + O(q^{10}) \) \( q - q^{5} + 2q^{11} + q^{13} + 4q^{17} - q^{19} - 4q^{23} + q^{25} - 5q^{31} - 5q^{37} - 2q^{41} - 9q^{43} + 2q^{47} - 12q^{53} - 2q^{55} + 8q^{59} - 14q^{61} - q^{65} + 9q^{67} - 2q^{71} + q^{73} - 3q^{79} + 18q^{83} - 4q^{85} + 4q^{89} + q^{95} + 10q^{97} + O(q^{100}) \)

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.j 1
3.b odd 2 1 2940.2.a.d 1
7.b odd 2 1 8820.2.a.y 1
7.c even 3 2 1260.2.s.d 2
21.c even 2 1 2940.2.a.h 1
21.g even 6 2 2940.2.q.h 2
21.h odd 6 2 420.2.q.a 2
84.n even 6 2 1680.2.bg.a 2
105.o odd 6 2 2100.2.q.a 2
105.x even 12 4 2100.2.bc.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.a 2 21.h odd 6 2
1260.2.s.d 2 7.c even 3 2
1680.2.bg.a 2 84.n even 6 2
2100.2.q.a 2 105.o odd 6 2
2100.2.bc.c 4 105.x even 12 4
2940.2.a.d 1 3.b odd 2 1
2940.2.a.h 1 21.c even 2 1
2940.2.q.h 2 21.g even 6 2
8820.2.a.j 1 1.a even 1 1 trivial
8820.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11} - 2 \)
\( T_{13} - 1 \)
\( T_{17} - 4 \)
\( T_{31} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( -1 + T \)
$17$ \( -4 + T \)
$19$ \( 1 + T \)
$23$ \( 4 + T \)
$29$ \( T \)
$31$ \( 5 + T \)
$37$ \( 5 + T \)
$41$ \( 2 + T \)
$43$ \( 9 + T \)
$47$ \( -2 + T \)
$53$ \( 12 + T \)
$59$ \( -8 + T \)
$61$ \( 14 + T \)
$67$ \( -9 + T \)
$71$ \( 2 + T \)
$73$ \( -1 + T \)
$79$ \( 3 + T \)
$83$ \( -18 + T \)
$89$ \( -4 + T \)
$97$ \( -10 + T \)
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