Properties

Label 8820.2.a.h
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 1260)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + O(q^{10}) \) \( q - q^{5} + 4 q^{13} - 6 q^{17} - 2 q^{19} + 6 q^{23} + q^{25} - 2 q^{31} + 2 q^{37} - 6 q^{41} - 4 q^{43} - 6 q^{53} - 12 q^{59} + 10 q^{61} - 4 q^{65} - 4 q^{67} + 12 q^{71} + 4 q^{73} + 8 q^{79} - 12 q^{83} + 6 q^{85} - 6 q^{89} + 2 q^{95} - 8 q^{97} + O(q^{100}) \)

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.h 1
3.b odd 2 1 8820.2.a.u 1
7.b odd 2 1 1260.2.a.j yes 1
21.c even 2 1 1260.2.a.d 1
28.d even 2 1 5040.2.a.w 1
35.c odd 2 1 6300.2.a.h 1
35.f even 4 2 6300.2.k.j 2
84.h odd 2 1 5040.2.a.e 1
105.g even 2 1 6300.2.a.i 1
105.k odd 4 2 6300.2.k.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.a.d 1 21.c even 2 1
1260.2.a.j yes 1 7.b odd 2 1
5040.2.a.e 1 84.h odd 2 1
5040.2.a.w 1 28.d even 2 1
6300.2.a.h 1 35.c odd 2 1
6300.2.a.i 1 105.g even 2 1
6300.2.k.i 2 105.k odd 4 2
6300.2.k.j 2 35.f even 4 2
8820.2.a.h 1 1.a even 1 1 trivial
8820.2.a.u 1 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11} \)
\( T_{13} - 4 \)
\( T_{17} + 6 \)
\( T_{31} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -4 + T \)
$17$ \( 6 + T \)
$19$ \( 2 + T \)
$23$ \( -6 + T \)
$29$ \( T \)
$31$ \( 2 + T \)
$37$ \( -2 + T \)
$41$ \( 6 + T \)
$43$ \( 4 + T \)
$47$ \( T \)
$53$ \( 6 + T \)
$59$ \( 12 + T \)
$61$ \( -10 + T \)
$67$ \( 4 + T \)
$71$ \( -12 + T \)
$73$ \( -4 + T \)
$79$ \( -8 + T \)
$83$ \( 12 + T \)
$89$ \( 6 + T \)
$97$ \( 8 + T \)
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