Properties

Label 8820.2.a.r
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 140)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + O(q^{10}) \) \( q + q^{5} - 3q^{11} + q^{13} - 3q^{17} - 2q^{19} + 6q^{23} + q^{25} + 9q^{29} - 8q^{31} - 10q^{37} + 2q^{43} - 3q^{47} - 3q^{55} + 12q^{59} - 8q^{61} + q^{65} + 8q^{67} - 14q^{73} + 5q^{79} - 12q^{83} - 3q^{85} + 12q^{89} - 2q^{95} - 17q^{97} + O(q^{100}) \)

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.r 1
3.b odd 2 1 980.2.a.c 1
7.b odd 2 1 1260.2.a.c 1
12.b even 2 1 3920.2.a.u 1
15.d odd 2 1 4900.2.a.p 1
15.e even 4 2 4900.2.e.l 2
21.c even 2 1 140.2.a.a 1
21.g even 6 2 980.2.i.d 2
21.h odd 6 2 980.2.i.h 2
28.d even 2 1 5040.2.a.h 1
35.c odd 2 1 6300.2.a.d 1
35.f even 4 2 6300.2.k.c 2
84.h odd 2 1 560.2.a.c 1
105.g even 2 1 700.2.a.d 1
105.k odd 4 2 700.2.e.c 2
168.e odd 2 1 2240.2.a.r 1
168.i even 2 1 2240.2.a.g 1
420.o odd 2 1 2800.2.a.y 1
420.w even 4 2 2800.2.g.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 21.c even 2 1
560.2.a.c 1 84.h odd 2 1
700.2.a.d 1 105.g even 2 1
700.2.e.c 2 105.k odd 4 2
980.2.a.c 1 3.b odd 2 1
980.2.i.d 2 21.g even 6 2
980.2.i.h 2 21.h odd 6 2
1260.2.a.c 1 7.b odd 2 1
2240.2.a.g 1 168.i even 2 1
2240.2.a.r 1 168.e odd 2 1
2800.2.a.y 1 420.o odd 2 1
2800.2.g.j 2 420.w even 4 2
3920.2.a.u 1 12.b even 2 1
4900.2.a.p 1 15.d odd 2 1
4900.2.e.l 2 15.e even 4 2
5040.2.a.h 1 28.d even 2 1
6300.2.a.d 1 35.c odd 2 1
6300.2.k.c 2 35.f even 4 2
8820.2.a.r 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{11} + 3 \)
\( T_{13} - 1 \)
\( T_{17} + 3 \)
\( T_{31} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( 3 + T \)
$13$ \( -1 + T \)
$17$ \( 3 + T \)
$19$ \( 2 + T \)
$23$ \( -6 + T \)
$29$ \( -9 + T \)
$31$ \( 8 + T \)
$37$ \( 10 + T \)
$41$ \( T \)
$43$ \( -2 + T \)
$47$ \( 3 + T \)
$53$ \( T \)
$59$ \( -12 + T \)
$61$ \( 8 + T \)
$67$ \( -8 + T \)
$71$ \( T \)
$73$ \( 14 + T \)
$79$ \( -5 + T \)
$83$ \( 12 + T \)
$89$ \( -12 + T \)
$97$ \( 17 + T \)
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