Properties

Label 8820.2.a.br
Level $8820$
Weight $2$
Character orbit 8820.a
Self dual yes
Analytic conductor $70.428$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \(x^{4} - 6 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} +O(q^{10})\) \( q - q^{5} + ( \beta_{1} + \beta_{2} ) q^{11} + ( -\beta_{1} + \beta_{2} ) q^{13} + ( 2 - \beta_{3} ) q^{17} + ( \beta_{1} + 2 \beta_{2} ) q^{19} + ( 2 \beta_{1} + \beta_{2} ) q^{23} + q^{25} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{29} + 5 \beta_{1} q^{31} + ( -2 - 2 \beta_{3} ) q^{37} + ( 8 + \beta_{3} ) q^{41} + ( -6 - \beta_{3} ) q^{43} + ( 4 + \beta_{3} ) q^{47} + ( -6 \beta_{1} - \beta_{2} ) q^{53} + ( -\beta_{1} - \beta_{2} ) q^{55} + ( -4 + 2 \beta_{3} ) q^{59} + ( \beta_{1} - 2 \beta_{2} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{65} + ( -4 + 2 \beta_{3} ) q^{67} + ( -3 \beta_{1} - \beta_{2} ) q^{71} + ( -5 \beta_{1} + \beta_{2} ) q^{73} + ( 2 + 2 \beta_{3} ) q^{79} + ( 4 - 3 \beta_{3} ) q^{83} + ( -2 + \beta_{3} ) q^{85} + 6 q^{89} + ( -\beta_{1} - 2 \beta_{2} ) q^{95} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + O(q^{10}) \) \( 4q - 4q^{5} + 8q^{17} + 4q^{25} - 8q^{37} + 32q^{41} - 24q^{43} + 16q^{47} - 16q^{59} - 16q^{67} + 8q^{79} + 16q^{83} - 8q^{85} + 24q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 6 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 8 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} + 4 \beta_{1}\)

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
−2.28825
−0.874032
0.874032
2.28825
0 0 0 −1.00000 0 0 0 0 0
1.2 0 0 0 −1.00000 0 0 0 0 0
1.3 0 0 0 −1.00000 0 0 0 0 0
1.4 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8820.2.a.br 4
3.b odd 2 1 8820.2.a.bs yes 4
7.b odd 2 1 8820.2.a.bs yes 4
21.c even 2 1 inner 8820.2.a.br 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8820.2.a.br 4 1.a even 1 1 trivial
8820.2.a.br 4 21.c even 2 1 inner
8820.2.a.bs yes 4 3.b odd 2 1
8820.2.a.bs yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11}^{4} - 24 T_{11}^{2} + 64 \)
\( T_{13}^{4} - 24 T_{13}^{2} + 64 \)
\( T_{17}^{2} - 4 T_{17} - 16 \)
\( T_{31}^{2} - 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T )^{4} \)
$7$ \( T^{4} \)
$11$ \( 64 - 24 T^{2} + T^{4} \)
$13$ \( 64 - 24 T^{2} + T^{4} \)
$17$ \( ( -16 - 4 T + T^{2} )^{2} \)
$19$ \( 1444 - 84 T^{2} + T^{4} \)
$23$ \( 4 - 36 T^{2} + T^{4} \)
$29$ \( 1024 - 96 T^{2} + T^{4} \)
$31$ \( ( -50 + T^{2} )^{2} \)
$37$ \( ( -76 + 4 T + T^{2} )^{2} \)
$41$ \( ( 44 - 16 T + T^{2} )^{2} \)
$43$ \( ( 16 + 12 T + T^{2} )^{2} \)
$47$ \( ( -4 - 8 T + T^{2} )^{2} \)
$53$ \( 3844 - 164 T^{2} + T^{4} \)
$59$ \( ( -64 + 8 T + T^{2} )^{2} \)
$61$ \( 1444 - 84 T^{2} + T^{4} \)
$67$ \( ( -64 + 8 T + T^{2} )^{2} \)
$71$ \( 64 - 56 T^{2} + T^{4} \)
$73$ \( 1600 - 120 T^{2} + T^{4} \)
$79$ \( ( -76 - 4 T + T^{2} )^{2} \)
$83$ \( ( -164 - 8 T + T^{2} )^{2} \)
$89$ \( ( -6 + T )^{4} \)
$97$ \( 5184 - 216 T^{2} + T^{4} \)
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