Properties

Label 8820.2.a.bg
Level $8820$
Weight $2$
Character orbit 8820.a
Self dual yes
Analytic conductor $70.428$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 980)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} +O(q^{10})\) \( q - q^{5} + ( 1 + 2 \beta ) q^{11} + ( -5 + \beta ) q^{13} + ( 5 + \beta ) q^{17} + ( 2 - 4 \beta ) q^{19} + ( -2 + \beta ) q^{23} + q^{25} + ( -1 - 4 \beta ) q^{29} + ( -6 + \beta ) q^{31} + ( -2 - \beta ) q^{37} + ( 2 - \beta ) q^{41} + ( 6 + 4 \beta ) q^{43} + ( 1 - 7 \beta ) q^{47} + ( 8 - 3 \beta ) q^{53} + ( -1 - 2 \beta ) q^{55} + ( 2 + \beta ) q^{59} + ( -8 + 2 \beta ) q^{61} + ( 5 - \beta ) q^{65} + ( -4 - 5 \beta ) q^{67} + ( 2 - 6 \beta ) q^{71} + ( -8 + 2 \beta ) q^{73} + ( -1 + 10 \beta ) q^{79} -8 q^{83} + ( -5 - \beta ) q^{85} + 12 \beta q^{89} + ( -2 + 4 \beta ) q^{95} + ( -3 - 9 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + O(q^{10}) \) \( 2 q - 2 q^{5} + 2 q^{11} - 10 q^{13} + 10 q^{17} + 4 q^{19} - 4 q^{23} + 2 q^{25} - 2 q^{29} - 12 q^{31} - 4 q^{37} + 4 q^{41} + 12 q^{43} + 2 q^{47} + 16 q^{53} - 2 q^{55} + 4 q^{59} - 16 q^{61} + 10 q^{65} - 8 q^{67} + 4 q^{71} - 16 q^{73} - 2 q^{79} - 16 q^{83} - 10 q^{85} - 4 q^{95} - 6 q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
0 0 0 −1.00000 0 0 0 0 0
1.2 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.bg 2
3.b odd 2 1 980.2.a.j 2
7.b odd 2 1 8820.2.a.bl 2
12.b even 2 1 3920.2.a.bx 2
15.d odd 2 1 4900.2.a.z 2
15.e even 4 2 4900.2.e.q 4
21.c even 2 1 980.2.a.k yes 2
21.g even 6 2 980.2.i.k 4
21.h odd 6 2 980.2.i.l 4
84.h odd 2 1 3920.2.a.bo 2
105.g even 2 1 4900.2.a.x 2
105.k odd 4 2 4900.2.e.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 3.b odd 2 1
980.2.a.k yes 2 21.c even 2 1
980.2.i.k 4 21.g even 6 2
980.2.i.l 4 21.h odd 6 2
3920.2.a.bo 2 84.h odd 2 1
3920.2.a.bx 2 12.b even 2 1
4900.2.a.x 2 105.g even 2 1
4900.2.a.z 2 15.d odd 2 1
4900.2.e.q 4 15.e even 4 2
4900.2.e.r 4 105.k odd 4 2
8820.2.a.bg 2 1.a even 1 1 trivial
8820.2.a.bl 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11}^{2} - 2 T_{11} - 7 \)
\( T_{13}^{2} + 10 T_{13} + 23 \)
\( T_{17}^{2} - 10 T_{17} + 23 \)
\( T_{31}^{2} + 12 T_{31} + 34 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -7 - 2 T + T^{2} \)
$13$ \( 23 + 10 T + T^{2} \)
$17$ \( 23 - 10 T + T^{2} \)
$19$ \( -28 - 4 T + T^{2} \)
$23$ \( 2 + 4 T + T^{2} \)
$29$ \( -31 + 2 T + T^{2} \)
$31$ \( 34 + 12 T + T^{2} \)
$37$ \( 2 + 4 T + T^{2} \)
$41$ \( 2 - 4 T + T^{2} \)
$43$ \( 4 - 12 T + T^{2} \)
$47$ \( -97 - 2 T + T^{2} \)
$53$ \( 46 - 16 T + T^{2} \)
$59$ \( 2 - 4 T + T^{2} \)
$61$ \( 56 + 16 T + T^{2} \)
$67$ \( -34 + 8 T + T^{2} \)
$71$ \( -68 - 4 T + T^{2} \)
$73$ \( 56 + 16 T + T^{2} \)
$79$ \( -199 + 2 T + T^{2} \)
$83$ \( ( 8 + T )^{2} \)
$89$ \( -288 + T^{2} \)
$97$ \( -153 + 6 T + T^{2} \)
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