Properties

Label 8820.2.a.bk
Level $8820$
Weight $2$
Character orbit 8820.a
Self dual yes
Analytic conductor $70.428$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 420)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} +O(q^{10})\) \( q + q^{5} + \beta q^{11} + ( -1 + \beta ) q^{13} -\beta q^{17} -7 q^{19} + \beta q^{23} + q^{25} + ( 6 - \beta ) q^{29} + ( -1 - 2 \beta ) q^{31} + ( -1 + \beta ) q^{37} + \beta q^{41} + ( -1 + \beta ) q^{43} + 6 q^{47} -2 \beta q^{53} + \beta q^{55} + ( 6 + \beta ) q^{59} + ( -4 + 2 \beta ) q^{61} + ( -1 + \beta ) q^{65} + ( -1 - \beta ) q^{67} + 3 \beta q^{71} + ( 5 + \beta ) q^{73} + 11 q^{79} + ( 6 + \beta ) q^{83} -\beta q^{85} + ( 6 + \beta ) q^{89} -7 q^{95} + ( 8 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{5} - 2 q^{13} - 14 q^{19} + 2 q^{25} + 12 q^{29} - 2 q^{31} - 2 q^{37} - 2 q^{43} + 12 q^{47} + 12 q^{59} - 8 q^{61} - 2 q^{65} - 2 q^{67} + 10 q^{73} + 22 q^{79} + 12 q^{83} + 12 q^{89} - 14 q^{95} + 16 q^{97} + O(q^{100}) \)

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.bk 2
3.b odd 2 1 2940.2.a.r 2
7.b odd 2 1 8820.2.a.bf 2
7.c even 3 2 1260.2.s.e 4
21.c even 2 1 2940.2.a.p 2
21.g even 6 2 2940.2.q.q 4
21.h odd 6 2 420.2.q.d 4
84.n even 6 2 1680.2.bg.t 4
105.o odd 6 2 2100.2.q.k 4
105.x even 12 4 2100.2.bc.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d 4 21.h odd 6 2
1260.2.s.e 4 7.c even 3 2
1680.2.bg.t 4 84.n even 6 2
2100.2.q.k 4 105.o odd 6 2
2100.2.bc.f 8 105.x even 12 4
2940.2.a.p 2 21.c even 2 1
2940.2.a.r 2 3.b odd 2 1
2940.2.q.q 4 21.g even 6 2
8820.2.a.bf 2 7.b odd 2 1
8820.2.a.bk 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11}^{2} - 18 \)
\( T_{13}^{2} + 2 T_{13} - 17 \)
\( T_{17}^{2} - 18 \)
\( T_{31}^{2} + 2 T_{31} - 71 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -18 + T^{2} \)
$13$ \( -17 + 2 T + T^{2} \)
$17$ \( -18 + T^{2} \)
$19$ \( ( 7 + T )^{2} \)
$23$ \( -18 + T^{2} \)
$29$ \( 18 - 12 T + T^{2} \)
$31$ \( -71 + 2 T + T^{2} \)
$37$ \( -17 + 2 T + T^{2} \)
$41$ \( -18 + T^{2} \)
$43$ \( -17 + 2 T + T^{2} \)
$47$ \( ( -6 + T )^{2} \)
$53$ \( -72 + T^{2} \)
$59$ \( 18 - 12 T + T^{2} \)
$61$ \( -56 + 8 T + T^{2} \)
$67$ \( -17 + 2 T + T^{2} \)
$71$ \( -162 + T^{2} \)
$73$ \( 7 - 10 T + T^{2} \)
$79$ \( ( -11 + T )^{2} \)
$83$ \( 18 - 12 T + T^{2} \)
$89$ \( 18 - 12 T + T^{2} \)
$97$ \( -8 - 16 T + T^{2} \)
show more
show less