Properties

Label 8820.2.a.bo
Level $8820$
Weight $2$
Character orbit 8820.a
Self dual yes
Analytic conductor $70.428$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 1260)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{5} +O(q^{10})\) \( q - q^{5} + ( 1 - \beta_{2} ) q^{11} + ( 2 - \beta_{2} ) q^{13} + ( -\beta_{1} + \beta_{2} ) q^{17} + \beta_{1} q^{19} + ( -2 + \beta_{1} + \beta_{2} ) q^{23} + q^{25} + ( -5 - 2 \beta_{1} + \beta_{2} ) q^{29} + q^{31} + ( -3 + \beta_{1} + \beta_{2} ) q^{37} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{41} + ( -1 - \beta_{1} + \beta_{2} ) q^{43} + ( -4 - 2 \beta_{1} ) q^{47} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -1 + \beta_{2} ) q^{55} + ( 5 + \beta_{2} ) q^{59} + ( -1 - \beta_{1} ) q^{61} + ( -2 + \beta_{2} ) q^{65} + ( -2 + \beta_{2} ) q^{67} + ( -7 + \beta_{2} ) q^{71} + ( 5 + \beta_{1} + \beta_{2} ) q^{73} + ( -6 - 3 \beta_{1} + 2 \beta_{2} ) q^{79} + ( 2 + \beta_{1} - 3 \beta_{2} ) q^{83} + ( \beta_{1} - \beta_{2} ) q^{85} + ( 1 + 2 \beta_{1} - 3 \beta_{2} ) q^{89} -\beta_{1} q^{95} + ( 5 + \beta_{1} - 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + O(q^{10}) \) \( 3 q - 3 q^{5} + 2 q^{11} + 5 q^{13} + 2 q^{17} - q^{19} - 6 q^{23} + 3 q^{25} - 12 q^{29} + 3 q^{31} - 9 q^{37} - 10 q^{41} - q^{43} - 10 q^{47} - 4 q^{53} - 2 q^{55} + 16 q^{59} - 2 q^{61} - 5 q^{65} - 5 q^{67} - 20 q^{71} + 15 q^{73} - 13 q^{79} + 2 q^{83} - 2 q^{85} - 2 q^{89} + q^{95} + 12 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x - 1\):

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

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
2.86620
−0.210756
−1.65544
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
\(n\): 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.bo 3
3.b odd 2 1 8820.2.a.bp 3
7.b odd 2 1 8820.2.a.bq 3
7.d odd 6 2 1260.2.s.g 6
21.c even 2 1 8820.2.a.bn 3
21.g even 6 2 1260.2.s.h yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1260.2.s.g 6 7.d odd 6 2
1260.2.s.h yes 6 21.g even 6 2
8820.2.a.bn 3 21.c even 2 1
8820.2.a.bo 3 1.a even 1 1 trivial
8820.2.a.bp 3 3.b odd 2 1
8820.2.a.bq 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11}^{3} - 2 T_{11}^{2} - 18 T_{11} - 18 \)
\( T_{13}^{3} - 5 T_{13}^{2} - 11 T_{13} - 3 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 24 T_{17} - 18 \)
\( T_{31} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( T^{3} \)
$11$ \( -18 - 18 T - 2 T^{2} + T^{3} \)
$13$ \( -3 - 11 T - 5 T^{2} + T^{3} \)
$17$ \( -18 - 24 T - 2 T^{2} + T^{3} \)
$19$ \( -21 - 17 T + T^{2} + T^{3} \)
$23$ \( -162 - 36 T + 6 T^{2} + T^{3} \)
$29$ \( -378 - 18 T + 12 T^{2} + T^{3} \)
$31$ \( ( -1 + T )^{3} \)
$37$ \( -191 - 21 T + 9 T^{2} + T^{3} \)
$41$ \( -774 - 78 T + 10 T^{2} + T^{3} \)
$43$ \( -43 - 25 T + T^{2} + T^{3} \)
$47$ \( -72 - 36 T + 10 T^{2} + T^{3} \)
$53$ \( 144 - 96 T + 4 T^{2} + T^{3} \)
$59$ \( -18 + 66 T - 16 T^{2} + T^{3} \)
$61$ \( 4 - 16 T + 2 T^{2} + T^{3} \)
$67$ \( 3 - 11 T + 5 T^{2} + T^{3} \)
$71$ \( 198 + 114 T + 20 T^{2} + T^{3} \)
$73$ \( 41 + 27 T - 15 T^{2} + T^{3} \)
$79$ \( -1453 - 109 T + 13 T^{2} + T^{3} \)
$83$ \( -522 - 156 T - 2 T^{2} + T^{3} \)
$89$ \( -126 - 174 T + 2 T^{2} + T^{3} \)
$97$ \( 116 - 24 T - 12 T^{2} + T^{3} \)
show more
show less