Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2940)
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.

\(f(q)\) \(=\) \( q + q^{5} +O(q^{10})\) \( q + q^{5} + 2 q^{11} -2 \beta q^{13} + ( 4 - 2 \beta ) q^{17} + ( 4 + \beta ) q^{19} + ( 6 + \beta ) q^{23} + q^{25} + ( 2 - 4 \beta ) q^{29} + 3 \beta q^{31} + ( -2 - 6 \beta ) q^{37} + 2 q^{41} + ( -4 + 4 \beta ) q^{43} + ( 2 + 6 \beta ) q^{47} + ( 6 - \beta ) q^{53} + 2 q^{55} -2 \beta q^{59} -7 \beta q^{61} -2 \beta q^{65} + ( -4 - 2 \beta ) q^{67} + ( 2 + 6 \beta ) q^{71} + ( -4 + 6 \beta ) q^{73} -10 q^{79} + ( 2 + 2 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} + ( 2 + 2 \beta ) q^{89} + ( 4 + \beta ) q^{95} + 4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} + 4q^{11} + 8q^{17} + 8q^{19} + 12q^{23} + 2q^{25} + 4q^{29} - 4q^{37} + 4q^{41} - 8q^{43} + 4q^{47} + 12q^{53} + 4q^{55} - 8q^{67} + 4q^{71} - 8q^{73} - 20q^{79} + 4q^{83} + 8q^{85} + 4q^{89} + 8q^{95} + 8q^{97} + O(q^{100}) \)

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.bm 2
3.b odd 2 1 2940.2.a.q yes 2
7.b odd 2 1 8820.2.a.bh 2
21.c even 2 1 2940.2.a.o 2
21.g even 6 2 2940.2.q.r 4
21.h odd 6 2 2940.2.q.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2940.2.a.o 2 21.c even 2 1
2940.2.a.q yes 2 3.b odd 2 1
2940.2.q.p 4 21.h odd 6 2
2940.2.q.r 4 21.g even 6 2
8820.2.a.bh 2 7.b odd 2 1
8820.2.a.bm 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8820))\):

\( T_{11} - 2 \)
\( T_{13}^{2} - 8 \)
\( T_{17}^{2} - 8 T_{17} + 8 \)
\( T_{31}^{2} - 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -8 + T^{2} \)
$17$ \( 8 - 8 T + T^{2} \)
$19$ \( 14 - 8 T + T^{2} \)
$23$ \( 34 - 12 T + T^{2} \)
$29$ \( -28 - 4 T + T^{2} \)
$31$ \( -18 + T^{2} \)
$37$ \( -68 + 4 T + T^{2} \)
$41$ \( ( -2 + T )^{2} \)
$43$ \( -16 + 8 T + T^{2} \)
$47$ \( -68 - 4 T + T^{2} \)
$53$ \( 34 - 12 T + T^{2} \)
$59$ \( -8 + T^{2} \)
$61$ \( -98 + T^{2} \)
$67$ \( 8 + 8 T + T^{2} \)
$71$ \( -68 - 4 T + T^{2} \)
$73$ \( -56 + 8 T + T^{2} \)
$79$ \( ( 10 + T )^{2} \)
$83$ \( -4 - 4 T + T^{2} \)
$89$ \( -4 - 4 T + T^{2} \)
$97$ \( ( -4 + T )^{2} \)
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