Properties

Label 8712.2.a.x
Level $8712$
Weight $2$
Character orbit 8712.a
Self dual yes
Analytic conductor $69.566$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8712.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.5656702409\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{5} + 2 q^{7} - 6 q^{17} - 4 q^{19} - q^{23} + 4 q^{25} - 8 q^{29} - 7 q^{31} + 6 q^{35} - q^{37} + 4 q^{41} - 6 q^{43} + 8 q^{47} - 3 q^{49} - 2 q^{53} + q^{59} - 4 q^{61} - 5 q^{67} - 3 q^{71} - 16 q^{73} - 2 q^{79} - 2 q^{83} - 18 q^{85} - 15 q^{89} - 12 q^{95} - 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
0
0 0 0 3.00000 0 2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(11\) \(-1\)

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 8712.2.a.x 1
3.b odd 2 1 968.2.a.a 1
11.b odd 2 1 792.2.a.g 1
12.b even 2 1 1936.2.a.l 1
24.f even 2 1 7744.2.a.b 1
24.h odd 2 1 7744.2.a.bk 1
33.d even 2 1 88.2.a.a 1
33.f even 10 4 968.2.i.j 4
33.h odd 10 4 968.2.i.i 4
44.c even 2 1 1584.2.a.q 1
88.b odd 2 1 6336.2.a.h 1
88.g even 2 1 6336.2.a.k 1
132.d odd 2 1 176.2.a.c 1
165.d even 2 1 2200.2.a.k 1
165.l odd 4 2 2200.2.b.a 2
231.h odd 2 1 4312.2.a.l 1
264.m even 2 1 704.2.a.l 1
264.p odd 2 1 704.2.a.b 1
528.s odd 4 2 2816.2.c.d 2
528.x even 4 2 2816.2.c.i 2
660.g odd 2 1 4400.2.a.a 1
660.q even 4 2 4400.2.b.b 2
924.n even 2 1 8624.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 33.d even 2 1
176.2.a.c 1 132.d odd 2 1
704.2.a.b 1 264.p odd 2 1
704.2.a.l 1 264.m even 2 1
792.2.a.g 1 11.b odd 2 1
968.2.a.a 1 3.b odd 2 1
968.2.i.i 4 33.h odd 10 4
968.2.i.j 4 33.f even 10 4
1584.2.a.q 1 44.c even 2 1
1936.2.a.l 1 12.b even 2 1
2200.2.a.k 1 165.d even 2 1
2200.2.b.a 2 165.l odd 4 2
2816.2.c.d 2 528.s odd 4 2
2816.2.c.i 2 528.x even 4 2
4312.2.a.l 1 231.h odd 2 1
4400.2.a.a 1 660.g odd 2 1
4400.2.b.b 2 660.q even 4 2
6336.2.a.h 1 88.b odd 2 1
6336.2.a.k 1 88.g even 2 1
7744.2.a.b 1 24.f even 2 1
7744.2.a.bk 1 24.h odd 2 1
8624.2.a.c 1 924.n even 2 1
8712.2.a.x 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8712))\):

\( T_{5} - 3 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 1 \) Copy content Toggle raw display
$29$ \( T + 8 \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T + 1 \) Copy content Toggle raw display
$41$ \( T - 4 \) Copy content Toggle raw display
$43$ \( T + 6 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T - 1 \) Copy content Toggle raw display
$61$ \( T + 4 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T + 3 \) Copy content Toggle raw display
$73$ \( T + 16 \) Copy content Toggle raw display
$79$ \( T + 2 \) Copy content Toggle raw display
$83$ \( T + 2 \) Copy content Toggle raw display
$89$ \( T + 15 \) Copy content Toggle raw display
$97$ \( T + 7 \) Copy content Toggle raw display
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