Properties

Label 8624.2.a.g
Level $8624$
Weight $2$
Character orbit 8624.a
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1078)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{3} + 2 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + 2 q^{5} + q^{9} - q^{11} - 2 q^{13} - 4 q^{15} - 2 q^{19} - q^{25} + 4 q^{27} + 6 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{37} + 4 q^{39} - 8 q^{41} - 12 q^{43} + 2 q^{45} + 12 q^{47} - 2 q^{53} - 2 q^{55} + 4 q^{57} + 10 q^{59} + 10 q^{61} - 4 q^{65} + 12 q^{67} - 4 q^{71} - 12 q^{73} + 2 q^{75} - 11 q^{81} - 18 q^{83} - 12 q^{87} - 8 q^{93} - 4 q^{95} + 12 q^{97} - q^{99}+O(q^{100}) \) 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 −2.00000 0 2.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)

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 8624.2.a.g 1
4.b odd 2 1 1078.2.a.l yes 1
7.b odd 2 1 8624.2.a.y 1
12.b even 2 1 9702.2.a.e 1
28.d even 2 1 1078.2.a.h 1
28.f even 6 2 1078.2.e.e 2
28.g odd 6 2 1078.2.e.a 2
84.h odd 2 1 9702.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.h 1 28.d even 2 1
1078.2.a.l yes 1 4.b odd 2 1
1078.2.e.a 2 28.g odd 6 2
1078.2.e.e 2 28.f even 6 2
8624.2.a.g 1 1.a even 1 1 trivial
8624.2.a.y 1 7.b odd 2 1
9702.2.a.e 1 12.b even 2 1
9702.2.a.t 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 8 \) Copy content Toggle raw display
$43$ \( T + 12 \) Copy content Toggle raw display
$47$ \( T - 12 \) Copy content Toggle raw display
$53$ \( T + 2 \) Copy content Toggle raw display
$59$ \( T - 10 \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 4 \) Copy content Toggle raw display
$73$ \( T + 12 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T + 18 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 12 \) Copy content Toggle raw display
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