Properties

Label 8624.2.a.h
Level $8624$
Weight $2$
Character orbit 8624.a
Self dual yes
Analytic conductor $68.863$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2 q^{3} + 4 q^{5} + q^{9} + O(q^{10}) \) \( q - 2 q^{3} + 4 q^{5} + q^{9} - q^{11} + 2 q^{13} - 8 q^{15} + 2 q^{17} - 8 q^{19} + 11 q^{25} + 4 q^{27} - 6 q^{29} - 2 q^{31} + 2 q^{33} - 10 q^{37} - 4 q^{39} + 10 q^{41} + 8 q^{43} + 4 q^{45} - 6 q^{47} - 4 q^{51} - 6 q^{53} - 4 q^{55} + 16 q^{57} + 6 q^{59} - 2 q^{61} + 8 q^{65} + 12 q^{67} + 8 q^{71} - 2 q^{73} - 22 q^{75} + 12 q^{79} - 11 q^{81} + 8 q^{83} + 8 q^{85} + 12 q^{87} - 8 q^{89} + 4 q^{93} - 32 q^{95} + 12 q^{97} - q^{99} + O(q^{100}) \)

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
0
0 −2.00000 0 4.00000 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.h 1
4.b odd 2 1 4312.2.a.k yes 1
7.b odd 2 1 8624.2.a.x 1
28.d even 2 1 4312.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4312.2.a.a 1 28.d even 2 1
4312.2.a.k yes 1 4.b odd 2 1
8624.2.a.h 1 1.a even 1 1 trivial
8624.2.a.x 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

\( T_{3} + 2 \)
\( T_{5} - 4 \)
\( T_{13} - 2 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

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