Properties

Label 8624.2.a.k
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 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - 2q^{9} + q^{11} - q^{13} - 6q^{17} - 2q^{19} + 6q^{23} - 5q^{25} + 5q^{27} + 9q^{29} + 4q^{31} - q^{33} + 2q^{37} + q^{39} - 6q^{41} + 4q^{43} + 6q^{47} + 6q^{51} + 2q^{57} + 3q^{59} + 11q^{61} - 11q^{67} - 6q^{69} + 2q^{73} + 5q^{75} - 5q^{79} + q^{81} + 6q^{83} - 9q^{87} - 18q^{89} - 4q^{93} - 13q^{97} - 2q^{99} + O(q^{100}) \)

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 −1.00000 0 0 0 0 0 −2.00000 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.k 1
4.b odd 2 1 1078.2.a.e 1
7.b odd 2 1 8624.2.a.u 1
7.c even 3 2 1232.2.q.d 2
12.b even 2 1 9702.2.a.br 1
28.d even 2 1 1078.2.a.c 1
28.f even 6 2 1078.2.e.k 2
28.g odd 6 2 154.2.e.c 2
84.h odd 2 1 9702.2.a.bs 1
84.n even 6 2 1386.2.k.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.c 2 28.g odd 6 2
1078.2.a.c 1 28.d even 2 1
1078.2.a.e 1 4.b odd 2 1
1078.2.e.k 2 28.f even 6 2
1232.2.q.d 2 7.c even 3 2
1386.2.k.e 2 84.n even 6 2
8624.2.a.k 1 1.a even 1 1 trivial
8624.2.a.u 1 7.b odd 2 1
9702.2.a.br 1 12.b even 2 1
9702.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

\( T_{3} + 1 \)
\( T_{5} \)
\( T_{13} + 1 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( 1 + T \)
$17$ \( 6 + T \)
$19$ \( 2 + T \)
$23$ \( -6 + T \)
$29$ \( -9 + T \)
$31$ \( -4 + T \)
$37$ \( -2 + T \)
$41$ \( 6 + T \)
$43$ \( -4 + T \)
$47$ \( -6 + T \)
$53$ \( T \)
$59$ \( -3 + T \)
$61$ \( -11 + T \)
$67$ \( 11 + T \)
$71$ \( T \)
$73$ \( -2 + T \)
$79$ \( 5 + T \)
$83$ \( -6 + T \)
$89$ \( 18 + T \)
$97$ \( 13 + T \)
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