Properties

Label 8624.2.a.bd
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

\(f(q)\) \(=\) \( q + 3q^{3} - 4q^{5} + 6q^{9} + O(q^{10}) \) \( q + 3q^{3} - 4q^{5} + 6q^{9} + q^{11} - q^{13} - 12q^{15} + 2q^{17} - 6q^{19} + 2q^{23} + 11q^{25} + 9q^{27} + q^{29} - 4q^{31} + 3q^{33} - 2q^{37} - 3q^{39} - 2q^{41} - 4q^{43} - 24q^{45} - 2q^{47} + 6q^{51} - 12q^{53} - 4q^{55} - 18q^{57} - 9q^{59} - 5q^{61} + 4q^{65} + 9q^{67} + 6q^{69} - 4q^{71} - 2q^{73} + 33q^{75} + 15q^{79} + 9q^{81} + 6q^{83} - 8q^{85} + 3q^{87} + 6q^{89} - 12q^{93} + 24q^{95} - 5q^{97} + 6q^{99} + O(q^{100}) \)

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 3.00000 0 −4.00000 0 0 0 6.00000 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.bd 1
4.b odd 2 1 1078.2.a.a 1
7.b odd 2 1 8624.2.a.d 1
7.c even 3 2 1232.2.q.a 2
12.b even 2 1 9702.2.a.cg 1
28.d even 2 1 1078.2.a.f 1
28.f even 6 2 1078.2.e.g 2
28.g odd 6 2 154.2.e.d 2
84.h odd 2 1 9702.2.a.bb 1
84.n even 6 2 1386.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.d 2 28.g odd 6 2
1078.2.a.a 1 4.b odd 2 1
1078.2.a.f 1 28.d even 2 1
1078.2.e.g 2 28.f even 6 2
1232.2.q.a 2 7.c even 3 2
1386.2.k.a 2 84.n even 6 2
8624.2.a.d 1 7.b odd 2 1
8624.2.a.bd 1 1.a even 1 1 trivial
9702.2.a.bb 1 84.h odd 2 1
9702.2.a.cg 1 12.b even 2 1

Hecke kernels

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

\( T_{3} - 3 \)
\( T_{5} + 4 \)
\( T_{13} + 1 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( 4 + T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( 1 + T \)
$17$ \( -2 + T \)
$19$ \( 6 + T \)
$23$ \( -2 + T \)
$29$ \( -1 + T \)
$31$ \( 4 + T \)
$37$ \( 2 + T \)
$41$ \( 2 + T \)
$43$ \( 4 + T \)
$47$ \( 2 + T \)
$53$ \( 12 + T \)
$59$ \( 9 + T \)
$61$ \( 5 + T \)
$67$ \( -9 + T \)
$71$ \( 4 + T \)
$73$ \( 2 + T \)
$79$ \( -15 + T \)
$83$ \( -6 + T \)
$89$ \( -6 + T \)
$97$ \( 5 + T \)
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