Properties

Label 8624.2.a.a
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 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{3} + q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + q^{5} + 6 q^{9} + q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 6 q^{19} + 5 q^{23} - 4 q^{25} - 9 q^{27} + 10 q^{29} + q^{31} - 3 q^{33} - 5 q^{37} - 12 q^{39} + 2 q^{41} + 8 q^{43} + 6 q^{45} + 8 q^{47} + 6 q^{51} - 6 q^{53} + q^{55} + 18 q^{57} + 3 q^{59} + 2 q^{61} + 4 q^{65} + 3 q^{67} - 15 q^{69} - q^{71} - 10 q^{73} + 12 q^{75} - 6 q^{79} + 9 q^{81} + 12 q^{83} - 2 q^{85} - 30 q^{87} + 15 q^{89} - 3 q^{93} - 6 q^{95} + 5 q^{97} + 6 q^{99}+O(q^{100}) \) 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 −3.00000 0 1.00000 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.a 1
4.b odd 2 1 539.2.a.c 1
7.b odd 2 1 1232.2.a.l 1
12.b even 2 1 4851.2.a.j 1
28.d even 2 1 77.2.a.a 1
28.f even 6 2 539.2.e.f 2
28.g odd 6 2 539.2.e.c 2
44.c even 2 1 5929.2.a.f 1
56.e even 2 1 4928.2.a.bj 1
56.h odd 2 1 4928.2.a.a 1
84.h odd 2 1 693.2.a.c 1
140.c even 2 1 1925.2.a.h 1
140.j odd 4 2 1925.2.b.e 2
308.g odd 2 1 847.2.a.b 1
308.s odd 10 4 847.2.f.h 4
308.t even 10 4 847.2.f.i 4
924.n even 2 1 7623.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 28.d even 2 1
539.2.a.c 1 4.b odd 2 1
539.2.e.c 2 28.g odd 6 2
539.2.e.f 2 28.f even 6 2
693.2.a.c 1 84.h odd 2 1
847.2.a.b 1 308.g odd 2 1
847.2.f.h 4 308.s odd 10 4
847.2.f.i 4 308.t even 10 4
1232.2.a.l 1 7.b odd 2 1
1925.2.a.h 1 140.c even 2 1
1925.2.b.e 2 140.j odd 4 2
4851.2.a.j 1 12.b even 2 1
4928.2.a.a 1 56.h odd 2 1
4928.2.a.bj 1 56.e even 2 1
5929.2.a.f 1 44.c even 2 1
7623.2.a.j 1 924.n even 2 1
8624.2.a.a 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

\( T_{3} + 3 \) Copy content Toggle raw display
\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 6 \) Copy content Toggle raw display
$23$ \( T - 5 \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T + 5 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T - 8 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 3 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 3 \) Copy content Toggle raw display
$71$ \( T + 1 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T + 6 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T - 5 \) Copy content Toggle raw display
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