Properties

Label 8624.2.a.j
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 11)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - q^{5} - 2 q^{9} + O(q^{10}) \) \( q - q^{3} - q^{5} - 2 q^{9} - q^{11} - 4 q^{13} + q^{15} + 2 q^{17} + q^{23} - 4 q^{25} + 5 q^{27} + 7 q^{31} + q^{33} + 3 q^{37} + 4 q^{39} + 8 q^{41} + 6 q^{43} + 2 q^{45} + 8 q^{47} - 2 q^{51} - 6 q^{53} + q^{55} + 5 q^{59} - 12 q^{61} + 4 q^{65} + 7 q^{67} - q^{69} + 3 q^{71} - 4 q^{73} + 4 q^{75} + 10 q^{79} + q^{81} - 6 q^{83} - 2 q^{85} - 15 q^{89} - 7 q^{93} + 7 q^{97} + 2 q^{99} + O(q^{100}) \)

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 −1.00000 0 −1.00000 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.j 1
4.b odd 2 1 539.2.a.a 1
7.b odd 2 1 176.2.a.b 1
12.b even 2 1 4851.2.a.t 1
21.c even 2 1 1584.2.a.g 1
28.d even 2 1 11.2.a.a 1
28.f even 6 2 539.2.e.h 2
28.g odd 6 2 539.2.e.g 2
35.c odd 2 1 4400.2.a.i 1
35.f even 4 2 4400.2.b.h 2
44.c even 2 1 5929.2.a.h 1
56.e even 2 1 704.2.a.h 1
56.h odd 2 1 704.2.a.c 1
77.b even 2 1 1936.2.a.i 1
84.h odd 2 1 99.2.a.d 1
112.j even 4 2 2816.2.c.j 2
112.l odd 4 2 2816.2.c.f 2
140.c even 2 1 275.2.a.b 1
140.j odd 4 2 275.2.b.a 2
168.e odd 2 1 6336.2.a.br 1
168.i even 2 1 6336.2.a.bu 1
252.s odd 6 2 891.2.e.b 2
252.bi even 6 2 891.2.e.k 2
308.g odd 2 1 121.2.a.d 1
308.s odd 10 4 121.2.c.a 4
308.t even 10 4 121.2.c.e 4
364.h even 2 1 1859.2.a.b 1
420.o odd 2 1 2475.2.a.a 1
420.w even 4 2 2475.2.c.a 2
476.e even 2 1 3179.2.a.a 1
532.b odd 2 1 3971.2.a.b 1
616.g odd 2 1 7744.2.a.x 1
616.o even 2 1 7744.2.a.k 1
644.h odd 2 1 5819.2.a.a 1
812.c even 2 1 9251.2.a.d 1
924.n even 2 1 1089.2.a.b 1
1540.b odd 2 1 3025.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 28.d even 2 1
99.2.a.d 1 84.h odd 2 1
121.2.a.d 1 308.g odd 2 1
121.2.c.a 4 308.s odd 10 4
121.2.c.e 4 308.t even 10 4
176.2.a.b 1 7.b odd 2 1
275.2.a.b 1 140.c even 2 1
275.2.b.a 2 140.j odd 4 2
539.2.a.a 1 4.b odd 2 1
539.2.e.g 2 28.g odd 6 2
539.2.e.h 2 28.f even 6 2
704.2.a.c 1 56.h odd 2 1
704.2.a.h 1 56.e even 2 1
891.2.e.b 2 252.s odd 6 2
891.2.e.k 2 252.bi even 6 2
1089.2.a.b 1 924.n even 2 1
1584.2.a.g 1 21.c even 2 1
1859.2.a.b 1 364.h even 2 1
1936.2.a.i 1 77.b even 2 1
2475.2.a.a 1 420.o odd 2 1
2475.2.c.a 2 420.w even 4 2
2816.2.c.f 2 112.l odd 4 2
2816.2.c.j 2 112.j even 4 2
3025.2.a.a 1 1540.b odd 2 1
3179.2.a.a 1 476.e even 2 1
3971.2.a.b 1 532.b odd 2 1
4400.2.a.i 1 35.c odd 2 1
4400.2.b.h 2 35.f even 4 2
4851.2.a.t 1 12.b even 2 1
5819.2.a.a 1 644.h odd 2 1
5929.2.a.h 1 44.c even 2 1
6336.2.a.br 1 168.e odd 2 1
6336.2.a.bu 1 168.i even 2 1
7744.2.a.k 1 616.o even 2 1
7744.2.a.x 1 616.g odd 2 1
8624.2.a.j 1 1.a even 1 1 trivial
9251.2.a.d 1 812.c even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

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