Properties

Label 8624.2.a.w
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 44)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 3 q^{5} - 2 q^{9} + O(q^{10}) \) \( q + q^{3} + 3 q^{5} - 2 q^{9} + q^{11} + 4 q^{13} + 3 q^{15} - 6 q^{17} + 8 q^{19} + 3 q^{23} + 4 q^{25} - 5 q^{27} + 5 q^{31} + q^{33} - q^{37} + 4 q^{39} + 10 q^{43} - 6 q^{45} - 6 q^{51} - 6 q^{53} + 3 q^{55} + 8 q^{57} + 3 q^{59} + 4 q^{61} + 12 q^{65} + q^{67} + 3 q^{69} - 15 q^{71} + 4 q^{73} + 4 q^{75} - 2 q^{79} + q^{81} + 6 q^{83} - 18 q^{85} + 9 q^{89} + 5 q^{93} + 24 q^{95} + 7 q^{97} - 2 q^{99} + O(q^{100}) \)

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 3.00000 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.w 1
4.b odd 2 1 2156.2.a.a 1
7.b odd 2 1 176.2.a.a 1
21.c even 2 1 1584.2.a.p 1
28.d even 2 1 44.2.a.a 1
28.f even 6 2 2156.2.i.b 2
28.g odd 6 2 2156.2.i.c 2
35.c odd 2 1 4400.2.a.v 1
35.f even 4 2 4400.2.b.k 2
56.e even 2 1 704.2.a.f 1
56.h odd 2 1 704.2.a.i 1
77.b even 2 1 1936.2.a.c 1
84.h odd 2 1 396.2.a.c 1
112.j even 4 2 2816.2.c.e 2
112.l odd 4 2 2816.2.c.k 2
140.c even 2 1 1100.2.a.b 1
140.j odd 4 2 1100.2.b.c 2
168.e odd 2 1 6336.2.a.j 1
168.i even 2 1 6336.2.a.i 1
252.s odd 6 2 3564.2.i.a 2
252.bi even 6 2 3564.2.i.j 2
308.g odd 2 1 484.2.a.a 1
308.s odd 10 4 484.2.e.b 4
308.t even 10 4 484.2.e.a 4
364.h even 2 1 7436.2.a.d 1
420.o odd 2 1 9900.2.a.h 1
420.w even 4 2 9900.2.c.g 2
616.g odd 2 1 7744.2.a.m 1
616.o even 2 1 7744.2.a.bc 1
924.n even 2 1 4356.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 28.d even 2 1
176.2.a.a 1 7.b odd 2 1
396.2.a.c 1 84.h odd 2 1
484.2.a.a 1 308.g odd 2 1
484.2.e.a 4 308.t even 10 4
484.2.e.b 4 308.s odd 10 4
704.2.a.f 1 56.e even 2 1
704.2.a.i 1 56.h odd 2 1
1100.2.a.b 1 140.c even 2 1
1100.2.b.c 2 140.j odd 4 2
1584.2.a.p 1 21.c even 2 1
1936.2.a.c 1 77.b even 2 1
2156.2.a.a 1 4.b odd 2 1
2156.2.i.b 2 28.f even 6 2
2156.2.i.c 2 28.g odd 6 2
2816.2.c.e 2 112.j even 4 2
2816.2.c.k 2 112.l odd 4 2
3564.2.i.a 2 252.s odd 6 2
3564.2.i.j 2 252.bi even 6 2
4356.2.a.j 1 924.n even 2 1
4400.2.a.v 1 35.c odd 2 1
4400.2.b.k 2 35.f even 4 2
6336.2.a.i 1 168.i even 2 1
6336.2.a.j 1 168.e odd 2 1
7436.2.a.d 1 364.h even 2 1
7744.2.a.m 1 616.g odd 2 1
7744.2.a.bc 1 616.o even 2 1
8624.2.a.w 1 1.a even 1 1 trivial
9900.2.a.h 1 420.o odd 2 1
9900.2.c.g 2 420.w even 4 2

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{5} - 3 \)
\( T_{13} - 4 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( -3 + T \)
$7$ \( T \)
$11$ \( -1 + T \)
$13$ \( -4 + T \)
$17$ \( 6 + T \)
$19$ \( -8 + T \)
$23$ \( -3 + T \)
$29$ \( T \)
$31$ \( -5 + T \)
$37$ \( 1 + T \)
$41$ \( T \)
$43$ \( -10 + T \)
$47$ \( T \)
$53$ \( 6 + T \)
$59$ \( -3 + T \)
$61$ \( -4 + T \)
$67$ \( -1 + T \)
$71$ \( 15 + T \)
$73$ \( -4 + T \)
$79$ \( 2 + T \)
$83$ \( -6 + T \)
$89$ \( -9 + T \)
$97$ \( -7 + T \)
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