Properties

Label 3920.2.a.s
Level $3920$
Weight $2$
Character orbit 3920.a
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} - 3q^{9} + O(q^{10}) \) \( q - q^{5} - 3q^{9} - 4q^{11} + 2q^{13} - 2q^{17} + 4q^{19} - 4q^{23} + q^{25} - 2q^{29} - 8q^{31} + 6q^{37} + 6q^{41} + 8q^{43} + 3q^{45} + 4q^{47} + 6q^{53} + 4q^{55} - 4q^{59} + 2q^{61} - 2q^{65} - 8q^{67} + 6q^{73} + 9q^{81} - 16q^{83} + 2q^{85} + 6q^{89} - 4q^{95} + 14q^{97} + 12q^{99} + O(q^{100}) \)

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 0 0 −1.00000 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

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 3920.2.a.s 1
4.b odd 2 1 1960.2.a.g 1
7.b odd 2 1 80.2.a.a 1
20.d odd 2 1 9800.2.a.x 1
21.c even 2 1 720.2.a.e 1
28.d even 2 1 40.2.a.a 1
28.f even 6 2 1960.2.q.h 2
28.g odd 6 2 1960.2.q.i 2
35.c odd 2 1 400.2.a.e 1
35.f even 4 2 400.2.c.d 2
56.e even 2 1 320.2.a.c 1
56.h odd 2 1 320.2.a.d 1
77.b even 2 1 9680.2.a.q 1
84.h odd 2 1 360.2.a.a 1
105.g even 2 1 3600.2.a.h 1
105.k odd 4 2 3600.2.f.t 2
112.j even 4 2 1280.2.d.j 2
112.l odd 4 2 1280.2.d.a 2
140.c even 2 1 200.2.a.c 1
140.j odd 4 2 200.2.c.b 2
168.e odd 2 1 2880.2.a.t 1
168.i even 2 1 2880.2.a.bg 1
252.s odd 6 2 3240.2.q.x 2
252.bi even 6 2 3240.2.q.k 2
280.c odd 2 1 1600.2.a.k 1
280.n even 2 1 1600.2.a.o 1
280.s even 4 2 1600.2.c.m 2
280.y odd 4 2 1600.2.c.k 2
308.g odd 2 1 4840.2.a.f 1
364.h even 2 1 6760.2.a.i 1
420.o odd 2 1 1800.2.a.v 1
420.w even 4 2 1800.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 28.d even 2 1
80.2.a.a 1 7.b odd 2 1
200.2.a.c 1 140.c even 2 1
200.2.c.b 2 140.j odd 4 2
320.2.a.c 1 56.e even 2 1
320.2.a.d 1 56.h odd 2 1
360.2.a.a 1 84.h odd 2 1
400.2.a.e 1 35.c odd 2 1
400.2.c.d 2 35.f even 4 2
720.2.a.e 1 21.c even 2 1
1280.2.d.a 2 112.l odd 4 2
1280.2.d.j 2 112.j even 4 2
1600.2.a.k 1 280.c odd 2 1
1600.2.a.o 1 280.n even 2 1
1600.2.c.k 2 280.y odd 4 2
1600.2.c.m 2 280.s even 4 2
1800.2.a.v 1 420.o odd 2 1
1800.2.f.a 2 420.w even 4 2
1960.2.a.g 1 4.b odd 2 1
1960.2.q.h 2 28.f even 6 2
1960.2.q.i 2 28.g odd 6 2
2880.2.a.t 1 168.e odd 2 1
2880.2.a.bg 1 168.i even 2 1
3240.2.q.k 2 252.bi even 6 2
3240.2.q.x 2 252.s odd 6 2
3600.2.a.h 1 105.g even 2 1
3600.2.f.t 2 105.k odd 4 2
3920.2.a.s 1 1.a even 1 1 trivial
4840.2.a.f 1 308.g odd 2 1
6760.2.a.i 1 364.h even 2 1
9680.2.a.q 1 77.b even 2 1
9800.2.a.x 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3920))\):

\( T_{3} \)
\( T_{11} + 4 \)
\( T_{13} - 2 \)
\( T_{17} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( -2 + T \)
$17$ \( 2 + T \)
$19$ \( -4 + T \)
$23$ \( 4 + T \)
$29$ \( 2 + T \)
$31$ \( 8 + T \)
$37$ \( -6 + T \)
$41$ \( -6 + T \)
$43$ \( -8 + T \)
$47$ \( -4 + T \)
$53$ \( -6 + T \)
$59$ \( 4 + T \)
$61$ \( -2 + T \)
$67$ \( 8 + T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( T \)
$83$ \( 16 + T \)
$89$ \( -6 + T \)
$97$ \( -14 + T \)
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