Properties

Label 3920.2.a.bk
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 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{3} + q^{5} + 6q^{9} + O(q^{10}) \) \( q + 3q^{3} + q^{5} + 6q^{9} + 2q^{11} + 3q^{15} + 4q^{17} - 6q^{19} - 3q^{23} + q^{25} + 9q^{27} + 9q^{29} - 4q^{31} + 6q^{33} - 4q^{37} + 7q^{41} + 5q^{43} + 6q^{45} + 8q^{47} + 12q^{51} - 2q^{53} + 2q^{55} - 18q^{57} + 10q^{59} - q^{61} + 9q^{67} - 9q^{69} - 2q^{71} + 4q^{73} + 3q^{75} - 10q^{79} + 9q^{81} - 7q^{83} + 4q^{85} + 27q^{87} - q^{89} - 12q^{93} - 6q^{95} - 14q^{97} + 12q^{99} + O(q^{100}) \)

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

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.bk 1
4.b odd 2 1 490.2.a.e 1
7.b odd 2 1 3920.2.a.b 1
7.d odd 6 2 560.2.q.i 2
12.b even 2 1 4410.2.a.h 1
20.d odd 2 1 2450.2.a.q 1
20.e even 4 2 2450.2.c.a 2
28.d even 2 1 490.2.a.k 1
28.f even 6 2 70.2.e.a 2
28.g odd 6 2 490.2.e.f 2
84.h odd 2 1 4410.2.a.r 1
84.j odd 6 2 630.2.k.f 2
140.c even 2 1 2450.2.a.b 1
140.j odd 4 2 2450.2.c.s 2
140.s even 6 2 350.2.e.l 2
140.x odd 12 4 350.2.j.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 28.f even 6 2
350.2.e.l 2 140.s even 6 2
350.2.j.f 4 140.x odd 12 4
490.2.a.e 1 4.b odd 2 1
490.2.a.k 1 28.d even 2 1
490.2.e.f 2 28.g odd 6 2
560.2.q.i 2 7.d odd 6 2
630.2.k.f 2 84.j odd 6 2
2450.2.a.b 1 140.c even 2 1
2450.2.a.q 1 20.d odd 2 1
2450.2.c.a 2 20.e even 4 2
2450.2.c.s 2 140.j odd 4 2
3920.2.a.b 1 7.b odd 2 1
3920.2.a.bk 1 1.a even 1 1 trivial
4410.2.a.h 1 12.b even 2 1
4410.2.a.r 1 84.h odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -3 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( -2 + T \)
$13$ \( T \)
$17$ \( -4 + T \)
$19$ \( 6 + T \)
$23$ \( 3 + T \)
$29$ \( -9 + T \)
$31$ \( 4 + T \)
$37$ \( 4 + T \)
$41$ \( -7 + T \)
$43$ \( -5 + T \)
$47$ \( -8 + T \)
$53$ \( 2 + T \)
$59$ \( -10 + T \)
$61$ \( 1 + T \)
$67$ \( -9 + T \)
$71$ \( 2 + T \)
$73$ \( -4 + T \)
$79$ \( 10 + T \)
$83$ \( 7 + T \)
$89$ \( 1 + T \)
$97$ \( 14 + T \)
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