Properties

Label 3920.2.a.k
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 140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - q^{5} - 2q^{9} + O(q^{10}) \) \( q - q^{3} - q^{5} - 2q^{9} - 6q^{11} + 2q^{13} + q^{15} - 6q^{17} - 8q^{19} - 3q^{23} + q^{25} + 5q^{27} + 3q^{29} - 2q^{31} + 6q^{33} + 8q^{37} - 2q^{39} - 3q^{41} - 5q^{43} + 2q^{45} + 6q^{51} + 12q^{53} + 6q^{55} + 8q^{57} - q^{61} - 2q^{65} + 7q^{67} + 3q^{69} - 10q^{73} - q^{75} + 4q^{79} + q^{81} - 3q^{83} + 6q^{85} - 3q^{87} - 3q^{89} + 2q^{93} + 8q^{95} - 10q^{97} + 12q^{99} + O(q^{100}) \)

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

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.k 1
4.b odd 2 1 980.2.a.g 1
7.b odd 2 1 3920.2.a.w 1
7.c even 3 2 560.2.q.f 2
12.b even 2 1 8820.2.a.p 1
20.d odd 2 1 4900.2.a.i 1
20.e even 4 2 4900.2.e.m 2
28.d even 2 1 980.2.a.e 1
28.f even 6 2 980.2.i.f 2
28.g odd 6 2 140.2.i.a 2
84.h odd 2 1 8820.2.a.a 1
84.n even 6 2 1260.2.s.c 2
140.c even 2 1 4900.2.a.q 1
140.j odd 4 2 4900.2.e.n 2
140.p odd 6 2 700.2.i.b 2
140.w even 12 4 700.2.r.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.i.a 2 28.g odd 6 2
560.2.q.f 2 7.c even 3 2
700.2.i.b 2 140.p odd 6 2
700.2.r.a 4 140.w even 12 4
980.2.a.e 1 28.d even 2 1
980.2.a.g 1 4.b odd 2 1
980.2.i.f 2 28.f even 6 2
1260.2.s.c 2 84.n even 6 2
3920.2.a.k 1 1.a even 1 1 trivial
3920.2.a.w 1 7.b odd 2 1
4900.2.a.i 1 20.d odd 2 1
4900.2.a.q 1 140.c even 2 1
4900.2.e.m 2 20.e even 4 2
4900.2.e.n 2 140.j odd 4 2
8820.2.a.a 1 84.h odd 2 1
8820.2.a.p 1 12.b even 2 1

Hecke kernels

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

\( T_{3} + 1 \)
\( T_{11} + 6 \)
\( T_{13} - 2 \)
\( T_{17} + 6 \)

Hecke characteristic polynomials

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