Properties

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

Related objects

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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: \(1\)
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

\(f(q)\) \(=\) \( q + 2q^{3} + q^{5} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + q^{5} + q^{9} - 3q^{11} - q^{13} + 2q^{15} - 6q^{17} + q^{19} - 9q^{23} + q^{25} - 4q^{27} + 6q^{29} - 8q^{31} - 6q^{33} - 7q^{37} - 2q^{39} + 3q^{41} - 2q^{43} + q^{45} - 9q^{47} - 12q^{51} + 9q^{53} - 3q^{55} + 2q^{57} + 8q^{61} - q^{65} - 8q^{67} - 18q^{69} - 4q^{73} + 2q^{75} + 10q^{79} - 11q^{81} - 6q^{85} + 12q^{87} + 6q^{89} - 16q^{93} + q^{95} - 10q^{97} - 3q^{99} + O(q^{100}) \)

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 2.00000 0 1.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.bh 1
4.b odd 2 1 490.2.a.a 1
7.b odd 2 1 3920.2.a.e 1
7.c even 3 2 560.2.q.b 2
12.b even 2 1 4410.2.a.x 1
20.d odd 2 1 2450.2.a.bf 1
20.e even 4 2 2450.2.c.q 2
28.d even 2 1 490.2.a.d 1
28.f even 6 2 490.2.e.g 2
28.g odd 6 2 70.2.e.d 2
84.h odd 2 1 4410.2.a.bg 1
84.n even 6 2 630.2.k.d 2
140.c even 2 1 2450.2.a.v 1
140.j odd 4 2 2450.2.c.e 2
140.p odd 6 2 350.2.e.b 2
140.w even 12 4 350.2.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.d 2 28.g odd 6 2
350.2.e.b 2 140.p odd 6 2
350.2.j.d 4 140.w even 12 4
490.2.a.a 1 4.b odd 2 1
490.2.a.d 1 28.d even 2 1
490.2.e.g 2 28.f even 6 2
560.2.q.b 2 7.c even 3 2
630.2.k.d 2 84.n even 6 2
2450.2.a.v 1 140.c even 2 1
2450.2.a.bf 1 20.d odd 2 1
2450.2.c.e 2 140.j odd 4 2
2450.2.c.q 2 20.e even 4 2
3920.2.a.e 1 7.b odd 2 1
3920.2.a.bh 1 1.a even 1 1 trivial
4410.2.a.x 1 12.b even 2 1
4410.2.a.bg 1 84.h odd 2 1

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke Characteristic Polynomials

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