Properties

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

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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 280)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.y 1
4.b odd 2 1 1960.2.a.e 1
7.b odd 2 1 3920.2.a.m 1
7.d odd 6 2 560.2.q.h 2
20.d odd 2 1 9800.2.a.bc 1
28.d even 2 1 1960.2.a.i 1
28.f even 6 2 280.2.q.a 2
28.g odd 6 2 1960.2.q.k 2
84.j odd 6 2 2520.2.bi.a 2
140.c even 2 1 9800.2.a.r 1
140.s even 6 2 1400.2.q.e 2
140.x odd 12 4 1400.2.bh.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a 2 28.f even 6 2
560.2.q.h 2 7.d odd 6 2
1400.2.q.e 2 140.s even 6 2
1400.2.bh.b 4 140.x odd 12 4
1960.2.a.e 1 4.b odd 2 1
1960.2.a.i 1 28.d even 2 1
1960.2.q.k 2 28.g odd 6 2
2520.2.bi.a 2 84.j odd 6 2
3920.2.a.m 1 7.b odd 2 1
3920.2.a.y 1 1.a even 1 1 trivial
9800.2.a.r 1 140.c even 2 1
9800.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3920))\):

\( T_{3} - 1 \)
\( T_{11} + 2 \)
\( T_{13} \)
\( T_{17} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( 2 + T \)
$13$ \( T \)
$17$ \( 4 + T \)
$19$ \( 2 + T \)
$23$ \( 1 + T \)
$29$ \( -9 + T \)
$31$ \( -4 + T \)
$37$ \( -4 + T \)
$41$ \( 1 + T \)
$43$ \( 9 + T \)
$47$ \( T \)
$53$ \( 10 + T \)
$59$ \( 10 + T \)
$61$ \( 9 + T \)
$67$ \( 5 + T \)
$71$ \( 14 + T \)
$73$ \( 12 + T \)
$79$ \( 14 + T \)
$83$ \( -11 + T \)
$89$ \( -15 + T \)
$97$ \( -18 + T \)
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