Properties

Label 5040.2.a.h
Level $5040$
Weight $2$
Character orbit 5040.a
Self dual yes
Analytic conductor $40.245$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.2446026187\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q - q^{5} - q^{7} + O(q^{10}) \) \( q - q^{5} - q^{7} + 3q^{11} - q^{13} + 3q^{17} - 2q^{19} - 6q^{23} + q^{25} + 9q^{29} - 8q^{31} + q^{35} - 10q^{37} - 2q^{43} - 3q^{47} + q^{49} - 3q^{55} + 12q^{59} + 8q^{61} + q^{65} - 8q^{67} + 14q^{73} - 3q^{77} - 5q^{79} - 12q^{83} - 3q^{85} - 12q^{89} + q^{91} + 2q^{95} + 17q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 5040.2.a.h 1
3.b odd 2 1 560.2.a.c 1
4.b odd 2 1 1260.2.a.c 1
12.b even 2 1 140.2.a.a 1
15.d odd 2 1 2800.2.a.y 1
15.e even 4 2 2800.2.g.j 2
20.d odd 2 1 6300.2.a.d 1
20.e even 4 2 6300.2.k.c 2
21.c even 2 1 3920.2.a.u 1
24.f even 2 1 2240.2.a.g 1
24.h odd 2 1 2240.2.a.r 1
28.d even 2 1 8820.2.a.r 1
60.h even 2 1 700.2.a.d 1
60.l odd 4 2 700.2.e.c 2
84.h odd 2 1 980.2.a.c 1
84.j odd 6 2 980.2.i.h 2
84.n even 6 2 980.2.i.d 2
420.o odd 2 1 4900.2.a.p 1
420.w even 4 2 4900.2.e.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.a.a 1 12.b even 2 1
560.2.a.c 1 3.b odd 2 1
700.2.a.d 1 60.h even 2 1
700.2.e.c 2 60.l odd 4 2
980.2.a.c 1 84.h odd 2 1
980.2.i.d 2 84.n even 6 2
980.2.i.h 2 84.j odd 6 2
1260.2.a.c 1 4.b odd 2 1
2240.2.a.g 1 24.f even 2 1
2240.2.a.r 1 24.h odd 2 1
2800.2.a.y 1 15.d odd 2 1
2800.2.g.j 2 15.e even 4 2
3920.2.a.u 1 21.c even 2 1
4900.2.a.p 1 420.o odd 2 1
4900.2.e.l 2 420.w even 4 2
5040.2.a.h 1 1.a even 1 1 trivial
6300.2.a.d 1 20.d odd 2 1
6300.2.k.c 2 20.e even 4 2
8820.2.a.r 1 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5040))\):

\( T_{11} - 3 \)
\( T_{13} + 1 \)
\( T_{17} - 3 \)
\( T_{19} + 2 \)
\( T_{29} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( 1 + T \)
$11$ \( -3 + T \)
$13$ \( 1 + T \)
$17$ \( -3 + T \)
$19$ \( 2 + T \)
$23$ \( 6 + T \)
$29$ \( -9 + T \)
$31$ \( 8 + T \)
$37$ \( 10 + T \)
$41$ \( T \)
$43$ \( 2 + T \)
$47$ \( 3 + T \)
$53$ \( T \)
$59$ \( -12 + T \)
$61$ \( -8 + T \)
$67$ \( 8 + T \)
$71$ \( T \)
$73$ \( -14 + T \)
$79$ \( 5 + T \)
$83$ \( 12 + T \)
$89$ \( 12 + T \)
$97$ \( -17 + T \)
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