Properties

Label 5040.2.a.k
Level $5040$
Weight $2$
Character orbit 5040.a
Self dual yes
Analytic conductor $40.245$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5040,2,Mod(1,5040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + q^{7} - 4 q^{11} - 2 q^{13} - 2 q^{17} - 4 q^{19} - 8 q^{23} + q^{25} + 2 q^{29} - q^{35} + 6 q^{37} + 6 q^{41} + 4 q^{43} + q^{49} + 10 q^{53} + 4 q^{55} + 12 q^{59} + 14 q^{61} + 2 q^{65} + 12 q^{67} - 8 q^{71} + 10 q^{73} - 4 q^{77} - 16 q^{79} - 12 q^{83} + 2 q^{85} - 10 q^{89} - 2 q^{91} + 4 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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.k 1
3.b odd 2 1 1680.2.a.j 1
4.b odd 2 1 630.2.a.a 1
12.b even 2 1 210.2.a.e 1
15.d odd 2 1 8400.2.a.ce 1
20.d odd 2 1 3150.2.a.bp 1
20.e even 4 2 3150.2.g.q 2
24.f even 2 1 6720.2.a.j 1
24.h odd 2 1 6720.2.a.bq 1
28.d even 2 1 4410.2.a.t 1
60.h even 2 1 1050.2.a.c 1
60.l odd 4 2 1050.2.g.g 2
84.h odd 2 1 1470.2.a.j 1
84.j odd 6 2 1470.2.i.j 2
84.n even 6 2 1470.2.i.a 2
420.o odd 2 1 7350.2.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.e 1 12.b even 2 1
630.2.a.a 1 4.b odd 2 1
1050.2.a.c 1 60.h even 2 1
1050.2.g.g 2 60.l odd 4 2
1470.2.a.j 1 84.h odd 2 1
1470.2.i.a 2 84.n even 6 2
1470.2.i.j 2 84.j odd 6 2
1680.2.a.j 1 3.b odd 2 1
3150.2.a.bp 1 20.d odd 2 1
3150.2.g.q 2 20.e even 4 2
4410.2.a.t 1 28.d even 2 1
5040.2.a.k 1 1.a even 1 1 trivial
6720.2.a.j 1 24.f even 2 1
6720.2.a.bq 1 24.h odd 2 1
7350.2.a.w 1 420.o odd 2 1
8400.2.a.ce 1 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5040))\):

\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17} + 2 \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display
\( T_{29} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 14 \) Copy content Toggle raw display
$67$ \( T - 12 \) Copy content Toggle raw display
$71$ \( T + 8 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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