Properties

Label 5040.2.k.a
Level $5040$
Weight $2$
Character orbit 5040.k
Analytic conductor $40.245$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5040,2,Mod(1889,5040)]
 
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, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5040.1889");
 
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.k (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + (\beta_{2} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + (\beta_{2} + \beta_1) q^{7} + \beta_{2} q^{11} + (\beta_{2} - 2 \beta_1) q^{13} - 2 \beta_{3} q^{17} - 6 q^{23} - 5 q^{25} + 2 \beta_{2} q^{29} + ( - 4 \beta_{2} + 3 \beta_1) q^{35} + 3 \beta_{2} q^{37} + ( - 3 \beta_{2} + 6 \beta_1) q^{41} + 6 \beta_{2} q^{43} + 2 \beta_{3} q^{47} + (3 \beta_{3} - 2) q^{49} + 6 q^{53} + ( - \beta_{2} + 2 \beta_1) q^{55} + (3 \beta_{2} - 6 \beta_1) q^{59} + 6 \beta_{3} q^{61} + 5 \beta_{2} q^{65} + 4 \beta_{2} q^{71} + ( - 2 \beta_{2} + 4 \beta_1) q^{73} + (\beta_{3} - 3) q^{77} + 4 q^{79} - 4 \beta_{3} q^{83} - 10 q^{85} + (3 \beta_{2} - 6 \beta_1) q^{89} + ( - 3 \beta_{3} - 5) q^{91} + (4 \beta_{2} - 8 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{23} - 20 q^{25} - 8 q^{49} + 24 q^{53} - 12 q^{77} + 16 q^{79} - 40 q^{85} - 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1889.1
−1.58114 0.707107i
1.58114 + 0.707107i
−1.58114 + 0.707107i
1.58114 0.707107i
0 0 0 2.23607i 0 −1.58114 2.12132i 0 0 0
1889.2 0 0 0 2.23607i 0 1.58114 + 2.12132i 0 0 0
1889.3 0 0 0 2.23607i 0 −1.58114 + 2.12132i 0 0 0
1889.4 0 0 0 2.23607i 0 1.58114 2.12132i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5040.2.k.a 4
3.b odd 2 1 5040.2.k.d 4
4.b odd 2 1 630.2.d.d yes 4
5.b even 2 1 5040.2.k.d 4
7.b odd 2 1 inner 5040.2.k.a 4
12.b even 2 1 630.2.d.a 4
15.d odd 2 1 inner 5040.2.k.a 4
20.d odd 2 1 630.2.d.a 4
20.e even 4 2 3150.2.b.c 8
21.c even 2 1 5040.2.k.d 4
28.d even 2 1 630.2.d.d yes 4
35.c odd 2 1 5040.2.k.d 4
60.h even 2 1 630.2.d.d yes 4
60.l odd 4 2 3150.2.b.c 8
84.h odd 2 1 630.2.d.a 4
105.g even 2 1 inner 5040.2.k.a 4
140.c even 2 1 630.2.d.a 4
140.j odd 4 2 3150.2.b.c 8
420.o odd 2 1 630.2.d.d yes 4
420.w even 4 2 3150.2.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.d.a 4 12.b even 2 1
630.2.d.a 4 20.d odd 2 1
630.2.d.a 4 84.h odd 2 1
630.2.d.a 4 140.c even 2 1
630.2.d.d yes 4 4.b odd 2 1
630.2.d.d yes 4 28.d even 2 1
630.2.d.d yes 4 60.h even 2 1
630.2.d.d yes 4 420.o odd 2 1
3150.2.b.c 8 20.e even 4 2
3150.2.b.c 8 60.l odd 4 2
3150.2.b.c 8 140.j odd 4 2
3150.2.b.c 8 420.w even 4 2
5040.2.k.a 4 1.a even 1 1 trivial
5040.2.k.a 4 7.b odd 2 1 inner
5040.2.k.a 4 15.d odd 2 1 inner
5040.2.k.a 4 105.g even 2 1 inner
5040.2.k.d 4 3.b odd 2 1
5040.2.k.d 4 5.b even 2 1
5040.2.k.d 4 21.c even 2 1
5040.2.k.d 4 35.c odd 2 1

Hecke kernels

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

\( T_{11}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 10 \) Copy content Toggle raw display
\( T_{23} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 4T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T + 6)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 160)^{2} \) Copy content Toggle raw display
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