Properties

Label 4000.2.a.a
Level $4000$
Weight $2$
Character orbit 4000.a
Self dual yes
Analytic conductor $31.940$
Analytic rank $1$
Dimension $4$
CM discriminant -20
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4000,2,Mod(1,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4000.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4000.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: \(31.9401608085\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2} - \beta_1 - 2) q^{3} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{7} + ( - \beta_{2} + 2 \beta_1 + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2} - \beta_1 - 2) q^{3} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{7} + ( - \beta_{2} + 2 \beta_1 + 5) q^{9} + ( - 8 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{21} + ( - 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 3) q^{23} + (2 \beta_{3} + \beta_{2} - 7 \beta_1 - 7) q^{27} + (2 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 3) q^{29} + ( - 6 \beta_{3} + \beta_{2} + 3) q^{41} + ( - 5 \beta_{3} - 5 \beta_{2} + \beta_1 - 6) q^{43} + ( - 5 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 8) q^{47} + (2 \beta_{3} + 3 \beta_{2} + 16) q^{49} + (3 \beta_{2} + 4 \beta_1 - 6) q^{61} + (8 \beta_{3} + 21 \beta_{2} + 7 \beta_1 + 7) q^{63} + ( - 10 \beta_{2} - 2) q^{67} + ( - 2 \beta_{3} + 8 \beta_{2} + 12 \beta_1 + 5) q^{69} + ( - 4 \beta_{3} - 4 \beta_{2} + 14 \beta_1 + 14) q^{81} + (3 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 3) q^{83} + ( - \beta_{3} - 16 \beta_{2} + 6 \beta_1 + 1) q^{87} + ( - 8 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{3} - 8 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{3} - 8 q^{7} + 22 q^{9} - 8 q^{21} - 16 q^{23} - 30 q^{27} + 6 q^{29} + 10 q^{41} - 14 q^{43} - 22 q^{47} + 58 q^{49} - 30 q^{61} - 14 q^{63} + 12 q^{67} + 4 q^{69} + 64 q^{81} - 4 q^{83} + 36 q^{87} - 6 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{20} + \zeta_{20}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) 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
1.17557
1.90211
−1.90211
−1.17557
0 −3.45965 0 0 0 −3.78704 0 8.96917 0
1.2 0 −3.34458 0 0 0 5.21586 0 8.18619 0
1.3 0 −1.89149 0 0 0 −4.74373 0 0.577740 0
1.4 0 2.69572 0 0 0 −4.68510 0 4.26689 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.2.a.a 4
4.b odd 2 1 4000.2.a.j yes 4
5.b even 2 1 4000.2.a.j yes 4
5.c odd 4 2 4000.2.c.a 8
8.b even 2 1 8000.2.a.bt 4
8.d odd 2 1 8000.2.a.y 4
20.d odd 2 1 CM 4000.2.a.a 4
20.e even 4 2 4000.2.c.a 8
40.e odd 2 1 8000.2.a.bt 4
40.f even 2 1 8000.2.a.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.a 4 1.a even 1 1 trivial
4000.2.a.a 4 20.d odd 2 1 CM
4000.2.a.j yes 4 4.b odd 2 1
4000.2.a.j yes 4 5.b even 2 1
4000.2.c.a 8 5.c odd 4 2
4000.2.c.a 8 20.e even 4 2
8000.2.a.y 4 8.d odd 2 1
8000.2.a.y 4 40.f even 2 1
8000.2.a.bt 4 8.b even 2 1
8000.2.a.bt 4 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4000))\):

\( T_{3}^{4} + 6T_{3}^{3} + T_{3}^{2} - 44T_{3} - 59 \) Copy content Toggle raw display
\( T_{7}^{4} + 8T_{7}^{3} - 11T_{7}^{2} - 218T_{7} - 439 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + T^{2} - 44 T - 59 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 8 T^{3} - 11 T^{2} - 218 T - 439 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 16 T^{3} + 21 T^{2} + \cdots - 1399 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} - 109 T^{2} + \cdots + 281 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} - 145 T^{2} + \cdots + 4705 \) Copy content Toggle raw display
$43$ \( T^{4} + 14 T^{3} - 119 T^{2} + \cdots - 7339 \) Copy content Toggle raw display
$47$ \( T^{4} + 22 T^{3} - 51 T^{2} + \cdots - 20899 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 30 T^{3} + 235 T^{2} + \cdots - 3895 \) Copy content Toggle raw display
$67$ \( (T^{2} - 6 T - 116)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 4 T^{3} - 159 T^{2} + \cdots - 1279 \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} - 409 T^{2} + \cdots + 24881 \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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