↑

Properties

Label 5004.2.a.i
Level $5004$
Weight $2$
Character orbit 5004.a
Self dual yes
Analytic conductor $39.957$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5004,2,Mod(1,5004)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5004.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5004, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5004 = 2^{2} \cdot 3^{2} \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5004.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.9571411714\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1668)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + (\beta_{2} - \beta_1 - 3) q^{11} + (2 \beta_{2} + 2) q^{13} + (\beta_{2} - 2) q^{17} + ( - \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_{2} - \beta_1 + 3) q^{23}+ \cdots + ( - \beta_{2} - 4 \beta_1 + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{5} + q^{7} - 10 q^{11} + 4 q^{13} - 7 q^{17} - 2 q^{19} + 10 q^{23} - q^{25} + 2 q^{29} + 7 q^{31} + 17 q^{35} + 13 q^{37} + 3 q^{43} - 23 q^{47} + 10 q^{49} - 20 q^{53} - 6 q^{55} - 13 q^{59}+ \cdots + 25 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 4x - 1 \) : Copy content Toggle raw display

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

Display \(\nu^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\nu^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.86081
2.11491
−0.254102
0 0 0 −3.32340 0 −4.18421 0 0 0
1.2 0 0 0 −0.357926 0 2.75698 0 0 0
1.3 0 0 0 1.68133 0 2.42723 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(139\) \( +1 \)

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 5004.2.a.i 3
3.b odd 2 1 1668.2.a.e ✓ 3
12.b even 2 1 6672.2.a.be 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1668.2.a.e ✓ 3 3.b odd 2 1
5004.2.a.i 3 1.a even 1 1 trivial
6672.2.a.be 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5004))\):

\( T_{5}^{3} + 2T_{5}^{2} - 5T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{3} - T_{7}^{2} - 15T_{7} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 2 T^{2} + \cdots - 2 \) Copy content Toggle raw display
$7$ \( T^{3} - T^{2} + \cdots + 28 \) Copy content Toggle raw display
$11$ \( T^{3} + 10 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$13$ \( T^{3} - 4 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$17$ \( T^{3} + 7 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 10 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{3} - 2 T^{2} + \cdots + 128 \) Copy content Toggle raw display
$31$ \( T^{3} - 7 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$37$ \( T^{3} - 13 T^{2} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{3} - 43T + 106 \) Copy content Toggle raw display
$43$ \( T^{3} - 3 T^{2} + \cdots + 188 \) Copy content Toggle raw display
$47$ \( T^{3} + 23 T^{2} + \cdots + 406 \) Copy content Toggle raw display
$53$ \( T^{3} + 20 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$59$ \( T^{3} + 13 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$61$ \( T^{3} + 5 T^{2} + \cdots + 178 \) Copy content Toggle raw display
$67$ \( T^{3} - 3 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$71$ \( T^{3} + 14 T^{2} + \cdots - 536 \) Copy content Toggle raw display
$73$ \( T^{3} - 2 T^{2} + \cdots - 98 \) Copy content Toggle raw display
$79$ \( T^{3} + 18 T^{2} + \cdots - 772 \) Copy content Toggle raw display
$83$ \( T^{3} + 24 T^{2} + \cdots - 698 \) Copy content Toggle raw display
$89$ \( T^{3} + 23 T^{2} + \cdots - 2738 \) Copy content Toggle raw display
$97$ \( T^{3} - 25 T^{2} + \cdots + 346 \) Copy content Toggle raw display
show more
show less