Properties

Label 800.4.a.e
Level $800$
Weight $4$
Character orbit 800.a
Self dual yes
Analytic conductor $47.202$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,4,Mod(1,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
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)
 
\(f(q)\) \(=\) \( q - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 27 q^{9} - 92 q^{13} + 104 q^{17} + 130 q^{29} + 396 q^{37} + 230 q^{41} - 343 q^{49} + 572 q^{53} - 830 q^{61} + 592 q^{73} + 729 q^{81} + 1670 q^{89} + 1816 q^{97}+O(q^{100}) \) 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 0 0 0 0 −27.0000 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.4.a.e 1
4.b odd 2 1 CM 800.4.a.e 1
5.b even 2 1 800.4.a.g 1
5.c odd 4 2 160.4.c.a 2
8.b even 2 1 1600.4.a.bb 1
8.d odd 2 1 1600.4.a.bb 1
15.e even 4 2 1440.4.f.e 2
20.d odd 2 1 800.4.a.g 1
20.e even 4 2 160.4.c.a 2
40.e odd 2 1 1600.4.a.z 1
40.f even 2 1 1600.4.a.z 1
40.i odd 4 2 320.4.c.e 2
40.k even 4 2 320.4.c.e 2
60.l odd 4 2 1440.4.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.a 2 5.c odd 4 2
160.4.c.a 2 20.e even 4 2
320.4.c.e 2 40.i odd 4 2
320.4.c.e 2 40.k even 4 2
800.4.a.e 1 1.a even 1 1 trivial
800.4.a.e 1 4.b odd 2 1 CM
800.4.a.g 1 5.b even 2 1
800.4.a.g 1 20.d odd 2 1
1440.4.f.e 2 15.e even 4 2
1440.4.f.e 2 60.l odd 4 2
1600.4.a.z 1 40.e odd 2 1
1600.4.a.z 1 40.f even 2 1
1600.4.a.bb 1 8.b even 2 1
1600.4.a.bb 1 8.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 92 \) 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
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 92 \) Copy content Toggle raw display
$17$ \( T - 104 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 130 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 396 \) Copy content Toggle raw display
$41$ \( T - 230 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 572 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 830 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 592 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 1670 \) Copy content Toggle raw display
$97$ \( T - 1816 \) Copy content Toggle raw display
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