Properties

Label 6400.2.a.cl
Level $6400$
Weight $2$
Character orbit 6400.a
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $4$
CM discriminant -40
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6400,2,Mod(1,6400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6400, 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("6400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.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: \(51.1042572936\)
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: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 640)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{7} - 3 q^{9} + \beta_{2} q^{11} + \beta_{3} q^{13} + \beta_{2} q^{19} - 3 \beta_1 q^{23} - \beta_{3} q^{37} + 2 q^{41} + \beta_1 q^{47} + q^{49} + 3 \beta_{3} q^{53} - \beta_{2} q^{59} + 3 \beta_1 q^{63} - 4 \beta_{3} q^{77} + 9 q^{81} + 14 q^{89} - 2 \beta_{2} q^{91} - 3 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 8 q^{41} + 4 q^{49} + 36 q^{81} + 56 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 8\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 2\beta_1 \) 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
−0.874032
2.28825
−2.28825
0.874032
0 0 0 0 0 −2.82843 0 −3.00000 0
1.2 0 0 0 0 0 −2.82843 0 −3.00000 0
1.3 0 0 0 0 0 2.82843 0 −3.00000 0
1.4 0 0 0 0 0 2.82843 0 −3.00000 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.cl 4
4.b odd 2 1 inner 6400.2.a.cl 4
5.b even 2 1 inner 6400.2.a.cl 4
5.c odd 4 2 1280.2.c.l 4
8.b even 2 1 inner 6400.2.a.cl 4
8.d odd 2 1 inner 6400.2.a.cl 4
16.e even 4 2 3200.2.d.w 4
16.f odd 4 2 3200.2.d.w 4
20.d odd 2 1 inner 6400.2.a.cl 4
20.e even 4 2 1280.2.c.l 4
40.e odd 2 1 CM 6400.2.a.cl 4
40.f even 2 1 inner 6400.2.a.cl 4
40.i odd 4 2 1280.2.c.l 4
40.k even 4 2 1280.2.c.l 4
80.i odd 4 2 640.2.f.d 4
80.j even 4 2 640.2.f.d 4
80.k odd 4 2 3200.2.d.w 4
80.q even 4 2 3200.2.d.w 4
80.s even 4 2 640.2.f.d 4
80.t odd 4 2 640.2.f.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.d 4 80.i odd 4 2
640.2.f.d 4 80.j even 4 2
640.2.f.d 4 80.s even 4 2
640.2.f.d 4 80.t odd 4 2
1280.2.c.l 4 5.c odd 4 2
1280.2.c.l 4 20.e even 4 2
1280.2.c.l 4 40.i odd 4 2
1280.2.c.l 4 40.k even 4 2
3200.2.d.w 4 16.e even 4 2
3200.2.d.w 4 16.f odd 4 2
3200.2.d.w 4 80.k odd 4 2
3200.2.d.w 4 80.q even 4 2
6400.2.a.cl 4 1.a even 1 1 trivial
6400.2.a.cl 4 4.b odd 2 1 inner
6400.2.a.cl 4 5.b even 2 1 inner
6400.2.a.cl 4 8.b even 2 1 inner
6400.2.a.cl 4 8.d odd 2 1 inner
6400.2.a.cl 4 20.d odd 2 1 inner
6400.2.a.cl 4 40.e odd 2 1 CM
6400.2.a.cl 4 40.f even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6400))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 40 \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display
\( T_{31} \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) 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
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T - 14)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
show more
show less