Properties

Label 1728.4.a.q
Level $1728$
Weight $4$
Character orbit 1728.a
Self dual yes
Analytic conductor $101.955$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,4,Mod(1,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, 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("1728.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.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: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
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)
 
\(f(q)\) \(=\) \( q + 17 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 17 q^{7} - 89 q^{13} - 107 q^{19} - 125 q^{25} + 308 q^{31} + 433 q^{37} + 520 q^{43} - 54 q^{49} + 901 q^{61} - 1007 q^{67} - 271 q^{73} + 503 q^{79} - 1513 q^{91} + 1853 q^{97}+O(q^{100}) \) 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 17.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.a.q 1
3.b odd 2 1 CM 1728.4.a.q 1
4.b odd 2 1 1728.4.a.p 1
8.b even 2 1 108.4.a.c 1
8.d odd 2 1 432.4.a.g 1
12.b even 2 1 1728.4.a.p 1
24.f even 2 1 432.4.a.g 1
24.h odd 2 1 108.4.a.c 1
72.j odd 6 2 324.4.e.d 2
72.n even 6 2 324.4.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.a.c 1 8.b even 2 1
108.4.a.c 1 24.h odd 2 1
324.4.e.d 2 72.j odd 6 2
324.4.e.d 2 72.n even 6 2
432.4.a.g 1 8.d odd 2 1
432.4.a.g 1 24.f even 2 1
1728.4.a.p 1 4.b odd 2 1
1728.4.a.p 1 12.b even 2 1
1728.4.a.q 1 1.a even 1 1 trivial
1728.4.a.q 1 3.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} - 17 \) Copy content Toggle raw display
\( T_{11} \) 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
$7$ \( T - 17 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 89 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 107 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 308 \) Copy content Toggle raw display
$37$ \( T - 433 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 520 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 901 \) Copy content Toggle raw display
$67$ \( T + 1007 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 271 \) Copy content Toggle raw display
$79$ \( T - 503 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 1853 \) Copy content Toggle raw display
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