Properties

Label 3024.1.cd.c
Level $3024$
Weight $1$
Character orbit 3024.cd
Analytic conductor $1.509$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,1,Mod(1999,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1999");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3024.cd (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.448084224.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{7} - q^{13} + (\zeta_{6} + 1) q^{19} + \zeta_{6} q^{25} + ( - \zeta_{6}^{2} + 1) q^{31} + \zeta_{6}^{2} q^{37} + (\zeta_{6}^{2} + \zeta_{6}) q^{43} + \zeta_{6}^{2} q^{49} - \zeta_{6}^{2} q^{61} - \zeta_{6} q^{73} - 2 \zeta_{6} q^{91} + q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{7} - 4 q^{13} + 3 q^{19} + q^{25} + 3 q^{31} - 2 q^{37} - q^{49} + q^{61} - q^{73} - 2 q^{91} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1999.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 0.500000 0.866025i 0 0 0
2431.1 0 0 0 0 0 0.500000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
28.g odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.1.cd.c yes 2
3.b odd 2 1 CM 3024.1.cd.c yes 2
4.b odd 2 1 3024.1.cd.b 2
7.c even 3 1 3024.1.cd.b 2
12.b even 2 1 3024.1.cd.b 2
21.h odd 6 1 3024.1.cd.b 2
28.g odd 6 1 inner 3024.1.cd.c yes 2
84.n even 6 1 inner 3024.1.cd.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3024.1.cd.b 2 4.b odd 2 1
3024.1.cd.b 2 7.c even 3 1
3024.1.cd.b 2 12.b even 2 1
3024.1.cd.b 2 21.h odd 6 1
3024.1.cd.c yes 2 1.a even 1 1 trivial
3024.1.cd.c yes 2 3.b odd 2 1 CM
3024.1.cd.c yes 2 28.g odd 6 1 inner
3024.1.cd.c yes 2 84.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3024, [\chi])\):

\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{19}^{2} - 3T_{19} + 3 \) Copy content Toggle raw display
\( T_{31}^{2} - 3T_{31} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 3 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 1)^{2} \) Copy content Toggle raw display
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