Properties

Label 2003.1.b.b
Level $2003$
Weight $1$
Character orbit 2003.b
Self dual yes
Analytic conductor $1.000$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -2003
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2003,1,Mod(2002,2003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2003, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2003.2002");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2003 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2003.b (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.999627220304\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.16096216216081.1
Artin image: $D_9$
Artin field: Galois closure of 9.1.16096216216081.1

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + q^{4} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + q^{4} + (\beta_{2} + 1) q^{9} - \beta_1 q^{12} + ( - \beta_{2} + \beta_1) q^{13} + q^{16} - q^{19} + q^{25} + ( - \beta_1 - 1) q^{27} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} + \beta_1 - 1) q^{39} + ( - \beta_{2} + \beta_1) q^{47} - \beta_1 q^{48} + q^{49} + ( - \beta_{2} + \beta_1) q^{52} - q^{53} + \beta_1 q^{57} + \beta_{2} q^{59} + q^{64} + \beta_{2} q^{73} - \beta_1 q^{75} - q^{76} + \beta_{2} q^{79} + (\beta_1 + 1) q^{81} - q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{4} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{4} + 3 q^{9} + 3 q^{16} - 3 q^{19} + 3 q^{25} - 3 q^{27} + 3 q^{36} - 3 q^{39} + 3 q^{49} - 3 q^{53} + 3 q^{64} - 3 q^{76} + 3 q^{81} - 3 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2003\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2002.1
1.87939
−0.347296
−1.53209
0 −1.87939 1.00000 0 0 0 0 2.53209 0
2002.2 0 0.347296 1.00000 0 0 0 0 −0.879385 0
2002.3 0 1.53209 1.00000 0 0 0 0 1.34730 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
2003.b odd 2 1 CM by \(\Q(\sqrt{-2003}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2003.1.b.b 3
2003.b odd 2 1 CM 2003.1.b.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2003.1.b.b 3 1.a even 1 1 trivial
2003.1.b.b 3 2003.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 3T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2003, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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