Properties

Label 2004.1.g.c
Level $2004$
Weight $1$
Character orbit 2004.g
Self dual yes
Analytic conductor $1.000$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -2004
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2004,1,Mod(2003,2004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2004.2003");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2004.g (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: \(1.00012628532\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.334668.2
Artin image: $D_8$
Artin field: Galois closure of 8.2.96577152768.2

$q$-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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - \beta q^{5} - q^{6} - q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} + q^{4} - \beta q^{5} - q^{6} - q^{8} + q^{9} + \beta q^{10} + q^{12} - \beta q^{15} + q^{16} + \beta q^{17} - q^{18} - \beta q^{20} - q^{24} + q^{25} + q^{27} + \beta q^{30} - q^{32} - \beta q^{34} + q^{36} + \beta q^{40} + \beta q^{41} - \beta q^{43} - \beta q^{45} + q^{48} + q^{49} - q^{50} + \beta q^{51} + \beta q^{53} - q^{54} - \beta q^{60} + q^{64} + \beta q^{67} + \beta q^{68} - q^{72} + q^{75} + \beta q^{79} - \beta q^{80} + q^{81} - \beta q^{82} - 2 q^{85} + \beta q^{86} + \beta q^{90} - q^{96} - 2 q^{97} - q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 2 q^{16} - 2 q^{18} - 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{32} + 2 q^{36} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 2 q^{54} + 2 q^{64} - 2 q^{72} + 2 q^{75} + 2 q^{81} - 4 q^{85} - 2 q^{96} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2004\mathbb{Z}\right)^\times\).

\(n\) \(673\) \(1003\) \(1337\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2003.1
1.41421
−1.41421
−1.00000 1.00000 1.00000 −1.41421 −1.00000 0 −1.00000 1.00000 1.41421
2003.2 −1.00000 1.00000 1.00000 1.41421 −1.00000 0 −1.00000 1.00000 −1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
2004.g odd 2 1 CM by \(\Q(\sqrt{-501}) \)
12.b even 2 1 inner
167.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2004.1.g.c 2
3.b odd 2 1 2004.1.g.d yes 2
4.b odd 2 1 2004.1.g.d yes 2
12.b even 2 1 inner 2004.1.g.c 2
167.b odd 2 1 inner 2004.1.g.c 2
501.c even 2 1 2004.1.g.d yes 2
668.b even 2 1 2004.1.g.d yes 2
2004.g odd 2 1 CM 2004.1.g.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.1.g.c 2 1.a even 1 1 trivial
2004.1.g.c 2 12.b even 2 1 inner
2004.1.g.c 2 167.b odd 2 1 inner
2004.1.g.c 2 2004.g odd 2 1 CM
2004.1.g.d yes 2 3.b odd 2 1
2004.1.g.d yes 2 4.b odd 2 1
2004.1.g.d yes 2 501.c even 2 1
2004.1.g.d yes 2 668.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2004, [\chi])\):

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{179} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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