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Properties

Label 2004.3.e.a
Level $2004$
Weight $3$
Character orbit 2004.e
Analytic conductor $54.605$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2004,3,Mod(1669,2004)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2004.1669");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2004.e (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: no
Analytic conductor: \(54.6050449778\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 168 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 168 q^{9} - 20 q^{11} - 40 q^{19} - 332 q^{25} - 52 q^{29} - 56 q^{31} - 24 q^{33} - 92 q^{47} + 512 q^{49} - 24 q^{57} + 112 q^{61} - 100 q^{65} + 72 q^{75} + 56 q^{77} + 504 q^{81} + 140 q^{85} + 72 q^{87} - 28 q^{89} + 204 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1669.1 0 −1.73205 0 − 8.72551i 0 −2.23221 0 3.00000 0
1669.2 0 −1.73205 0 − 8.37642i 0 −9.71534 0 3.00000 0
1669.3 0 −1.73205 0 − 8.13040i 0 0.301794 0 3.00000 0
1669.4 0 −1.73205 0 − 8.07963i 0 9.85629 0 3.00000 0
1669.5 0 −1.73205 0 − 7.57323i 0 1.74806 0 3.00000 0
1669.6 0 −1.73205 0 − 4.99429i 0 −4.87137 0 3.00000 0
1669.7 0 −1.73205 0 − 4.61753i 0 −13.6351 0 3.00000 0
1669.8 0 −1.73205 0 − 4.23508i 0 11.5054 0 3.00000 0
1669.9 0 −1.73205 0 − 3.88503i 0 4.36546 0 3.00000 0
1669.10 0 −1.73205 0 − 3.44886i 0 −0.789497 0 3.00000 0
1669.11 0 −1.73205 0 − 2.48369i 0 9.17063 0 3.00000 0
1669.12 0 −1.73205 0 − 2.33215i 0 7.05567 0 3.00000 0
1669.13 0 −1.73205 0 − 2.30437i 0 −9.25748 0 3.00000 0
1669.14 0 −1.73205 0 − 0.499440i 0 −3.50232 0 3.00000 0
1669.15 0 −1.73205 0 0.499440i 0 −3.50232 0 3.00000 0
1669.16 0 −1.73205 0 2.30437i 0 −9.25748 0 3.00000 0
1669.17 0 −1.73205 0 2.33215i 0 7.05567 0 3.00000 0
1669.18 0 −1.73205 0 2.48369i 0 9.17063 0 3.00000 0
1669.19 0 −1.73205 0 3.44886i 0 −0.789497 0 3.00000 0
1669.20 0 −1.73205 0 3.88503i 0 4.36546 0 3.00000 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1669.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 2004.3.e.a ✓ 56
167.b odd 2 1 inner 2004.3.e.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.3.e.a ✓ 56 1.a even 1 1 trivial
2004.3.e.a ✓ 56 167.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(2004, [\chi])\).