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Properties

Label 2004.2.h.a
Level $2004$
Weight $2$
Character orbit 2004.h
Analytic conductor $16.002$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2004,2,Mod(1001,2004)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2004.1001");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2004.h (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: \(16.0020205651\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 8 q^{9} - 4 q^{19} + 4 q^{21} + 52 q^{25} + 12 q^{27} + 4 q^{31} - 8 q^{33} + 32 q^{49} + 24 q^{57} + 28 q^{61} - 26 q^{63} - 54 q^{75} - 24 q^{81} + 8 q^{85} + 14 q^{87} - 20 q^{93} + 36 q^{97} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1001.1 0 −1.72468 − 0.159598i 0 −1.90394 0 −0.264728 0 2.94906 + 0.550512i 0
1001.2 0 −1.72468 − 0.159598i 0 1.90394 0 −0.264728 0 2.94906 + 0.550512i 0
1001.3 0 −1.72468 + 0.159598i 0 −1.90394 0 −0.264728 0 2.94906 − 0.550512i 0
1001.4 0 −1.72468 + 0.159598i 0 1.90394 0 −0.264728 0 2.94906 − 0.550512i 0
1001.5 0 −1.48918 − 0.884505i 0 −1.46580 0 2.74616 0 1.43530 + 2.63437i 0
1001.6 0 −1.48918 − 0.884505i 0 1.46580 0 2.74616 0 1.43530 + 2.63437i 0
1001.7 0 −1.48918 + 0.884505i 0 −1.46580 0 2.74616 0 1.43530 − 2.63437i 0
1001.8 0 −1.48918 + 0.884505i 0 1.46580 0 2.74616 0 1.43530 − 2.63437i 0
1001.9 0 −1.39778 − 1.02284i 0 −3.89630 0 −0.835962 0 0.907577 + 2.85942i 0
1001.10 0 −1.39778 − 1.02284i 0 3.89630 0 −0.835962 0 0.907577 + 2.85942i 0
1001.11 0 −1.39778 + 1.02284i 0 −3.89630 0 −0.835962 0 0.907577 − 2.85942i 0
1001.12 0 −1.39778 + 1.02284i 0 3.89630 0 −0.835962 0 0.907577 − 2.85942i 0
1001.13 0 −1.34740 − 1.08836i 0 −3.01275 0 −3.17416 0 0.630964 + 2.93290i 0
1001.14 0 −1.34740 − 1.08836i 0 3.01275 0 −3.17416 0 0.630964 + 2.93290i 0
1001.15 0 −1.34740 + 1.08836i 0 −3.01275 0 −3.17416 0 0.630964 − 2.93290i 0
1001.16 0 −1.34740 + 1.08836i 0 3.01275 0 −3.17416 0 0.630964 − 2.93290i 0
1001.17 0 −0.650751 − 1.60516i 0 −0.493180 0 −3.79661 0 −2.15305 + 2.08911i 0
1001.18 0 −0.650751 − 1.60516i 0 0.493180 0 −3.79661 0 −2.15305 + 2.08911i 0
1001.19 0 −0.650751 + 1.60516i 0 −0.493180 0 −3.79661 0 −2.15305 − 2.08911i 0
1001.20 0 −0.650751 + 1.60516i 0 0.493180 0 −3.79661 0 −2.15305 − 2.08911i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1001.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
167.b odd 2 1 inner
501.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2004.2.h.a ✓ 56
3.b odd 2 1 inner 2004.2.h.a ✓ 56
167.b odd 2 1 inner 2004.2.h.a ✓ 56
501.c even 2 1 inner 2004.2.h.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.h.a ✓ 56 1.a even 1 1 trivial
2004.2.h.a ✓ 56 3.b odd 2 1 inner
2004.2.h.a ✓ 56 167.b odd 2 1 inner
2004.2.h.a ✓ 56 501.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(2004, [\chi])\).