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Properties

Label 4004.2.m.c
Level $4004$
Weight $2$
Character orbit 4004.m
Analytic conductor $31.972$
Analytic rank $0$
Dimension $36$
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] = [4004,2,Mod(2157,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4004.2157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.m (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: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2157.1 0 −3.27485 0 − 2.17871i 0 1.00000i 0 7.72466 0
2157.2 0 −3.27485 0 2.17871i 0 − 1.00000i 0 7.72466 0
2157.3 0 −3.10315 0 1.71558i 0 1.00000i 0 6.62954 0
2157.4 0 −3.10315 0 − 1.71558i 0 − 1.00000i 0 6.62954 0
2157.5 0 −2.75761 0 1.98236i 0 1.00000i 0 4.60441 0
2157.6 0 −2.75761 0 − 1.98236i 0 − 1.00000i 0 4.60441 0
2157.7 0 −2.10949 0 − 3.39046i 0 1.00000i 0 1.44995 0
2157.8 0 −2.10949 0 3.39046i 0 − 1.00000i 0 1.44995 0
2157.9 0 −1.64159 0 − 4.02832i 0 − 1.00000i 0 −0.305195 0
2157.10 0 −1.64159 0 4.02832i 0 1.00000i 0 −0.305195 0
2157.11 0 −1.08723 0 3.34996i 0 1.00000i 0 −1.81792 0
2157.12 0 −1.08723 0 − 3.34996i 0 − 1.00000i 0 −1.81792 0
2157.13 0 −1.07570 0 − 3.26058i 0 1.00000i 0 −1.84286 0
2157.14 0 −1.07570 0 3.26058i 0 − 1.00000i 0 −1.84286 0
2157.15 0 −0.819032 0 1.69723i 0 1.00000i 0 −2.32919 0
2157.16 0 −0.819032 0 − 1.69723i 0 − 1.00000i 0 −2.32919 0
2157.17 0 −0.722868 0 0.383791i 0 − 1.00000i 0 −2.47746 0
2157.18 0 −0.722868 0 − 0.383791i 0 1.00000i 0 −2.47746 0
2157.19 0 −0.189991 0 − 1.01340i 0 − 1.00000i 0 −2.96390 0
2157.20 0 −0.189991 0 1.01340i 0 1.00000i 0 −2.96390 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2157.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4004.2.m.c ✓ 36
13.b even 2 1 inner 4004.2.m.c ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.m.c ✓ 36 1.a even 1 1 trivial
4004.2.m.c ✓ 36 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 2 T_{3}^{17} - 35 T_{3}^{16} - 70 T_{3}^{15} + 474 T_{3}^{14} + 954 T_{3}^{13} - 3141 T_{3}^{12} + \cdots + 324 \) acting on \(S_{2}^{\mathrm{new}}(4004, [\chi])\). Copy content Toggle raw display