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Properties

Label 4032.2.v.e
Level $4032$
Weight $2$
Character orbit 4032.v
Analytic conductor $32.196$
Analytic rank $0$
Dimension $40$
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] = [4032,2,Mod(1583,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.1583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.v (of order \(4\), degree \(2\), not 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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{7} - 24 q^{13} + 32 q^{19} - 8 q^{37} - 32 q^{43} + 40 q^{49} - 48 q^{55} - 24 q^{61} + 64 q^{85} + 24 q^{91} + 64 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1583.1 0 0 0 −2.96859 − 2.96859i 0 −1.00000 0 0 0
1583.2 0 0 0 −2.62814 − 2.62814i 0 −1.00000 0 0 0
1583.3 0 0 0 −2.51504 − 2.51504i 0 −1.00000 0 0 0
1583.4 0 0 0 −2.12043 − 2.12043i 0 −1.00000 0 0 0
1583.5 0 0 0 −1.65702 − 1.65702i 0 −1.00000 0 0 0
1583.6 0 0 0 −1.17902 − 1.17902i 0 −1.00000 0 0 0
1583.7 0 0 0 −0.925496 − 0.925496i 0 −1.00000 0 0 0
1583.8 0 0 0 −0.667815 − 0.667815i 0 −1.00000 0 0 0
1583.9 0 0 0 −0.111394 − 0.111394i 0 −1.00000 0 0 0
1583.10 0 0 0 −0.0893433 − 0.0893433i 0 −1.00000 0 0 0
1583.11 0 0 0 0.0893433 + 0.0893433i 0 −1.00000 0 0 0
1583.12 0 0 0 0.111394 + 0.111394i 0 −1.00000 0 0 0
1583.13 0 0 0 0.667815 + 0.667815i 0 −1.00000 0 0 0
1583.14 0 0 0 0.925496 + 0.925496i 0 −1.00000 0 0 0
1583.15 0 0 0 1.17902 + 1.17902i 0 −1.00000 0 0 0
1583.16 0 0 0 1.65702 + 1.65702i 0 −1.00000 0 0 0
1583.17 0 0 0 2.12043 + 2.12043i 0 −1.00000 0 0 0
1583.18 0 0 0 2.51504 + 2.51504i 0 −1.00000 0 0 0
1583.19 0 0 0 2.62814 + 2.62814i 0 −1.00000 0 0 0
1583.20 0 0 0 2.96859 + 2.96859i 0 −1.00000 0 0 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1583.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.v.e 40
3.b odd 2 1 inner 4032.2.v.e 40
4.b odd 2 1 1008.2.v.e ✓ 40
12.b even 2 1 1008.2.v.e ✓ 40
16.e even 4 1 1008.2.v.e ✓ 40
16.f odd 4 1 inner 4032.2.v.e 40
48.i odd 4 1 1008.2.v.e ✓ 40
48.k even 4 1 inner 4032.2.v.e 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.v.e ✓ 40 4.b odd 2 1
1008.2.v.e ✓ 40 12.b even 2 1
1008.2.v.e ✓ 40 16.e even 4 1
1008.2.v.e ✓ 40 48.i odd 4 1
4032.2.v.e 40 1.a even 1 1 trivial
4032.2.v.e 40 3.b odd 2 1 inner
4032.2.v.e 40 16.f odd 4 1 inner
4032.2.v.e 40 48.k even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{40} + 784 T_{5}^{36} + 224304 T_{5}^{32} + 29085120 T_{5}^{28} + 1705059168 T_{5}^{24} + \cdots + 65536 \) Copy content Toggle raw display
\( T_{11}^{40} + 3664 T_{11}^{36} + 4437680 T_{11}^{32} + 2045565248 T_{11}^{28} + 297547122016 T_{11}^{24} + \cdots + 426337261060096 \) Copy content Toggle raw display