Properties

Label 4020.2.g.b
Level $4020$
Weight $2$
Character orbit 4020.g
Analytic conductor $32.100$
Analytic rank $0$
Dimension $24$
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] = [4020,2,Mod(1609,4020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4020.1609");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4020.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: no
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1609.1 0 1.00000i 0 −2.22339 0.237781i 0 1.39060i 0 −1.00000 0
1609.2 0 1.00000i 0 −2.07549 0.832066i 0 0.944795i 0 −1.00000 0
1609.3 0 1.00000i 0 −1.71974 1.42915i 0 1.37984i 0 −1.00000 0
1609.4 0 1.00000i 0 −1.63494 + 1.52544i 0 3.48382i 0 −1.00000 0
1609.5 0 1.00000i 0 −1.47707 + 1.67877i 0 3.23400i 0 −1.00000 0
1609.6 0 1.00000i 0 −0.880059 2.05560i 0 1.30492i 0 −1.00000 0
1609.7 0 1.00000i 0 −0.131247 + 2.23221i 0 5.13753i 0 −1.00000 0
1609.8 0 1.00000i 0 0.853331 + 2.06684i 0 2.56410i 0 −1.00000 0
1609.9 0 1.00000i 0 1.39369 1.74860i 0 3.23145i 0 −1.00000 0
1609.10 0 1.00000i 0 1.45456 1.69831i 0 0.243187i 0 −1.00000 0
1609.11 0 1.00000i 0 2.21022 0.339036i 0 3.30262i 0 −1.00000 0
1609.12 0 1.00000i 0 2.23014 0.162710i 0 0.770377i 0 −1.00000 0
1609.13 0 1.00000i 0 −2.22339 + 0.237781i 0 1.39060i 0 −1.00000 0
1609.14 0 1.00000i 0 −2.07549 + 0.832066i 0 0.944795i 0 −1.00000 0
1609.15 0 1.00000i 0 −1.71974 + 1.42915i 0 1.37984i 0 −1.00000 0
1609.16 0 1.00000i 0 −1.63494 1.52544i 0 3.48382i 0 −1.00000 0
1609.17 0 1.00000i 0 −1.47707 1.67877i 0 3.23400i 0 −1.00000 0
1609.18 0 1.00000i 0 −0.880059 + 2.05560i 0 1.30492i 0 −1.00000 0
1609.19 0 1.00000i 0 −0.131247 2.23221i 0 5.13753i 0 −1.00000 0
1609.20 0 1.00000i 0 0.853331 2.06684i 0 2.56410i 0 −1.00000 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1609.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4020.2.g.b 24
5.b even 2 1 inner 4020.2.g.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.g.b 24 1.a even 1 1 trivial
4020.2.g.b 24 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 84 T_{7}^{22} + 2910 T_{7}^{20} + 54892 T_{7}^{18} + 622067 T_{7}^{16} + 4393594 T_{7}^{14} + 19452614 T_{7}^{12} + 53425378 T_{7}^{10} + 89447705 T_{7}^{8} + 87863794 T_{7}^{6} + \cdots + 492804 \) acting on \(S_{2}^{\mathrm{new}}(4020, [\chi])\). Copy content Toggle raw display