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Properties

Label 4000.2.d.c
Level $4000$
Weight $2$
Character orbit 4000.d
Analytic conductor $31.940$
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] = [4000,2,Mod(2001,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4000.2001");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4000.d (of order \(2\), degree \(1\), 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: \(31.9401608085\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 1000)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2001.1 0 − 1.69676i 0 0 0 4.40040 0 0.120998 0
2001.2 0 1.69676i 0 0 0 4.40040 0 0.120998 0
2001.3 0 − 1.21236i 0 0 0 4.63239 0 1.53019 0
2001.4 0 1.21236i 0 0 0 4.63239 0 1.53019 0
2001.5 0 − 3.08659i 0 0 0 3.24242 0 −6.52702 0
2001.6 0 3.08659i 0 0 0 3.24242 0 −6.52702 0
2001.7 0 − 2.07677i 0 0 0 −2.55636 0 −1.31298 0
2001.8 0 2.07677i 0 0 0 −2.55636 0 −1.31298 0
2001.9 0 − 2.52102i 0 0 0 −0.987050 0 −3.35556 0
2001.10 0 2.52102i 0 0 0 −0.987050 0 −3.35556 0
2001.11 0 − 0.207209i 0 0 0 2.73181 0 2.95706 0
2001.12 0 0.207209i 0 0 0 2.73181 0 2.95706 0
2001.13 0 − 2.62662i 0 0 0 −0.269237 0 −3.89913 0
2001.14 0 2.62662i 0 0 0 −0.269237 0 −3.89913 0
2001.15 0 − 1.83526i 0 0 0 1.31564 0 −0.368171 0
2001.16 0 1.83526i 0 0 0 1.31564 0 −0.368171 0
2001.17 0 − 0.939252i 0 0 0 1.90117 0 2.11781 0
2001.18 0 0.939252i 0 0 0 1.90117 0 2.11781 0
2001.19 0 − 0.513027i 0 0 0 −1.12889 0 2.73680 0
2001.20 0 0.513027i 0 0 0 −1.12889 0 2.73680 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2001.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.2.d.c 40
4.b odd 2 1 1000.2.d.c ✓ 40
5.b even 2 1 inner 4000.2.d.c 40
5.c odd 4 1 4000.2.f.c 20
5.c odd 4 1 4000.2.f.d 20
8.b even 2 1 inner 4000.2.d.c 40
8.d odd 2 1 1000.2.d.c ✓ 40
20.d odd 2 1 1000.2.d.c ✓ 40
20.e even 4 1 1000.2.f.c 20
20.e even 4 1 1000.2.f.d 20
40.e odd 2 1 1000.2.d.c ✓ 40
40.f even 2 1 inner 4000.2.d.c 40
40.i odd 4 1 4000.2.f.c 20
40.i odd 4 1 4000.2.f.d 20
40.k even 4 1 1000.2.f.c 20
40.k even 4 1 1000.2.f.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.d.c ✓ 40 4.b odd 2 1
1000.2.d.c ✓ 40 8.d odd 2 1
1000.2.d.c ✓ 40 20.d odd 2 1
1000.2.d.c ✓ 40 40.e odd 2 1
1000.2.f.c 20 20.e even 4 1
1000.2.f.c 20 40.k even 4 1
1000.2.f.d 20 20.e even 4 1
1000.2.f.d 20 40.k even 4 1
4000.2.d.c 40 1.a even 1 1 trivial
4000.2.d.c 40 5.b even 2 1 inner
4000.2.d.c 40 8.b even 2 1 inner
4000.2.d.c 40 40.f even 2 1 inner
4000.2.f.c 20 5.c odd 4 1
4000.2.f.c 20 40.i odd 4 1
4000.2.f.d 20 5.c odd 4 1
4000.2.f.d 20 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4000, [\chi])\):

\( T_{3}^{20} + 36 T_{3}^{18} + 538 T_{3}^{16} + 4348 T_{3}^{14} + 20735 T_{3}^{12} + 59708 T_{3}^{10} + \cdots + 256 \) Copy content Toggle raw display
\( T_{7}^{20} - 73 T_{7}^{18} + 2133 T_{7}^{16} - 32388 T_{7}^{14} + 279482 T_{7}^{12} - 1409126 T_{7}^{10} + \cdots + 119961 \) Copy content Toggle raw display