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Properties

Label 3920.2.k.f
Level $3920$
Weight $2$
Character orbit 3920.k
Analytic conductor $31.301$
Analytic rank $0$
Dimension $32$
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] = [3920,2,Mod(2351,3920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3920.2351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.k (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: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 16 q^{9} - 32 q^{25} - 16 q^{29} - 64 q^{53} + 64 q^{57} + 16 q^{65} + 64 q^{81} + 48 q^{85} + 64 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1 0 −3.31796 0 1.00000i 0 0 0 8.00884 0
2351.2 0 −3.31796 0 − 1.00000i 0 0 0 8.00884 0
2351.3 0 −2.38802 0 − 1.00000i 0 0 0 2.70262 0
2351.4 0 −2.38802 0 1.00000i 0 0 0 2.70262 0
2351.5 0 −2.31556 0 1.00000i 0 0 0 2.36182 0
2351.6 0 −2.31556 0 − 1.00000i 0 0 0 2.36182 0
2351.7 0 −1.62129 0 1.00000i 0 0 0 −0.371430 0
2351.8 0 −1.62129 0 − 1.00000i 0 0 0 −0.371430 0
2351.9 0 −1.58591 0 − 1.00000i 0 0 0 −0.484899 0
2351.10 0 −1.58591 0 1.00000i 0 0 0 −0.484899 0
2351.11 0 −0.655965 0 1.00000i 0 0 0 −2.56971 0
2351.12 0 −0.655965 0 − 1.00000i 0 0 0 −2.56971 0
2351.13 0 −0.583510 0 − 1.00000i 0 0 0 −2.65952 0
2351.14 0 −0.583510 0 1.00000i 0 0 0 −2.65952 0
2351.15 0 −0.110764 0 1.00000i 0 0 0 −2.98773 0
2351.16 0 −0.110764 0 − 1.00000i 0 0 0 −2.98773 0
2351.17 0 0.110764 0 − 1.00000i 0 0 0 −2.98773 0
2351.18 0 0.110764 0 1.00000i 0 0 0 −2.98773 0
2351.19 0 0.583510 0 1.00000i 0 0 0 −2.65952 0
2351.20 0 0.583510 0 − 1.00000i 0 0 0 −2.65952 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2351.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.2.k.f ✓ 32
4.b odd 2 1 inner 3920.2.k.f ✓ 32
7.b odd 2 1 inner 3920.2.k.f ✓ 32
28.d even 2 1 inner 3920.2.k.f ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.2.k.f ✓ 32 1.a even 1 1 trivial
3920.2.k.f ✓ 32 4.b odd 2 1 inner
3920.2.k.f ✓ 32 7.b odd 2 1 inner
3920.2.k.f ✓ 32 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 28T_{3}^{14} + 294T_{3}^{12} - 1484T_{3}^{10} + 3773T_{3}^{8} - 4568T_{3}^{6} + 2172T_{3}^{4} - 352T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(3920, [\chi])\). Copy content Toggle raw display