Properties

Label 3040.2.e.a
Level $3040$
Weight $2$
Character orbit 3040.e
Analytic conductor $24.275$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3040,2,Mod(911,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("3040.911");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.e (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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(80\)
Twist minimal: no (minimal twist has level 760)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - 80 q^{9} - 8 q^{19} - 80 q^{25} - 112 q^{49} - 8 q^{57} - 32 q^{73} + 96 q^{81} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
911.1 0 3.34453i 0 1.00000i 0 3.87119i 0 −8.18585 0
911.2 0 3.34453i 0 1.00000i 0 3.87119i 0 −8.18585 0
911.3 0 3.23880i 0 1.00000i 0 0.457390i 0 −7.48983 0
911.4 0 3.23880i 0 1.00000i 0 0.457390i 0 −7.48983 0
911.5 0 2.99395i 0 1.00000i 0 3.07859i 0 −5.96377 0
911.6 0 2.99395i 0 1.00000i 0 3.07859i 0 −5.96377 0
911.7 0 2.70495i 0 1.00000i 0 3.65554i 0 −4.31673 0
911.8 0 2.70495i 0 1.00000i 0 3.65554i 0 −4.31673 0
911.9 0 2.56963i 0 1.00000i 0 2.10549i 0 −3.60298 0
911.10 0 2.56963i 0 1.00000i 0 2.10549i 0 −3.60298 0
911.11 0 2.46872i 0 1.00000i 0 2.44811i 0 −3.09460 0
911.12 0 2.46872i 0 1.00000i 0 2.44811i 0 −3.09460 0
911.13 0 2.38464i 0 1.00000i 0 4.57467i 0 −2.68651 0
911.14 0 2.38464i 0 1.00000i 0 4.57467i 0 −2.68651 0
911.15 0 2.25164i 0 1.00000i 0 0.940149i 0 −2.06991 0
911.16 0 2.25164i 0 1.00000i 0 0.940149i 0 −2.06991 0
911.17 0 2.07809i 0 1.00000i 0 0.0366154i 0 −1.31848 0
911.18 0 2.07809i 0 1.00000i 0 0.0366154i 0 −1.31848 0
911.19 0 2.06995i 0 1.00000i 0 4.61911i 0 −1.28468 0
911.20 0 2.06995i 0 1.00000i 0 4.61911i 0 −1.28468 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 911.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3040.2.e.a 80
4.b odd 2 1 760.2.e.a 80
8.b even 2 1 760.2.e.a 80
8.d odd 2 1 inner 3040.2.e.a 80
19.b odd 2 1 inner 3040.2.e.a 80
76.d even 2 1 760.2.e.a 80
152.b even 2 1 inner 3040.2.e.a 80
152.g odd 2 1 760.2.e.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.e.a 80 4.b odd 2 1
760.2.e.a 80 8.b even 2 1
760.2.e.a 80 76.d even 2 1
760.2.e.a 80 152.g odd 2 1
3040.2.e.a 80 1.a even 1 1 trivial
3040.2.e.a 80 8.d odd 2 1 inner
3040.2.e.a 80 19.b odd 2 1 inner
3040.2.e.a 80 152.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(3040, [\chi])\).