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Properties

Label 3040.2.j.f
Level $3040$
Weight $2$
Character orbit 3040.j
Analytic conductor $24.275$
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] = [3040,2,Mod(2431,3040)]
 
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, 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("3040.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.j (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: \(24.2745222145\)
Analytic rank: \(0\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 40 q^{5} + 32 q^{9} + 8 q^{17} + 40 q^{25} - 32 q^{45} - 64 q^{49} + 8 q^{57} + 16 q^{61} - 8 q^{73} - 8 q^{81} - 8 q^{85} - 32 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1 0 −2.99724 0 −1.00000 0 − 0.311594i 0 5.98347 0
2431.2 0 −2.99724 0 −1.00000 0 0.311594i 0 5.98347 0
2431.3 0 −2.83493 0 −1.00000 0 − 5.04828i 0 5.03680 0
2431.4 0 −2.83493 0 −1.00000 0 5.04828i 0 5.03680 0
2431.5 0 −2.74517 0 −1.00000 0 1.31597i 0 4.53595 0
2431.6 0 −2.74517 0 −1.00000 0 − 1.31597i 0 4.53595 0
2431.7 0 −2.18082 0 −1.00000 0 0.543154i 0 1.75597 0
2431.8 0 −2.18082 0 −1.00000 0 − 0.543154i 0 1.75597 0
2431.9 0 −1.97124 0 −1.00000 0 − 0.761303i 0 0.885789 0
2431.10 0 −1.97124 0 −1.00000 0 0.761303i 0 0.885789 0
2431.11 0 −1.60474 0 −1.00000 0 − 3.93171i 0 −0.424824 0
2431.12 0 −1.60474 0 −1.00000 0 3.93171i 0 −0.424824 0
2431.13 0 −1.36977 0 −1.00000 0 − 4.25145i 0 −1.12372 0
2431.14 0 −1.36977 0 −1.00000 0 4.25145i 0 −1.12372 0
2431.15 0 −0.409433 0 −1.00000 0 0.496765i 0 −2.83236 0
2431.16 0 −0.409433 0 −1.00000 0 − 0.496765i 0 −2.83236 0
2431.17 0 −0.337590 0 −1.00000 0 − 2.01484i 0 −2.88603 0
2431.18 0 −0.337590 0 −1.00000 0 2.01484i 0 −2.88603 0
2431.19 0 −0.262587 0 −1.00000 0 − 4.46898i 0 −2.93105 0
2431.20 0 −0.262587 0 −1.00000 0 4.46898i 0 −2.93105 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.b odd 2 1 inner
76.d 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.j.f ✓ 40
4.b odd 2 1 inner 3040.2.j.f ✓ 40
19.b odd 2 1 inner 3040.2.j.f ✓ 40
76.d even 2 1 inner 3040.2.j.f ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.j.f ✓ 40 1.a even 1 1 trivial
3040.2.j.f ✓ 40 4.b odd 2 1 inner
3040.2.j.f ✓ 40 19.b odd 2 1 inner
3040.2.j.f ✓ 40 76.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 38 T_{3}^{18} + 597 T_{3}^{16} - 5016 T_{3}^{14} + 24352 T_{3}^{12} - 68792 T_{3}^{10} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(3040, [\chi])\). Copy content Toggle raw display