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Properties

Label 4014.2.d.a
Level $4014$
Weight $2$
Character orbit 4014.d
Analytic conductor $32.052$
Analytic rank $0$
Dimension $72$
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] = [4014,2,Mod(4013,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4014.4013");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.d (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: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(72\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4013.1 − 1.00000i 0 −1.00000 3.51050 0 3.72470 1.00000i 0 − 3.51050i
4013.2 1.00000i 0 −1.00000 3.51050 0 3.72470 − 1.00000i 0 3.51050i
4013.3 − 1.00000i 0 −1.00000 −0.969046 0 −3.46872 1.00000i 0 0.969046i
4013.4 1.00000i 0 −1.00000 −0.969046 0 −3.46872 − 1.00000i 0 − 0.969046i
4013.5 − 1.00000i 0 −1.00000 −2.31709 0 −2.47546 1.00000i 0 2.31709i
4013.6 1.00000i 0 −1.00000 −2.31709 0 −2.47546 − 1.00000i 0 − 2.31709i
4013.7 − 1.00000i 0 −1.00000 0.125404 0 3.10184 1.00000i 0 − 0.125404i
4013.8 1.00000i 0 −1.00000 0.125404 0 3.10184 − 1.00000i 0 0.125404i
4013.9 − 1.00000i 0 −1.00000 3.59603 0 4.66381 1.00000i 0 − 3.59603i
4013.10 1.00000i 0 −1.00000 3.59603 0 4.66381 − 1.00000i 0 3.59603i
4013.11 − 1.00000i 0 −1.00000 −2.58876 0 −1.62417 1.00000i 0 2.58876i
4013.12 1.00000i 0 −1.00000 −2.58876 0 −1.62417 − 1.00000i 0 − 2.58876i
4013.13 − 1.00000i 0 −1.00000 2.29842 0 2.05558 1.00000i 0 − 2.29842i
4013.14 1.00000i 0 −1.00000 2.29842 0 2.05558 − 1.00000i 0 2.29842i
4013.15 − 1.00000i 0 −1.00000 −2.37383 0 3.81286 1.00000i 0 2.37383i
4013.16 1.00000i 0 −1.00000 −2.37383 0 3.81286 − 1.00000i 0 − 2.37383i
4013.17 − 1.00000i 0 −1.00000 0.707934 0 0.916361 1.00000i 0 − 0.707934i
4013.18 1.00000i 0 −1.00000 0.707934 0 0.916361 − 1.00000i 0 0.707934i
4013.19 − 1.00000i 0 −1.00000 −1.71459 0 2.72316 1.00000i 0 1.71459i
4013.20 1.00000i 0 −1.00000 −1.71459 0 2.72316 − 1.00000i 0 − 1.71459i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4013.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
223.b odd 2 1 inner
669.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4014.2.d.a ✓ 72
3.b odd 2 1 inner 4014.2.d.a ✓ 72
223.b odd 2 1 inner 4014.2.d.a ✓ 72
669.c even 2 1 inner 4014.2.d.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4014.2.d.a ✓ 72 1.a even 1 1 trivial
4014.2.d.a ✓ 72 3.b odd 2 1 inner
4014.2.d.a ✓ 72 223.b odd 2 1 inner
4014.2.d.a ✓ 72 669.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(4014, [\chi])\).