The mod p reductions of the modular forms f, where
p∈Z is a prime number and f is a modular form
over a number field (or the rationals or integers).
Because of denominators, the list of reductions might
be empty. (In some cases when f is defined over a field
of large degree, the algorithm that I've implemented is
definitely not close to optimal. I know a much better algorithm,
but haven't implemented it yet.)
The p-adic embeddings of the modular form f.
The complex embeddings of the modular form f.
We compute various reductions and embeddings
of a degree 3 newform in S
2(Γ
0(47)).
> M := ModularForms(Gamma0(47),2);
> f := Newform(M,1);
> Degree(f);
4
> f;
q + a*q^2 + (a^3 - a^2 - 6*a + 4)*q^3 + (a^2 - 2)*q^4 + (-4*a^3 +
2*a^2 + 20*a - 10)*q^5 + (-a^2 - a + 1)*q^6 + (3*a^3 - a^2 - 16*a +
7)*q^7 + O(q^8)
> Parent(f);
Space of modular forms on Gamma_0(47) of weight 2 and dimension 4 over
Number Field with defining polynomial x^4 - x^3 - 5*x^2 + 5*x - 1 over
the Rational Field.
> Reductions(f,3);
[* [*
q + 2*q^2 + 2*q^3 + 2*q^4 + q^6 + q^7 + O(q^8)
*], [*
q + $.1^4*q^2 + $.1^25*q^3 + $.1^15*q^4 + $.1^20*q^5 + $.1^3*q^6 +
$.1^3*q^7 + O(q^8),
q + $.1^10*q^2 + $.1^17*q^3 + $.1^5*q^4 + $.1^24*q^5 + $.1*q^6 +
$.1*q^7 + O(q^8),
q + $.1^12*q^2 + $.1^23*q^3 + $.1^19*q^4 + $.1^8*q^5 + $.1^9*q^6 +
$.1^9*q^7 + O(q^8)
*] *]
The reductions are genuine modular forms, so we can compute
them to higher precision later.
> f3 := Reductions(f,3)[1][1];
> Type(f3);
ModFrmElt
> Parent(f3);
Space of modular forms on Gamma_0(47) of weight 2 and dimension 4 over
Finite field of size 3.
> PowerSeries(f3,15);
q + 2*q^2 + 2*q^3 + 2*q^4 + q^6 + q^7 + q^9 + q^12 + 2*q^14 + O(q^15)
> f3^2;
q^2 + q^3 + 2*q^4 + q^7 + O(q^8)
> pAdicEmbeddings(f,3)[1][1];
q + (1383893738 + O(3^20))*q^2 + (288495368 + O(3^20))*q^3 +
(1448516780 + O(3^20))*q^4 + (291254407*3 + O(3^20))*q^5 + (654373882
+ O(3^20))*q^6 - (443261663 + O(3^20))*q^7 + O(q^8)
The p-adic precision can be increased by recreating
p-adic field with higher precision.
> _ := pAdicField(3 : Precision := 30);
> pAdicEmbeddings(f,3)[1][1];
q + (27072777983102 + O(3^30))*q^2 - (7262683411915 + O(3^30))*q^3 +
(102359491392536 + O(3^30))*q^4 - (26022742991723*3 + O(3^30))*q^5 +
(76458862719010 + O(3^30))*q^6 + (31185356420881 + O(3^30))*q^7 +
O(q^8)
> ComplexEmbeddings(f)[1][1];
q + 0.28384129078391142728240587955711710388*q^2 +
2.23925429984775850028724717074345372615990081162*q^3 -
1.91943412164612303785432431586190778394853582709*q^4 -
4.25351411923443363456996356947846775230044737382*q^5 +
0.635592830862211610571918436304790680059056881712*q^6 +
2.44657723781885227971790463962077981836158867144*q^7 + O(q^8)
V2.28, 13 July 2023