Example ModFrm_Congruences (H144E18)

We verify that the newform corresponding to the first elliptic curve of rank 2 is congruent modulo 5 to some Galois-conjugate newform corresponding to the winding quotient of J₀(389) (there is also a congruence modulo 2 to some conjugate form).

> M := ModularForms(Gamma0(389),2);
> f := Newform(M,1);
> Degree(f);
1
> g := Newform(M,5);
> Degree(g);
20
> CongruenceGroup(Parent(f),Parent(g),30);
Abelian Group isomorphic to Z/20
Defined on 1 generator
Relations:
    20*$.1 = 0

The congruence can be seen directly by computing the reductions of f and g modulo 5:

> fmod5 := Reductions(f,5);
> gmod5 := Reductions(g,5);  // takes a few seconds.
> #gmod5;
7
> #fmod5;
1
> [gbar : gbar in gmod5 | #gbar eq 1];
[ [*
q + 4*q^2 + q^3 + 4*q^4 + q^5 + 4*q^6 + O(q^8)
*], [*
q + 3*q^2 + 3*q^3 + 2*q^4 + 2*q^5 + 4*q^6 + O(q^8)
*] ]
> fmod5[1][1];
q + 3*q^2 + 3*q^3 + 2*q^4 + 2*q^5 + 4*q^6 + O(q^8)

---

[Next][Prev] [Right] [Left] [Up] [Index] [Root]