Twists#

This section is about twists of newforms by Dirichlet characters. A newform is specified by giving a space of modular symbols that contains a single Galois-orbit (over \({\mathbb{Q}}\) or some extension of \({\mathbb{Q}}\)) of newforms. Spaces of this kind are obtained using NewformDecomposition.

To prove that \(f_2 = f_1^\chi\) holds for two newforms, the program compares their Hecke eigenvalues up to an appropriate Sturm bound. (For instance, a bound of this kind is given in Lemma 1.4 of [Buzzard and Stein, 2002]).

IsTwist(M1, M2, p): ModSym, ModSym, RngIntElt -> BoolElt, GrpDrchElt#: Given two spaces M1 and M2 of modular symbols that specify newforms \(f_1\) and \(f_2\) as above, and a prime \(p\), this determines whether some Galois conjugate (over \({\mathbb{Q}}\)) of \(f_2\) is the twist of \(f_1\) by a nontrivial Dirichlet character of \(p\)-power conductor. If so, a character \(\chi\) such that \(f_2 = f_1^\chi\) is also returned.

IsMinimalTwist(M, p): ModSym, RngIntElt -> BoolElt, ModSym, GrpDrchElt#: Given a space M of modular symbols that specifies a newform \(f\) as above, and a prime \(p\), this determines whether some Galois conjugate (over \({\mathbb{Q}}\)) of \(f\) is a twist of some newform of lower level by some Dirichlet character of \(p\)-power conductor. If so, it returns false, together with the newform of lower level (specified by a space of modular symbols), and the Dirichlet character.

Example Twists (ex-1f6206)#

We exhibit a newform that is a twist of itself, namely the only newform of level \(9\) and weight \(4\). The newform is specified by the space of modular symbols on \(\Gamma_0(9)\) of weight \(4\) (with sign \(1\)).

> M9 := CuspidalSubspace(ModularSymbols(9, 4, 1));
> newforms := NewformDecomposition(NewSubspace(M9));
> newforms;
[
    Modular symbols space for Gamma_0(9) of weight 4 and dimension 1
    over Rational Field
]
> f := newforms[1];
> Eigenform(f, 20);
q - 8*q^4 + 20*q^7 - 70*q^13 + 64*q^16 + 56*q^19 + O(q^20)

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Note that here the coefficients for primes congruent to \(2\) mod \(3\) are all zero.

> bool, chi := IsTwist(f, f, 3);
> bool;
true
> Parent(chi);
Group of Dirichlet characters of modulus 3 over Rational Field
> Conductor(chi), Order(chi);
3 2

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However, \(f\) is not a twist of any newform with lower level:

> bool := IsMinimalTwist(f, 3);
> bool;
true

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