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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 Q or some extension of Q) of newforms.
Spaces of this kind are obtained using NewformDecomposition.
To prove that f2 = f1χ 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
[BS02]).
Given two spaces M1 and M2 of modular symbols that specify
newforms f1 and f2 as above,
and a prime p, this determines whether some Galois conjugate (over Q)
of f2 is the twist of f1 by a nontrivial Dirichlet character
of p-power conductor. If so, a character χ such that
f2 = f1χ is also returned.
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 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.
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 Γ 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)
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
However, f is not a twist of any newform with lower level:
> bool := IsMinimalTwist(f, 3);
> bool;
true
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