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**f69d0e06f67b9ac6f57e7d8e6ba3b3d69e650352**authored by Maarten Derickx on**02 November 2020, 22:43:35 UTC**, committed by Maarten Derickx on**02 November 2020, 22:43:35 UTC****1 parent**0460952

rank_zero_of_X1mn.m

```
function TwistOfSimpleModularSymbolsSpace(S,chi);
//On input of a irreducible new modular symbols space S and a character chi of prime power
//modulus output the modular symbols space corresponding to the twist of M by the primitive character associated to chi
//i.e. if f is the modular form associated to S then outputs the modular symbols space
//corresponding to the modular form f_chi
//the sign of M should be 1 or -1, and the sign of the output will be the same
if Conductor(chi) eq 1 then;
return S;
end if;
chi := AssociatedPrimitiveCharacter(chi);
chi_S := DirichletCharacter(S);
m := Modulus(chi_S);
n := Modulus(chi);
assert IsPrimePower(n);
p := PrimeDivisors(n)[1];
sign := Sign(S);
k := Weight(S);
chi_t := Extend(chi_S*chi^2,m*n^2);
Mt := ModularSymbols(chi_t,k,sign);
St := CuspidalSubspace(Mt);
for Si in NewformDecomposition(St) do;
Snew := AssociatedNewSpace(Si);
tf,chi_i := IsTwist(S,Snew,p);
if tf and AssociatedPrimitiveCharacter(chi_i) eq chi then;
return Snew;
end if;
end for;
print "Did not find a twist while we should have!!!!";
assert false;
end function;
function PostiveRankNewFactors(m,n);
//Let m,n be two integer and let
//G be the congruence subgroup given by the matrices of the form
//
// [a b]
// [c d]
//
//with a,d congruent to 1 modulo mn
//and c congruent to 0 modulo m^2n
//Then this function returns one modular symbols space
//for every isogeny class of simple abelian varieties that occurs as an isogeny factor
//of J(G) and obtains positive rank over Q(zeta_m)
pr_new_factors := [];
D := FullDirichletGroup(m*n);
Chi := FullDirichletGroup(m);
for chi in Elements(Chi) do;
for d in GaloisConjugacyRepresentatives(D) do;
d1 := Extend(d,m^2*n);
//if IsOdd(d) then;
// continue;
//end if;
M := ModularSymbols(d1,2,1);
S := CuspidalSubspace(M);
for Si in NewformDecomposition(S) do;
Snew := AssociatedNewSpace(Si);
St := TwistOfSimpleModularSymbolsSpace(Snew,chi);
if Dimension(St) ne Dimension(WindingSubmodule(St)) then;
Append(~pr_new_factors,Snew);
end if;
end for;
end for;
end for;
return pr_new_factors;
end function;
function IsX1mnRankZero(m,n);
return #PostiveRankNewFactors(m,n) eq 0;
end function;
```

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