
// Make newform 1050.3.q.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_q();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1050_3_q_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_q();"
function MakeCharacter_1050_q()
    N := 1050;
    order := 6;
    char_gens := [701, 127, 451];
    v := [6, 3, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_1050_q_Hecke(Kf)
    return MakeCharacter_1050_q();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1050_3_q_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_1050_3_q_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<11,R![9682651996416, -10143730662912, 26588869767936, 15929692886016, 26420738609664, -11416664448, 516442695552, 676772544, 6945139504, -7989152, 49322848, 129616, 252064, 352, 604, 4, 1]>],Snew);
    return Vf;
end function;