
// Make newform 8019.2.a.i in Magma, downloaded from the LMFDB on 04 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8019_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8019_2_a_i();".
// 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_8019_a();"
function MakeCharacter_8019_a()
    N := 8019;
    order := 1;
    char_gens := [5105, 2917];
    v := [1, 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_8019_a_Hecke(Kf)
    return MakeCharacter_8019_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8019_2_a_i();".
// 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_8019_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8019_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![6813, -227448, 1485630, 5039448, -55963647, -35647170, 831592327, 91213842, -6711174231, -332761690, 33164257128, 3579410436, -108083491406, -20337005412, 244817717736, 64843949248, -400375536783, -133399763340, 486131758193, 191558934384, -447238333767, -201228269162, 316276364934, 159350345028, -173519231826, -97034220306, 74170250253, 46030051812, -24667808409, -17141278464, 6326897097, 5026671684, -1224820419, -1159186412, 170396508, 208883442, -14821123, -29046432, 292830, 3051688, 120186, -233988, -18743, 12342, 1407, -400, -57, 6, 1]>],Snew);
    return Vf;
end function;