
// Make newform 8019.2.a.l in Magma, downloaded from the LMFDB on 18 April 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_l();".
// 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_l();".
// 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_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8019_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-576, -31104, 783072, 4940544, -57865572, -275325888, 1254088061, 5374822956, -11601954744, -49245795522, 57308512539, 255495593862, -173629401444, -844564373028, 353177629869, 1922168407139, -512141389683, -3174258244386, 551533634877, 3945077245500, -453802998519, -3787465448034, 290972350737, 2862178950015, -147348250759, -1725793380792, 59429424510, 838207900854, -19170700686, -329962572234, 4947272646, 105615644589, -1017527955, -27499297076, 165401580, 5809699842, -20948580, -990133086, 2021040, 134781867, -143289, -14429982, 7030, 1186818, -213, -72291, 3, 3069, 0, -81, 0, 1]>],Snew);
    return Vf;
end function;