
// Make newform 75.4.g.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_75_g();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_75_4_g_b();".
// 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_75_g();"
function MakeCharacter_75_g()
    N := 75;
    order := 5;
    char_gens := [26, 52];
    v := [5, 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_75_g_Hecke(Kf)
    return MakeCharacter_75_g();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_75_4_g_b();".
// 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_75_4_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_75_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1769380096, -20184326272, 89525309120, -2139832544, 285460183840, -135978086344, 93338725268, 70316056540, 176884160821, 54921921330, 100983902135, 23424791075, 24367245765, 2909227545, 2474205980, 159189399, 191341802, 14783120, 13441499, 655690, 852818, 26209, 33580, 803, 1285, 21, 43, 0, 1]>],Snew);
    return Vf;
end function;