
// Make newform 756.2.bo.b in Magma, downloaded from the LMFDB on 13 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_756_bo();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_bo_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_756_bo();"
function MakeCharacter_756_bo()
    N := 756;
    order := 9;
    char_gens := [379, 29, 325];
    v := [9, 1, 9];
    // 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_756_bo_Hecke(Kf)
    return MakeCharacter_756_bo();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_bo_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_756_2_bo_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_756_bo();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2804931592849, 41206751004198, 282740594453739, 1159048875069097, 3225349766613927, 6864713135690253, 12284147760201569, 18701800388182170, 24657270825482448, 29994910379761309, 34546740683653401, 38140908759836811, 41979960443647514, 42539547308618139, 38521566456314169, 30970535317439812, 21149783304231960, 12272832525146637, 8122796644945904, 6928170535642773, 5411335199079789, 3087834091400299, 1187924382760323, 283067886287679, 293050165578692, 466105594453182, 448494694584045, 298625861736112, 148273920220830, 54818372479113, 14862574047704, 2648454656172, 230165806215, 68307300463, 108266959089, 67991153412, 27672214562, 8184381654, 1817859555, 383716312, 148714752, 59466336, 19410275, 5159988, 735909, -56045, 2922, 13599, 3071, 816, 300, -2, -9, 3, 1]>],Snew);
    return Vf;
end function;