
// Make newform 1728.3.h.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_h_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_h_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1728_3_h_g();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 2, 0, 0, 0], [0, 0, 2, 0, 0, 0, -1, 0], [0, 2, 0, 2, 0, 2, 0, -4], [0, -2, 0, 2, 0, -2, 0, -4], [0, -6, 0, 6, 0, 6, 0, 0], [0, 6, 0, 6, 0, -6, 0, 0]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1728_h();"
function MakeCharacter_1728_h()
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_h_Hecke();"
function MakeCharacter_1728_h_Hecke(Kf)
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 3;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0], [0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 0, 0], [0, 0, 0, 0, 8, 0, 0, 0], [0, 0, 0, 8, 0, 0, 0, 0], [0, 0, 19, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0], [0, 34, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9], [0, 0, 13, 0, 0, 0, 0, 0], [0, 19, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -14, 0, 0], [-25, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 81, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 12], [0, 0, 0, 0, 0, 0, -1, 0], [-71, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -26, 0, 0, 0], [0, 0, 0, 31, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 21], [0, 0, 88, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 25, 0], [0, 0, 0, 64, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4], [0, 0, 0, 0, 0, 0, -1, 0], [0, 77, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -30, 0, 0, 0], [0, 0, 0, 151, 0, 0, 0, 0], [0, 0, 152, 0, 0, 0, 0, 0], [0, 101, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -23, 0, 0], [0, 0, 0, 0, 26, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -14], [0, 0, 101, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -65, 0, 0], [-265, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, 0, 0], [0, 0, 0, 15, 0, 0, 0, 0], [0, -53, 0, 0, 0, 0, 0, 0], [0, 0, 0, 232, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -22], [0, 0, 176, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 14, 0], [0, 0, 0, 0, 0, 76, 0, 0], [-25, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -22], [0, 0, 0, 0, 0, 0, -46, 0], [0, 0, 0, 0, 0, 84, 0, 0], [0, 0, 0, 0, -33, 0, 0, 0], [0, 0, 0, -89, 0, 0, 0, 0], [0, 0, -184, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 16, 0], [0, -226, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 49, 0, 0, 0], [0, 130, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -95, 0, 0], [481, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -108, 0, 0, 0], [0, 413, 0, 0, 0, 0, 0, 0], [-503, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 36], [0, 0, -293, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0], [0, 0, 0, 0, 0, -101, 0, 0], [0, 0, 0, 127, 0, 0, 0, 0], [0, 0, 147, 0, 0, 0, 0, 0], [0, -533, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -74, 0, 0], [0, 0, 0, 0, 19, 0, 0, 0], [0, 0, -40, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 48, 0], [431, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 65], [0, 0, -413, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 97, 0, 0], [-218, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 168, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 38], [0, 0, 0, 0, 0, 0, 85, 0], [794, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 27, 0, 0, 0], [0, 0, 0, -303, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 59], [0, 0, 0, 0, 0, -150, 0, 0], [0, 0, 0, -231, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5], [0, -326, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 65, 0, 0], [0, 0, 0, 0, -143, 0, 0, 0], [0, 0, 0, 0, 0, 0, -71, 0], [0, -605, 0, 0, 0, 0, 0, 0], [0, 0, -101, 0, 0, 0, 0, 0], [0, -67, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -93, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -114], [0, 0, 0, 0, 0, 0, -51, 0], [0, -451, 0, 0, 0, 0, 0, 0], [-623, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -101], [0, 0, 0, 0, 0, 0, -68, 0], [0, 0, 0, 0, 0, 140, 0, 0], [986, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -311, 0, 0, 0, 0], [0, 0, -275, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 113, 0], [0, 821, 0, 0, 0, 0, 0, 0], [0, 0, 0, -553, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 26, 0], [0, -898, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -62, 0, 0], [0, 0, 0, 0, -36, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 50], [0, 0, -179, 0, 0, 0, 0, 0], [-215, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 35, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 82], [0, -562, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 231, 0, 0, 0], [0, 0, -213, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 264, 0, 0, 0, 0], [0, 0, -56, 0, 0, 0, 0, 0], [0, 514, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -215, 0, 0], [0, 0, 0, -167, 0, 0, 0, 0], [0, 0, 251, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 129, 0], [-529, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -272, 0, 0, 0], [0, 197, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 86, 0, 0, 0], [0, 0, 0, 0, 0, 0, -26, 0], [0, -110, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -259, 0, 0, 0], [0, 0, 0, -225, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 49], [0, 0, -123, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -22, 0, 0], [0, 0, 395, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 122, 0], [0, -341, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -281, 0, 0], [0, 0, 595, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 41, 0], [0, 1613, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 24, 0, 0], [0, 1171, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 38, 0, 0], [0, 0, 0, 408, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 128, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 155, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 127], [0, 0, 0, 0, 0, 0, -11, 0], [0, 0, 0, 665, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 127], [0, 0, 0, 0, 0, 0, -44, 0], [0, 0, 0, 0, 0, -129, 0, 0], [0, 0, 0, 127, 0, 0, 0, 0], [0, 0, 600, 0, 0, 0, 0, 0], [-911, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -224], [0, 0, 72, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 269, 0, 0], [-311, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 255, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -162, 0], [0, -562, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -77, 0, 0, 0], [0, 0, 0, 561, 0, 0, 0, 0], [0, 0, 992, 0, 0, 0, 0, 0], [0, 0, 0, 817, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -188], [0, 0, 504, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 79, 0], [0, 0, 0, 0, 0, -342, 0, 0], [0, 0, 0, 0, -42, 0, 0, 0], [0, 0, 589, 0, 0, 0, 0, 0], [0, -1445, 0, 0, 0, 0, 0, 0], [-719, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 353, 0, 0], [23, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 50], [0, 1555, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -259, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -53], [0, 0, 0, 0, 0, 0, -97, 0], [-335, 0, 0, 0, 0, 0, 0, 0], [0, 0, 789, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -249, 0], [0, 0, 0, 0, 0, 388, 0, 0], [0, 0, 0, 0, 323, 0, 0, 0], [0, 0, 0, -969, 0, 0, 0, 0], [0, 0, 141, 0, 0, 0, 0, 0], [-983, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -194], [0, 0, 0, 0, 364, 0, 0, 0], [0, 0, 0, 63, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 129], [0, 0, 0, 0, 0, 0, 101, 0], [0, 1387, 0, 0, 0, 0, 0, 0], [47, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 61, 0, 0, 0], [0, 0, 0, 281, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 47], [0, 0, 0, 0, 0, 42, 0, 0], [-337, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1432, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, -222, 0, 0], [0, 0, 0, 0, -250, 0, 0, 0], [0, 0, -1275, 0, 0, 0, 0, 0], [0, 0, 0, 792, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 210, 0], [0, 0, 0, 1064, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -201], [0, 0, 741, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 257, 0], [0, 0, 0, 0, 0, -333, 0, 0], [0, 0, 0, -1129, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 107], [0, 0, 67, 0, 0, 0, 0, 0], [0, 2054, 0, 0, 0, 0, 0, 0], [0, 0, 0, -80, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -121, 0], [0, 1507, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -377, 0, 0], [673, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 245, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 86], [0, 0, 0, 0, 0, -276, 0, 0], [0, 0, 0, 0, 0, 0, 0, -160], [0, -2563, 0, 0, 0, 0, 0, 0], [0, 0, 0, 391, 0, 0, 0, 0], [0, 0, 827, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -29, 0], [0, 0, 0, 0, 0, -127, 0, 0], [0, 0, 0, 969, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -76], [0, 1718, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -372, 0, 0], [0, 0, -792, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -99, 0], [0, 0, 0, 0, 0, 325, 0, 0], [-2351, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 513, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 137], [0, 0, -1499, 0, 0, 0, 0, 0], [0, -1570, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 184, 0, 0, 0], [-1990, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1479, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 173], [0, 0, -373, 0, 0, 0, 0, 0], [0, 0, -1411, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 182, 0], [0, -1742, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -353, 0, 0, 0], [0, 0, 0, 0, 0, 0, -292, 0], [0, -86, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 235, 0, 0, 0], [0, 0, -1939, 0, 0, 0, 0, 0], [0, 2779, 0, 0, 0, 0, 0, 0], [-626, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1625, 0, 0, 0, 0], [-166, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -497, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -27], [0, 0, -40, 0, 0, 0, 0, 0], [74, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -354], [0, 0, 0, 0, 0, 717, 0, 0], [0, 0, 0, -1040, 0, 0, 0, 0], [0, 0, 0, 0, 0, -144, 0, 0], [0, 0, -205, 0, 0, 0, 0, 0], [0, 2875, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -603, 0, 0], [2759, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -380, 0, 0, 0], [0, 0, 0, -1639, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -421, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -82], [0, 0, 0, 0, 0, 0, 110, 0], [0, 0, 0, 0, 0, 0, 0, -446], [0, 0, 816, 0, 0, 0, 0, 0], [0, 0, 0, 0, 486, 0, 0, 0], [0, 0, 0, -1599, 0, 0, 0, 0], [0, 0, 0, 0, -27, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 354], [0, -3523, 0, 0, 0, 0, 0, 0], [-1847, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 154, 0, 0, 0], [0, 0, 0, -200, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_h_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1728_3_h_g(:prec:=8)
    chi := MakeCharacter_1728_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1728_3_h_g();".
// 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_1728_3_h_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-24, 0, 1]>,<7,R![-3, 0, 1]>,<11,R![-72, 0, 1]>],Snew);
    return Vf;
end function;