
// Make newform 1728.3.b.h in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_b_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_b_h();".
// 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_b_h();".
// 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, -2, 0], [0, 0, 6, 0, 0, 0, -3, 0], [-1, -2, 0, 2, 2, -2, 0, -4], [0, 6, 0, -6, 0, -6, 1, 0], [-5, 2, 0, -2, 10, 2, 0, 4], [0, 4, 2, 4, 0, 4, -1, -8], [0, 12, 0, 12, 0, -12, 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_b();"
function MakeCharacter_1728_b()
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    v := [1, 1, 2];
    // 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_b_Hecke();"
function MakeCharacter_1728_b_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, 1, 0, 0, 0, 0], [0, 2, 0, 0, -1, 0, 0, 0], [0, 0, 2, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, -1, 0, 0], [0, 7, 0, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -12, 0, -7, 0, 0], [0, 0, -4, 0, 0, 0, 12, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [25, 0, 0, 0, 0, 0, 0, -7], [0, 29, 0, 0, 0, 0, 0, 0], [0, 0, 24, 0, 0, 0, -5, 0], [0, 0, 0, 0, 0, 0, 0, 0], [95, 0, 0, 0, 0, 0, 0, -2], [0, 0, 0, -10, 0, -17, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 11, 0, 0, 0, 10, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -52, 0, 0, 11, 0, 0, 0], [0, 0, 43, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -30, 0, -23, 0, 0], [0, 77, 0, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 33, 0, -2, 0, 0], [0, 0, 43, 0, 0, 0, -14, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-145, 0, 0, 0, 0, 0, 0, 13], [0, 0, 0, 13, 0, -22, 0, 0], [0, -68, 0, 0, -35, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 115, 0, 0, 0, 0, 0, 0], [0, 0, -10, 0, 0, 0, 30, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [382, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 0, 18, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -55, 0, 11, 0, 0], [0, -13, 0, 0, -55, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 75, 0, -15, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [265, 0, 0, 0, 0, 0, 0, 17], [0, 0, 0, -18, 0, 37, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-190, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 0, -35, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 187, 0, 0, 19, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 80, 0, 3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-359, 0, 0, 0, 0, 0, 0, 20], [0, 0, 14, 0, 0, 0, -42, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-335, 0, 0, 0, 0, 0, 0, -28], [0, 112, 0, 0, 85, 0, 0, 0], [0, 0, 30, 0, 0, 0, -90, 0], [0, 0, 0, 0, 0, 0, 0, 0], [265, 0, 0, 0, 0, 0, 0, 38], [0, 0, 0, -20, 0, 57, 0, 0], [0, 17, 0, 0, -91, 0, 0, 0], [0, 0, -94, 0, 0, 0, -35, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -485, 0, 0, 0, 0, 0, 0], [0, 0, -93, 0, 0, 0, 44, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 95, 0, 82, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 22, 0, 87, 0, 0], [0, 0, -36, 0, 0, 0, -55, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-290, 0, 0, 0, 0, 0, 0, 0], [0, 0, -159, 0, 0, 0, 40, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [599, 0, 0, 0, 0, 0, 0, -5], [0, -365, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -322, 0, 0, -25, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 13, 0, 98, 0, 0], [0, 0, -103, 0, 0, 0, -56, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-95, 0, 0, 0, 0, 0, 0, 68], [0, 0, 0, 25, 0, -5, 0, 0], [0, 0, 10, 0, 0, 0, -30, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -115, 0, 28, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 368, 0, 0, 11, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -323, 0, 0, 55, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [431, 0, 0, 0, 0, 0, 0, -65], [0, 0, 0, 175, 0, -35, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 187, 0, 92, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 255, 0, -51, 0, 0], [0, -37, 0, 0, 161, 0, 0, 0], [0, 0, 20, 0, 0, 0, -60, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -253, 0, 0, -175, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [745, 0, 0, 0, 0, 0, 0, -58], [0, 0, 0, 120, 0, -67, 0, 0], [0, 0, -241, 0, 0, 0, 70, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 493, 0, 0, 31, 0, 0, 0], [0, 0, 318, 0, 0, 0, 11, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 418, 0, 0, 145, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-241, 0, 0, 0, 0, 0, 0, 100], [0, 0, 0, 325, 0, -65, 0, 0], [0, 0, -33, 0, 0, 0, 104, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1390, 0, 0, 0, 0, 0, 0, 0], [0, -527, 0, 0, -65, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -195, 0, -172, 0, 0], [0, -437, 0, 0, 91, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -287, 0, 0, -209, 0, 0, 0], [0, 0, 352, 0, 0, 0, 19, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -260, 0, -73, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [71, 0, 0, 0, 0, 0, 0, 115], [0, 0, 0, 0, 0, 0, 0, 0], [1150, 0, 0, 0, 0, 0, 0, 0], [0, 0, 109, 0, 0, 0, 110, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -275, 0, -142, 0, 0], [0, 0, -80, 0, 0, 0, 240, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1151, 0, 0, 0, 0, 0, 0, 35], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 170, 0, -77, 0, 0], [0, -592, 0, 0, 35, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1201, 0, 0, 0, 0, 0, 0, -35], [0, 0, 198, 0, 0, 0, -119, 0], [0, 0, 0, 125, 0, -25, 0, 0], [0, 1171, 0, 0, 0, 0, 0, 0], [0, 0, 326, 0, 0, 0, -95, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1055, 0, 0, 0, 0, 0, 0, -77], [0, 0, 0, 345, 0, -69, 0, 0], [0, -473, 0, 0, -221, 0, 0, 0], [0, 0, 141, 0, 0, 0, -130, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1129, 0, 0, 0, 0, 0, 0, 70], [0, -355, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 175, 0, -35, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1301, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1315, 0, 0, 0, 0, 0, 0], [0, 0, -404, 0, 0, 0, -65, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -475, 0, 0, 0, 0, 0, 0], [0, 0, -56, 0, 0, 0, 168, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1229, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-478, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -375, 0, 75, 0, 0], [0, 0, -432, 0, 0, 0, 91, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 70, 0, 0, 0, -210, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 392, 0, 0, 299, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 217, 0, 0, 319, 0, 0, 0], [0, 0, 517, 0, 0, 0, 14, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-841, 0, 0, 0, 0, 0, 0, -140], [0, 0, 0, -377, 0, -102, 0, 0], [0, 0, -503, 0, 0, 0, -46, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 352, 0, 237, 0, 0], [1415, 0, 0, 0, 0, 0, 0, 88], [0, -667, 0, 0, -259, 0, 0, 0], [0, 0, 79, 0, 0, 0, -170, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -145, 0, 29, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -575, 0, 115, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-985, 0, 0, 0, 0, 0, 0, 148], [0, -812, 0, 0, 85, 0, 0, 0], [1585, 0, 0, 0, 0, 0, 0, -82], [0, -902, 0, 0, -119, 0, 0, 0], [0, 0, -424, 0, 0, 0, -115, 0], [0, 0, 0, 0, 0, 0, 0, 0], [551, 0, 0, 0, 0, 0, 0, 175], [0, 0, 84, 0, 0, 0, -252, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 868, 0, 0, 205, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1825, 0, 0, 0, 0, 0, 0, -43], [0, 0, 0, 333, 0, 308, 0, 0], [0, -821, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 595, 0, -119, 0, 0], [0, 0, -631, 0, 0, 0, 50, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 453, 0, 0, 0, -154, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 155, 0, -31, 0, 0], [0, -442, 0, 0, -385, 0, 0, 0], [0, 0, 0, 442, 0, 267, 0, 0], [0, 0, -452, 0, 0, 0, 161, 0], [0, 0, 0, 0, 0, 0, 0, 0], [3790, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 437, 0, 282, 0, 0], [0, 742, 0, 0, -215, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_b_h();".
// 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_b_h(:prec:=8)
    chi := MakeCharacter_1728_b();
    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_b_h();".
// 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_b_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![441, 0, 54, 0, 1]>,<7,R![2209, 0, 194, 0, 1]>,<11,R![441, 0, -342, 0, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;