
// Make newform 1728.3.e.v in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1728_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1728_e_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_e_v();".
// 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_e_v();".
// 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 := [196, 0, -188, 0, 77, 0, -14, 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, -90, 0, 63, 0, -14, 0, 1], [0, -338, 0, 245, 0, -56, 0, 5], [0, -106, 0, 0, 0, 7, 0, -1], [0, 250, 0, -245, 0, 84, 0, -9], [22, 0, -31, 0, 10, 0, -1, 0], [-106, 0, 67, 0, -14, 0, 1, 0], [-150, 0, 105, 0, -26, 0, 3, 0]];
    Rf_basisdens := [1, 14, 56, 7, 56, 2, 4, 4];
    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_e();"
function MakeCharacter_1728_e()
    N := 1728;
    order := 2;
    char_gens := [703, 325, 1217];
    v := [2, 2, 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_e_Hecke();"
function MakeCharacter_1728_e_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, 0, 0, -1, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, -1, -1, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, -1, 0, 0, 0, 0], [0, -3, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 1], [0, 0, 0, -3, 1, 0, 0, 0], [-4, 0, 0, 0, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, -3, -1], [0, 0, 0, -2, -6, 0, 0, 0], [0, 14, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, -1], [0, 10, -18, 0, 0, 0, 0, 0], [-12, 0, 0, 0, 0, -8, 0, 0], [0, 0, 0, 4, 6, 0, 0, 0], [0, -13, -26, 0, 0, 0, 0, 0], [-27, 0, 0, 0, 0, 11, 0, 0], [0, 0, 0, 1, -10, 0, 0, 0], [0, -33, -19, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 22, -4], [-5, 0, 0, 0, 0, 8, 0, 0], [0, 0, 0, 0, 0, 0, -8, -5], [0, 0, 0, 7, 10, 0, 0, 0], [0, 16, -41, 0, 0, 0, 0, 0], [-44, 0, 0, 0, 0, -7, 0, 0], [0, 0, 0, 0, 0, 0, 3, 5], [0, 0, 0, -4, 9, 0, 0, 0], [0, 50, -19, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 4], [0, 0, 0, -6, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, -14, 1], [0, 0, 0, 1, 19, 0, 0, 0], [148, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 13, -10, 0, 0, 0], [0, -46, -58, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -9, 6], [0, 48, -3, 0, 0, 0, 0, 0], [-152, 0, 0, 0, 0, 18, 0, 0], [0, 75, -26, 0, 0, 0, 0, 0], [123, 0, 0, 0, 0, -7, 0, 0], [0, 0, 0, 0, 0, 0, 15, 4], [0, 0, 0, 0, -29, 0, 0, 0], [0, 0, 0, -17, -4, 0, 0, 0], [0, 0, 0, -13, -2, 0, 0, 0], [0, -102, 18, 0, 0, 0, 0, 0], [178, 0, 0, 0, 0, 24, 0, 0], [0, 0, 0, 0, 0, 0, -18, 12], [0, -69, 40, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, -30, 0, 0], [0, 0, -82, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 65, -11], [0, 123, -44, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -41, -17], [0, 0, 0, -9, 21, 0, 0, 0], [-28, 0, 0, 0, 0, 26, 0, 0], [0, 0, 0, 0, 0, 0, 61, -11], [0, 0, 0, 15, -40, 0, 0, 0], [0, 0, 0, 0, 0, 0, -47, 1], [0, 0, 0, -25, -46, 0, 0, 0], [0, -75, -80, 0, 0, 0, 0, 0], [-203, 0, 0, 0, 0, -9, 0, 0], [0, 0, 0, 0, 0, 0, -4, -19], [0, 0, 0, 12, -22, 0, 0, 0], [278, 0, 0, 0, 0, -14, 0, 0], [0, 91, -13, 0, 0, 0, 0, 0], [-304, 0, 0, 0, 0, 23, 0, 0], [0, 0, 0, 0, 0, 0, 53, 3], [0, 22, -10, 0, 0, 0, 0, 0], [0, 0, 0, 21, 9, 0, 0, 0], [310, 0, 0, 0, 0, -9, 0, 0], [0, 0, 0, 20, 72, 0, 0, 0], [0, 5, 98, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 32, 11], [-346, 0, 0, 0, 0, 21, 0, 0], [0, 0, 0, 0, 0, 0, 32, 0], [397, 0, 0, 0, 0, 16, 0, 0], [0, -16, 146, 0, 0, 0, 0, 0], [148, 0, 0, 0, 0, -14, 0, 0], [0, -6, 206, 0, 0, 0, 0, 0], [-317, 0, 0, 0, 0, 34, 0, 0], [0, 0, 0, -14, -33, 0, 0, 0], [0, -36, -34, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -89, -7], [-379, 0, 0, 0, 0, 16, 0, 0], [0, 0, 0, 0, 0, 0, 36, -3], [0, 0, 0, 1, 41, 0, 0, 0], [0, -97, 193, 0, 0, 0, 0, 0], [0, -24, 126, 0, 0, 0, 0, 0], [0, 0, 0, -37, 22, 0, 0, 0], [0, -118, -49, 0, 0, 0, 0, 0], [0, 0, 0, -30, -54, 0, 0, 0], [0, -39, -52, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -13, -10], [0, 0, 0, 0, 0, 0, 123, 3], [0, 0, 0, -1, -58, 0, 0, 0], [-100, 0, 0, 0, 0, -16, 0, 0], [0, 0, 0, 25, -62, 0, 0, 0], [0, 0, 0, 0, 0, 0, -46, -9], [0, 87, 273, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 154, -18], [0, 0, 0, -13, -18, 0, 0, 0], [-258, 0, 0, 0, 0, 34, 0, 0], [0, 42, -25, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 70, 28], [0, 290, 2, 0, 0, 0, 0, 0], [-455, 0, 0, 0, 0, -9, 0, 0], [0, 0, 0, 9, -38, 0, 0, 0], [-306, 0, 0, 0, 0, -96, 0, 0], [0, 0, 0, 0, 0, 0, -27, -21], [0, 0, 0, -31, -58, 0, 0, 0], [0, 0, 0, -20, -49, 0, 0, 0], [0, 0, 0, 0, 0, 0, -51, -17], [0, 0, 0, -42, 72, 0, 0, 0], [0, 96, 100, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -53, 6], [0, 144, 165, 0, 0, 0, 0, 0], [136, 0, 0, 0, 0, 9, 0, 0], [47, 0, 0, 0, 0, 124, 0, 0], [0, 0, 0, 0, 0, 0, 55, -21], [0, -240, -30, 0, 0, 0, 0, 0], [0, 0, 0, -30, 28, 0, 0, 0], [0, 0, 0, 0, 0, 0, 137, -4], [56, 0, 0, 0, 0, 24, 0, 0], [0, -88, 292, 0, 0, 0, 0, 0], [0, 0, 0, 8, -13, 0, 0, 0], [184, 0, 0, 0, 0, -29, 0, 0], [0, 0, 0, 17, -6, 0, 0, 0], [0, 57, -70, 0, 0, 0, 0, 0], [0, 0, 0, 13, 41, 0, 0, 0], [-402, 0, 0, 0, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, -88, -16], [-65, 0, 0, 0, 0, -37, 0, 0], [0, 0, 0, 0, 0, 0, -55, 5], [0, 0, 0, 22, -92, 0, 0, 0], [0, 0, 0, 0, 0, 0, 39, 18], [0, 0, 0, 0, 0, 0, 54, -20], [0, 0, 0, -15, -128, 0, 0, 0], [0, 0, 0, 0, 0, 0, -35, -31], [0, 0, 0, -19, -29, 0, 0, 0], [0, 92, -66, 0, 0, 0, 0, 0], [166, 0, 0, 0, 0, 20, 0, 0], [0, -457, -30, 0, 0, 0, 0, 0], [1002, 0, 0, 0, 0, -32, 0, 0], [0, 0, 0, 0, 0, 0, 67, 47], [0, 0, 0, 42, -124, 0, 0, 0], [0, 56, -196, 0, 0, 0, 0, 0], [758, 0, 0, 0, 0, 70, 0, 0], [0, 0, 0, 0, 0, 0, -11, -23], [0, 0, 0, 9, 80, 0, 0, 0], [0, -33, -130, 0, 0, 0, 0, 0], [0, 0, 0, -41, 68, 0, 0, 0], [0, -12, 122, 0, 0, 0, 0, 0], [0, 0, 0, 55, 151, 0, 0, 0], [0, 0, 0, 0, 0, 0, 144, 12], [-115, 0, 0, 0, 0, -88, 0, 0], [0, 0, 0, 0, 0, 0, -170, -33], [0, 510, 61, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -58, 26], [0, 0, 0, -7, 11, 0, 0, 0], [0, -61, -333, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -16, -62], [0, -66, -258, 0, 0, 0, 0, 0], [0, 0, 0, -16, -111, 0, 0, 0], [558, 0, 0, 0, 0, 87, 0, 0], [79, 0, 0, 0, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 159, -37], [0, 42, -117, 0, 0, 0, 0, 0], [-54, 0, 0, 0, 0, 216, 0, 0], [0, 208, -350, 0, 0, 0, 0, 0], [-1046, 0, 0, 0, 0, 70, 0, 0], [0, 0, 0, -51, -139, 0, 0, 0], [0, 0, 0, 0, 0, 0, -123, 7], [0, 0, 0, -13, -8, 0, 0, 0], [0, 0, 0, 0, 0, 0, -19, -22], [0, 0, 0, -45, -37, 0, 0, 0], [-22, 0, 0, 0, 0, 75, 0, 0], [0, 0, 0, -27, -23, 0, 0, 0], [0, -145, -347, 0, 0, 0, 0, 0], [-676, 0, 0, 0, 0, -153, 0, 0], [0, 0, 0, 0, 0, 0, -186, -22], [0, -309, 190, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 162, 41], [1512, 0, 0, 0, 0, 27, 0, 0], [0, 0, 0, -41, -126, 0, 0, 0], [197, 0, 0, 0, 0, -75, 0, 0], [0, -63, -504, 0, 0, 0, 0, 0], [46, 0, 0, 0, 0, -2, 0, 0], [0, 158, -313, 0, 0, 0, 0, 0], [0, 0, 0, 46, 30, 0, 0, 0], [0, 0, 0, 0, 0, 0, 119, 46], [0, 208, 18, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 274, -38], [-881, 0, 0, 0, 0, -53, 0, 0], [-1270, 0, 0, 0, 0, -88, 0, 0], [0, 0, 0, 0, 0, 0, 248, 4], [0, 536, 156, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 140, 5], [0, 0, 0, -94, -123, 0, 0, 0], [-782, 0, 0, 0, 0, 55, 0, 0], [567, 0, 0, 0, 0, 99, 0, 0], [0, 533, 91, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 19, -65], [0, 0, 0, -107, 150, 0, 0, 0], [0, 391, -155, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -210, -56], [0, 0, 0, -1, 226, 0, 0, 0], [419, 0, 0, 0, 0, -95, 0, 0], [0, 0, 0, 0, 0, 0, 29, 53], [0, 0, 0, 69, -165, 0, 0, 0], [0, -258, 267, 0, 0, 0, 0, 0], [0, -202, 328, 0, 0, 0, 0, 0], [-543, 0, 0, 0, 0, -144, 0, 0], [0, 0, 0, 95, 102, 0, 0, 0], [0, 0, 0, 0, 0, 0, 130, 32], [0, -368, -162, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -219, 33], [-2120, 0, 0, 0, 0, 63, 0, 0], [0, 0, 0, -61, -198, 0, 0, 0], [0, 0, 0, 0, 0, 0, -230, 22], [0, 0, 0, 37, -30, 0, 0, 0], [0, 489, -175, 0, 0, 0, 0, 0], [1132, 0, 0, 0, 0, -138, 0, 0], [0, 0, 0, 0, 0, 0, 320, -4], [0, 99, -412, 0, 0, 0, 0, 0], [0, 0, 0, 37, -54, 0, 0, 0], [0, -514, -122, 0, 0, 0, 0, 0], [-94, 0, 0, 0, 0, -196, 0, 0], [0, 0, 0, 30, 116, 0, 0, 0], [0, 0, 0, 95, 94, 0, 0, 0], [0, 0, 0, 0, 0, 0, 344, -4], [0, 0, 0, 84, -130, 0, 0, 0], [0, -68, 648, 0, 0, 0, 0, 0], [-2326, 0, 0, 0, 0, -12, 0, 0], [0, 0, 0, 0, 0, 0, 41, 17], [0, -153, 235, 0, 0, 0, 0, 0], [0, -398, -120, 0, 0, 0, 0, 0], [0, -654, 98, 0, 0, 0, 0, 0], [0, 0, 0, 80, 98, 0, 0, 0], [0, 0, 0, -72, 63, 0, 0, 0], [1026, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -103, -51], [0, 661, 322, 0, 0, 0, 0, 0], [0, 0, 0, 93, -19, 0, 0, 0], [0, 60, -191, 0, 0, 0, 0, 0], [0, 0, 0, 79, -28, 0, 0, 0], [0, -301, 458, 0, 0, 0, 0, 0], [-960, 0, 0, 0, 0, -238, 0, 0], [0, 0, 0, 0, 0, 0, -63, -31], [0, -342, 706, 0, 0, 0, 0, 0], [201, 0, 0, 0, 0, -34, 0, 0], [0, 0, 0, 0, 0, 0, 127, -46], [0, 354, -287, 0, 0, 0, 0, 0], [-450, 0, 0, 0, 0, 75, 0, 0], [0, 0, 0, -116, 164, 0, 0, 0], [0, 0, 0, 0, 0, 0, 224, 23], [1681, 0, 0, 0, 0, -74, 0, 0], [0, 0, 0, 67, -195, 0, 0, 0], [0, -714, 213, 0, 0, 0, 0, 0], [-1322, 0, 0, 0, 0, -108, 0, 0], [1718, 0, 0, 0, 0, -125, 0, 0], [0, 0, 0, 0, 0, 0, -360, -60], [0, 0, 0, -134, 236, 0, 0, 0], [0, 0, 0, 0, 0, 0, 187, 99], [0, 0, 0, 0, 0, 0, 209, -5], [0, 0, 0, 121, -126, 0, 0, 0], [0, 0, 0, 0, 0, 0, 275, -49], [1150, 0, 0, 0, 0, 135, 0, 0], [0, 0, 0, -50, -76, 0, 0, 0], [-1003, 0, 0, 0, 0, 204, 0, 0], [0, 0, 0, 98, -235, 0, 0, 0], [-917, 0, 0, 0, 0, -192, 0, 0], [0, 0, 0, -44, -317, 0, 0, 0], [0, -229, 335, 0, 0, 0, 0, 0], [-1610, 0, 0, 0, 0, -63, 0, 0], [-119, 0, 0, 0, 0, -215, 0, 0], [0, -44, -442, 0, 0, 0, 0, 0], [0, -570, -236, 0, 0, 0, 0, 0], [0, 0, 0, -126, -197, 0, 0, 0], [0, 617, 500, 0, 0, 0, 0, 0], [1476, 0, 0, 0, 0, 135, 0, 0], [0, 0, 0, 92, 54, 0, 0, 0], [0, 564, -70, 0, 0, 0, 0, 0], [1375, 0, 0, 0, 0, 229, 0, 0], [0, 0, 0, 0, 0, 0, -493, 48], [0, 0, 0, 49, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 74, 102], [0, 0, 0, 0, 0, 0, -173, 79], [0, 120, 473, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -50, -26], [0, 370, -197, 0, 0, 0, 0, 0], [-24, 0, 0, 0, 0, -183, 0, 0], [0, 0, 0, 0, 0, 0, -289, 83], [0, 0, 0, 130, 213, 0, 0, 0], [0, 0, 0, 0, 0, 0, 150, -39], [0, 87, 947, 0, 0, 0, 0, 0], [0, 0, 0, -52, -182, 0, 0, 0], [-278, 0, 0, 0, 0, 78, 0, 0], [0, 0, 0, 0, 0, 0, -1, -78], [0, 0, 0, -62, 53, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1728_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1728_3_e_v();".
// 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_e_v(:prec:=8)
    chi := MakeCharacter_1728_e();
    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_e_v();".
// 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_e_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1728_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![193, 0, 58, 0, 1]>,<7,R![193, 0, -122, 0, 1]>],Snew);
    return Vf;
end function;