
// Make newform 7728.2.a.cg in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7728_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_7728_2_a_cg();".
// 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_7728_2_a_cg();".
// 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 := [2, -17, 31, 7, -13, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-4, -1, 1, 0, 0, 0], [-6, 21, 5, -12, -1, 1], [4, 32, -5, -14, 0, 1], [12, -39, -7, 14, 1, -1], [-26, 98, 9, -40, -2, 3]];
    Rf_basisdens := [1, 1, 2, 2, 2, 2];
    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_7728_a();"
function MakeCharacter_7728_a()
    N := 7728;
    order := 1;
    char_gens := [4831, 5797, 5153, 6625, 6721];
    v := [1, 1, 1, 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;

function MakeCharacter_7728_a_Hecke(Kf)
    return MakeCharacter_7728_a();
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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0], [0, 0, 1, 0, 0, 1], [2, 0, -1, 0, 0, 0], [-1, 1, -1, 1, 0, -1], [-1, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 0], [0, 0, 1, 0, 0, 0], [0, 1, 0, 0, -1, 0], [1, 0, 0, -1, -1, 1], [0, -1, 0, -1, 1, 2], [1, -2, 1, -1, -1, 1], [-1, 1, 0, 0, 0, 1], [-3, 1, 0, 0, 0, 1], [3, 0, 0, 1, -1, -2], [-2, 0, 0, 1, 0, 2], [-2, 1, 0, 0, 1, 1], [4, 1, 0, -1, -1, -1], [-2, -1, 0, -1, 1, 3], [0, 0, -1, 0, 0, -2], [2, -2, 1, -1, 0, 2], [6, 0, -1, 1, 0, -1], [3, -2, -1, -1, 1, 2], [-3, 1, 0, 0, 2, 2], [-6, -1, 0, 0, -1, 1], [4, -2, 2, -2, 2, 3], [2, 2, 0, 0, 0, -3], [1, 0, 2, -2, -1, 1], [1, -2, -2, 0, -1, 0], [-1, 2, 0, 1, -1, -1], [-4, 0, 0, 0, 2, -1], [3, 1, 0, 1, 2, -1], [-1, 1, 0, 0, 0, 4], [9, 1, 0, 1, 0, -1], [-5, 1, -2, 2, -2, -1], [3, 1, -1, 1, 0, -1], [-4, 0, 1, 0, 2, -2], [0, 0, 2, 0, 2, 1], [6, -1, 0, 1, -1, -1], [-3, 1, -2, 3, 2, 1], [3, 1, -2, 0, 2, 0], [-2, -3, -2, 1, 3, 0], [-3, -1, 0, -3, -2, 0], [1, 0, 0, 0, 1, 0], [6, 1, 1, 0, -3, -5], [8, 1, -3, -1, -1, -4], [3, -1, 2, -4, -2, -1], [2, -2, 4, -1, -4, 4], [2, 0, 2, 1, 0, 2], [7, 1, -3, 0, 2, -2], [2, -1, 3, -2, -3, 0], [-5, -1, 2, -3, -4, -1], [9, -4, 0, -3, 1, 5], [-2, 2, 1, 1, 0, -2], [6, 0, 1, 0, 2, -2], [5, 0, -2, -1, 1, 0], [4, -2, -2, -3, 0, -3], [-4, 3, 1, -2, 1, 1], [0, 2, -2, 2, -4, -2], [2, -1, 2, -2, 1, -2], [15, 1, 1, 0, -2, 1], [-9, -2, 1, -2, -5, 0], [-2, 2, 0, 0, 0, -3], [5, -1, -4, -1, 0, -1], [16, 1, 0, 0, 1, -3], [4, 1, 2, 2, 1, -4], [6, -1, 3, -2, 1, 7], [-4, -1, 0, 0, 1, 2], [-4, -2, 4, -4, -4, -1], [9, -2, 2, 2, 1, -1], [10, 1, -2, 3, -3, -3], [0, -4, 0, 1, 4, 3], [-2, 5, 2, 1, -1, -3], [-6, -3, 2, -1, 1, 5], [0, 1, -3, 3, -1, -1], [3, 0, 2, -2, 1, 4], [10, -3, 2, 1, 5, 3], [-2, 3, 1, -2, -1, -1], [0, 1, 0, 1, -3, -2], [7, 2, 0, 0, -5, -3], [7, 2, 1, 1, -1, 7], [-2, 0, 3, 0, -2, 0], [-10, 2, 0, 1, 0, -1], [4, -4, 4, -1, 0, -2], [4, 4, 4, -1, -4, 0], [6, -3, -1, -2, 3, -3], [7, -4, 0, 1, 7, 1], [-2, 0, 3, -5, 2, 3], [12, -1, 0, -1, 3, 5], [2, 1, -4, -2, -1, 2], [4, -3, 0, -2, -1, 3], [-2, -2, -2, 1, -2, 2], [18, 0, -4, -1, 0, 4], [-7, 4, -1, 0, 3, 0], [8, -1, -1, -1, 5, 1], [-5, 4, -3, 2, 3, 4], [3, 1, -4, 1, 2, -5], [14, 4, 0, -1, -4, -8], [-10, 2, -4, 2, -2, -4], [10, 0, -3, 4, 0, -1], [9, 3, -6, 0, -2, -2], [8, 1, -2, 1, 1, 1], [10, -2, 2, 2, -2, 2], [19, 0, -4, 3, -3, -2], [-12, 0, 4, -3, 2, 1], [0, -1, -2, 0, 1, 9], [18, 2, -2, 1, 0, -2], [8, -2, 1, -3, 0, 0], [4, 1, 0, 0, -1, -6], [4, 3, -2, 0, -1, -3], [4, 2, 1, -1, 0, 8], [6, 3, 0, 5, 1, -5], [-5, -2, 4, -4, -3, -1], [7, 1, -3, 2, 2, -1], [16, -3, 1, -2, 3, 5], [-8, -2, 2, -4, 2, -1], [-16, 3, 2, -1, -5, -5], [13, 2, 0, 0, 1, 2], [-1, 0, -4, 3, 1, 3], [-2, -1, 0, 2, -3, 1], [8, 1, 4, 1, 1, -3], [-4, 3, -2, 3, 5, 0], [-5, -3, 4, -2, -4, -7], [4, -1, -2, 0, -1, 1], [-8, -2, 5, -1, 2, -7], [-1, 6, 0, 2, 1, 2], [-18, -2, -4, 2, 2, -8], [-4, 7, -4, 1, 3, 1], [17, -7, 2, -2, 0, 3], [-10, -2, 0, 1, 6, -4], [8, -7, 6, -5, 3, 9], [-3, 3, -2, 3, -2, -9], [26, 3, 3, 0, 3, 2], [-22, -3, 0, 2, 1, 4], [7, -2, 3, -1, -1, 2], [0, 4, -2, -1, -2, -5], [-12, -1, -2, 0, -1, -1], [0, 1, 4, -4, -3, 2], [8, -4, 2, 2, 2, 0], [15, -3, 0, -3, 2, 10], [-3, -1, -6, -2, -2, -1], [-8, 4, 7, 2, -2, 6], [4, -1, -2, 0, 1, 11], [-12, 1, -1, 0, -1, -6], [1, -5, 2, 0, 2, 0], [2, 0, 4, -1, 2, 1], [-2, -4, 2, -6, -2, -4], [23, -3, 2, -1, 6, 7], [12, -5, 1, 1, 1, 1], [22, -1, -4, 0, -1, 3], [24, -4, 3, -5, 2, 8], [12, 0, 0, 0, -2, 9], [-10, 4, 0, 3, -6, -9], [-6, -5, 8, -3, -3, 3], [-10, 4, -2, 6, -2, -9], [27, -1, -7, 2, 0, 2], [-14, 2, 0, 3, 2, -11], [2, -4, 0, 3, 6, 1], [3, 4, 0, 1, -9, -2], [-20, -2, 0, 2, 2, 2], [-18, 4, -5, 5, 0, -8], [-19, 0, 0, 0, 1, -5], [14, -3, 2, -1, 7, 7], [-5, -2, 0, 1, -1, 0], [-6, 6, -1, 0, -6, -4], [4, -4, -4, -3, 4, 2], [-5, -7, 7, -2, 4, 3], [-23, 2, 0, -1, -3, -6], [-6, -5, 1, -2, 3, 10], [-9, 0, 2, 2, 1, 3], [-4, -5, -4, -3, -3, 3], [0, 7, 1, -1, -9, -7], [-8, -2, -3, -2, 0, 4], [5, 2, -2, -1, 1, -2], [-16, 4, -4, -1, 0, 2], [-11, 5, 2, 1, -8, 2], [-16, -2, 4, 1, 2, 0], [-15, -3, 6, -2, 4, -5], [-8, 0, 5, -6, -2, 1], [-25, 7, 2, 4, -6, -5], [-5, 6, -8, 0, -3, -1], [5, -4, -4, -3, 1, -1], [-6, 5, -8, 1, -5, -10], [-7, -5, 8, -1, 6, -1], [38, 2, 1, 0, -4, 0], [1, 5, -6, 1, 2, -2], [-8, 4, -9, 3, -4, -9], [-8, -4, 4, 3, 4, 2], [0, 4, 0, -4, -6, -1], [4, 6, 2, 4, 2, -2], [-19, -5, -1, -1, 6, 5], [-19, -2, 2, -1, -3, -10], [-8, 0, 1, -1, 8, 2], [4, -1, 0, -2, -1, 1], [-10, -2, 4, -4, -6, -2], [-2, 3, -7, 3, -1, -3], [11, 2, -5, 4, 7, -3], [-19, -1, 8, -1, -2, 7], [-6, -1, -5, -1, 9, -3], [-24, 5, 3, 1, 1, -6], [1, -1, -9, 0, 0, -9], [-10, -2, 7, 3, 4, -3], [-13, -4, -3, -2, -7, -8], [-6, -3, 0, -4, -5, 4], [0, 2, -6, -1, -4, 0], [26, -2, 4, -5, -4, 4], [18, -7, 0, -7, 3, 5], [11, -5, -6, 2, -4, 2], [-23, 2, 3, -1, -5, -2], [-2, -5, 2, -5, -3, -5], [4, 1, 3, -3, 1, 7], [6, 4, -1, -2, -8, -9], [2, -5, -7, 2, 13, 7], [-13, -4, 6, -4, 3, 12], [-18, 5, 2, 1, -9, -8], [-30, 7, 2, 4, -3, -4], [0, -1, 0, -4, -7, -6], [29, 0, 4, -1, 1, 4], [-3, 7, 0, -1, -4, -5], [10, -1, 2, -2, 5, -3], [4, 2, 2, -2, 6, 2], [-7, -4, 2, -2, 1, 4], [11, -3, -6, 4, 4, -1], [8, 0, 1, -6, 2, -4], [-20, 2, 2, 2, -2, 6], [4, 1, 6, -1, 7, -4], [-12, -1, -2, -1, 3, -5], [-3, 10, 0, 0, 1, -2], [24, 0, -3, 3, 6, 2], [-22, 6, -6, -1, 6, 0], [-16, -4, 8, -2, 2, 2], [-9, 3, -1, -4, -8, -1], [-2, -1, 0, 4, -1, -3], [-9, -2, 3, 5, 3, 7], [6, 3, -2, -1, 7, -1], [-6, 3, -4, 10, 1, -8], [30, -8, 5, 0, 2, -2], [-3, -4, 2, -2, 1, -6], [-10, -2, -4, -5, -2, 0], [-2, -1, -5, -3, 1, 5], [-5, 2, -10, 2, 3, -10], [-3, 0, 2, 1, 3, -5], [17, 1, 3, 5, 4, 0], [-4, -3, 3, 3, 1, 5], [-14, -5, 3, 0, -3, 12], [-20, 3, 0, 0, -1, -8], [-1, -2, -2, -1, 1, -1], [18, 3, -4, -4, -5, 12], [-18, -7, -1, -3, 5, 19], [7, 2, 4, 1, -7, -1], [13, -5, 9, 0, 6, 8], [-8, -4, 9, -4, -2, 8], [-20, 2, 2, -7, 2, 1], [-18, -7, 0, 0, 5, 12], [9, -5, 9, 5, 4, -1], [1, -7, 7, 0, 2, 5], [18, 1, -6, 5, 1, 7], [-42, -1, -2, 2, 3, -3], [-17, -3, 0, 3, 8, 7], [0, 1, -5, -1, 1, -6], [-19, 10, -8, 2, -3, -2], [7, -4, 6, -1, 7, -6], [-5, 5, 9, 1, -4, 1], [13, 7, 0, 7, 6, -9], [0, -2, 5, 0, 0, -6], [10, -7, 4, 0, -1, 2], [-30, 6, -4, 3, 0, 0], [4, -4, 3, -7, -2, -9], [-2, 5, 4, -1, -1, -3], [-4, 0, 3, 6, 4, 1], [-42, 4, -4, 0, -2, 2], [24, 3, 4, -3, 1, -1], [16, 3, -1, 5, -5, -9], [6, -3, 0, -2, -9, -8], [-37, -2, -2, -2, -1, 2], [8, -1, 6, 1, -11, 3], [-28, -2, 3, -5, 6, 3], [-7, -2, 1, 3, -1, 1], [-18, 1, 6, 0, 1, 1], [-10, 5, 6, -1, -1, 3], [-18, 5, 0, -3, -11, -5], [-30, 12, 0, 2, 0, -1], [-34, -3, -4, -1, -1, -1], [6, -3, 6, -3, -7, -1], [9, -1, 0, -1, 10, -5], [24, -6, 3, -1, 0, 4], [8, 2, -2, 1, -4, 7], [-7, 2, 10, -1, 3, 10], [-8, 6, -1, 4, 4, -9], [28, 1, -6, -2, 1, -11], [-16, 0, -3, 5, 0, 7], [10, -2, 4, 2, -6, -2], [32, -9, 8, -1, 13, 7], [-30, 8, 6, 1, -10, -9], [-7, -1, -9, 0, 0, 1], [-45, -6, 4, 2, 1, 8], [20, -1, 4, -8, -5, 0], [16, 2, -5, 2, -6, -7], [-10, 4, 0, -2, -4, -2], [-2, -9, -2, -3, 1, 5], [-12, 4, 0, -2, 10, -1], [-8, 6, -2, -3, -2, 7], [9, 0, 0, 7, -1, 1], [-28, 2, -2, 4, 4, -1], [-5, -2, 8, 0, -1, 8], [-31, 3, 2, 2, -4, -2], [8, -7, -6, -4, 1, -9], [23, 3, -11, 2, 6, 0], [2, -3, -6, -3, 5, 13], [-1, 7, -8, 6, -10, -15], [-16, -1, 0, 1, -5, -13], [-29, 4, -6, 2, -1, -8], [-3, 6, -9, -2, -7, 4], [-5, 8, -12, 6, -1, -8], [0, -1, 0, 3, 3, 11], [3, -7, 7, 0, 10, 12], [16, -2, 0, -2, -8, 8], [1, 10, -3, -4, -9, -2], [8, 5, 8, 6, -3, -1], [-5, -8, 3, 1, 11, 8], [-11, 6, -8, 13, -1, -11], [-9, 12, -10, 9, 5, -7], [7, 4, -11, 5, 7, -8], [24, -9, 2, -9, -1, 11], [36, -10, 4, -6, 0, -11], [-31, 7, -4, 6, 0, -10], [-1, -1, 0, -5, 4, -7], [6, -2, 0, 8, 6, -4], [-34, -1, -4, 0, -9, -6], [-20, 2, 1, 3, -4, -4], [-1, 1, -4, 2, 10, -1], [10, -6, 2, -5, 8, 7], [31, 3, -6, 3, 6, -9], [28, 6, -2, 1, 4, -7], [4, -5, -1, 1, 1, 13], [24, 0, 8, -3, -4, 14], [8, -10, -2, 0, -4, -7], [14, -11, 2, -5, 5, 5], [45, 5, 2, 2, 4, 4], [21, -7, 7, -2, -6, 1], [-6, -1, -2, 1, 1, -9], [-3, 3, -4, 1, 2, -7], [22, -3, -4, 1, 9, 2], [18, -9, -1, -5, 5, 0], [1, -7, 4, -2, 0, 1], [-16, 0, 8, -1, -6, -2], [0, 5, -6, -2, 3, 10], [-44, 3, -2, 2, -7, 1], [-3, -6, 0, -1, 9, 19], [6, 1, -2, -8, -7, -6], [5, -1, 8, -8, -6, 5], [0, -1, 12, -4, 9, 18], [9, -1, -6, 0, 10, 10], [-6, -6, 0, -2, 8, -4], [4, 0, 1, 2, 0, 14], [30, 3, -6, -5, -1, -3], [22, -1, 10, 1, -7, 1], [0, 5, 2, 9, -5, -19], [27, 2, -4, 1, -5, -5], [-20, -5, 10, -1, 3, 19], [-15, -3, 4, -3, 10, 17], [2, -2, 2, -1, 4, 15], [-15, -1, 2, -4, 10, 2], [13, 1, -1, 2, 6, 3], [10, -4, 4, -2, 2, 0], [-8, 3, 2, -2, 3, -9], [-10, 1, 0, -6, -11, -9], [10, 7, 2, 2, 9, 10], [10, -6, 12, 0, -4, 4], [15, -5, 8, -1, 0, -3], [7, 9, -6, 1, 0, -7], [14, 9, 2, 1, 1, -13], [-14, 8, -6, 5, -8, -8], [-27, 3, -6, 5, 2, -8], [3, 2, -5, -5, 3, -3], [10, 1, -5, 5, 9, 3], [16, 11, -12, 3, 1, -7], [28, -11, 2, -1, -5, 1], [0, -3, 4, -2, 5, -9], [7, -9, -6, 1, 10, 3], [-6, -5, -6, 6, 9, 1], [-8, -1, -5, 2, 5, 3], [37, -1, -5, 1, -6, -10], [15, -10, 10, -6, -5, 9], [4, 3, 7, 6, 5, 10], [6, 11, 8, -2, -7, -2], [32, 0, 0, -4, -4, 6], [-44, -3, 8, 1, 3, 9], [-9, -2, 9, -1, 5, -6], [16, -4, 2, -7, 4, 7], [10, 4, 4, -3, -6, 5], [-1, 4, -8, 5, 5, -14], [0, -10, 6, -3, 6, 12], [15, 8, -4, -1, -9, -11], [20, 0, -3, 8, 2, -8], [-30, 7, -2, 5, -9, -2], [4, 7, -8, 8, 5, 2], [-44, -6, 7, -1, 4, 5], [-18, 3, -2, -2, -3, -3], [-30, -8, 6, -8, -10, -2], [10, 8, 5, 1, 6, 6], [-19, -1, -1, 0, 0, -13], [-22, -6, 3, -5, 10, 16], [-22, -4, 2, 7, 4, 9], [-14, 10, -4, 6, 6, 4], [42, -7, 5, 1, -1, -1], [-2, 12, -14, 7, 0, -10], [0, -1, -2, 4, -1, -3], [8, -4, 14, -5, -10, 7], [-17, 7, -4, -5, -6, 3], [26, 9, -4, 0, -3, -21], [4, -11, 4, -4, -3, -5], [24, 3, 3, 1, -11, -25], [-10, 1, -6, 1, 15, -1], [6, -4, 7, -4, 10, 16], [10, 6, 5, 6, -6, -4], [16, 6, -8, 2, 10, -8], [-30, -5, -2, -5, -5, -1], [-9, 4, -2, 0, -3, 10], [-8, -4, 0, 1, 10, -6], [2, -1, -2, -10, -5, -7], [14, 3, -2, 4, -11, -13], [1, 8, 3, 0, 7, 4], [39, 2, -4, -1, -1, 16], [18, -6, -4, -10, 6, 0], [-22, 2, -4, -8, -6, 3], [37, 1, -1, 11, -8, -10], [-7, 3, -9, 2, 10, 2], [-5, -9, 12, -7, 6, 13], [1, 3, 0, 10, 2, 5], [-8, -5, -4, -4, 5, -7], [28, -1, -11, 4, 5, 14], [-22, 5, -5, 9, -3, -9], [25, -1, -10, -1, 8, -10], [1, -9, -2, 3, 6, 1], [29, 4, -2, 4, 9, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7728_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7728_2_a_cg();".
// 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_7728_2_a_cg(:prec:=6)
    chi := MakeCharacter_7728_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(3067) - 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_7728_2_a_cg();".
// 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_7728_2_a_cg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-32, -60, 100, 5, -24, 0, 1]>,<11,R![-64, 464, 100, -145, -33, 5, 1]>,<13,R![-3160, -660, 934, 81, -58, -2, 1]>,<17,R![-896, 1936, -1412, 372, -7, -10, 1]>],Snew);
    return Vf;
end function;