
// Make newform 7728.2.a.ch in Magma, downloaded from the LMFDB on 25 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_ch();".
// 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_ch();".
// 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, -23, 9, 23, -11, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-4, 6, 9, -12, -2, 1], [10, 5, -12, -2, 1, 0], [6, -53, -17, 40, 5, -3], [0, -61, -24, 52, 7, -4], [16, -87, -37, 66, 9, -5]];
    Rf_basisdens := [1, 2, 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, -1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, 1, -1, 0, 0], [1, 1, 0, 0, 0, -1], [0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0], [1, 0, 1, -1, 0, 1], [1, 1, 0, 0, 0, 1], [0, 1, 0, -1, 0, 0], [1, -2, 0, -1, -1, 1], [1, -1, 0, 1, 0, -1], [2, 1, 0, 1, 1, 0], [2, -1, 0, 0, 1, 0], [3, 0, -1, 2, 1, -1], [1, 0, 0, 0, 1, -1], [0, 1, 1, 2, 0, 0], [0, 1, 0, 0, 2, 0], [1, 1, -2, 0, 0, -1], [-3, -1, 0, -2, 0, 1], [-2, -2, 1, 0, 0, 2], [6, 1, 0, 2, 2, 0], [1, -2, 1, -3, 0, 1], [3, 0, 1, 0, -1, 1], [6, 1, -2, 3, 1, 0], [-3, 0, -1, -4, -2, 1], [-2, 0, 0, -3, 0, 2], [9, 1, 1, 1, 2, 1], [-2, 2, 0, -1, 1, 0], [1, -1, 1, 0, 1, 1], [7, 2, 0, 3, 1, -1], [-5, 1, -1, 1, -2, 1], [3, -3, 0, 0, -1, 1], [-6, 1, -2, 1, -1, 0], [-3, -1, 0, 0, -1, 3], [-1, 2, 1, 0, 1, -1], [-1, 3, -1, 2, 1, -1], [1, 1, 0, 2, -2, -1], [-1, 3, -1, 1, -2, -3], [4, 2, -1, 0, 2, -2], [-4, 0, 1, 0, -1, 0], [-2, -1, 2, -3, -3, 2], [10, 0, 1, 1, 0, -2], [7, 1, -2, 1, 3, -3], [-5, -1, 3, -2, -1, -1], [-3, -2, 0, -2, 2, 1], [6, 2, 2, 3, 0, 0], [-6, -3, 0, 0, -1, 0], [3, -4, 0, 4, -2, 1], [-9, 2, 2, 0, 0, -1], [1, -2, 0, -1, 1, 1], [-9, -2, 2, 3, 0, 1], [6, 0, -1, 4, -1, 0], [-9, 2, 2, -1, 1, 1], [2, 1, -2, -2, -2, 0], [3, -1, 2, 2, -2, -1], [-6, -1, 1, -4, -1, 0], [12, 1, 1, -1, 2, 0], [10, -1, -1, 0, 0, -2], [3, -1, -1, -2, -2, 1], [-5, 2, 3, 1, 0, 1], [0, 3, -3, 2, -1, 0], [0, 0, 1, 4, -1, 2], [7, 7, -1, -1, 2, -1], [-8, 2, 1, -4, 1, 0], [2, 3, 2, 4, -2, 0], [-1, -2, -3, -1, 2, 1], [-3, 0, 2, -4, 0, -1], [7, -4, 3, 1, 0, -1], [-1, 1, -3, 3, -2, -3], [8, 0, -4, -1, -3, 2], [-2, 2, 3, -4, 2, 0], [3, -6, 2, 1, 0, 1], [5, 3, 4, 2, 0, 1], [0, 2, 3, -4, -2, 2], [-10, -2, -2, 4, 0, 0], [19, 1, -1, 2, 1, 1], [-13, -3, 0, -4, -2, 1], [-8, -1, -3, 6, -2, 0], [5, 1, -4, 3, 2, -3], [2, 2, -2, 7, 5, -4], [4, -3, 2, 5, -1, 0], [7, -1, 4, 4, 2, -1], [0, -3, -1, -5, -2, 2], [9, 0, -2, 0, 4, -1], [3, -6, 2, -6, 0, 5], [11, 2, 0, 0, 0, -3], [5, 0, 0, 1, -3, 3], [14, -1, -2, 3, -2, 0], [1, -1, 0, 0, -2, 1], [1, 2, -1, 1, 0, -5], [-2, -3, -4, -2, 0, -2], [-1, -2, -2, -2, 4, -1], [-7, 2, 2, -2, -4, 5], [-12, 0, -1, 6, -1, -2], [-4, 2, 2, -4, 0, 2], [7, 1, 0, 0, -3, -1], [-8, -8, 1, -2, -3, 2], [9, 0, -4, 2, 0, 1], [-3, -7, -1, 0, 0, -1], [17, 1, 2, 1, -4, -1], [-2, -1, 2, -5, 1, -2], [-4, 0, 1, 4, -2, 0], [6, 4, -4, 12, 4, -4], [-8, -1, -3, 2, 1, -4], [-4, -1, -1, 5, -2, -4], [-4, -5, -2, -2, -4, 0], [1, 4, -2, 10, 6, -5], [0, -3, 0, 0, -6, 0], [-24, 1, 0, 1, 0, -4], [8, -1, -4, 6, 2, -6], [1, 4, 1, -2, 6, 1], [1, 1, -2, -2, -4, -3], [4, 2, 2, -7, -1, 0], [19, 4, 0, 6, 1, -1], [9, 2, 2, -4, -4, 1], [3, -1, 1, 5, -4, -1], [8, -2, 5, -8, 2, 4], [-30, 2, 2, 0, 1, 0], [-3, -4, 0, -1, -1, 5], [-5, -2, 3, 0, 0, 5], [1, 1, 2, -10, 2, 1], [-2, -4, -1, 1, 4, -4], [10, 1, -2, 0, 1, 0], [9, 2, 1, -4, 2, 3], [9, 0, 1, -9, 0, -1], [7, 1, 1, -10, 1, -1], [-17, -1, -3, 0, 4, 1], [11, 3, -6, 4, 6, -1], [13, -8, 3, -4, 1, 1], [-11, 2, 0, 0, 4, -3], [-2, 2, -1, 8, -4, 0], [-2, 2, 1, -8, -3, 0], [6, -1, -1, 7, -2, -6], [-10, -1, -2, 1, -2, 2], [2, 3, 0, -2, -5, -2], [-5, 0, -2, -5, 2, -5], [-6, 3, -2, 4, 2, 0], [-3, -8, 1, -13, -2, 7], [2, -2, 0, 0, 2, -4], [13, 1, -2, 1, 1, 3], [16, 3, 0, 4, 1, 0], [-7, 1, -2, 10, -2, -1], [8, 5, -4, 4, 0, -8], [2, -5, -3, 3, -4, 2], [-10, -1, 2, -11, 1, 2], [-12, -1, -1, 1, 6, 2], [-10, -4, 2, -4, -2, 6], [-24, -2, 1, 6, -3, 2], [18, -2, 4, 4, 4, 0], [8, 3, -2, 4, 2, 4], [-4, -3, 2, -2, -2, 8], [-20, -4, -4, 1, -2, -6], [8, 3, -5, 7, 6, -2], [-5, -5, -4, 2, -4, -1], [-1, -1, 5, 1, 6, -1], [-17, -4, -4, 5, 5, 3], [1, -4, 2, -9, 0, 1], [14, -1, -1, -1, -6, 0], [4, -3, -5, -2, -1, 2], [4, -4, -2, 12, 0, -4], [-13, 2, -3, -4, 4, -1], [-4, 0, 2, -5, -3, 4], [15, -9, 4, 0, 0, -1], [-10, -3, -1, 6, -5, -2], [1, -3, 0, 6, 0, -1], [11, 0, 4, -8, -8, 1], [-30, -3, -3, 8, 1, 0], [21, 2, 2, 2, 3, 1], [4, 1, -1, 1, 2, 0], [-32, 4, -4, -1, 1, 2], [-19, 3, 0, -6, 0, -1], [-8, -4, 2, -8, -6, 8], [11, -1, 6, -2, 2, 5], [-22, -5, -1, 4, -3, 2], [-17, -2, 0, 12, 0, 1], [7, 3, -4, -7, 3, -3], [6, 1, -3, -2, 0, -4], [-11, 4, 3, 0, -1, -7], [-19, -1, -4, 7, 6, -1], [-13, 0, -7, 2, -1, -3], [-4, 0, 0, -3, 3, 0], [-5, 6, -6, 1, 1, 1], [-3, -1, -6, -3, -4, 1], [-8, 0, -5, 8, -1, -2], [-10, -2, -3, 12, -2, 2], [23, 1, -4, 9, 1, 3], [5, 6, 3, -5, 0, 3], [-11, 0, -2, 4, 4, 3], [-7, 1, 3, -9, -2, -5], [-27, 1, -3, 10, 0, -3], [5, 3, -3, -2, 5, -1], [3, -8, 2, 6, -1, -1], [18, -1, -4, 0, -4, 0], [-18, 3, -4, -2, 2, 0], [1, -1, 1, 6, 12, -1], [30, 0, -2, 8, 2, -4], [25, 5, -6, 3, 5, -5], [3, -3, -2, -2, 1, -3], [-16, 4, 4, -12, 4, 0], [-15, 1, 1, 1, -6, -3], [6, 3, 3, -6, -7, 4], [4, 7, 0, 3, 2, -2], [-2, -4, 5, -6, -3, 8], [25, 2, 1, 7, 4, -1], [-1, -6, 0, -4, 2, 3], [-13, 0, -4, 8, 2, -7], [-13, -1, 0, -6, -4, 3], [20, 1, -2, 1, 1, 0], [13, -2, -1, 0, -3, 3], [9, 7, -6, 2, 8, -3], [-19, -5, -7, -6, -2, -3], [-6, 2, -1, -3, 4, -4], [23, 0, 4, 8, 0, 5], [-11, 5, -1, -2, -7, -3], [20, -8, -1, 7, -4, 0], [13, 2, -7, 7, 4, -5], [-1, 4, -1, -9, 4, -1], [-14, -9, 3, 0, -1, 0], [-6, -8, 3, -12, -7, 6], [-4, -1, 2, -6, -4, -2], [-10, -2, -6, 4, -4, -2], [-5, -3, 5, -10, 1, 7], [-13, 2, 1, -4, 7, 1], [-23, 3, 0, -12, 6, -1], [-2, -2, -6, 0, 8, -2], [8, 4, 3, 1, 8, 2], [-1, 3, -4, -2, 2, -1], [-23, 5, -1, 6, 5, -5], [1, 2, -3, 14, 10, -7], [5, -8, 4, 4, 0, 5], [12, 8, -4, 10, 2, -2], [-3, 2, 2, -12, 1, -3], [-4, 1, -2, -2, -2, 0], [-6, 3, 2, 9, -1, -2], [4, -4, -1, -4, 2, 6], [19, 0, 7, 9, 4, 1], [-13, -3, -8, 2, 2, -1], [-16, 2, 4, -2, -3, -4], [-37, -2, 6, -4, -4, -1], [28, 2, 6, -4, 0, 4], [21, -1, -1, 14, 3, -1], [19, -2, -2, -7, 3, -1], [-14, -2, 6, -11, -3, 0], [8, -2, 0, 2, 10, -4], [21, 6, -6, -1, 0, -1], [-10, 3, 4, -7, 6, 4], [-3, -4, 0, -1, 3, -9], [2, 1, -6, 5, 6, -2], [10, 0, -10, 4, 2, 0], [1, -2, -4, 7, 7, 3], [-19, 8, -8, -1, -5, -3], [-23, 5, -4, 8, -2, 3], [2, -1, -1, -7, -8, 0], [0, -3, 4, -5, 6, 0], [-2, 8, -4, 2, 1, 2], [6, 3, -9, 14, -1, -8], [-17, 1, -4, 4, -6, -3], [12, -5, 2, -2, -6, 0], [13, -1, 2, -16, -3, 5], [3, -3, -3, 9, -6, -1], [20, -2, 2, 8, 5, -6], [4, 1, 1, 2, -5, 10], [-14, 2, 0, -14, -1, 6], [-9, -3, -2, -10, -7, 1], [23, -7, 0, 0, -2, 5], [-8, 3, -2, 1, -8, -2], [3, -4, 0, 8, -6, 5], [-1, 2, -1, -1, -6, -1], [-24, -2, 13, 0, 0, 4], [2, -4, -7, 1, 0, -6], [-22, -2, -6, 6, 6, -2], [1, -7, -8, -2, 0, 3], [22, -4, 4, -14, 2, 2], [5, 0, 5, 7, 0, -1], [-7, -5, 3, -6, 3, 11], [-5, 1, 8, -2, -4, 5], [4, 3, 4, 13, 0, 2], [4, 3, 2, -3, 1, -6], [4, 7, -4, 4, 2, -4], [-1, -1, 12, -8, 0, -1], [10, 6, 5, 6, -2, 0], [-19, 5, 3, -7, 8, -5], [-25, -7, -2, 0, 4, -1], [9, 9, 6, 4, -2, -1], [-9, -3, -2, 2, 5, 1], [-10, -7, -2, -8, -2, -2], [-6, -5, 11, -7, -6, 2], [-29, -6, 4, -10, -3, 13], [-19, -5, 0, -5, 2, 7], [-24, -3, 2, 0, -6, 6], [-14, -7, 10, -5, -4, 4], [-6, -8, -2, 4, 0, 4], [-10, -12, 1, 0, -4, 0], [-18, 9, -5, -11, 2, 0], [-2, -11, -5, -8, 3, 4], [5, 5, 1, -12, 7, -3], [-3, 6, 1, 3, -10, -1], [-14, -2, -3, 3, 0, -2], [17, -5, 3, 8, 0, 3], [-1, 3, 2, 0, 0, 7], [27, -3, 3, 3, -6, -3], [-13, 0, 0, -1, 4, -3], [29, -4, 0, -7, 9, 1], [-9, -7, -1, -3, -6, 5], [-2, -6, 6, -6, -9, 0], [10, -1, -6, 11, -5, -4], [-14, 1, 0, -14, -10, 8], [1, 0, 0, 13, -7, -1], [-8, 0, -3, 16, 2, 2], [11, 2, -7, 12, 1, -3], [-21, 3, -2, 2, 6, -5], [3, -1, 5, -6, -1, -5], [30, -6, 9, -8, 1, 0], [23, -3, -5, 2, -5, 3], [-47, -7, 0, 2, 2, 5], [-5, 0, 4, -23, 1, -1], [-16, 2, -4, -2, -2, 2], [-35, -5, 4, 8, -5, 5], [16, 3, 2, 6, 6, -6], [37, 6, 1, -4, -5, 3], [15, 8, -6, 11, 11, -7], [5, 6, 4, -13, 1, -3], [-5, -12, 5, -4, -7, 5], [-16, 0, -9, 0, 2, 4], [-35, 3, -1, -11, -2, -5], [-16, 9, -6, -9, 3, 2], [-1, 9, -2, -8, -1, -3], [-10, 2, 4, 2, -12, -4], [-23, 6, -3, -5, -2, -7], [-8, 3, 2, 0, 2, -8], [-3, -4, -9, 12, -5, -3], [-19, 8, 0, -9, 10, -1], [-24, -2, 1, 4, 1, 0], [7, -2, -6, -7, -2, 3], [-16, 4, -4, 20, -4, -2], [15, 8, -6, 12, 8, -11], [7, -13, 1, -9, 2, 5], [-3, 7, -2, 10, 6, -5], [4, 5, -2, -13, -7, 4], [34, 5, -7, 2, 7, -4], [-19, -7, 0, -2, 4, 9], [-29, -1, 0, -8, 1, 1], [20, -4, -1, 3, 2, -2], [-9, 3, -11, 1, 0, -5], [-27, 0, -3, -18, -1, 1], [-31, -2, -6, 8, 6, 1], [6, -9, 4, -1, 4, 6], [-18, 7, -2, 2, 0, -2], [-3, -10, -4, 9, -3, -7], [-9, 2, -3, -3, 14, -5], [9, -4, -3, 12, -9, -3], [-22, 5, -2, 15, 0, -6], [4, 1, 12, -1, -3, 4], [50, 2, -8, 2, -2, -2], [-18, -4, -3, 12, 4, 2], [21, -13, 4, 8, -6, 3], [17, 3, 0, -4, 4, 9], [0, -6, -1, -4, 4, 10], [-27, 6, -2, -9, -3, -3], [-17, 11, 4, -2, 10, -5], [1, -3, 0, 10, -1, -3], [-4, -3, -3, -3, 2, 10], [0, -9, 0, 5, -1, 4], [8, -1, -1, -12, -5, -2], [0, -2, 2, 4, 6, -8], [-4, 5, 0, 4, -10, 2], [12, 7, -6, 10, 6, -12], [-5, -10, 3, 5, 0, 1], [-30, 4, 6, 2, 6, -2], [3, -7, 6, 16, 7, 1], [36, -6, 5, -8, 3, 0], [5, -3, 4, 8, -2, 7], [3, -4, 6, 6, 0, 3], [-13, -1, 0, -1, -3, 5], [-8, -1, -2, 3, -13, 0], [-16, 2, 4, 8, -6, -2], [-11, -1, 2, 0, 4, 3], [18, -2, 3, 8, 8, -4], [2, -7, 10, 6, 2, 4], [14, -2, -1, 8, -9, 2], [32, -1, 2, -2, 4, -6], [20, -7, 3, -2, 8, -4], [22, 16, 2, -3, 1, 4], [-6, 4, 4, -3, -1, 0], [-32, -3, -3, -19, -4, 4], [-30, -3, -2, 7, -8, 4], [-24, 8, 0, -10, 0, -12], [33, -7, 0, 2, 0, 11], [7, -4, -5, 8, 3, -11], [-24, 3, 1, 9, 0, -2], [-32, -7, 1, -15, -6, 6], [-12, -5, 5, 2, -7, -4], [-47, 0, -4, 16, 0, -1], [21, 4, 6, 5, -7, 3], [-31, 5, -6, -16, -2, -1], [7, -9, 2, -11, -2, 5], [-25, 0, -13, 15, 4, -1], [7, 2, 9, -17, 2, 7], [-24, 11, 6, 10, 4, -6], [30, 6, 0, 6, 2, 2], [-16, 11, 4, 4, 6, -2], [-4, -1, -7, -10, 3, -2], [-26, 5, -6, 4, -12, -2], [11, 6, 8, -13, -4, 3], [-42, 0, -2, 14, -4, 0], [37, 3, 3, -14, 0, -3], [13, 0, 2, -4, 4, 9], [56, -7, -4, -6, 2, 2], [12, -1, 1, -3, 14, 0], [15, -1, -4, 18, 1, 3], [-5, 0, -5, -12, 2, -1], [-4, 1, 12, 2, 2, -2], [8, -4, 4, 2, 8, 6], [5, -3, 4, -20, -8, 5], [5, 1, -2, 4, 10, -3], [19, -1, 2, 6, 0, -3], [2, 10, -8, 10, 8, -2], [24, 8, -1, 8, -2, -6], [26, -12, 4, -6, 3, 4], [3, 2, -4, -4, 2, -5], [4, 13, -6, 2, -2, -12], [-4, 1, 4, -4, -4, -6], [10, -2, 5, -2, -1, 4], [3, 8, 0, 16, 13, -5], [18, 0, -6, -14, 8, -4], [-29, 11, -1, 11, 4, -11], [44, 6, -6, -1, 1, -2], [29, 6, -4, 1, 7, 1], [3, 1, 2, 0, 13, -5], [-7, 2, -3, -4, -5, 1], [-2, -9, 4, -10, -4, 10], [10, -3, 11, -19, 0, 6], [20, -6, 4, 16, -2, 2], [-1, 5, -6, 19, 9, -1], [20, -4, 1, 6, -13, -2], [-47, -7, 7, 2, 3, 5]];
    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_ch();".
// 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_ch(: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_ch();".
// 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_ch( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-16, 12, 92, 17, -18, -2, 1]>,<11,R![-2368, -416, 560, 75, -41, -3, 1]>,<13,R![-2264, 92, 1094, -11, -66, 0, 1]>,<17,R![-7168, -3200, 1252, 284, -63, -6, 1]>],Snew);
    return Vf;
end function;