
// Make newform 4020.2.a.g in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4020_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_4020_2_a_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_4020_2_a_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 := [3, 9, -3, -25, -15, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [2, 8, -11, -14, -1, 1], [-12, -16, 34, 30, 1, -2], [-16, -17, 48, 43, 2, -3], [21, 22, -70, -59, -2, 4]];
    Rf_basisdens := [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_4020_a();"
function MakeCharacter_4020_a()
    N := 4020;
    order := 1;
    char_gens := [2011, 2681, 3217, 1141];
    v := [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_4020_a_Hecke(Kf)
    return MakeCharacter_4020_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], [-1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0], [0, 0, -1, 0, 1, 0], [1, -1, -1, 1, 0, 1], [1, 1, 0, 1, 1, 1], [-1, -1, 0, 0, 0, -1], [1, -1, 1, -1, 0, 0], [3, 0, 1, 1, -1, -2], [-1, 2, 0, -1, 0, -1], [1, -1, -1, -1, 0, 2], [2, 2, -1, -1, 0, 1], [-1, 2, 3, 0, 0, -1], [4, 1, -2, -3, 0, 1], [3, 1, 0, 4, 0, 1], [3, -1, -1, -1, -2, -1], [3, 1, 0, -2, -2, -1], [1, 0, 0, 0, 0, 0], [1, -1, 2, 0, 2, -2], [0, -1, 1, -3, -3, -1], [0, -2, -2, -2, 0, -3], [4, 1, 1, 0, 0, 0], [6, -3, 1, -4, -4, 0], [2, -1, -1, 3, 1, 2], [3, 3, 3, 3, 0, -2], [1, 2, 3, 2, 2, 1], [6, 0, -2, -1, -1, 1], [2, -3, -1, 0, -2, 4], [0, 1, -2, -2, -1, 2], [3, 1, -1, 0, -1, 1], [3, -5, 2, 1, 1, -1], [3, 3, 2, 1, 1, 0], [-1, -3, -5, -2, -1, 2], [7, -3, 0, -2, -2, 1], [5, 2, -4, -3, -2, 2], [1, -2, -1, 1, -3, -2], [-5, 1, 3, -2, -1, -5], [8, -2, 2, 3, 3, -1], [-4, -3, 1, 4, 4, 1], [7, 0, -2, 5, 0, 4], [3, -4, -3, 1, 1, 0], [8, 3, -1, -7, -3, 1], [-3, 1, 2, 3, -1, 2], [6, 4, -5, -2, -1, -2], [2, 1, 2, 4, 1, 2], [6, 1, -1, 0, -4, 6], [-2, 0, -1, -5, 0, -3], [-2, 3, 2, -2, 5, -4], [3, 1, 4, 7, 3, -2], [-1, 0, -1, 0, 0, 3], [7, 3, 2, 4, 2, 1], [2, -2, 5, 5, 4, -1], [2, 2, -4, 1, -3, 1], [6, 5, 0, -7, 0, 2], [6, -2, -1, -3, -2, 1], [6, -4, 1, -3, -4, -1], [-8, 3, 6, -1, 2, 0], [2, -2, -4, -1, 1, 1], [-1, 0, -4, -3, 0, 4], [-8, 2, 0, -1, 5, 5], [-1, -4, -6, -1, 2, 1], [-13, 2, 3, -4, 0, -3], [-1, -4, 5, 4, 0, -5], [-6, 1, 3, -2, -2, -9], [-6, 2, 4, -6, 2, -3], [-6, -5, -4, 1, -4, 2], [-6, 4, 6, -3, 1, -2], [1, 1, -5, -1, 0, -4], [-2, 5, -3, 0, 2, 0], [-10, 2, 4, 7, 1, -1], [-2, -6, -3, -9, -8, -4], [12, 1, 1, 4, 0, 4], [1, 4, -3, 1, -1, 1], [-13, -1, 2, -7, 1, -3], [6, -8, -1, -3, -8, 3], [10, -4, -5, 0, -3, 0], [7, -6, -1, -2, 0, 3], [-3, -2, 5, 6, -2, -7], [9, 3, 1, -3, 0, -4], [-7, -3, 2, -4, 0, -2], [-1, 5, 3, 4, 3, -3], [8, -3, 5, -1, -3, -6], [-10, 2, 3, 3, 0, -6], [-8, 9, 4, 0, 1, 0], [-21, 0, 0, 3, 0, -4], [5, -3, -3, 4, 5, 9], [1, -5, -3, 0, -1, 1], [-4, 2, 1, 2, 1, 8], [-13, 9, 2, 0, 4, 1], [10, -7, 0, 0, -5, -2], [9, -6, -2, 0, -7, -4], [1, -6, -4, -7, -6, -4], [1, -2, 1, 5, 3, 1], [-14, -4, -4, -1, 3, 2], [-5, -8, 0, 4, -5, -1], [19, -2, 0, -1, -8, 2], [11, 4, -4, -6, -3, 3], [-13, -4, -6, 0, -3, 7], [-4, -5, -1, 9, 7, 1], [-8, 0, 3, 12, 5, -2], [2, -3, 2, -1, 0, 2], [-7, 0, 2, -7, 0, 0], [8, 2, -2, 7, 1, 3], [-7, -4, 4, -2, -7, -7], [3, 8, -4, 4, 5, 2], [-2, 4, 9, 4, 7, 0], [-6, 1, 12, 7, 0, -1], [16, 1, 6, -1, -2, 3], [1, -6, -8, -12, 1, 1], [2, 1, 2, -11, -2, -3], [-1, -4, 4, 4, -1, 1], [3, 4, -2, 8, -1, -2], [-9, -2, -2, 4, -3, 1], [4, 11, 6, 0, -1, -9], [-3, 4, -2, 6, 11, 7], [-8, 7, -7, 2, 0, 0], [2, 2, -8, -2, 4, 0], [-10, 4, 1, -3, 4, -1], [-5, 8, 1, 3, 9, -3], [14, 1, 8, 0, 1, 0], [-6, -2, 10, -3, -9, -8], [2, -8, 3, 2, -1, -10], [-19, 0, 5, 5, -3, -2], [-11, -5, -2, -7, 1, -2], [20, 3, 6, -8, -13, -2], [3, -5, -1, 0, -7, 9], [-2, 7, 3, 5, 1, -6], [6, 0, -7, 4, 5, 10], [14, -12, -3, 1, -6, 4], [-25, -2, -4, 8, 1, 0], [-6, 2, 9, 6, -3, 0], [0, -10, 3, 13, 4, 1], [-4, -13, -1, 7, -5, 0], [13, -13, -1, 7, -4, 2], [3, 2, -1, -11, -1, 0], [4, 3, -5, 7, 3, 6], [-15, 2, -2, -11, 0, 0], [-4, 5, 10, -4, -5, -13], [5, -2, -8, 9, 4, 10], [-13, -4, 3, -6, -6, -6], [-6, -2, 0, -6, 0, 2], [-6, -1, -4, -2, 5, 12], [3, -6, -5, 0, -4, 13], [18, 6, 9, -4, -5, -4], [-8, 1, -2, -2, 3, -1], [-6, 2, -4, -6, -4, 4], [-5, -11, 4, 4, 0, -5], [-10, 6, -6, 6, 0, 8], [-22, -6, 7, 2, -5, -2], [-16, -4, -3, -4, -3, -2], [4, -9, 0, 1, -2, -5], [-28, -6, 1, 9, 2, -3], [6, 1, 3, -1, 3, -1], [-6, -6, 0, -4, -2, -2], [4, 7, -12, -1, 4, 8], [0, -7, 7, 3, -9, -4], [6, 0, -13, -7, -2, -3], [-2, 3, -1, 12, 6, -1], [0, -3, -13, -11, -5, -3], [-13, 14, 4, -5, 10, -6], [-1, 6, 4, 4, -5, 0], [-10, 11, 0, -8, 3, -10], [-5, -5, 6, -7, -11, -13], [-12, -6, -7, -5, -8, 4], [-2, -3, -2, 3, -6, -9], [14, -2, -2, -2, -10, 5], [-2, -1, -3, 3, -7, 5], [-6, 5, 4, 2, 3, -14], [-11, 2, 6, 6, 7, 3], [-4, 6, 10, 6, 4, -3], [7, 5, 7, 13, -2, 2], [3, 14, 1, -11, 5, 3], [-3, -7, -6, -9, 1, -1], [-1, -12, -12, 4, 3, 10], [21, 0, 7, -4, 2, -7], [1, -5, 6, 4, -10, 0], [0, 3, -3, -3, -3, -7], [4, 7, -6, -5, -4, 7], [13, -1, 2, 14, 0, 9], [-7, -3, -7, -2, 1, 5], [-4, -7, 1, -13, 1, -6], [11, 14, 5, -2, 10, -9], [-5, 7, -10, 2, 8, 4], [-4, -11, 3, 4, -10, -6], [15, 0, 2, 5, -2, 8], [-28, -4, 6, -8, 6, -1], [3, -1, 0, 6, 10, 9], [16, -7, -2, 4, -11, 2], [6, -13, -2, 10, 5, 10], [4, 0, 1, -4, -11, 2], [-35, 0, 1, -13, -3, -6], [7, 7, 3, 12, -1, 0], [-25, -3, 3, 7, 8, -8], [3, -7, 9, -8, -9, -11], [-1, -3, -14, -21, -5, 1], [22, 0, -3, 2, 5, 0], [-4, 14, 0, -10, 0, -5], [-20, -5, -3, -11, -11, -7], [-3, 3, -3, -9, -12, -2], [11, 7, 7, -9, -14, -4], [-7, 13, 3, 4, 9, 1], [-1, 9, 10, 0, 8, 2], [8, 2, 13, -4, -1, 2], [-2, -3, 5, -2, -4, -2], [-3, 2, 13, 0, -4, -6], [-38, -8, 4, 6, 4, 0], [7, 0, 3, -2, -8, 1], [8, -12, 1, 8, 1, 6], [7, -14, -2, 4, -5, 2], [-21, 4, -5, -12, 0, -7], [21, 3, -9, 6, 9, 17], [6, 6, 3, 1, 10, -3], [-12, 11, 6, 7, 0, 9], [5, 7, 3, 11, 2, 4], [14, 4, -7, -12, -5, 0], [-3, 8, 0, 1, -2, 0], [-11, -2, 10, 10, 3, -7], [7, -4, 9, -5, -5, -6], [-26, -4, -1, 3, 14, 6], [31, -10, -6, 6, 5, 7], [-22, -3, 14, 4, -11, -10], [20, -4, -8, 12, 10, 14], [2, -6, -5, -8, -1, 4], [5, 5, -10, 5, -3, 8], [31, -22, -3, -8, -10, 1], [-29, -2, 6, 9, 10, 3], [-7, -9, -3, -3, 2, 6], [39, -2, 8, -4, -1, -5], [8, 11, -2, -2, 7, 0], [-14, -11, -10, 4, 9, 11], [-14, -10, 11, -15, -12, -12], [16, -8, 1, 6, -11, -2], [3, -7, 3, 10, -9, 5], [-4, 2, 9, 12, -3, -6], [-9, 12, 6, 3, 6, -12], [9, -3, 6, 2, -2, -11], [0, 3, -3, 0, 6, 1], [18, -6, -13, -9, -2, -2], [22, 9, 4, -11, -6, 0], [-14, 9, 1, 10, 12, 10], [4, -2, -4, -9, 3, 4], [17, -4, -5, 2, 2, -1], [-3, -15, -4, 9, -1, -6], [-3, 2, -4, -8, -5, 5], [-5, 1, -1, -15, -4, -8], [-2, 16, 10, 4, 4, -6], [16, -10, -11, -10, 3, 2], [-5, 9, 4, -12, -8, -8], [-23, -6, -12, -17, -8, 6], [-4, -5, 10, 4, -1, -16], [3, 4, 16, -1, -10, -6], [7, -9, 3, 9, 4, 0], [1, 2, 12, 5, 2, -8], [-20, 9, -5, -3, -3, -3], [-37, 9, 2, 1, 9, -6], [28, 0, -2, 20, 4, 8], [25, -1, -9, -8, 5, 11], [-1, 16, -7, -7, 9, 11], [-14, 9, -2, 5, 16, 15], [12, -14, 6, -5, 1, -11], [-10, -7, -5, -4, -8, 10], [17, -3, 1, 2, -1, -5], [-4, 13, -6, -2, 9, -3], [3, -14, 3, 3, -9, -15], [-21, 15, -3, 2, 1, -3], [11, 7, -14, 0, 10, 3], [-13, -14, -13, 9, -3, 8], [18, -3, 8, 5, -4, -5], [13, 0, -4, 12, 5, 5], [6, 9, -2, -20, 3, -2], [2, 18, 6, -1, 9, -15], [4, -19, 10, -4, -5, -1], [6, -9, 0, -5, -8, 9], [23, -5, -2, -1, -13, -2], [20, 6, 11, 19, 10, -1], [25, -10, -2, 6, -3, 7], [-2, 10, 2, 7, 3, -11], [4, -5, 0, -17, 2, 5], [1, -5, 3, 2, -11, -5], [6, -4, -7, -6, 7, 2], [26, 6, -14, -11, -1, 5], [8, 10, -7, 3, 12, 19], [8, -13, -1, 4, -2, -6], [14, 7, 9, 1, 9, -3], [-14, 7, 11, -9, -1, -16], [27, -6, -4, -4, -7, -3], [-25, -8, -5, 16, 10, 4], [23, 10, 4, -5, -16, -1], [1, 4, -13, 2, 8, 11], [13, -7, 5, 0, -7, 5], [-1, -10, -5, -13, -11, -10], [12, -11, 9, -5, -9, -6], [18, 8, -8, -4, 8, -3], [12, 8, -6, 15, 1, 6], [18, 4, 7, 1, 2, 15], [5, -16, 12, -4, -3, -2], [-11, 11, 0, -3, 1, -14], [-1, -8, 7, 6, -2, 8], [12, 3, 3, 11, -5, 5], [28, 6, -1, 2, 9, 1], [-30, -3, 16, 2, -7, -5], [-4, -14, -15, -3, -6, 5], [-25, -8, 15, 18, 8, 1], [1, 17, -3, -5, 2, 8], [18, -5, 5, 4, 8, 3], [-9, -20, -3, 6, -18, 7], [16, 10, 0, -2, 2, -8], [17, -7, -12, 2, 8, 2], [15, 5, 8, 0, 2, 0], [42, -9, -2, 6, -7, 8], [-21, 12, 9, -1, -3, -16], [-14, 9, -3, -10, -2, 8], [-6, -12, -11, -16, -15, -8], [27, -1, -7, -1, -2, -4], [9, -15, 9, 8, 1, -9], [-24, 6, 11, 3, 4, -10], [12, 14, 1, -13, -4, 5], [37, -11, -6, 14, -10, 5], [29, 6, 5, 9, 3, 8], [3, -20, 1, 12, 0, 3], [-10, 21, -7, 6, 10, 4], [2, -1, 0, 11, 2, -7], [7, 1, -1, -15, 0, -4], [-36, -9, -11, 0, 2, 1], [42, -3, -5, 19, -3, 7], [1, -2, -14, -8, 7, 21], [-2, 5, 3, -18, -16, -10], [23, -7, -5, -4, -5, -3], [6, 21, 3, 3, 21, -1], [-10, -3, 5, 6, 4, -2], [-15, 0, -4, -8, 7, 8], [0, 17, 14, -5, 8, 2], [24, -15, -8, -15, -16, -1], [25, 8, 3, -8, 8, -3], [8, -19, 2, -3, 0, 5], [-13, 0, -5, -9, 11, 2], [11, -1, 12, 3, -13, -9], [3, -9, -2, 10, -6, 1], [40, -3, 1, -8, -8, -2], [7, 15, 12, -5, 5, -2], [-23, -2, -7, 11, -1, 9], [30, -5, 15, 3, 3, 3], [10, -8, 6, 7, -13, -5], [-15, -12, -6, 3, -12, 7], [6, -6, -4, 18, -2, 0], [46, -7, -2, 5, 4, 3], [25, -5, 14, -16, -12, -3], [-21, 15, 3, -6, 9, -4], [-19, 10, -8, 18, 13, 9], [-3, -10, -8, 12, -3, 11], [12, -14, -7, -9, -16, -4], [16, 7, 3, 2, 4, 2], [9, 3, 15, 9, 4, 2], [12, 13, 3, 9, 13, 19], [32, 3, -1, 12, 6, 14], [-3, -12, -9, 13, -9, 3], [-26, 15, 7, 8, 6, 0], [-10, 11, 26, 13, -6, -3], [-17, 9, 0, -4, -2, -1], [0, 18, 3, -12, 13, -8], [-13, -4, -5, 0, -8, -1], [-17, 6, 0, 7, 22, -4], [21, 8, -12, -8, 9, 8], [28, 4, 6, 6, 2, -2], [-10, 4, 0, -4, -6, -3], [26, 2, -2, 2, 4, 19], [-8, -11, 19, 3, -5, -13], [-22, 1, -4, -10, 7, 0], [-38, 2, -5, -16, -3, -10], [-2, -16, -20, 5, -5, 3], [31, -2, 7, 3, 3, -10], [-23, 15, -4, 13, 23, 8], [11, 13, 0, -2, 6, -6], [-29, -13, -8, -1, -5, 13], [15, 15, -1, 2, 13, 4], [0, 3, 6, 0, -15, 0], [26, 1, 13, -17, -13, -3], [40, 12, -3, -20, 1, 4], [-9, -5, 24, 2, 4, -9], [-47, 10, 4, -14, 9, -3], [-9, 8, 13, 17, 11, 4], [-20, -5, 4, -2, 13, -6], [-6, -11, 2, 18, 1, -2], [-24, -16, -5, 7, 6, 8], [3, 17, -14, 16, 20, 19], [-18, 7, -5, -6, 6, -3], [13, 3, -16, -2, -6, 7], [-28, -11, -19, 2, -2, 2], [-1, 4, -9, -30, -4, 3], [17, 10, -15, 5, 9, 26], [-24, -2, -9, -5, 0, -17], [23, 11, 22, -2, -2, 2], [11, -14, -16, -8, -5, 3], [16, -12, 9, 5, 4, 7], [6, 22, -13, 5, 8, 7], [-20, 6, 1, -10, -5, -22], [9, 14, -4, -12, -5, -3], [6, -4, -2, -9, 9, 0], [-17, 8, -1, -8, 2, -11], [-24, 26, 10, -10, 12, -2], [-30, -13, -1, 3, 1, -12], [27, -11, 2, 7, -13, 6], [-3, -17, -8, 11, -1, 8], [13, -1, -13, -3, 2, 0], [-10, -1, -7, -8, 22, -9], [-23, 10, 5, 11, 15, 11], [-56, 1, 2, 5, 2, 1], [-2, 10, 11, 15, -12, 13], [-18, 10, 10, 11, -7, 5], [41, -21, 1, -12, -13, -3], [-2, -8, -1, 11, -10, 2], [10, 0, 3, 14, 13, 2], [13, 0, -6, 7, 8, 8], [-28, 13, 12, 1, 6, -6], [-29, 14, 11, 6, 20, 15], [41, -7, 0, -1, 5, -5], [-2, 1, -17, -4, 4, 0], [-23, 1, 5, 9, 20, -3], [5, 11, -11, -14, 7, 1], [-16, 17, 4, 18, 7, -6], [-3, 12, 0, -4, 1, -13], [-44, 11, 3, -18, -2, -20], [-5, -2, 7, -8, 0, -13], [21, -21, -9, 11, -14, 5], [-12, 3, -1, 5, 1, -14], [-17, 5, 7, -11, 0, -4], [-11, 6, 13, -16, -6, -11], [3, -3, -2, -5, 3, 2], [14, -7, 12, -12, -13, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4020_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4020_2_a_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_4020_2_a_g(:prec:=6)
    chi := MakeCharacter_4020_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(2999) - 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_4020_2_a_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_4020_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4020_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![176, -264, 51, 60, -18, -3, 1]>],Snew);
    return Vf;
end function;