
// Make newform 4004.2.a.g in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4004_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_4004_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_4004_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 := [-10, -13, 15, 14, -9, -2, 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], [10, 5, -9, -1, 1, 0], [4, -6, -12, 8, 2, -1], [-8, 6, 13, -8, -2, 1], [-29, -3, 42, -13, -6, 2]];
    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_4004_a();"
function MakeCharacter_4004_a()
    N := 4004;
    order := 1;
    char_gens := [2003, 3433, 365, 925];
    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_4004_a_Hecke(Kf)
    return MakeCharacter_4004_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], [0, -1, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [0, 1, -1, -1, 0, 0], [-2, 1, 0, 0, 0, 1], [1, 0, 1, 0, 1, 0], [-3, 1, 1, 0, 0, 0], [1, 0, -2, 0, 2, 0], [-1, 0, 1, -1, 0, -1], [3, -1, 0, 1, 3, 1], [-3, 0, 0, 1, -1, 0], [-2, 3, 0, 0, -2, -1], [1, 0, 0, 1, 2, 2], [0, 1, -1, -2, 0, -2], [-4, 0, 1, 0, 0, 2], [-1, 2, 2, -1, -2, -1], [1, -1, -1, 2, 1, -1], [0, -2, 1, 3, 2, 1], [-2, 4, 0, -1, -1, -1], [0, -4, 1, 3, 2, 3], [-1, 2, -1, -4, -1, -2], [-4, 1, 0, 1, -1, -1], [-6, -2, 1, 0, -2, -2], [2, -2, 2, 4, 1, 2], [-1, 5, -2, 1, 1, -1], [-4, 1, -1, -3, -3, -2], [-1, -2, 0, 0, 1, -2], [1, -2, -3, 0, 1, -1], [-3, 1, 2, -1, -5, 1], [5, -1, 2, 0, 2, -1], [-8, 1, 2, -1, -5, -2], [0, 1, -2, 1, -2, -2], [5, 0, -1, -2, 1, -3], [-8, 0, -1, -1, 1, 2], [-1, 3, 0, -3, 2, -1], [-1, -5, 0, 0, 4, 4], [1, 0, -2, 0, -4, -3], [4, -5, 5, 3, -1, 3], [-5, -5, 0, 0, 0, -1], [7, 1, -2, -2, 1, 3], [-3, -1, -1, -1, -2, 3], [0, -4, 0, 2, -3, 2], [-3, -3, 5, 3, -4, 3], [-10, -1, 0, 0, -2, -3], [-8, -2, 2, 0, -4, -5], [-3, 3, 3, 3, -1, 1], [-8, 1, -5, 0, -2, 0], [-3, -2, 2, 3, 2, 2], [-2, 1, 2, -1, 2, 0], [0, -5, 1, 0, 3, 3], [-5, -3, 3, 2, -5, -1], [0, 0, 2, 0, -1, 0], [-4, 4, -2, -3, 0, -1], [0, -7, 0, -1, 5, 2], [-1, 0, -1, -2, 0, -3], [-8, -5, 1, 0, 0, 3], [-8, 5, -2, -3, -1, -2], [-5, 4, -1, -4, 2, 1], [-2, 3, -1, -4, 5, -1], [-3, -6, 2, 5, 4, 6], [0, 9, -1, -1, 3, -2], [-9, -2, 0, -1, -7, -3], [-5, -4, 2, 3, 5, 1], [-7, -2, 5, 1, -5, 1], [0, 4, -6, 3, 3, -3], [1, -6, 3, 2, -1, -1], [-3, -6, 3, 7, 5, 0], [-3, -5, 5, 0, -3, -5], [0, 1, -3, -3, 6, 0], [-6, 5, -4, -2, 5, 0], [-10, -2, 1, 2, -8, -4], [2, -2, -3, -3, 4, 2], [10, -2, 3, -3, -1, 2], [14, -8, -1, -1, 10, 4], [5, 6, -4, 3, 5, 0], [9, -9, -1, 1, 8, 5], [-12, 0, -1, 3, 3, 5], [-3, 2, -3, 2, -5, -1], [-7, 2, 3, 0, -2, 1], [17, -11, 1, 5, 0, 3], [-2, 2, 4, 1, -1, 6], [9, -2, 5, 4, -1, 1], [7, -1, -2, -1, -4, -4], [-2, 6, 1, 1, 1, -2], [4, 5, 0, -3, 4, 4], [-6, 1, 4, 3, 3, 2], [3, 0, 0, -1, -2, -3], [4, 8, -7, -3, 6, -4], [17, -1, -5, -2, 3, 2], [4, -2, 2, 7, -3, 6], [-5, -1, -1, -1, 3, 3], [17, -7, -7, -3, -1, 3], [1, 2, -9, -6, 2, -3], [11, -2, -4, 4, 3, 8], [0, 0, -7, 1, 1, 0], [-3, 10, -4, -6, 0, 1], [-18, 3, 3, -5, 0, -6], [1, 10, -3, -5, 2, -2], [7, 4, 0, 0, 4, -3], [7, 4, 5, 0, -4, -3], [-11, 4, 5, 3, -14, 2], [15, -9, 0, 0, 9, 0], [-6, -1, -4, -4, -9, -6], [6, 3, -5, -10, -2, -1], [-10, 6, 3, 5, -2, 5], [15, -5, 8, 2, -2, 0], [-9, 3, -9, -1, 3, 1], [-7, 3, -3, -5, -3, 1], [16, -6, -1, 0, 5, 0], [-1, 8, 1, -4, -1, 3], [-18, 5, 7, 6, -7, -4], [21, -13, 6, 4, 6, 3], [11, 2, -5, -2, -4, -6], [1, -14, 5, 1, -8, 3], [0, -1, -9, -2, 6, -6], [-1, 3, 0, -4, -4, -7], [-13, 4, 1, -1, -1, 4], [3, -6, 6, 0, -9, 2], [-6, -11, 0, 4, -1, 1], [26, -2, -1, 4, 3, -2], [7, 4, 3, -4, -7, 1], [-6, -3, 0, 3, -6, -5], [-21, 3, -4, -4, -5, 0], [-2, 6, -1, -1, -4, 2], [15, 3, 1, -4, -6, 3], [-8, 6, -1, -1, -6, 2], [2, 2, 6, 3, 5, 7], [-13, -8, 6, 0, -11, 1], [9, -5, -3, 5, -4, 5], [6, 0, 2, 6, 4, 7], [-14, 4, -5, -5, -3, 3], [11, 8, -6, 3, 3, 4], [-5, 0, -7, -3, 6, -4], [5, -2, -10, -4, -1, -3], [4, 7, -5, -2, 6, -5], [5, -10, -6, -3, 4, -3], [13, 4, -1, -5, -3, 4], [-26, 1, 8, 3, -12, -4], [-14, -10, 1, -2, -6, -4], [14, 2, -7, -9, 0, -2], [-14, -11, 7, 7, 2, 8], [6, -15, 7, 12, 7, 4], [4, 12, -6, -11, -5, 0], [-8, 12, -5, -9, -2, -5], [-12, -1, -3, 1, -2, 4], [7, 0, -4, 1, 14, -1], [5, -8, 0, -2, -3, 3], [10, -2, -7, 0, 1, -4], [-13, 7, -2, 4, 0, 5], [13, 9, -7, -3, 10, -3], [-7, 1, 2, -3, -7, -3], [19, -11, 0, 4, 4, 8], [11, 3, -6, 2, 3, -6], [13, -17, 9, 7, 3, 1], [9, 0, 5, -1, 2, 2], [-18, 9, -3, 5, 1, -2], [6, 4, 10, 3, -5, 7], [24, -5, 6, 2, 0, -4], [-7, 6, 0, 9, 9, 4], [0, -2, -1, 2, 0, -8], [-3, -9, -7, -2, -6, -5], [-15, 3, 6, 2, -18, 1], [-7, -5, 9, 7, -4, 6], [14, -4, -4, 4, 0, 2], [9, -4, 3, 6, 18, 3], [-19, -7, 9, 3, -6, 9], [13, 6, 4, 1, 1, -3], [8, -8, 9, 6, 0, 0], [-13, 9, -3, -11, 2, -7], [18, 1, -1, 4, 6, 11], [-4, 12, 4, -6, -2, -2], [0, 1, 7, 0, 2, 2], [10, 1, 4, 4, 12, 0], [14, -8, -3, 0, 14, -2], [1, -11, 6, 8, 0, -1], [-5, -5, -1, -5, 1, 7], [-24, 7, 7, 3, -1, -6], [-22, 14, -2, -8, -1, -2], [-19, -7, 1, 5, -3, 1], [9, 8, 0, 5, -4, 1], [8, -6, -2, -3, 6, -1], [9, -2, -5, -1, 2, 3], [6, -17, 1, 1, -4, 8], [6, 7, 1, 2, -10, -5], [1, -18, -7, 5, 6, 8], [11, -2, -6, 4, 4, -1], [-19, -3, 0, -2, -12, 2], [0, -6, 2, -3, 4, 2], [-5, 11, 1, -15, -14, -3], [0, -10, 0, -2, -5, -3], [23, -10, -6, 3, 2, 4], [-1, -9, 1, 12, 14, 2], [-13, -20, 4, 5, -6, -1], [-7, 0, 4, 0, 4, 11], [21, -11, 8, 3, 19, 10], [-16, 9, 7, 0, 7, -5], [-10, -6, 15, 6, 0, 8], [5, 5, 8, -7, -13, -1], [-9, -5, 1, 7, 15, 1], [-9, 7, 5, -12, -7, 0], [-15, 15, -13, -9, -4, -3], [7, -7, -1, 13, 7, -2], [-15, 3, 3, 0, -8, 1], [10, -7, -7, -1, 4, -3], [-22, 4, -10, -5, -17, -9], [-1, 1, -5, -7, -4, -15], [3, 14, 0, -4, -4, -9], [-22, 7, -12, -6, -15, -5], [-5, 7, -3, -6, -17, -3], [9, -5, 4, -2, 5, 5], [16, 3, -1, 9, 11, 10], [20, 2, -10, -11, 6, -5], [0, 1, 5, -1, -5, -3], [-11, 11, -4, -10, 0, -2], [-34, 3, 7, -1, -17, -8], [-5, 16, 4, -6, -5, -3], [-1, -1, -10, -2, 16, -7], [-25, 2, -6, -5, 1, 2], [-18, 0, 0, 2, -7, -6], [12, 0, -11, 3, 6, 4], [17, -1, 1, 4, -16, 7], [15, 10, -13, -4, 4, -5], [-29, 11, -1, 0, -13, 2], [18, 6, 8, -4, -4, 2], [-6, 11, 5, -2, -10, -5], [4, 1, 7, -5, -9, 6], [-5, -16, 5, 12, 5, 1], [27, 5, -8, 2, 9, 5], [-37, 11, 5, -9, -14, -11], [13, -12, 7, 0, 6, 3], [2, -9, -1, 1, -3, 10], [-10, -4, 2, 9, 10, 8], [-21, 20, -4, -9, -10, -8], [-9, 5, -4, 0, -8, -11], [7, 3, -8, -9, -6, -5], [21, -17, 13, 13, 14, 8], [2, 2, 4, -1, 6, 4], [-13, 10, -2, -7, -15, 5], [0, 7, 0, -1, 6, -8], [-20, 11, -4, -3, 10, -2], [-20, -6, 8, 11, 9, 6], [-8, 11, 1, 6, -3, -5], [-12, -10, -1, 8, 3, -2], [-2, 14, 6, 7, -23, 1], [-5, -15, 0, 0, 6, 4], [9, -1, -9, -5, 13, 7], [3, -14, 4, 7, 0, 4], [-5, -5, -9, 3, -6, -1], [23, -10, -4, -2, -13, -5], [-12, 2, 2, -3, -7, 2], [-13, 13, 11, -5, -1, -11], [-11, 2, 11, 2, -1, 1], [19, -16, -6, 10, 12, 9], [-6, 16, -2, 1, 2, 5], [-26, 0, 12, 6, 1, 6], [-30, 8, 8, 3, -7, -2], [-2, 3, 4, 2, 4, 5], [-16, 17, -7, -14, -19, -1], [-2, -15, 3, 14, 2, -5], [-2, 0, -10, -3, 7, -1], [-30, 24, -2, -4, 7, -2], [-15, 19, -3, 5, -9, 3], [3, -6, -9, 4, 7, 7], [-18, 14, -6, -5, -16, -4], [-24, 12, -7, 0, -9, 5], [-5, 12, 2, -2, 4, -11], [33, -13, 5, 13, -7, -1], [-7, 9, 6, 0, 20, 1], [-16, -5, -11, 4, 1, -4], [24, -7, -4, -5, -2, 0], [9, 9, -12, 2, 5, 0], [-24, -7, 2, 7, 2, 4], [0, 15, 2, 2, -17, -6], [26, -16, -5, 5, 4, 4], [-19, 9, -5, 5, 10, -1], [8, 3, -8, -5, -12, -6], [-16, 15, -1, -7, -1, 2], [-4, 4, -18, -6, 5, 2], [-8, 12, 3, -7, -11, -6], [-2, -14, 1, -8, 14, 6], [-7, -8, 2, 3, 11, 12], [-35, 17, -2, 13, 3, 1], [-17, 12, -4, 2, -12, -2], [-7, -15, -9, -2, 0, -8], [2, 19, 2, -1, -7, -12], [1, -6, 5, -3, -8, -5], [12, -1, 3, 5, 14, 4], [23, -6, -14, 8, 14, 4], [-3, 2, -3, 12, 2, 7], [16, 4, 5, -2, -6, 6], [-9, 18, 6, 3, -4, -2], [-2, 0, -4, 3, 9, 3], [-23, 24, -1, -6, -7, -9], [-19, 0, 3, -7, -15, 5], [-8, 8, 3, 13, 2, 8], [-40, 7, -5, -1, -9, -4], [-11, 6, -1, -3, 10, -8], [-15, 2, 13, 4, -11, 7], [5, 4, -10, 8, 9, 1], [-2, 3, 13, -2, -1, -3], [1, 5, -3, -4, -9, -10], [6, -10, -8, -8, 18, -3], [16, 6, -8, 9, 6, 4], [-7, 9, -18, -10, 8, -3], [-27, 5, 4, 8, -5, 3], [18, -8, 5, 0, -2, 0], [-32, -9, 5, 2, -8, 9], [-12, -4, 2, -2, -15, 2], [25, 8, -4, -14, -14, -9], [-30, 4, -7, -12, -13, -16], [17, 4, 7, 2, 14, -6], [5, -17, 2, 5, -16, 3], [-17, -1, 4, -7, 11, -5], [-2, 22, -16, -8, -1, -9], [9, 10, -4, -8, 15, 3], [-7, 24, 2, 1, -6, 2], [10, -9, 11, -6, -1, 1], [-7, 0, 3, 8, 16, -1], [22, 0, 0, 1, 6, -6], [-29, 4, -3, -10, -5, -3], [-23, 13, -1, 4, -14, 1], [10, 3, -1, -11, -6, 0], [-45, -9, 4, -2, -10, 1], [-3, -17, 2, -4, 19, 5], [-14, 8, -1, -16, 2, 1], [-34, 11, -6, -7, -18, -4], [7, -20, 7, 14, 19, 15], [2, -35, 14, 17, 3, 12], [-6, 7, -3, 0, -20, 3], [7, -12, 8, 6, 6, 5], [-8, -25, 7, 2, -9, -7], [-13, 17, 0, -3, -12, -9], [-27, 10, -6, -4, -14, -8], [8, -22, -6, 7, 13, 6], [18, -4, -9, 10, -3, 4], [-1, 13, -13, -12, 14, -7], [-10, -14, 0, 10, 3, -4], [15, 10, -9, 11, 2, 3], [0, 9, -7, -8, 10, -4], [7, 36, -12, -6, 15, -5], [24, -3, -1, 4, -6, 2], [13, -24, 9, 13, 9, 13], [-24, 17, -4, -5, -6, 2], [9, -9, 1, -4, 23, -7], [-11, -9, 11, 0, -3, 7], [-27, -9, 0, 0, 12, 6], [5, 1, 6, -14, -6, -11], [29, -1, 3, 0, 19, -2], [18, 8, -7, -7, 4, 6], [22, -11, 5, 13, 20, 10], [42, -15, -3, 19, 13, 14], [-31, 11, 4, -1, -5, -5], [-4, -8, 9, -10, -11, -8], [28, -5, 0, 5, 0, 5], [16, 7, 1, 6, 22, 13], [-11, -25, 11, 7, -10, 11], [6, 1, 2, 6, -4, 0], [1, -24, 5, 10, -9, -3], [-4, 11, -6, -15, -2, 0], [-28, 10, 0, -20, -9, -11], [-53, 10, 5, -12, -26, -13], [-11, 13, -15, -16, 7, -15], [-27, -16, -5, 0, -10, -14], [1, -8, 14, 15, 2, 10], [42, -23, -4, -11, 12, 4], [-33, 11, 1, 12, -12, -6], [-13, 5, 8, -6, 1, -5], [-4, 21, -7, -13, -9, -6], [23, 24, -8, -8, 8, -5], [16, -13, 5, 6, -8, 1], [-28, -17, 7, 9, 9, 8], [-29, 7, 0, -11, -7, -10], [-28, 10, -3, -2, -13, -8], [3, 8, -12, -16, 6, -7], [28, -8, 9, 12, 8, -2], [-36, 16, 6, 0, -3, -8], [1, 19, -17, -9, 1, -9], [-26, 8, -1, -5, -10, 5], [15, -20, 0, 7, 9, 7], [-3, -28, 10, 22, 22, 13], [11, -11, 1, 1, 20, 17], [19, -24, 4, 15, 28, 16], [-11, 5, 1, -13, -14, -8], [14, 15, 4, 1, 17, 10], [9, -10, 5, -3, -4, -5], [-11, -9, 4, -13, -13, -5], [26, -27, -2, 20, 19, 18], [-11, 18, -8, -3, -8, -11], [32, -7, -11, 13, -3, -1], [-12, 7, 1, 0, 21, -6], [-14, -12, -1, 7, 6, -6], [-8, 27, -11, -24, -16, -10], [12, -26, 8, 9, 11, 5], [7, -1, -8, 7, 5, 7], [17, 3, -12, 11, 5, -9], [-3, -13, 19, 20, -3, 16], [-14, 4, 16, -2, -3, -10], [-36, 22, -17, -9, 1, -5], [-15, -6, -6, 2, -13, 6], [7, -1, 16, 9, 19, 7], [34, -1, 12, 4, 10, -7], [19, 1, 7, -12, -7, -6], [3, -13, 5, 5, 10, -11], [30, -10, -4, -12, 6, -6], [-26, 1, 2, 5, 10, 2], [1, -5, 3, -6, 10, -8], [-18, 7, 13, 13, -10, -1], [-1, 6, -6, -8, -13, -3], [-8, -9, -1, 0, -1, 3], [-17, 28, 7, -7, -14, -12], [11, -3, -19, -7, -1, -5], [41, -23, -5, -8, -9, 0], [-18, -8, 2, -5, -12, 4], [-4, 13, -2, -1, -14, -14], [-25, 6, -16, -9, -10, -11], [25, -7, 6, -5, 10, -5], [11, -1, 8, 7, -11, 1], [-18, 16, -19, -8, 1, -11], [-22, 28, 9, -2, -19, -16], [-18, -3, 9, 5, -5, 14], [18, 3, -20, -5, 12, -6], [-10, 5, 3, -13, -21, 2], [10, 13, 16, -3, 3, 7], [25, -2, -8, 0, 3, 16], [30, 8, -13, 8, 7, 2], [7, -8, -13, 10, 13, 8], [-15, -8, -7, 8, 13, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4004_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_4004_2_a_g(:prec:=6)
    chi := MakeCharacter_4004_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_4004_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_4004_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-10, 13, 15, -14, -9, 2, 1]>],Snew);
    return Vf;
end function;