
// Make newform 4014.2.a.t in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4014_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_4014_2_a_t();".
// 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_4014_2_a_t();".
// 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, -16, 13, 14, -8, -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], [3, -5, -1, 1, 0, 0], [5, -6, -7, 1, 1, 0], [-5, 13, 6, -8, -1, 1], [-8, 13, 7, -8, -1, 1]];
    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_4014_a();"
function MakeCharacter_4014_a()
    N := 4014;
    order := 1;
    char_gens := [893, 2233];
    v := [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_4014_a_Hecke(Kf)
    return MakeCharacter_4014_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 := [[-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1], [1, 0, 0, 0, -1, 0], [0, 0, 0, -1, 1, 0], [0, -1, 0, 0, 0, 1], [0, 0, -1, 1, 0, 0], [-1, 0, 1, -1, -1, 0], [-4, 1, 0, 0, 0, 0], [-3, 1, 2, 0, -2, 2], [-2, -1, 0, 2, -1, 1], [-1, 2, 1, 0, 0, 1], [-4, -1, -1, 0, 0, 0], [3, 2, -2, 1, 2, -1], [-2, 0, 2, -1, -3, 1], [-3, -3, -1, 0, 0, 0], [-1, -1, 0, 2, -2, 3], [0, -4, -1, 2, -2, -1], [4, 0, 0, -1, 3, 1], [0, 0, -3, 0, 3, -1], [-3, -2, 2, -2, 1, 1], [3, -3, -1, 0, -1, 1], [-2, 1, -2, 1, -1, -1], [-2, 6, 1, 0, 0, 1], [-3, -1, 3, 2, -4, 2], [-3, 5, -2, -2, 2, 0], [0, 2, 0, -2, 6, -4], [-6, 1, 2, -2, -3, 1], [2, 0, 0, 0, -2, -1], [-1, -8, 2, 1, -3, 2], [3, -1, 2, -5, 3, -1], [3, 3, -2, -1, 5, -2], [-4, -2, 1, -3, -3, 0], [-1, 1, 4, 3, -2, 1], [-1, 1, -5, 5, 2, 0], [-2, 5, -2, 3, 3, -1], [2, 5, 2, -3, 2, -2], [-2, 0, -2, 4, 0, -2], [-5, 2, 0, 3, 2, -5], [9, 0, 0, -2, -1, 3], [-6, 2, 1, 3, -4, -1], [-9, 6, -1, -2, 2, -4], [-7, -4, 3, 1, -1, 0], [3, -7, 2, 2, -2, 0], [-2, 6, -1, -4, 4, -6], [3, 0, -6, 2, 7, -4], [-2, -4, 3, 0, -1, -3], [-1, 0, 0, 0, 0, 0], [-3, 1, 3, -2, 7, -1], [10, 0, -6, -1, 4, -7], [-11, 0, 5, 0, 0, 4], [1, 1, 4, -4, 3, 1], [1, 2, -4, 0, 4, 2], [-6, -7, -1, 2, -3, -2], [0, 1, 4, -2, 2, -1], [-6, 1, -3, 2, 5, -3], [-8, -9, 3, -1, -6, 2], [-4, -2, 6, -4, 0, 2], [6, -1, -3, 5, 2, 1], [0, -3, -2, 1, 4, 4], [12, -2, -6, 2, 4, -2], [4, 4, -2, 2, 3, 1], [-10, 3, -1, -2, -4, 1], [-5, -7, 3, 1, -5, 8], [-10, 6, 3, -4, -1, 2], [-2, 0, -4, 2, -3, 4], [5, -9, 2, 1, -8, 6], [-7, 1, -1, -4, 3, -4], [-6, 4, -3, 9, -6, -1], [-3, 0, -4, 1, 5, -4], [14, 2, -3, 2, 4, -5], [8, -2, -6, 0, 4, -6], [-8, 0, -2, -5, -3, 2], [2, -3, 3, -2, -2, 2], [-5, 3, -7, 3, 5, -2], [1, -2, -5, 7, -1, 0], [-4, -6, -2, 2, -7, 0], [-10, 5, -5, 6, 2, -2], [-6, 3, -2, 0, 0, -7], [-6, 6, -1, -4, 4, -2], [-1, -4, -3, -1, -8, -4], [-10, -8, 2, 4, -6, 0], [-7, -3, 6, -7, -6, 3], [-12, -5, -1, -1, 1, 3], [-12, 4, 5, -1, -6, 5], [4, 1, 1, 4, 3, 0], [-6, 2, 3, -1, -5, 1], [-13, -9, 2, 2, -5, 2], [0, 6, 5, -2, -2, -2], [6, 1, -2, 7, 2, -3], [-2, -1, 4, -1, -2, -3], [9, 4, -3, 2, 3, -4], [7, 1, 2, 2, 4, 5], [-5, -8, 2, 1, 0, 0], [-7, 10, 7, -5, -5, 8], [5, 7, 0, -9, 8, -1], [-2, -6, 2, 6, -8, -2], [-8, 8, -1, 1, 7, 2], [0, 2, 10, -7, -3, 1], [-14, 0, -6, 6, 0, -8], [9, 5, 7, 3, -3, -2], [12, -6, -6, 10, -7, 3], [9, 0, -2, 5, -2, 0], [2, -6, 2, -2, -4, 1], [9, 6, -7, -7, 10, -9], [-16, 1, -1, 1, -2, 6], [19, 2, 2, 0, -5, 6], [7, -6, -5, 0, 0, -8], [11, -7, 2, 2, 1, 3], [-1, 5, -4, 4, 8, -4], [13, 1, -5, -4, 9, -5], [-4, 2, 0, -8, 6, 0], [6, 3, -10, -1, 6, -8], [-7, -5, 0, -2, -1, 3], [-10, 2, 0, 6, -4, 6], [-5, 6, 3, -8, 2, 0], [5, -6, -9, 7, 5, -10], [7, 8, 1, -6, 3, 0], [4, -3, 5, -9, 0, 4], [-11, -14, 6, 2, -6, 1], [6, 11, 4, -6, 8, 1], [7, -5, -9, 5, 8, -8], [11, -3, -3, 2, 7, 6], [-11, 0, 5, -5, -2, 6], [16, -5, 1, -5, 9, -2], [-6, 6, 4, -2, 5, 8], [-6, 4, 9, -7, 4, -4], [5, 13, -6, -3, 8, 0], [7, -8, 0, -2, 5, -6], [-11, 0, 12, -5, -5, 6], [-11, 4, 3, 1, 6, -7], [2, -1, 7, 1, -11, -2], [13, 4, 4, -7, -2, 2], [0, 9, 9, -15, 10, -5], [-10, -6, 11, -3, -3, -2], [9, 7, -3, 1, 7, -7], [-6, 3, 5, 1, -13, 7], [9, 3, -6, 1, -2, 6], [16, -4, -3, 8, 4, 4], [-14, 10, -2, -2, -6, -3], [8, -6, 2, 7, -3, 3], [-7, -1, 9, -4, -6, 12], [22, 1, 3, 0, -3, 6], [-3, -7, -4, 6, -2, -1], [4, 7, 7, 1, -2, -1], [-3, -5, 4, -9, 2, 5], [-8, -4, -2, -6, 10, 2], [-2, 9, -2, 1, -4, 0], [5, -1, 6, 3, -12, 10], [3, -10, 0, -1, 0, 11], [10, -3, -4, -1, 10, -2], [2, 3, 9, -7, -4, 7], [16, -1, -3, -5, 5, -14], [0, -5, 7, -4, -2, 1], [0, 4, 0, 2, -8, 6], [12, 13, -5, 3, 13, -2], [-11, -5, -3, 4, -3, -3], [8, 5, 2, -8, 6, -7], [-21, 5, -1, 2, -1, -6], [7, 17, -2, -2, 2, 2], [23, 1, -1, 2, -1, -5], [19, -4, -2, 3, 5, -9], [13, 2, 7, -11, 6, -2], [-4, 6, -8, -6, 10, -10], [-6, 4, 14, -4, -10, 5], [4, -4, -10, 14, -2, 6], [14, 2, -3, -1, 14, -7], [1, 2, 4, -1, -17, 7], [-21, -3, 7, 6, -6, -2], [4, -2, 7, -2, -9, 2], [18, -15, 4, 5, -7, 9], [-4, -1, 1, 2, 2, 2], [5, 9, -10, 8, 3, -3], [9, -1, -5, 12, -3, -4], [2, -11, 9, -4, -15, 18], [20, -10, 0, -8, 14, -6], [-1, -14, 3, -2, -17, 6], [-7, -3, 0, 2, -2, -4], [-2, 10, -2, 5, -11, 2], [11, -10, 7, -8, -6, 12], [-21, 2, 6, 4, -2, 1], [3, 11, -3, 6, 5, -9], [3, -2, 2, 3, -15, -1], [0, -10, 8, -2, -6, 2], [3, 9, -4, 7, -10, 2], [14, -12, 2, 0, -7, 8], [-2, 8, -9, -5, 3, -7], [19, -3, -6, 1, 2, 13], [-8, 10, -6, 0, -5, 9], [-6, -1, -5, 0, -10, -1], [-6, 4, -12, 16, 1, -7], [2, -18, 4, -6, -10, 8], [-12, -18, 1, -4, -11, 7], [-2, -8, 2, 4, -2, 10], [-6, 4, 11, -7, 6, -5], [18, -7, 9, -17, 2, 5], [30, -8, 4, -6, 3, 7], [33, 0, -10, 3, 5, -9], [9, -2, -5, 10, -10, 5], [-5, -4, 11, 1, -1, -2], [-26, -14, 2, 4, -7, 5], [0, 1, 8, 6, -1, 11], [10, 0, -9, -1, 0, 8], [-2, 10, -1, -4, 8, -4], [-28, 8, 2, -4, 0, -4], [-1, 1, -7, -12, 8, 0], [26, -2, -8, 0, 6, 6], [5, -4, -4, 7, -2, -4], [5, -4, -14, 5, 3, -13], [13, -16, 1, -7, 6, 1], [14, 2, 9, -10, 11, 4], [0, 5, 5, -1, -9, -5], [17, 4, -2, -1, 6, -12], [16, -14, 0, -2, 2, -4], [-10, -17, 10, -2, -4, 6], [21, 11, -6, 6, 1, -6], [9, -6, -2, -12, 7, 0], [-20, -2, 12, -8, 4, -2], [-1, -7, -6, -9, 4, -4], [-1, 1, 1, -7, 10, -6], [12, -10, 0, 4, -12, -2], [11, 17, -12, 1, 13, -10], [15, 14, 0, 3, 11, -2], [17, 8, -5, 1, 0, 0], [-18, -10, -15, 7, 2, 5], [-18, -6, -7, 4, -14, 5], [1, 4, 1, 0, 6, 2], [-17, 10, 9, 1, -6, -3], [-9, 11, -8, 11, -9, -3], [1, 14, 2, 6, -1, -6], [3, 8, 0, -7, 1, 0], [-11, -19, 12, -1, -8, -1], [20, -5, 6, -18, 3, 3], [-12, 2, -7, -1, 20, -10], [17, -7, -14, 3, 16, 2], [7, 10, 7, -9, 1, 6], [-7, -9, -9, 2, -5, -8], [-26, -2, 6, -4, 11, -1], [-8, -5, 4, 4, -11, 13], [-8, -4, -4, 6, 0, -4], [-8, 3, 8, -13, -2, -3], [3, 13, 4, -11, 12, -10], [15, -18, 7, -5, -16, 4], [-36, 10, -5, 5, 7, 6], [20, 8, -11, 9, 7, -3], [-1, 3, -14, 9, 12, -1], [9, -6, 10, -12, -10, 7], [-19, 2, 6, -13, 12, -4], [18, -15, 1, 0, 9, -11], [-5, 5, -3, 3, -5, 8], [9, 8, 4, 5, 17, -7], [1, -12, -2, 1, -3, 1], [-21, -8, 13, -5, -17, 8], [-12, 22, -4, -10, 11, -10], [9, -7, 5, 13, -13, -1], [-17, 8, -8, 4, 22, -21], [14, 12, -8, -2, 8, 8], [-8, 9, -4, 8, -6, -8], [-15, -5, 5, -16, -2, 2], [-20, -6, 7, -6, 5, -4], [5, -8, 11, 7, -4, -4], [-10, 3, 11, 6, -17, 8], [18, 0, -10, 12, -8, 2], [-6, -14, 7, 3, 1, 6], [10, -4, -3, -4, 16, -7], [-14, -5, 2, -6, 12, 2], [-27, -7, 6, 2, 3, -4], [-18, -2, 14, -6, -20, 17], [9, 11, -5, 1, 5, 10], [12, -18, 1, 0, -7, 8], [8, -6, -1, 12, -8, -1], [1, -27, 6, 1, -18, -2], [9, 3, 8, -2, 5, -14], [18, 7, -10, 5, 17, -15], [-10, 12, -1, -4, -1, 10], [9, 26, -9, -3, 18, -22], [10, 3, -10, -9, 12, 1], [-4, -12, 12, 1, 4, -7], [-34, -9, 11, -5, -3, 9], [16, -6, -8, -5, 3, -6], [1, 3, -4, 5, 12, -2], [-2, 8, 10, -2, -10, 0], [19, -9, -6, -3, 12, -16], [8, -4, -4, -6, 0, -10], [14, 11, -3, -6, 19, -21], [-16, -1, 3, 6, -9, 1], [-13, 10, -4, 2, -16, 6], [-10, 2, -6, 8, 4, -10], [-21, 2, 3, -1, 0, 6], [-12, -4, -6, 1, -8, -1], [-2, 0, -10, 2, 17, -3], [12, 1, -6, 0, 9, -19], [-24, 20, 2, -16, 12, -16], [-22, 7, 24, -6, -21, 9], [18, 0, -12, -3, 14, -13], [-10, 12, -8, 4, 15, -17], [3, -13, -16, 1, 5, -2], [1, -12, 0, -4, -11, 1], [-3, -6, 6, -4, 11, -6], [-1, -4, -12, 15, 4, -13], [17, -3, 4, -6, -11, -6], [-32, 12, 3, -4, -19, 6], [7, -16, 0, -6, 4, 9], [-15, -9, -12, 13, -5, -5], [18, 11, -5, 2, 1, -7], [5, -5, -16, 0, 11, 6], [13, -7, -6, 4, -8, 15], [0, -2, 6, -5, 4, 1], [1, 6, 3, -3, -10, -13], [20, -4, -7, 4, -14, -3], [16, -23, -3, 13, -3, 2], [20, -2, -8, 10, 2, -8], [-12, 9, 8, 4, -8, -11], [2, 5, 9, 4, 1, 9], [-3, -12, 12, 9, -2, 1], [-15, -3, -1, -5, -14, -2], [10, -3, -10, 6, 13, 7], [21, -9, -14, 9, -14, -5], [18, -19, -9, 11, -4, 8], [-50, 8, -6, 0, 14, -5], [-22, 6, 4, 2, 4, 16], [-2, -4, -10, 10, -19, 15], [-3, 19, -5, -9, 3, 7], [12, -1, -2, 6, -17, 17], [16, 12, 9, -11, 0, 12], [-24, 8, -7, 2, 15, -6], [4, 6, 1, 0, -9, 7], [11, -12, -3, -3, -11, 4], [15, 0, 9, 0, -13, -6], [-13, 3, -1, 5, -16, -2], [20, 1, 5, -7, 14, -1], [-5, -13, -10, 8, -18, -8], [-3, -12, -8, -3, 20, 4], [38, 17, -10, 0, 13, -11], [13, 6, 17, -12, -9, 18], [-16, -8, -16, 20, -2, -2], [6, -4, -14, 18, -12, 11], [18, 5, 12, -9, -4, -8], [-21, -7, 4, 12, -16, -4], [21, -24, -3, 7, 6, -6], [23, -6, -6, -3, 7, -10], [-2, -10, -1, 7, -7, -5], [-11, -10, 16, -16, 1, 9], [11, -13, -4, 0, 14, -9], [23, -17, 11, -1, -16, -2], [-32, 10, -7, 13, 6, -3], [6, -11, -6, -12, 0, 7], [10, -6, -2, 4, -18, 20], [4, -2, -14, 1, -5, 9], [-5, 15, -7, -2, 6, -6], [-9, -23, 4, -6, -19, 7], [14, 14, -3, -6, 12, -26], [-33, 19, -3, -5, 18, -14], [-6, 9, 12, -1, 10, 11], [-6, 3, 12, -2, -18, 11], [15, -4, -20, 21, -7, -1], [10, -8, -2, -9, 1, -5], [6, 2, -3, -7, 0, 1], [-10, 8, 4, 2, 14, 10], [1, -9, 0, -7, -2, -11], [2, -1, -14, 1, 16, -4], [0, -26, 8, -1, -14, 17], [-13, 11, 8, -5, -18, 22], [-9, -15, -6, -3, 15, 3], [15, 0, -12, 15, -28, 7], [-32, 14, 7, -8, 2, 2], [-34, 0, 18, -18, -20, 12], [27, 9, 4, 10, -16, 3], [-12, -19, 7, -1, -19, -3], [-21, 0, -12, 12, -4, 1], [-4, -3, -5, 12, -13, 9], [31, -13, 13, 3, -17, 2], [5, 21, -11, 11, 3, -3], [29, -3, 12, -4, -2, 5], [-18, -17, 20, -23, -14, 12], [-18, -7, 8, -3, -3, 15], [-33, -4, 11, -11, -18, 15], [2, -4, 7, -6, -18, 26], [10, -10, 8, 13, -6, 9], [1, -13, 21, -15, -21, 1], [-17, 0, -12, 8, 6, -16], [15, 16, 9, -6, 18, 3], [18, -24, 3, 21, -14, 7], [-17, 4, 2, -1, -3, 7], [-7, 4, -9, 7, -10, 7], [-23, 7, 12, -9, 10, 0], [-9, -3, 10, -6, -12, 13], [-17, 12, -5, 24, -6, -2], [0, 15, -2, -4, 12, 2], [-59, 4, 5, 1, -17, 16], [-19, 5, -5, -12, 10, -8], [3, 3, 14, -16, 2, 12], [-1, 0, -18, -10, 21, -12], [-13, 18, -1, 9, -3, -8], [-14, 11, 4, 2, 24, -8], [-10, -16, -14, 4, -9, 11], [38, 14, -8, 8, 10, -8], [4, -12, 6, -14, 6, 12], [8, 11, -6, -12, 14, -12], [32, -12, -26, 16, 24, -22], [2, 2, 6, 20, -10, 3], [-5, -16, -3, 8, 4, -6], [18, 14, -1, 7, 6, -5], [4, 3, -8, -2, 10, -19], [-5, 6, -4, -10, -10, 14], [5, -23, -4, 13, -19, 19], [13, 8, -10, -3, 17, 3], [-25, -14, 2, 3, -5, 21], [33, 1, -7, 1, 13, -2], [4, 22, 17, -21, 16, -1], [6, 0, -3, 8, -19, 18], [-2, -9, -5, -7, -9, -5], [-17, 28, -8, 16, -3, -4], [46, 6, -13, -2, 5, 3], [-49, 0, 0, 12, -9, -13], [-5, -19, -1, -1, -13, 9], [36, -20, 2, 10, -10, 2], [-6, -17, -6, 5, 0, -1], [23, -4, 7, -24, 8, 2], [43, 13, -16, 5, 20, -18], [-6, -16, 6, 12, -6, -6], [26, 14, 4, 0, 6, -5], [-27, 12, -11, 0, 31, -12], [-30, 17, 0, -11, 16, 3], [-6, -26, 2, -6, -20, 16], [-36, 0, 3, 6, -13, -8], [15, -12, -3, 7, 14, -4], [-50, -14, 2, 6, -2, -9], [21, 4, -7, -1, 1, -18], [2, 2, -6, 0, 20, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4014_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4014_2_a_t();".
// 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_4014_2_a_t(:prec:=6)
    chi := MakeCharacter_4014_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_4014_2_a_t();".
// 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_4014_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4014_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-183, 112, 110, -30, -19, 2, 1]>,<7,R![15, 22, -84, 36, 11, -8, 1]>,<11,R![7, -42, 75, -28, -22, 2, 1]>],Snew);
    return Vf;
end function;