
// Make newform 4056.2.a.bi in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4056_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_4056_2_a_bi();".
// 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_4056_2_a_bi();".
// 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 := [-127, -71, 103, 17, -19, -1, 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], [338, 160, -136, 15, 11, -4], [-325, -642, 195, 159, -17, -9], [1326, 268, -662, -121, 56, 10], [-1156, -482, 226, 174, -6, -13]];
    Rf_basisdens := [1, 1, 167, 167, 167, 167];
    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_4056_a();"
function MakeCharacter_4056_a()
    N := 4056;
    order := 1;
    char_gens := [1015, 2029, 2705, 3889];
    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_4056_a_Hecke(Kf)
    return MakeCharacter_4056_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, -1, 0, 1, 0, 0], [-1, 0, 1, 0, 1, 0], [-2, 0, 2, 1, 1, 1], [0, 0, 0, 0, 0, 0], [2, 1, -1, -1, 0, 0], [0, -1, 1, 3, 0, 0], [3, -1, -2, 0, 1, 0], [1, -1, 0, 0, -2, 0], [-1, -1, -2, 0, 0, 0], [1, 0, 1, -2, -1, 1], [3, 2, -1, -1, 1, -2], [1, 1, 2, 1, 1, 3], [2, -1, -5, -5, -2, -2], [4, 0, 1, -2, 0, 0], [0, 1, 1, -4, 0, -2], [4, -3, -1, 3, 0, 0], [2, 0, -2, -3, -2, 1], [-2, 1, 3, 6, 0, 1], [4, -1, -6, 1, 0, 2], [8, 1, -9, -7, 0, -3], [-3, 3, -2, 2, -1, 1], [10, 0, -4, -9, 0, -1], [8, -1, 0, 2, -2, -3], [-1, 0, 3, 10, 1, 2], [-1, 0, -4, -5, 1, 2], [9, -1, 2, -3, 1, 0], [-9, 2, 7, 7, -1, -2], [7, 2, 3, 0, 1, 1], [0, 5, 0, 2, 0, 1], [3, 1, -5, 1, -1, 3], [-1, 1, 4, -1, -3, 1], [-7, 2, 5, 7, -1, -2], [-1, 2, 3, -3, 2, 4], [2, 1, -11, 0, -1, 1], [-5, -3, -2, -3, 1, 3], [-6, 0, 4, -3, -2, 1], [-5, 5, 2, -1, 1, 1], [9, 1, -6, 4, 0, -2], [6, 1, -4, -5, 0, 0], [0, 0, 6, -1, 2, 1], [0, -2, 8, 3, 0, -1], [11, 3, 4, -9, 3, 1], [-1, -1, -4, 0, -1, 3], [1, 0, 7, 3, -2, -4], [2, -3, 5, 0, -4, 1], [-3, 2, 1, -11, 1, -2], [8, -1, -10, -4, 0, -1], [2, 1, -5, 2, 2, -3], [14, 1, -9, -11, 2, 0], [-5, 0, -3, 2, 3, -1], [-9, 2, 10, 8, 1, -3], [6, -2, -2, 5, -2, 0], [6, 4, 0, -1, 2, -5], [4, 0, 2, 1, 2, 1], [5, -3, 3, 0, -3, 0], [0, 0, 3, 3, -4, -1], [-1, 0, -3, -5, 5, 6], [-1, 1, 2, 8, -1, 4], [-6, -3, 3, 3, 0, -2], [10, 1, -3, -6, 5, -3], [0, 3, -5, -2, 4, -3], [-4, -2, 6, -1, 4, -3], [-3, 3, -7, -2, 1, -2], [-4, -1, 19, 13, 0, -1], [5, -6, -5, 5, -3, 0], [-5, 3, 2, -2, -2, -2], [-13, -3, 14, 12, 1, 3], [0, -2, 0, 0, 4, 6], [6, -2, 2, 9, -2, -1], [-18, -4, 10, 9, 2, 3], [-6, 1, 1, -9, -6, -3], [-1, 0, -3, -3, -1, -4], [9, 2, -3, -12, 3, -1], [-12, -6, -6, 5, -4, -1], [6, -1, -1, 5, -3, 4], [-6, 1, -7, 9, -2, 0], [12, -5, -5, 4, 0, 7], [-5, 0, 21, 8, 3, 2], [6, 3, -5, 1, -1, -4], [-8, -3, -5, -1, -8, 0], [-2, -2, -14, -7, -4, -5], [1, -1, -4, -19, -2, -3], [-6, -4, 6, 8, 2, 3], [-9, 3, 6, 7, 6, -1], [18, -1, -1, 1, 2, 4], [12, 1, -5, -6, -1, -7], [-3, 2, 3, -5, -3, -1], [-4, 4, -8, 0, 0, 0], [10, -1, -8, -4, -2, 5], [-11, -6, -11, -2, -3, -3], [0, 5, -8, -2, 0, -3], [4, -6, -2, -17, -1, -1], [-10, 4, 2, -4, 6, 10], [13, 1, -4, -8, -3, -4], [2, -2, 14, 8, -4, 2], [-3, 1, 16, 11, 5, 5], [-7, -5, 10, 18, 1, 0], [9, 8, 3, -1, -1, 2], [7, 2, -5, -10, 1, -9], [6, 1, -9, 2, -2, -1], [-6, 2, -3, -3, -2, -1], [13, 2, 1, -11, -3, 2], [-15, 1, 12, 4, 3, 0], [7, 5, 4, -8, 5, 3], [-7, -4, -9, 10, -6, -1], [24, 0, -10, -2, 2, -4], [-6, -8, 16, 19, 2, 3], [-6, 3, -1, 7, -5, 4], [-8, 4, -12, -6, 0, 3], [-1, 4, 5, -18, 3, 5], [22, 0, -18, -14, 2, -6], [8, -9, 7, 19, -4, 0], [7, 3, -22, -14, -5, -5], [-3, -7, -2, 2, -1, -6], [-14, 3, 1, 1, 0, 4], [15, 6, -15, -13, -1, -8], [-4, 9, 5, 7, -4, -3], [7, 5, -6, -1, -5, -2], [-14, 2, 2, 13, -8, -3], [3, -3, 0, -5, -5, 6], [-5, 1, 18, 0, 7, 7], [-14, 1, -4, 9, -4, 2], [19, 0, -1, -3, -7, -4], [-1, 4, 15, 11, 1, 2], [6, -5, -23, 0, 0, -1], [8, 6, 8, -10, -2, 2], [-5, -3, -16, -2, -9, -5], [6, 4, -8, -9, 6, -5], [5, 8, 7, 4, -3, -3], [-14, 3, 11, 3, 0, 2], [2, 5, 9, -4, 7, -3], [-9, 4, 9, 5, 7, -2], [22, 2, -8, -3, 0, -5], [8, 8, 2, 4, 2, 6], [1, -4, -15, -2, -5, 4], [19, -2, -25, -19, -3, -8], [-9, 2, -13, 2, -4, -1], [-1, -3, 4, -3, 3, -1], [16, 9, -17, -29, 0, -2], [-8, 2, 27, 8, 0, -4], [-19, 0, 7, 8, -5, 2], [5, -1, -8, -24, 0, 0], [-9, 5, 0, -6, 1, -8], [13, 9, 0, -3, -5, -1], [-13, -1, 10, 2, 1, 4], [-10, -10, -8, 10, -2, -4], [5, -11, -10, -6, 1, 2], [-2, 5, 1, 9, 8, 6], [-15, -1, -4, -7, 1, -5], [6, -9, 9, 9, 8, 4], [-12, -1, 15, 7, 14, 4], [-14, -1, -9, 18, -6, 3], [-21, -6, 5, 17, -7, 2], [-10, 3, 5, 10, -10, -5], [-6, -2, -21, -12, -4, -2], [3, -1, 0, -9, -3, 1], [-17, -10, -5, 15, 0, -8], [2, -6, 8, -7, 7, 3], [-20, 2, 17, 14, 2, 0], [-12, -2, 26, 13, 4, -1], [4, -6, 6, -12, -7, 0], [14, -5, 15, 7, -3, -2], [-15, -4, 21, 30, 5, 3], [4, -4, 14, 9, -4, -9], [12, 8, -7, -6, 2, 4], [-40, -2, 4, 3, 2, 3], [5, 12, 11, 8, 4, -1], [-1, -4, -5, -8, -5, -3], [-14, 3, -5, 1, -2, 4], [-6, 1, -13, 0, 2, 7], [-5, -6, -7, -5, -5, 4], [3, 0, -9, 8, -1, -3], [-1, 0, -7, 2, -3, -3], [-1, -5, -18, 13, -5, -1], [1, 7, 4, 12, 3, 10], [-34, 2, 6, 8, -6, -1], [-7, -2, 21, 14, -5, 1], [-5, -1, -4, 12, -9, -2], [22, -3, -16, -16, -2, -7], [-1, 14, -5, -4, -6, -3], [27, 6, -15, -7, 1, -4], [3, 3, 2, -8, -3, 4], [-18, 7, 1, 7, -2, 0], [-16, -1, 11, 15, 0, -9], [-20, 1, 25, 3, 2, 6], [4, -8, 20, 11, 0, 3], [-16, -12, 4, 5, -4, 6], [-5, -2, 23, 16, -1, -5], [21, 1, -6, -6, 0, -2], [-15, 0, 17, -19, -1, -3], [0, -9, 7, 17, 2, 2], [4, -4, -21, 3, 6, 3], [3, 16, -8, -17, 9, -8], [16, 7, 21, -16, 14, -1], [8, 1, -5, -12, -13, -1], [0, 8, -6, -4, 0, 4], [13, 1, -14, -9, -5, 1], [13, 7, -10, -12, 3, 0], [27, 6, -35, -28, -1, -5], [12, 2, 14, 0, -1, -4], [-23, -1, 12, -4, 1, -4], [-6, 9, 6, 5, 6, -2], [13, 1, -24, -30, -1, 3], [-1, 0, 13, -25, 0, 4], [14, -3, -20, 11, -4, -2], [-9, -1, 8, -7, 7, 12], [7, 2, -5, 11, -1, -4], [3, -9, 0, -3, 1, 3], [-22, 1, 10, -2, 12, 5], [24, 7, -19, -24, -7, -5], [19, -2, -13, -15, 7, 0], [-19, 2, 5, 26, -5, -3], [13, -13, -2, 2, -5, 2], [-24, 1, 7, 2, -4, -8], [-33, 3, 14, 7, 4, -7], [0, -10, -6, 8, 4, -2], [12, 10, -18, -23, -8, -5], [-1, -4, 12, 25, -9, -4], [14, 10, 10, 0, -4, -2], [-21, 5, 17, 21, 7, 9], [18, 2, -22, -31, 0, -3], [-22, 0, 14, -9, 3, 5], [-12, -10, -2, 11, -6, 3], [18, 14, -16, -25, -4, -1], [3, 4, 3, 10, -9, -5], [8, 2, 28, 17, 2, 1], [-7, -4, -26, 10, -3, -3], [6, -7, -28, -22, -2, 1], [7, 3, -2, 2, -3, 0], [-21, -9, 12, 18, -7, 2], [10, 4, -22, -7, 0, 2], [-21, 6, -11, -28, 1, -3], [-9, -7, 18, 12, 3, 4], [5, 8, 19, 17, 7, -2], [11, -5, 14, 6, -3, -5], [4, 3, 7, 0, -2, 6], [-4, -5, 21, -2, -2, -3], [-16, 3, 25, 8, 4, 3], [-18, 4, 10, -3, -5, -3], [11, 8, -3, -16, 3, -7], [-3, -7, 14, 19, -1, 11], [3, -4, -11, -1, -1, 4], [11, 8, -3, 1, 7, 2], [-2, -6, 4, -5, 6, -3], [-33, -5, 18, 14, -2, 10], [-5, -13, -2, 8, -14, 2], [6, 7, 9, 1, -2, -6], [5, 1, -14, 11, -3, -9], [-10, 9, 7, 9, -2, 4], [5, 3, 6, -7, 1, 3], [48, 1, -1, -9, 6, 0], [27, 5, 4, -34, 9, -1], [28, -5, -21, -10, 6, 6], [-7, 0, 13, -1, 3, -9], [-9, 16, 13, -8, -5, 1], [18, -7, -31, -1, 0, 0], [1, 2, 7, -27, 9, 3], [-10, 6, 8, 26, 13, 8], [34, 6, 8, 4, 1, 2], [20, 5, -8, -44, 4, -3], [-48, -1, 3, 20, 4, -1], [-8, 4, 22, 25, 12, 9], [0, 13, 3, -7, 0, -6], [17, 4, 7, -15, 5, 12], [1, 3, 27, -5, -1, -5], [21, -8, -35, -16, 5, -5], [0, 1, -3, -5, 0, -6], [16, 4, -9, -12, -8, -2], [45, -4, -9, -7, -3, -8], [9, 5, 30, 9, 9, 9], [-36, -3, 24, 12, 6, 9], [20, 6, -5, -6, -2, -10], [-5, -7, 16, -3, -1, 0], [-1, -6, 19, 8, 10, -1], [-7, 6, 6, 4, -3, 3], [-39, 8, 9, -13, 3, -8], [4, 3, 5, -21, -2, -1], [-7, -7, -8, 18, -9, 3], [10, -1, -13, 20, 0, -5], [1, 11, 27, 5, 5, 1], [-17, -6, 11, 4, 1, 9], [-25, 2, 17, -5, 5, 4], [-16, 2, 8, -16, -2, 4], [22, 0, -2, -20, 10, 6], [-14, 0, 37, 41, 4, 9], [-16, 6, 9, -13, -2, -3], [36, -3, -9, -15, 1, -2], [28, -6, 6, 11, -8, -9], [5, -3, 20, 5, -1, 0], [-1, -7, 28, 34, -6, -6], [8, -13, -7, 18, -14, 7], [-41, -4, 6, 41, -3, -2], [27, 14, 1, -8, 9, -11], [-6, 6, 10, 25, 8, 10], [2, -5, 16, -7, 8, -6], [-1, -1, 8, -1, 5, 0], [5, 5, 14, -1, 12, -1], [38, -7, -21, -36, 6, 3], [10, -11, 25, 32, -5, 5], [-3, -3, -23, -19, 7, 15], [-9, 15, 18, -13, 1, 12], [13, -2, 2, -18, 5, 1], [31, -6, 1, -5, -11, 4], [14, 8, -15, 10, 2, -8], [-11, 1, 22, 27, 11, 11], [20, -7, -3, -20, -4, -5], [44, 3, -35, -32, -4, -3], [32, 0, -14, -10, -10, -4], [23, -1, 0, 1, -9, -1], [18, -4, -28, -16, 2, -1], [0, 11, -7, -13, 10, -8], [-4, 11, 23, -13, 4, -2], [-9, 8, 15, -29, 5, 8], [-15, -6, 53, 38, 6, 7], [-2, -21, 18, 21, 0, -4], [5, 1, -2, 15, 3, 5], [2, -7, 31, 6, -2, 3], [9, 10, 7, -27, -5, 2], [33, -2, 3, 15, -1, 4], [51, 9, -24, 2, -10, -12], [-12, -16, 26, 9, -2, 4], [-21, -5, 2, 17, -4, -5], [-8, 11, 15, 12, 8, 7], [5, -8, -1, 10, 8, 1], [-20, 0, 18, 24, -10, -12], [-24, -3, 17, 3, -2, 0], [-32, 9, -9, 10, 4, 3], [38, -11, 19, 9, -4, -4], [-18, -2, -14, 8, -10, -2], [-23, 16, 18, 1, 1, -10], [16, -1, -27, 17, -8, -2], [-12, 3, -22, -3, -2, -2], [-33, 14, -7, 3, 1, 0], [-29, -6, 6, 31, -15, -2], [24, -3, -21, -5, -2, -2], [19, -12, -15, -7, -3, -2], [37, -5, -24, -15, -2, -23], [27, 10, -37, -32, 4, -5], [21, -9, -34, -10, -5, 6], [-13, -4, -9, 25, -5, 8], [-22, 0, 10, 15, -11, -5], [15, -1, 38, 35, 10, 13], [21, -4, -5, 1, -2, -4], [39, 11, -36, -27, -9, 6], [24, 4, 2, -19, 20, 3], [-26, -2, 24, 0, 4, -8], [2, -3, 3, -20, 0, 7], [10, 6, 1, 3, 4, 3], [16, 0, 4, 7, -10, -3], [33, -8, -5, 18, 1, 10], [8, -6, 10, -21, -1, 3], [11, 4, -27, -3, 3, 6], [4, -8, -2, -8, -10, 10], [38, 6, -24, -1, 6, -13], [5, -9, 36, 36, -1, 8], [0, -5, 19, 9, 14, 5], [-36, 0, 22, 32, -6, 0], [0, -7, -1, -20, -8, -7], [11, 11, -38, -11, -1, -3], [-9, -12, 1, -5, -5, -8], [15, 8, 13, -17, 1, 8], [-25, -10, -11, -8, -9, 6], [24, 8, -38, -15, -4, 5], [15, -11, 11, 6, 19, 6], [-11, 14, 11, -1, 17, -7], [6, 5, -18, -16, 10, 5], [30, -5, 23, 17, -7, -10], [15, 7, -23, -4, -19, -2], [13, 4, 19, 5, -7, 0], [-12, -5, 7, 8, -4, 15], [35, -12, -27, -10, 6, 5], [3, -1, -7, -31, -5, -1], [-21, -7, 28, -10, 9, 2], [-5, 4, 10, -29, 9, 12], [39, 0, -5, -25, -5, 6], [16, 1, 19, -9, 2, -3], [-1, 9, 54, 27, 7, 5], [-14, 2, -8, 22, 0, -11], [18, 7, -5, -15, 17, 0], [18, 3, -5, -1, 2, 4], [-23, 4, 3, -11, -2, -2], [41, -16, 9, -3, 5, -6], [38, -4, 14, -17, 8, -1], [27, -12, 5, 27, -3, 8], [26, -5, -9, -44, 0, 0], [-10, 6, 10, 15, 8, -7], [-10, -12, -28, 7, 0, -1], [-1, -5, -44, -16, -7, 2], [-4, 12, -4, 5, 9, 1], [11, -5, 5, -16, 11, 4], [-18, 10, 6, 22, 12, -7], [-38, 0, 6, 16, 2, 12], [0, 12, 50, 14, 8, -2], [15, -4, -3, -26, 10, -5], [16, 10, -42, -39, -6, -8], [2, 9, -7, -12, 10, 1], [33, -8, -7, 30, -9, -9], [-19, -6, 5, 18, 2, 1], [-29, -6, 33, 2, 1, 13], [4, 11, -3, -4, -6, -13], [-3, -5, -11, -15, -11, -11], [1, -9, -8, -4, -5, -8], [-52, -3, 31, 23, 5, -2], [-34, -5, 28, 41, 8, 10], [-53, -6, 25, 32, -11, 13], [-1, -6, 33, 11, 9, 4], [-20, -2, 34, 26, 10, -4], [16, 5, -13, 2, 6, 15], [-5, 15, -2, -28, 21, 1], [-4, -10, -18, -10, -9, -16], [-1, 6, 39, 28, -2, -3], [13, -6, -9, -3, -19, -18], [9, -5, -19, -13, 3, -11], [-11, -6, -13, -7, 3, 16], [-12, 7, 23, 46, 2, -1], [-32, -15, 19, 19, -3, 0], [37, 1, -18, -15, 1, -5], [-28, 4, 20, 17, -2, -13], [-16, -8, 26, 12, 16, 3], [16, -2, 4, 9, -14, -13], [-35, -1, -10, 4, 5, 4], [-12, 15, -7, -22, 6, 1], [7, 6, 35, -1, 18, 8], [3, -3, -10, -10, 1, 20], [6, -2, -25, 14, 16, -2], [-47, -3, 8, 26, 7, -6], [8, -16, 10, 1, 6, 3], [2, -1, -51, -21, 6, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4056_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4056_2_a_bi();".
// 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_4056_2_a_bi(:prec:=6)
    chi := MakeCharacter_4056_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_4056_2_a_bi();".
// 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_4056_2_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4056_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-127, -71, 103, 17, -19, -1, 1]>,<7,R![-91, 119, 63, -69, -15, 5, 1]>],Snew);
    return Vf;
end function;