
// Make newform 1040.2.da.b in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1040_da();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_da_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_da_b();".
// 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_1040_2_da_b();".
// 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 := [16, -32, 20, 4, -11, 2, 5, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [8, -8, 2, 3, -4, -1, 2, -1], [12, -12, 3, 5, -3, -1, 3, -1], [-4, 11, -6, -2, 4, 0, -2, 1], [32, -40, 11, 15, -11, -3, 7, -3], [22, -27, 7, 11, -7, -3, 5, -2], [-88, 116, -28, -45, 30, 11, -20, 7], [-104, 132, -30, -47, 32, 13, -22, 9]];
    Rf_basisdens := [1, 8, 4, 2, 4, 2, 8, 8];
    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_1040_da();"
function MakeCharacter_1040_da()
    N := 1040;
    order := 6;
    char_gens := [911, 261, 417, 561];
    v := [6, 6, 6, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_da_Hecke();"
function MakeCharacter_1040_da_Hecke(Kf)
    N := 1040;
    order := 6;
    char_gens := [911, 261, 417, 561];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0], [0, 1, 1, 0, -1, 0, -2, -1], [0, -1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, -1, 3, 0], [-1, 1, 1, 1, 0, 0, 0, 1], [0, 1, 1, 2, -2, -1, 1, -1], [-1, 3, 0, 0, 1, 0, -1, 1], [-1, 0, 0, 0, -1, 0, -4, 0], [-1, 2, 2, 1, 2, -2, -2, -2], [-1, 3, 1, 1, -2, -2, -4, -1], [2, -2, 0, 0, -4, 0, 2, 0], [0, 1, 1, 2, -1, 0, 0, 1], [2, -2, 0, 0, -1, 0, 0, 0], [1, -3, 0, 0, -1, 0, 1, -1], [-2, 0, -2, -2, 6, 2, 0, 4], [-4, 0, 1, -1, 1, -1, 0, 0], [0, 1, 2, 0, 1, -1, 2, -1], [-8, 3, 0, 2, 7, -1, 0, 1], [-1, 8, 0, 1, 0, 0, 0, 0], [-3, 0, 0, 0, 3, -6, 2, 0], [-4, 4, 1, 3, 5, -3, -2, -2], [4, 0, -1, 3, -3, 3, -2, 0], [-2, 2, 0, -2, 6, 2, -4, 0], [0, -2, 4, -2, 1, 0, 0, 4], [-3, -1, -2, 0, -1, -2, 3, 1], [-3, 5, 1, 3, 0, -6, -4, -1], [-14, 2, 1, -1, 1, -1, 2, 0], [0, -1, 1, 0, -9, 0, 2, -1], [6, 4, 1, -3, -9, 3, -8, -2], [9, -1, 0, -4, -7, 2, -3, -3], [1, 2, 2, -1, 0, 2, -6, -2], [0, -2, 1, 1, -1, 1, -4, 0], [4, -2, -4, 0, -5, 9, 3, 2], [9, 2, 0, 4, -11, -2, -2, -6], [5, -2, -4, 0, 1, 4, 0, 2], [0, 8, 1, 5, -5, -5, -4, -2], [0, 6, 5, 1, -1, 1, 0, 0], [1, -3, -6, 0, -7, 8, 5, 3], [4, 3, -1, 6, -5, 0, 0, -1], [-4, -16, 0, -6, 7, 3, 5, 0], [-1, -5, 3, 1, -10, -2, 12, -3], [-2, -2, -7, 1, -1, 1, 4, 0], [1, -8, 0, 4, -3, -2, 8, 4], [-8, 15, -1, 2, 3, 0, 0, -1], [4, 5, 1, -2, -1, 0, 0, 1], [-8, 1, 0, -10, 13, 5, -4, 3], [0, 2, 2, 0, -7, 0, -4, -2], [10, 11, 1, 4, -7, 0, 0, 1], [-5, -1, -2, 0, -3, -2, -5, 1], [4, 8, 0, -4, -4, 4, -12, 0], [8, 6, -1, -1, 1, -1, 8, 0], [-2, 4, -2, -6, 10, 6, -8, 4], [-6, 5, 10, 0, -9, 3, -18, -5], [-4, -5, 0, -2, 5, 1, 0, -3], [3, -2, -4, -3, -10, 6, -2, 4], [0, 5, -3, 0, 3, 0, -10, 3], [-11, 2, 0, 8, 7, -4, 2, -2], [10, -10, -6, 0, -5, 0, 0, -6], [-4, -18, 0, -6, 7, 3, 3, -6], [8, 4, -2, -2, -14, 2, -4, 4], [2, -5, -7, -2, 9, 4, 6, 7], [0, 2, 4, 0, 3, -3, -15, -2], [2, 4, 1, -3, -1, 3, -8, -2], [0, 2, 3, -1, 1, -1, 0, 0], [14, 6, -6, 0, 0, 0, 12, 0], [0, -4, -2, -4, 4, 4, 2, 4], [11, -6, -12, 0, 3, 8, 0, 6], [-8, 8, 9, -1, 1, -1, 0, 0], [-2, 12, 0, -6, 5, 3, -9, 0], [-17, -3, -3, 1, 8, 0, 0, -3], [-11, 2, 0, 7, 2, 0, 0, 0], [2, 8, -6, 2, -6, -2, 0, 12], [0, 11, 5, 0, -1, 0, -22, -5], [13, 5, 0, -4, -11, 2, -7, -5], [2, 8, 0, -8, 3, 0, 0, 0], [0, 0, 0, 0, 1, -1, 9, 0], [2, 8, 3, -1, 1, -1, 6, 0], [2, -6, -12, 0, 3, -1, 3, 6], [9, -13, -3, 3, -6, 0, 0, -3], [-3, -6, -12, 0, -7, 4, 10, 6], [-1, -5, -1, 1, 22, -2, 12, 1], [0, 8, -2, -2, 2, 2, -8, 4], [-17, 3, 1, -7, 12, 0, 0, 1], [4, 8, 0, 10, -9, -5, -3, -8], [-1, 7, 7, 1, -2, -2, -12, -7], [10, 12, 3, -1, 1, -1, 10, 0], [11, 0, 0, 0, 13, -2, 4, 0], [23, 2, 0, 9, -16, 0, 0, 0], [3, 2, 4, 0, 5, -2, 2, -2], [-10, -2, 3, 1, 19, -1, 0, -6], [0, 0, 0, -6, 6, -6, 6, 0], [-1, 15, 3, 9, -4, 0, 0, 3], [-6, 6, 12, 0, 1, -7, -1, -6], [5, -7, -9, -5, -8, 10, 4, 9], [2, -18, 2, 6, -10, -6, 22, -4], [9, -17, 0, -8, -5, 4, 1, -7], [-22, 14, 6, -8, 15, 0, 0, 6], [-18, -12, -6, -2, 2, -2, -4, 0], [8, 5, -3, -8, -5, 16, -26, 3], [-6, 10, 3, 7, 5, -7, -6, -6], [2, 8, 7, 3, -3, 3, -2, 0], [-7, 12, 4, 11, -2, 0, 0, 4], [-2, 10, 0, 10, -3, -5, 1, 2], [7, 3, 5, -7, -6, 14, -20, -5], [-4, 6, 14, -2, 2, -2, -6, 0], [0, 0, 7, 1, -1, -1, -6, -14], [-19, -10, 2, 7, 6, 0, 0, 2], [-4, 4, -3, 3, 5, -3, 2, 6], [8, -8, -5, 7, -7, 7, -10, 0], [9, -5, 1, -9, 8, 18, -8, -1], [7, 13, 0, 12, -13, -6, -5, -9], [6, 5, 1, -8, 1, 0, 0, 1], [20, -2, -4, 0, 23, -3, 7, 2], [-6, 12, 2, 6, 6, -6, -8, -4], [-13, 2, 4, 0, -13, 0, -16, -2], [-1, 18, 0, 12, -5, -6, -8, -10], [1, -1, -2, 0, 1, 0, -11, 1], [2, 1, 5, -2, 17, 4, -6, -5], [-2, 9, -3, 2, -15, -4, -14, 3], [2, 3, 0, -6, 1, 3, -2, 5], [4, 4, -2, -6, 1, 0, 0, -2], [5, 2, 0, -5, 16, 10, -14, 0], [-20, -2, 5, -11, 11, -11, 4, 0], [-3, 1, 2, 0, 9, -12, -7, -1], [2, -4, 2, 0, -1, 0, 0, 2], [-6, 0, 14, 2, -2, 2, -16, 0], [-7, -2, -4, 0, 5, -12, 14, 2], [-24, 5, 0, 6, 21, -3, 2, 3], [6, 12, -9, 7, -7, 7, 14, 0], [-12, 0, 0, 6, 18, -6, 6, 0], [-1, -35, 3, -7, 4, 0, 0, 3], [26, -5, -9, 4, -15, 0, 0, -9], [2, 2, 12, -2, 3, 4, -8, -12], [-3, -4, -8, 3, -16, -6, 14, 8], [14, 1, 2, 0, 11, 3, -16, -1], [-15, 29, -1, 3, 6, 0, 0, -1], [-2, -10, -20, 0, -1, -1, 17, 10], [8, -10, -20, 0, 13, -5, 23, 10], [-13, -9, 0, 8, 9, -4, 11, 5], [6, -20, -12, -6, 3, 12, 28, 12], [-10, -8, -4, 4, 16, -4, 16, 8], [-13, -9, -1, -11, 12, 0, 0, -1], [8, 2, 0, 6, -11, -3, -3, -10], [-6, -8, -9, 7, 5, -7, 24, 18], [-8, 10, 10, 8, 13, -16, -4, -10], [0, -3, -6, 0, -7, 7, -26, 3], [-12, -2, 1, 1, 23, -1, 2, -2], [14, -18, -12, 6, -6, 6, -12, 0], [0, -10, -11, -7, 7, -7, 8, 0], [6, -4, -1, 11, -23, -11, 16, 2], [10, -2, -4, 0, -1, 11, -9, 2], [0, -20, -8, 0, 13, 0, 40, 8], [16, -10, -22, 6, -6, 6, 6, 0], [3, -8, -6, -3, -22, 6, 10, 6], [1, -18, -2, -1, 12, 2, 34, 2], [-32, -6, -4, -4, 4, -4, 2, 0], [7, 4, 0, 16, -15, -8, -4, -20], [-6, 3, 6, 0, -13, 7, -12, -3], [-16, 2, -7, 5, -5, 5, 4, 0], [14, -14, -1, -5, -23, 5, 10, 2], [46, -7, 3, -8, -19, 0, 0, 3], [23, -15, 0, -8, -19, 4, 5, 3], [2, -26, -2, -4, 0, 4, 24, 4], [-18, 25, 0, -6, 21, 3, -8, 15], [-18, 3, 5, 8, 5, 0, 0, 5], [-10, 14, 3, 21, -1, -21, 4, -6], [4, 6, -14, -4, 9, 8, -20, 14], [9, -5, 0, 12, -15, -6, 7, -3], [-4, -26, -6, -2, 10, 2, 30, 12], [-12, 18, 3, 3, -3, 3, 12, 0], [8, 16, 11, 7, -23, -7, -20, -22], [-2, 13, 26, 0, 7, -9, -20, -13], [33, 12, 0, -12, -27, 6, -16, -8], [-8, 0, 0, 0, -1, -7, -23, 0], [-20, 0, 12, -4, 4, -4, -8, 0], [-7, 26, 0, 8, 3, -4, -10, -2], [2, -14, -2, -16, 7, 0, 0, -2], [14, -24, 1, -7, -21, 7, 16, -2], [5, 0, 10, -5, 12, 10, -10, -10], [-7, 13, 7, 7, -16, -14, -12, -7], [6, 8, 8, -4, -8, 4, -20, -16], [-24, 4, 10, -2, 2, -2, -4, 0], [10, -2, 0, -2, 2, -2, 0, 0], [12, 26, 1, 9, -33, -9, -18, -2], [-34, -9, -3, -8, 21, 0, 0, -3], [0, 37, 0, 10, -5, -5, -16, -5], [2, 14, 12, -6, 6, -6, 8, 0], [-2, -18, -4, -14, 18, 14, 8, 8], [20, 20, -4, 10, -15, 0, 0, -4], [-17, 10, 20, 0, -5, -12, -12, -10], [-18, -18, 0, 14, 11, -7, 21, 10], [7, 1, 2, 0, 13, -6, -39, -1], [8, 8, -7, 5, -5, 5, 10, 0], [6, 22, 4, -4, -1, 0, 0, 4], [-9, -15, 0, 4, 7, -2, 11, 3], [-8, -17, -1, 8, 15, -16, 50, 1], [-2, 6, -7, 5, -1, -5, 6, 14], [-6, 2, 0, -10, 11, 5, 5, 22], [4, -10, 7, 19, -27, -19, 22, -14], [6, -10, 1, -9, 9, -9, -2, 0], [34, -13, -26, 0, 15, 19, -16, 13], [-20, -17, 0, 14, 13, -7, 18, 5], [7, -22, 4, -31, 12, 0, 0, 4], [-6, 2, 13, -7, 7, -7, -4, 0], [-22, 16, -4, 8, 7, 0, 0, -4], [1, 12, 10, -1, 20, 2, -26, -10], [4, -18, -1, 3, -11, -3, 22, 2], [2, 12, 0, 26, -15, -13, 9, 4], [23, -19, -7, 17, -20, 0, 0, -7], [0, -17, 0, -26, 13, 13, -4, 1], [-4, 13, -5, 4, 15, -8, -18, 5], [-10, -20, 11, -1, 1, -1, -30, 0], [-13, 14, 18, 13, -2, -26, -2, -18], [-16, 6, 12, 0, 1, -17, -5, -6], [3, -8, -16, 0, 3, 0, 40, 8], [24, 24, 10, -14, -34, 14, -48, -20], [0, -26, 4, -4, 4, -4, -26, 0], [-40, 23, 0, 10, 35, -5, -10, -7], [-7, 18, 8, 3, 2, 0, 0, 8], [-12, 36, 1, 3, 21, -3, -34, -2], [-6, -8, -10, 6, 43, -12, 28, 10], [-18, 28, 6, 18, 18, -18, -16, -12], [-6, 6, -13, 11, 1, -11, 18, 26], [-6, -4, -8, 0, -3, -3, 45, 4], [-2, 29, 13, 2, 1, -4, -54, -13], [10, 0, -6, -10, 10, -10, 16, 0], [-6, 1, 13, 6, -47, -12, 10, -13], [32, 5, 0, -2, -31, 1, 0, 7], [-25, -31, 0, -4, 27, 2, 15, 3], [12, 26, 3, 3, -27, -3, -26, -6], [4, 7, -9, -4, 23, 8, -22, 9], [2, -26, 2, -2, -17, 4, 48, -2], [-30, 22, 14, -16, 23, 0, 0, 14], [-42, 4, 13, -15, 15, -15, 6, 0], [-10, 2, 13, -5, 5, -5, -6, 0], [2, 44, -1, -1, -3, 1, -44, 2], [-21, 12, 24, 0, -15, -6, -20, -12], [12, 11, -11, -2, -5, 0, 0, -11], [-39, 18, 0, 12, 33, -6, -4, -2], [0, -8, 2, 0, 5, 0, 16, -2], [9, 12, 6, -13, 2, 0, 0, 6], [3, -13, 1, -3, -26, 6, 20, -1], [6, -12, 0, -14, 1, 7, 5, 12], [-43, -19, 7, 15, 14, 0, 0, 7], [1, 7, 14, 0, -9, 10, 9, -7], [8, 24, 0, 8, -8, 8, 16, 0], [1, 5, 10, 0, 13, -12, -1, -5], [16, -4, -8, 22, -19, 0, 0, -8], [-17, 4, 8, 0, 15, -32, 2, -4], [-1, -6, -14, 1, 26, -2, 14, 14], [-4, -3, 9, 6, -1, 0, 0, 9], [9, 13, -7, 11, -10, 0, 0, -7], [2, -8, 5, 21, -25, -21, 24, -10], [8, 8, 14, -8, 1, 16, -32, -14], [0, 2, -1, -1, 1, -1, 4, 0], [5, -10, -20, 0, -13, 18, 8, 10], [13, 0, 0, 24, -25, -12, 6, -12], [29, -6, -4, -9, -10, 0, 0, -4], [12, 0, 11, -7, 7, -7, -4, 0], [-1, -5, -10, 0, -1, 0, -9, 5], [-18, -6, 7, -7, 7, -7, -6, 0], [4, -5, 23, -4, 11, 8, 2, -23], [-4, -16, 12, 0, 8, 0, 4, -24], [-11, 2, 10, 11, -8, -22, 18, -10], [-19, 9, 18, 0, -5, -14, 7, -9], [-1, 52, 0, 0, 1, 0, -28, -4], [-8, -7, -14, 0, -21, 13, 34, 7], [8, 8, 0, 12, -28, -12, 4, 0], [-2, -2, -4, 0, -7, 5, 41, 2], [20, 2, 0, 18, -29, -9, -3, -22], [-2, -18, 8, 0, 0, 0, -26, 0], [18, -4, 11, 19, -55, -19, 12, -22], [-21, 32, 6, 13, 4, 0, 0, 6], [-16, -43, 0, 14, 9, -7, 32, 7], [2, -2, 0, -2, -1, 1, -1, -2], [48, 13, 5, 10, -29, 0, 0, 5], [-9, 7, 14, 0, -27, 18, -25, -7], [12, 8, 2, 10, -34, -10, 0, -4], [-2, -7, -14, 0, -9, 7, 14, 7], [-12, 63, 0, 14, 5, -7, -28, -7], [-2, -9, 15, 2, -23, -4, 22, -15], [-2, 26, 2, 20, -9, 0, 0, 2], [-12, 16, 30, 0, 0, 0, -14, 0], [-30, 12, -8, 24, 3, 0, 0, -8], [-14, 0, -9, 3, 25, -3, 12, 18], [0, -16, 2, -10, 10, -10, -8, 0], [12, 18, -5, 7, -7, 7, 16, 0], [8, -20, 5, 3, -19, -3, 18, -10], [-13, -4, -8, 0, -5, -8, -30, 4], [-14, -21, 0, -10, 19, 5, 2, -7], [-1, -17, 19, 1, -2, -2, 36, -19], [-13, 5, 0, 4, 11, -2, -9, -17], [-20, -13, 11, -18, 19, 0, 0, 11], [15, 2, 4, 0, 27, -12, 24, -2], [61, 5, 0, 4, -63, -2, -3, -5], [-10, 16, 4, 0, 0, 0, 12, 0], [-8, -18, -13, -1, 1, -1, -4, 0], [-1, 24, 22, 1, -6, -2, -46, -22], [2, 14, 6, -2, -1, 4, -32, -6], [-31, 18, 6, -5, 18, 0, 0, 6], [-4, -38, 0, -6, 7, 3, 9, -14], [24, -20, -6, -4, -44, 4, 22, 12], [-7, 7, 17, 7, 8, -14, 0, -17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1040_da_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_da_b();".
// 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_1040_2_da_b(:prec:=8)
    chi := MakeCharacter_1040_da();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1040_2_da_b();".
// 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_1040_2_da_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1040_da();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -10, 106, 56, 55, 8, 10, 2, 1]>],Snew);
    return Vf;
end function;