
// Make newform 1040.2.da.c in Magma, downloaded from the LMFDB on 25 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_c();".
// 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_c();".
// 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, 1, 0], [-2, 0, 0, 0, 1, 0, 0, 0], [-2, 1, 2, -1, 1, 0, -1, -1], [2, -4, 0, -2, -1, 1, 1, 0], [0, 1, 2, 0, -1, 1, -2, -1], [1, -2, -2, -1, 2, 2, 2, 2], [1, 1, -1, -1, 0, 2, -4, 1], [2, -2, 0, -4, 0, 4, -2, 0], [1, 6, 0, 3, -2, 0, 0, 0], [2, -6, 0, 0, -1, 0, 0, 0], [-3, 1, 0, -4, 5, 2, -1, 3], [2, 0, 0, 0, -4, 0, 0, 0], [4, 4, 1, 3, -3, 3, 0, 0], [1, 2, 4, 0, 1, 0, 0, -2], [0, 7, 0, 2, -1, -1, -4, -3], [9, 4, 0, -1, -4, 0, 0, 0], [6, -3, -6, 0, 3, 3, 0, 3], [0, -4, 1, 3, -3, -3, 6, -2], [0, -4, 3, -1, 1, -1, -6, 0], [2, 10, 2, 0, -4, 0, -12, -4], [-6, 4, 2, 0, 3, 0, 0, 2], [-6, 2, 4, 0, -1, -5, 1, -2], [1, 1, 1, -1, -4, 2, -4, -1], [-6, -2, 3, 1, -1, 1, -6, 0], [-2, -5, 1, 2, -1, -4, 14, -1], [2, 4, 3, -1, -3, 1, -8, -6], [6, 6, 0, -2, -5, 1, -5, -2], [1, -4, 0, -1, -10, 2, 6, 0], [4, -2, -5, 3, -3, 3, 0, 0], [5, 1, 2, 0, -1, 6, -3, -1], [-2, -5, 0, -2, 3, 1, 0, -3], [5, -2, -4, 0, 5, 0, 12, 2], [0, 4, 1, 1, -1, -1, -4, -2], [0, 6, 3, -1, 1, -1, 4, 0], [1, 1, 2, 0, 3, -2, 7, -1], [8, 5, 1, 6, -7, 0, 0, 1], [1, -5, 0, -4, 1, 2, -1, -3], [-4, 8, 2, 4, 7, -8, -8, -2], [6, -6, -9, -1, 1, -1, 4, 0], [-6, -3, 0, -2, 7, 1, -2, -5], [-3, 12, -2, -1, 2, 0, 0, -2], [-15, -6, 0, 3, 6, 0, 0, 0], [-1, 16, 0, -4, 3, 2, -10, 0], [-1, 1, 3, 1, 0, -2, 0, -3], [8, -1, -3, -2, -3, 0, 0, -3], [-3, 5, 10, 0, 3, -6, -3, -5], [0, 0, 4, 8, -8, -8, 4, -8], [-8, -2, -7, 1, -1, 1, 4, 0], [-2, -4, -8, 0, 4, 0, 12, 16], [12, -5, -10, 0, 7, 5, 2, 5], [-7, -4, 0, -8, 11, 4, -2, 0], [0, 3, 1, 0, 19, 0, -6, -1], [-2, 1, 9, 2, -5, -4, 2, -9], [9, 6, 0, 8, -13, -4, 2, 2], [5, -7, 1, 3, -4, 0, 0, 1], [7, 3, 0, 0, -7, 0, -3, -3], [-4, -4, -2, 6, 2, -6, 12, 4], [-6, 5, 3, 6, 7, -12, 2, -3], [-1, -1, -2, 0, -1, 0, 3, 1], [-6, -4, -1, -1, 13, 1, 4, 2], [12, -6, -7, 5, -5, 5, -4, 0], [-6, 2, 0, -2, 2, -2, 4, 0], [0, -8, -4, -6, 6, 6, 6, 8], [0, -1, -2, 0, 3, -3, 18, 1], [0, -4, 9, -5, 5, -5, -8, 0], [4, -14, 0, -6, -1, 3, 9, 10], [13, -9, 3, -9, -2, 0, 0, 3], [22, -5, -1, 0, -11, 0, 0, -1], [2, -8, -4, 0, -4, 0, 12, 8], [-4, 5, 3, 4, 13, -8, -2, -3], [14, 0, 0, 6, -17, -3, 3, 0], [-23, 3, -1, 7, 8, 0, 0, -1], [6, -2, -4, 0, -9, 15, 3, 2], [-14, -8, 5, 1, -1, 1, -14, 0], [3, 5, 10, 0, 11, -8, -11, -5], [3, -3, 3, -3, 0, 0, 0, 3], [3, 0, 0, 0, 5, -2, -2, 0], [4, -8, -2, -4, -13, 8, 8, 2], [4, 16, 2, -6, -2, 6, -24, -4], [6, -26, -4, 0, -3, 0, 0, -4], [7, 5, 0, -8, -3, 4, -5, 3], [2, -6, 0, -2, 13, 4, 8, 0], [6, 0, -3, -3, 3, -3, 6, 0], [-3, -2, -4, 0, -3, 0, 12, 2], [-26, -7, 1, 0, 13, 0, 0, 1], [-3, 4, 8, 0, -3, 0, -6, -4], [2, -2, 5, 3, -7, -3, 0, -10], [-20, 4, 10, 0, 0, 0, -6, 0], [2, -2, 2, -12, 5, 0, 0, 2], [6, 2, 4, 0, 11, -5, 1, -2], [10, -2, -2, -10, -7, 20, -16, 2], [2, -10, 2, 2, -6, -2, 10, -4], [-1, 17, 0, 0, 1, 0, -9, -1], [14, 2, -2, -12, -1, 0, 0, -2], [-14, 4, 10, 6, -6, 6, -12, 0], [0, 7, -9, 0, -7, 0, -14, 9], [6, -26, -3, -7, -5, 7, 22, 6], [-26, -12, -3, -3, 3, -3, -6, 0], [12, -5, -7, 6, -9, 0, 0, -7], [22, -14, 0, -2, -21, 1, 7, 2], [7, -17, -11, -7, -22, 14, 20, 11], [8, 2, 0, 0, 0, 0, 2, 0], [4, 24, -5, 5, -13, -5, -14, 10], [31, -14, 2, -3, -14, 0, 0, 2], [0, 4, 5, 3, -3, -3, -6, -10], [-12, 12, 7, -5, 5, -5, 10, 0], [3, -15, 3, -3, 14, 6, 24, -3], [-1, 33, 0, 0, 1, 0, -15, 3], [5, -8, 2, -9, 2, 0, 0, 2], [-3, -7, -14, 0, 3, -6, 9, 7], [2, -4, 8, 8, -12, -8, 4, -16], [-2, -3, -6, 0, 11, -13, 14, 3], [3, -6, 0, -4, -1, 2, 8, 14], [-3, -1, -2, 0, -9, 6, 27, 1], [2, -1, -9, -2, -13, 4, -2, 9], [-3, 0, 12, 3, -12, -6, 6, -12], [1, -26, 0, 0, -1, 0, 12, -2], [6, 18, 0, 12, -9, 0, 0, 0], [0, 9, 15, 0, -17, 0, -18, -15], [20, -10, -13, 3, -3, 3, 0, 0], [-7, 1, 2, 0, -1, -6, -9, -1], [-27, -5, 5, 3, 12, 0, 0, 5], [22, -8, 2, -2, 2, -2, -8, 0], [-19, 2, 4, 0, -15, -4, -10, -2], [-3, 18, 0, 16, -5, -8, 4, 10], [10, -16, -3, 1, -1, 1, -14, 0], [-4, 28, 2, 0, 8, 0, -30, -4], [2, 8, -2, 12, -7, 0, 0, -2], [0, -25, -5, 6, -3, 0, 0, -5], [-7, 13, 9, 7, -2, -14, -12, -9], [-6, -11, -3, 6, 5, -12, 34, 3], [-8, 3, 6, 0, 7, -15, 24, -3], [9, -11, 7, -5, -2, 0, 0, 7], [15, 1, 2, 0, 3, 12, 3, -1], [-10, 0, 0, 0, -9, -1, -19, 0], [32, 14, 0, -6, -29, 3, -15, -10], [4, -14, -6, -4, 13, 8, 20, 6], [2, -8, -2, 10, -14, -10, 20, 4], [-21, 3, 3, -3, 12, 0, 0, 3], [-6, 16, 0, -10, 11, 5, -13, 0], [2, -8, 5, 9, -13, -9, 12, -10], [8, 10, 6, -8, -11, 16, -36, -6], [-13, 4, 8, 0, 5, -18, 0, -4], [-12, 6, 7, 11, 13, -11, -2, -14], [10, -14, -2, 0, 0, 0, -12, 0], [20, -18, -9, 3, -3, 3, -12, 0], [-10, 4, 5, 21, -1, -21, 12, -10], [3, -7, -14, 0, -5, 8, 5, 7], [4, -8, 4, -4, -19, 8, 8, -4], [4, 2, -12, 4, -4, 4, 10, 0], [3, -18, -12, -3, 12, 6, 30, 12], [7, -22, -18, -7, -4, 14, 30, 18], [12, 6, -2, -2, 2, -2, 10, 0], [-24, 13, 0, -14, 31, 7, -6, 15], [-2, 7, 14, 0, -5, 3, -12, -7], [0, 2, -9, -5, 5, -5, 16, 0], [2, 14, 13, 9, -13, -9, -18, -26], [-4, 25, -1, -6, 5, 0, 0, -1], [-8, 28, 0, 26, -5, -13, -1, 0], [-10, -18, 0, 6, 14, -6, 24, 0], [-13, 2, 0, 4, 11, -2, -2, -6], [-25, 8, -8, 9, 8, 0, 0, -8], [-14, 6, 9, 3, 25, -3, -12, -18], [5, -1, 9, -5, 10, 10, -8, -9], [-16, 8, 0, 26, 3, -13, 9, 0], [4, -14, 6, 2, -10, -2, 10, -12], [-36, -6, 1, 1, -1, 1, -8, 0], [8, 36, 3, 3, -19, -3, -36, -6], [-2, 1, 2, 0, 15, -17, -8, -1], [14, -19, 0, 6, -17, -3, 6, -13], [15, -11, -22, 0, 7, 8, 11, 11], [-4, -8, 12, -4, 4, -4, -16, 0], [-9, 0, 0, -4, 11, 2, 2, 8], [-5, 3, 11, 1, 2, 0, 0, 11], [10, -24, -9, -9, -11, 9, 24, 18], [3, 0, 18, -3, 8, 6, -6, -18], [-13, 3, -3, 13, 22, -26, 20, 3], [18, 8, 6, -14, -22, 14, -28, -12], [-16, -4, 6, -14, 14, -14, 4, 0], [-2, 10, -6, 0, 0, 0, 16, 0], [-4, -14, -1, 3, 5, -3, 18, 2], [12, 19, -7, 18, -15, 0, 0, -7], [10, -17, 0, -2, -9, 1, 4, -7], [14, -6, -6, 0, 0, 0, 0, 0], [2, -26, -14, -8, 4, 8, 32, 28], [18, 2, -10, 8, -13, 0, 0, -10], [6, 1, 2, 0, 3, 3, 18, -1], [-3, 7, 0, 4, 1, -2, 3, 9], [-17, 13, 26, 0, -5, -12, -9, -13], [-28, -4, 9, 5, -5, 5, -18, 0], [12, -16, 10, -6, -3, 0, 0, 10], [-31, 11, 0, 4, 29, -2, 1, 9], [5, 8, 0, -5, 20, 10, -26, 0], [10, 34, -1, 3, -23, -3, -30, 2], [7, 13, 0, -8, -3, 4, -9, 3], [20, 22, 5, -3, -37, 3, -30, -10], [22, 10, -9, 5, -5, 5, 14, 0], [-16, 9, 18, 0, -1, -15, -12, -9], [17, 24, 0, 0, -17, 0, -12, 0], [-4, -13, -5, 6, -1, 0, 0, -5], [-6, 14, 3, -1, 1, -1, 12, 0], [-11, 7, -1, 3, 4, 0, 0, -1], [4, 7, -19, -4, 15, 8, -22, 19], [8, 10, 1, 1, -17, -1, -10, -2], [22, -32, 0, 6, -25, -3, 15, -8], [-5, -19, 13, -3, 4, 0, 0, 13], [5, 36, 0, 8, -9, -4, -14, 0], [1, 2, 12, -1, 12, 2, -6, -12], [-2, -20, -3, -7, 7, -7, -10, 0], [-3, -10, -6, 3, -14, -6, 26, 6], [-6, 4, 8, 0, -9, 3, -27, -4], [-22, -1, -2, 0, -13, -9, -18, 1], [12, -16, -2, 6, -30, -6, 24, 4], [-12, -14, 6, 2, -2, 2, -22, 0], [10, 25, 0, -10, -5, 5, -22, -9], [13, 12, -6, 3, -8, 0, 0, -6], [8, -4, 13, -9, -7, 9, -18, -26], [0, -6, 0, 0, -11, 0, 12, 0], [6, 20, -4, -4, -8, 4, -20, 8], [-2, -14, -7, 9, -5, -9, 30, 14], [12, -2, -4, 0, 3, 9, -33, 2], [4, 1, 5, -4, 9, 8, -10, -5], [-18, 16, 6, 2, -2, 2, 8, 0], [11, 2, -2, -11, -18, 22, -26, 2], [-17, 16, 0, -4, 19, 2, 2, 24], [-9, 37, 0, -8, 13, 4, -15, 15], [8, 6, -3, 9, -25, -9, 6, 6], [-1, 2, -6, 1, 6, -2, -2, 6], [-1, 7, 9, 1, -20, -2, -12, -9], [-27, 9, 5, -5, 16, 0, 0, 5], [14, 20, 3, 7, -7, 7, 10, 0], [-2, -18, 15, -3, 3, -3, -30, 0], [-6, 20, 1, -3, 15, 3, -24, -2], [29, -10, -20, 0, 5, 24, -12, 10], [-3, 4, -4, 15, -6, 0, 0, -4], [0, -15, 0, -6, 3, 3, 6, 3], [-1, -13, 7, 1, -12, -2, 28, -7], [-3, 18, 0, -21, 12, 0, 0, 0], [8, -16, 0, -8, -9, 16, 16, 0], [20, -2, 0, -22, -9, 11, -13, -6], [3, -21, -11, 5, -4, 0, 0, -11], [6, 2, 4, 0, 27, -21, 15, -2], [-16, 8, 8, 0, 0, 0, 0, 0], [1, 13, 26, 0, -1, 2, -7, -13], [-19, -15, 9, -21, 20, 0, 0, 9], [2, 3, 6, 0, -9, 11, 20, -3], [5, 2, 10, -5, 6, 10, -14, -10], [-19, -14, 8, -9, 14, 0, 0, 8], [-41, 23, -5, 9, 16, 0, 0, -5], [-6, 8, -17, 3, 9, -3, 12, 34], [2, -10, -24, -2, 7, 4, 16, 24], [40, 10, -11, -3, 3, -3, 24, 0], [14, -1, -2, 0, 11, 3, 18, 1], [-29, 2, 0, -20, 39, 10, -2, 18], [19, 22, -4, 9, -14, 0, 0, -4], [-4, 20, 19, 5, -5, 5, -4, 0], [14, -4, -8, 0, -13, 27, 9, 4], [-42, -2, 9, -13, 13, -13, 2, 0], [6, 3, 11, -6, -1, 12, -18, -11], [8, -8, -4, 0, -16, 0, 12, 8], [12, -15, -3, -12, -29, 24, 6, 3], [-24, 14, 28, 0, -9, -15, -15, -14], [10, 3, 0, 30, -25, -15, 6, -15], [-2, -9, -18, 0, -5, 3, 42, 9], [4, -32, 8, -12, 4, 12, 12, -16], [-14, -2, -4, 0, -29, 15, 3, 2], [-5, -29, 0, -8, 9, 4, 7, -7], [-2, 18, 0, 0, 0, 0, 18, 0], [2, 4, -7, -3, -1, 3, 0, 14], [48, 9, -3, 18, -33, 0, 0, -3], [3, -8, 0, -4, -1, 2, 2, 0], [-11, -13, 0, -20, 21, 10, 1, 9], [26, 9, 1, 8, -17, 0, 0, 1], [-11, 1, 2, 0, -5, -6, -15, -1], [0, -24, -10, -26, 26, 26, 8, 20], [-12, 7, 14, 0, -5, -7, -46, -7], [-23, 16, 0, -8, 27, 4, -14, -4], [4, -17, -1, -4, -3, 8, 26, 1], [25, -35, -7, 15, -20, 0, 0, -7], [16, 4, -12, 2, -2, 2, 14, 0], [16, -38, -2, -2, -7, 0, 0, -2], [-6, 16, -3, 29, -17, -29, 16, 6], [-32, -8, 10, 6, -6, 6, -24, 0], [4, 18, 33, -3, 3, -3, -12, 0], [-20, -12, -3, 3, 37, -3, 18, 6], [16, 5, 10, 0, -5, 21, -24, -5], [10, -17, 0, -18, -1, 9, 6, 13], [-5, 19, 3, 5, 34, -10, -28, -3], [-15, 3, 0, 12, 9, -6, -3, -15], [65, 4, -10, 3, -34, 0, 0, -10], [-8, -1, -2, 0, 7, -15, 18, 1], [26, -20, 0, -10, -21, 5, -1, -12], [-38, 16, 24, -4, 4, -4, -4, 0], [-20, -26, -15, -3, 3, -3, -8, 0], [-8, 1, 15, 8, 19, -16, 14, -15], [-13, 11, 9, 13, -4, -26, 4, -9], [3, 10, 2, 9, -6, 0, 0, 2], [5, -7, 0, -4, -3, 2, 3, 3], [-8, 8, 4, 30, -14, -30, 18, -8], [-6, -12, 0, 6, -7, -12, 36, 0]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : 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;