
// Make newform 1040.2.cd.n in Magma, downloaded from the LMFDB on 27 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1040_cd();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_cd_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_cd_n();".
// 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_cd_n();".
// 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 := [1, 2, 2, -8, 14, -2, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [63, 378, -276, 944, -127, 4, 16, 64], [-67, -402, 1448, -1009, 130, 16, 64, -63], [268, -1263, 1545, -1068, 118, -64, 63, -67], [306, 1198, -1614, 1304, -70, 65, -59, 83], [-269, 619, 662, -2218, 441, 69, -43, -172], [-173, -1038, 1244, -2572, 369, -92, -49, -196], [2, 1, -8, 14, -2, 0, 0, 1]];
    Rf_basisdens := [1, 319, 319, 319, 319, 319, 319, 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_1040_cd();"
function MakeCharacter_1040_cd()
    N := 1040;
    order := 4;
    char_gens := [911, 261, 417, 561];
    v := [4, 4, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_cd_Hecke();"
function MakeCharacter_1040_cd_Hecke(Kf)
    N := 1040;
    order := 4;
    char_gens := [911, 261, 417, 561];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 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], [1, 0, -1, 0, -1, 1, 0, 0], [0, -1, 0, 1, 0, 1, 0, 0], [0, 0, 0, 2, 1, 1, 0, 1], [-1, 0, 1, 0, -1, 0, -1, 0], [2, 0, 0, 1, -1, 1, -1, -1], [2, 1, -2, 0, 1, 0, 1, 1], [2, 0, -2, 0, -1, 1, 0, 0], [2, 1, 2, 1, 1, 0, -1, -1], [0, 1, -2, -1, 1, -1, 2, 0], [-1, 0, -1, 1, 0, -1, 1, 0], [-6, 0, 0, 1, 0, 0, 0, 1], [2, 2, 2, -2, 2, -3, 1, -2], [0, 1, 0, -3, 1, -4, 3, -1], [-2, 0, 0, 0, -3, 3, 0, 3], [-3, -1, 3, 0, -3, 1, -2, -1], [2, 0, 2, -5, 0, -4, 4, 0], [0, 0, 0, 1, -2, -2, 0, 3], [0, -1, 0, 1, -1, 1, 2, 0], [2, 1, 2, -7, 1, -3, 2, -1], [0, -3, 0, 1, -1, 1, 0, 0], [0, -2, 2, 5, -5, 5, -6, 0], [-4, 0, 0, 0, 3, 1, 0, -3], [5, -1, 5, -2, -1, 3, -2, 1], [0, -2, 2, 5, -5, 5, -6, 0], [0, 2, 0, -4, 4, -4, 2, 0], [1, -1, 1, -3, -1, 1, 0, 1], [0, 2, 0, 3, 2, -1, -1, -2], [-4, 1, 4, 0, 5, -4, 1, 1], [2, 2, -2, 0, 0, 5, 5, 2], [1, 3, -1, 0, 0, -4, -4, 3], [4, 0, 0, 2, -2, 2, 0, 4], [-2, 0, 0, 1, -1, -1, 0, 2], [0, 2, 4, -7, 7, -7, 8, 0], [8, 2, -8, 0, -2, 3, 1, 2], [2, 5, -2, 0, 2, 0, 2, 5], [4, 1, 4, 4, 1, 2, -3, -1], [0, 1, -8, 1, -1, 1, 0, 0], [-6, 0, 0, -4, -7, 1, 0, 3], [-3, 4, -3, -4, 4, -4, 0, -4], [4, 0, 0, 4, 4, 6, 0, 0], [0, 3, -4, 1, -1, 1, -4, 0], [10, 0, 0, -4, -2, 4, 0, -2], [0, -4, -6, -5, 5, -5, 0, 0], [0, -2, 6, -1, 1, -1, 0, 0], [2, 0, 0, -2, 2, -6, 0, -4], [2, 0, 0, 8, 6, 2, 0, 2], [-12, 0, 0, 0, 1, 1, 0, -1], [0, 1, 4, 7, -7, 7, -2, 0], [-3, -1, -3, 4, -1, -3, 4, 1], [3, -2, 3, -8, -2, 0, 2, 2], [-5, 5, 5, 0, 10, -7, 3, 5], [-4, -2, 4, 0, 2, -7, -5, -2], [0, -6, -10, 9, -9, 9, -4, 0], [-9, 3, 9, 0, -1, 1, 0, 3], [5, 4, -5, 0, 1, 5, 6, 4], [0, -2, 2, -10, 10, -10, 4, 0], [-6, -3, 6, 0, 8, -12, -4, -3], [7, 0, -7, 0, -2, 4, 2, 0], [-6, 4, 6, 0, -4, 3, -1, 4], [0, -7, 0, 9, -7, 6, 1, 7], [0, 1, -8, 5, -5, 5, 4, 0], [-2, 0, 0, 0, 3, -13, 0, -3], [0, 6, 6, -7, 7, -7, 0, 0], [2, -3, -2, 0, -7, 0, -7, -3], [0, 7, 6, 1, -1, 1, -10, 0], [-5, 2, -5, -7, 2, -11, 9, -2], [6, -2, -6, 0, 8, -5, 3, -2], [-3, -4, -3, -1, -4, 8, -4, 4], [13, 0, 13, -4, 0, -4, 4, 0], [6, 0, 0, 7, 2, 2, 0, 5], [8, -4, 8, 11, -4, 12, -8, 4], [-5, 2, -5, -13, 2, -6, 4, -2], [4, 3, -4, 0, -3, -2, -5, 3], [-1, -9, 1, 0, 2, -5, -3, -9], [2, 0, 0, 2, -1, -5, 0, 3], [6, 0, 0, -2, 10, -6, 0, -12], [-12, 0, 0, 5, 2, 0, 0, 3], [11, -3, -11, 0, -3, -1, -4, -3], [8, -6, -8, 0, -10, 13, 3, -6], [0, 12, -2, -5, 5, -5, -4, 0], [-12, 5, -12, 0, 5, -14, 9, -5], [3, -2, 3, -3, -2, -7, 9, 2], [1, -2, 1, -14, -2, -12, 14, 2], [6, 0, 0, -4, -10, 18, 0, 6], [1, -6, -1, 0, -5, 3, -2, -6], [-6, -3, -6, 0, -3, 6, -3, 3], [0, 2, 0, 7, -7, 7, -8, 0], [9, 0, 9, -16, 0, -10, 10, 0], [0, -5, 12, 9, -9, 9, -8, 0], [-1, -1, 1, 0, -6, 2, -4, -1], [8, 2, 8, 7, 2, -2, 0, -2], [0, -9, 2, 9, -9, 9, 2, 0], [0, 0, 4, 3, -3, 3, -12, 0], [8, -2, -8, 0, 13, -11, 2, -2], [-2, 2, 2, 0, -7, 8, 1, 2], [7, 0, -7, 0, 6, 2, 8, 0], [10, 0, 0, -11, -12, 4, 0, 1], [13, 2, -13, 0, 1, -3, -2, 2], [3, 3, -3, 0, 25, -11, 14, 3], [-2, 6, -2, -13, 6, -17, 11, -6], [-14, 0, 0, 13, 0, 0, 0, 13], [20, 9, 20, -31, 9, -14, 5, -9], [22, 0, 0, 13, 2, 16, 0, 11], [0, -10, -2, 7, -7, 7, 2, 0], [0, -1, -8, 11, -11, 11, -8, 0], [0, 13, -2, -3, 3, -3, 6, 0], [0, 0, 12, 13, -13, 13, -16, 0], [0, -4, -12, 5, -5, 5, 6, 0], [18, 0, 0, 14, 10, 10, 0, 4], [4, 6, 4, -3, 6, 1, -7, -6], [4, 0, 0, -9, -11, 9, 0, 2], [0, 10, -18, -3, 3, -3, 12, 0], [-20, -4, -20, 11, -4, 4, 0, 4], [1, 4, -1, 0, 17, -4, 13, 4], [0, -14, -6, -2, 2, -2, 10, 0], [-4, 0, 0, -4, 11, -17, 0, -15], [-19, -5, 19, 0, -4, -8, -12, -5], [-8, 6, 8, 0, 20, -13, 7, 6], [0, 0, -14, 5, -5, 5, -8, 0], [-6, -1, 6, 0, -1, 0, -1, -1], [4, 1, 4, -4, 1, 0, -1, -1], [1, -1, 1, 2, -1, 1, 0, 1], [0, -11, 2, -3, 3, -3, 4, 0], [-15, -8, -15, -9, -8, 3, 5, 8], [0, -8, -6, 2, -2, 2, 0, 0], [-3, -5, -3, 14, -5, 11, -6, 5], [-36, 0, 0, -6, -6, -2, 0, 0], [-4, -3, 4, 0, 2, 5, 7, -3], [4, 0, 0, 15, 11, 13, 0, 4], [-3, -5, -3, 11, -5, 18, -13, 5], [-16, 0, 0, 6, -1, 5, 0, 7], [0, 6, 0, -7, 7, -7, -10, 0], [-11, 6, -11, -14, 6, -10, 4, -6], [1, -2, -1, 0, 24, -12, 12, -2], [-11, 8, 11, 0, -24, 16, -8, 8], [-24, 0, 0, -1, 6, -16, 0, -7], [-8, 0, 0, 10, 15, -15, 0, -5], [14, -7, -14, 0, -25, 12, -13, -7], [0, 7, -10, -5, 5, -5, 18, 0], [5, 0, 5, 15, 0, 7, -7, 0], [-12, 4, -12, -22, 4, -21, 17, -4], [13, 1, 13, -5, 1, -11, 10, -1], [18, 0, 0, -10, -11, 17, 0, 1], [-18, 0, 0, -15, -8, -14, 0, -7], [3, 5, 3, -9, 5, -10, 5, -5], [0, 11, 6, 1, -1, 1, -6, 0], [-17, -1, 17, 0, 5, -1, 4, -1], [0, -12, -26, -7, 7, -7, 10, 0], [4, 0, 0, 18, 5, 13, 0, 13], [0, -8, -8, -7, 7, -7, -8, 0], [0, -5, -12, 9, -9, 9, -18, 0], [-8, -3, -8, 3, -3, 8, -5, 3], [-9, 12, -9, -7, 12, -14, 2, -12], [-1, 5, 1, 0, 0, -8, -8, 5], [28, 0, 0, -8, -4, 2, 0, -4], [0, 14, 18, 3, -3, 3, -8, 0], [12, -9, -12, 0, -9, 0, -9, -9], [-1, 6, -1, -22, 6, -16, 10, -6], [-1, -3, -1, 2, -3, 15, -12, 3], [-20, 0, 0, 14, 11, 9, 0, 3], [15, 1, 15, -12, 1, 7, -8, -1], [0, 3, -32, 7, -7, 7, -10, 0], [-4, 0, 0, 2, -8, 14, 0, 10], [0, 7, 28, -3, 3, -3, 0, 0], [0, -3, -10, 1, -1, 1, 14, 0], [28, 0, 0, 0, 12, -6, 0, -12], [-13, 3, -13, -10, 3, -15, 12, -3], [-15, -4, -15, 20, -4, 14, -10, 4], [9, 3, 9, 6, 3, -1, -2, -3], [-15, 3, 15, 0, 0, 5, 5, 3], [15, -5, -15, 0, -5, 9, 4, -5], [0, -8, 0, -7, 7, -7, 4, 0], [14, 0, 0, -1, 2, 10, 0, -3], [-12, 0, 0, 0, 4, 18, 0, -4], [0, -4, 24, -2, 2, -2, 12, 0], [4, 9, -4, 0, -8, -4, -12, 9], [11, -10, -11, 0, -12, 10, -2, -10], [-18, -7, -18, 15, -7, 10, -3, 7], [0, 0, 2, 2, -2, 2, -22, 0], [26, 0, 0, 6, 1, 15, 0, 5], [0, -20, -10, 5, -5, 5, -4, 0], [11, -12, -11, 0, 22, -14, 8, -12], [0, -4, -8, 9, -9, 9, -2, 0], [0, -7, 2, 15, -15, 15, -20, 0], [-12, 0, 0, 13, 6, 8, 0, 7], [0, 3, 0, 0, -23, 20, -3, 3], [16, 0, 0, -6, 17, -7, 0, -23], [6, -8, 6, 22, -8, 21, -13, 8], [22, 7, -22, 0, 0, -6, -6, 7], [2, -2, -2, 0, 18, -9, 9, -2], [46, 0, 0, 2, -3, 5, 0, 5], [2, 0, 0, -24, -4, -12, 0, -20], [-2, 2, 2, 0, 10, -13, -3, 2], [14, 3, -14, 0, 0, -5, -5, 3], [-13, 4, -13, -6, 4, -8, 4, -4], [12, -6, 12, 8, -6, 9, -3, 6], [-20, 9, -20, 0, 9, -22, 13, -9], [-8, 0, 0, -15, -6, -8, 0, -9], [-16, 8, 16, 0, 25, -24, 1, 8], [11, 10, 11, -32, 10, -14, 4, -10], [18, 0, 0, -2, -8, 10, 0, 6], [0, 14, 4, -17, 17, -17, 8, 0], [0, -21, 2, 5, -5, 5, 18, 0], [13, 1, 13, -25, 1, -6, 5, -1], [7, 6, 7, -18, 6, -18, 12, -6], [-7, 5, 7, 0, -6, 5, -1, 5], [-27, -8, 27, 0, 5, -19, -14, -8], [9, -8, -9, 0, -6, -8, -14, -8], [0, -4, 22, 9, -9, 9, -6, 0], [3, -12, -3, 0, 10, -12, -2, -12], [-12, 0, 0, 9, 12, -4, 0, -3], [-10, -2, 10, 0, -9, 2, -7, -2], [-4, 0, 0, 4, 21, -31, 0, -17], [-8, -14, 8, 0, 13, -11, 2, -14], [-20, -1, 20, 0, -17, 12, -5, -1], [-7, 2, -7, -25, 2, -8, 6, -2], [10, 0, 0, -2, -6, 8, 0, 4], [0, 1, 6, -3, 3, -3, -2, 0], [0, -10, 12, -7, 7, -7, 30, 0], [44, 0, 0, -15, -4, 0, 0, -11], [1, 3, 1, 11, 3, -2, -1, -3], [-21, -5, 21, 0, 1, -3, -2, -5], [0, 0, 0, -18, -21, 29, 0, 3], [1, 1, -1, 0, -20, 16, -4, 1], [-22, 0, 0, 19, 4, 10, 0, 15], [15, -3, -15, 0, -11, 11, 0, -3], [0, -12, -8, 1, -1, 1, 14, 0], [12, -13, -12, 0, 6, -5, 1, -13], [2, -5, -2, 0, -20, 22, 2, -5], [10, -14, 10, 4, -14, 3, 11, 14], [0, -12, -2, -3, 3, -3, 26, 0], [-14, -3, -14, -5, -3, 3, 0, 3], [0, 8, -38, -4, 4, -4, 4, 0], [21, -10, -21, 0, 5, 5, 10, -10], [0, -11, 24, 17, -17, 17, -8, 0], [-20, -14, -20, 32, -14, 17, -3, 14], [0, -9, 14, 19, -19, 19, -18, 0], [2, 0, 0, 10, 14, -2, 0, -4], [14, 0, 0, -12, 8, -16, 0, -20], [4, 0, 0, 0, 17, -13, 0, -17], [0, 10, 0, 3, -3, 3, -10, 0], [18, 12, -18, 0, -1, 12, 11, 12], [25, -7, -25, 0, 17, -13, 4, -7], [-12, 0, 0, 13, 5, 7, 0, 8], [22, 0, 0, 6, 2, 14, 0, 4], [30, 0, 0, 0, 3, 5, 0, -3], [15, 10, -15, 0, -5, 18, 13, 10], [-7, -3, 7, 0, 10, -5, 5, -3], [17, 3, 17, 9, 3, -3, 0, -3], [-12, 0, 0, -1, -4, 10, 0, 3], [-23, -2, -23, -22, -2, -2, 4, 2], [26, 0, 0, 6, 13, -17, 0, -7], [2, 0, 0, -12, 0, 4, 0, -12], [8, 0, -8, 0, -26, 9, -17, 0], [0, 12, 0, -31, 12, -16, 4, -12], [6, 0, 0, 1, -14, 34, 0, 15], [0, 7, 54, -5, 5, -5, 2, 0], [-6, -12, 6, 0, -10, -5, -15, -12], [0, 11, 58, -9, 9, -9, 2, 0], [4, 19, 4, -29, 19, -24, 5, -19], [-11, -14, -11, -1, -14, 6, 8, 14], [-7, 0, 7, 0, -26, 16, -10, 0], [11, 1, -11, 0, 5, 1, 6, 1], [16, 0, 0, -9, 6, 0, 0, -15], [0, -4, -44, 11, -11, 11, 2, 0], [11, 5, -11, 0, 11, -11, 0, 5], [-3, 4, -3, 24, 4, 0, -4, -4], [0, -11, -28, -7, 7, -7, 18, 0], [21, -5, 21, 14, -5, 3, 2, 5], [0, 8, 36, -6, 6, -6, -10, 0], [0, 3, -8, -19, 19, -19, 28, 0], [0, 22, -18, -1, 1, -1, 0, 0], [-12, 0, 0, 6, -20, 10, 0, 26], [10, 5, 10, -28, 5, -8, 3, -5], [4, 0, 0, -24, -29, -3, 0, 5], [0, -5, 36, -11, 11, -11, 14, 0], [3, 17, 3, -44, 17, -33, 16, -17], [-12, 21, 12, 0, -3, 0, -3, 21], [0, 6, -8, -13, 13, -13, 24, 0], [-2, -6, 2, 0, 27, -32, -5, -6], [-9, -8, 9, 0, 1, -10, -9, -8], [-28, 0, -28, 27, 0, 11, -11, 0], [25, -7, 25, 6, -7, 19, -12, 7], [0, 0, 0, -4, -17, 25, 0, 13], [0, -4, 14, -17, 17, -17, -16, 0], [-1, -4, 1, 0, 2, 10, 12, -4], [0, -22, -26, -1, 1, -1, 6, 0], [2, -2, 2, -11, -2, 22, -20, 2], [-18, 0, 0, 11, -2, 4, 0, 13], [-20, 0, 0, -25, 4, -36, 0, -29], [4, 10, 4, -9, 10, 5, -15, -10], [-34, 0, 0, -13, 1, -13, 0, -14], [-3, -6, 3, 0, 7, -4, 3, -6], [-22, -12, 22, 0, -30, 1, -29, -12], [-14, 0, 0, 4, -2, -14, 0, 6], [-12, 0, 0, 0, -14, 12, 0, 14], [-8, 17, -8, -12, 17, -22, 5, -17], [0, -2, 50, 5, -5, 5, -16, 0], [-16, 0, 0, -2, 7, -7, 0, -9], [11, -18, 11, 16, -18, 34, -16, 18], [-14, 0, 0, -5, -31, 23, 0, 26], [16, 0, 0, -8, -8, 2, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1040_cd_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_cd_n();".
// 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_cd_n(:prec:=8)
    chi := MakeCharacter_1040_cd();
    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_cd_n();".
// 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_cd_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1040_cd();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -16, 32, 4, 8, -20, 18, -6, 1]>,<7,R![52, -4, -20, 0, 1]>,<19,R![100, -520, 1352, -1524, 1004, -396, 98, -14, 1]>],Snew);
    return Vf;
end function;