
// Make newform 1008.2.r.l in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1008_r_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_1008_2_r_l();".
// 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_1008_2_r_l();".
// 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 := [81, -27, 9, 9, -15, 3, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [54, 9, -18, 3, -3, -2, 2, 1], [27, -9, -9, 15, -3, -1, 1, -1], [-81, 72, 9, -21, 12, -4, -5, 2], [-81, 72, 36, -21, 12, -4, -5, 2], [27, -27, 0, 9, -6, 2, 1, -1], [-216, -9, 72, -12, 15, 8, -8, -1]];
    Rf_basisdens := [1, 1, 27, 27, 27, 27, 9, 27];
    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_1008_r();"
function MakeCharacter_1008_r()
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    v := [3, 3, 1, 3];
    // 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_1008_r_Hecke();"
function MakeCharacter_1008_r_Hecke(Kf)
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 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], [0, -1, 0, 0, 0, 0, 1, 0], [-1, 1, -1, 1, -1, 0, -1, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 2, 0, 1, 0, 1, 0], [1, 0, 1, 0, 0, 0, 0, 1], [0, 0, 0, 1, -1, -1, 1, 0], [1, -2, 0, -1, -1, 0, 1, 0], [0, 0, 0, 1, 0, 0, -1, 1], [0, -1, 3, 1, -1, -2, 0, 2], [-1, 1, -1, -1, -1, 0, 1, 2], [1, -1, 0, -1, 0, 0, 0, 0], [2, 1, 2, -1, -1, 0, 1, 2], [0, 2, 1, -2, 1, 1, -1, -1], [0, 1, 1, -1, 2, -1, 1, 1], [0, 1, 0, 4, -3, 1, 3, 0], [-3, 0, -3, -3, 0, 0, 3, 1], [0, 1, -6, -1, 1, 3, 0, -3], [-4, 1, -4, -2, -1, 0, 2, -1], [1, 1, 0, 0, 1, 0, -1, 0], [-5, 2, 0, -1, 3, 0, -3, 0], [0, 3, 5, -3, 2, -3, -1, 3], [0, -1, 2, 1, -2, 0, -1, 0], [0, 1, 0, 3, -2, -1, 2, 0], [0, -1, -3, 1, 1, -1, 2, 1], [0, -3, 1, 3, -4, 5, -1, -5], [-4, 4, -4, 1, -4, 0, -1, -1], [9, 0, 0, 2, -2, 0, 2, 0], [-7, 0, 0, -1, 1, 1, -1, 0], [-4, 4, -4, 2, -4, 0, -2, 0], [2, -1, 0, -3, 2, -3, -2, 0], [7, 0, 7, -3, 0, 0, 3, 2], [0, -4, -3, 4, -5, -1, -1, 1], [-11, -3, -11, -2, 3, 0, 2, 0], [-6, 1, -6, 1, -1, 0, -1, -5], [0, -1, 9, 1, -3, 0, -2, 0], [9, -7, 9, -4, 7, 0, 4, -3], [0, 2, 0, -5, 7, -2, -7, 0], [-1, -2, -1, 0, 2, 0, 0, 1], [0, -2, 9, 2, 1, 0, 3, 0], [5, -6, 0, -3, -3, 1, 3, 0], [-11, -1, 0, 1, -2, 2, 2, 0], [0, -6, 5, 6, -3, -2, 3, 2], [5, 2, 5, -5, -2, 0, 5, 1], [3, -1, 0, -2, 1, -2, -1, 0], [0, 3, 0, 8, -5, 1, 5, 0], [3, -6, 3, -2, 6, 0, 2, -1], [0, 0, 9, 0, 4, 5, 4, -5], [0, 3, 1, -3, -3, 1, -6, -1], [-1, 0, -1, -2, 0, 0, 2, -3], [7, -3, 0, -7, 4, -3, -4, 0], [7, -2, 7, 0, 2, 0, 0, -5], [0, 5, -8, -5, 3, 4, -2, -4], [0, 0, 0, 5, -5, 5, 5, 0], [14, 0, 14, 4, 0, 0, -4, 1], [0, -7, 3, 7, -4, 5, 3, -5], [5, 1, 0, -2, 3, 0, -3, 0], [6, -3, 0, 1, -4, 1, 4, 0], [0, 1, -8, -1, -6, 0, -7, 0], [0, 9, 0, -9, 8, 2, -1, -2], [1, -7, 1, -3, 7, 0, 3, 0], [-15, -2, -15, -5, 2, 0, 5, 2], [8, 10, 0, 1, 9, 3, -9, 0], [-2, -5, -2, -9, 5, 0, 9, 7], [0, 3, 1, -3, -1, -3, -4, 3], [0, 8, 12, -8, 1, -1, -7, 1], [0, 5, -8, -5, 10, 2, 5, -2], [-11, 6, -11, 5, -6, 0, -5, -3], [-3, 5, -3, 2, -5, 0, -2, 6], [0, -2, -12, 2, 2, 4, 4, -4], [0, -1, 2, 1, 3, -4, 4, 4], [0, 8, 0, 9, -1, -2, 1, 0], [0, -6, -1, 6, -5, -4, 1, 4], [5, 5, 5, -5, -5, 0, 5, 0], [-5, 11, 0, 6, 5, 1, -5, 0], [-3, 5, -3, 9, -5, 0, -9, 0], [0, -9, 5, 9, -4, 5, 5, -5], [13, -1, 0, -3, 2, 4, -2, 0], [-6, -9, -6, -13, 9, 0, 13, 0], [2, -8, 2, -4, 8, 0, 4, 3], [-2, 0, -2, -7, 0, 0, 7, 3], [0, -3, 7, 3, -6, 1, -3, -1], [19, 3, 0, 2, 1, 0, -1, 0], [7, -8, 0, 0, -8, 8, 8, 0], [0, -7, -3, 7, -9, -4, -2, 4], [0, -8, 12, 8, -4, -1, 4, 1], [8, -2, 0, -5, 3, 3, -3, 0], [0, -7, 19, 7, -4, -2, 3, 2], [0, 7, -6, -7, 2, 6, -5, -6], [-4, 0, -4, 1, 0, 0, -1, -7], [1, -3, 0, -5, 2, -7, -2, 0], [0, 11, 3, -11, 8, -5, -3, 5], [-5, -1, 0, 10, -11, -2, 11, 0], [-9, 3, -9, -5, -3, 0, 5, 5], [8, -4, 8, -2, 4, 0, 2, -7], [10, 1, 0, -1, 2, 5, -2, 0], [-8, -2, -8, 3, 2, 0, -3, 1], [9, -8, 0, -4, -4, 6, 4, 0], [5, -9, 0, -6, -3, -2, 3, 0], [5, 2, 0, 8, -6, 1, 6, 0], [0, 3, -18, -3, 3, -8, 0, 8], [31, 1, 0, 0, 1, -2, -1, 0], [-2, -3, -2, -1, 3, 0, 1, -12], [0, 8, -5, -8, 10, 4, 2, -4], [-11, 6, -11, 1, -6, 0, -1, 11], [10, -1, 0, -2, 1, 2, -1, 0], [0, 2, -6, -2, -1, -2, -3, 2], [-13, 8, 0, 11, -3, 3, 3, 0], [-6, -8, -6, -2, 8, 0, 2, 4], [0, -12, -16, 12, -1, 2, 11, -2], [11, 14, 11, 12, -14, 0, -12, -3], [-15, -2, 0, 6, -8, -5, 8, 0], [-2, -5, -2, -7, 5, 0, 7, -4], [0, -7, 2, 7, -9, 10, -2, -10], [-12, 7, 0, 9, -2, 7, 2, 0], [0, -9, 18, 9, -3, -2, 6, 2], [-5, -6, -5, -3, 6, 0, 3, 3], [25, 2, 0, -7, 9, 9, -9, 0], [-14, 9, -14, 2, -9, 0, -2, 6], [0, 9, -15, -9, 6, 3, -3, -3], [7, -2, 7, 3, 2, 0, -3, 8], [0, -10, 10, 10, -13, 3, -3, -3], [0, -1, -15, 1, 3, -2, 4, 2], [6, 8, 0, 8, 0, 3, 0, 0], [0, 2, 2, -2, 5, 3, 3, -3], [-6, -4, 0, -6, 2, -6, -2, 0], [0, 6, 6, -6, 6, 0, 0, 0], [-13, 3, 0, -6, 9, 2, -9, 0], [0, -2, 5, 2, 1, 0, 3, 0], [-23, 8, -23, 8, -8, 0, -8, 5], [5, -7, 0, 4, -11, -3, 11, 0], [-20, -3, -20, 8, 3, 0, -8, -2], [13, 1, 13, -2, -1, 0, 2, 3], [28, 0, 0, -1, 1, -4, -1, 0], [6, 4, 6, 2, -4, 0, -2, -6], [28, -8, 28, -1, 8, 0, 1, 1], [31, -2, 0, 0, -2, -1, 2, 0], [5, -5, 5, 5, 5, 0, -5, -12], [0, 1, 0, -10, -1, 0, 10, -6], [-20, 8, 0, 11, -3, -3, 3, 0], [20, -5, 0, -6, 1, 4, -1, 0], [0, -3, 31, 3, 2, -9, 5, 9], [12, -4, 12, -16, 4, 0, 16, -2], [-7, -7, 0, -4, -3, -4, 3, 0], [5, 16, 0, 5, 11, 3, -11, 0], [0, 6, 21, -6, 6, 3, 0, -3], [0, 3, -1, -3, 2, -7, -1, 7], [0, 5, 0, -5, -1, 5, -6, -5], [5, 12, 5, 10, -12, 0, -10, -2], [16, 10, 0, 7, 3, -4, -3, 0], [-12, -3, -12, -4, 3, 0, 4, 0], [9, -9, 0, -15, 6, 0, -6, 0], [-25, -1, 0, -3, 2, -5, -2, 0], [22, -6, 22, -5, 6, 0, 5, -5], [0, 6, 1, -6, 6, 0, 0, 0], [0, -1, -9, 1, -1, -2, 0, 2], [-2, -5, 0, 1, -6, 1, 6, 0], [0, -15, -7, 15, -7, 4, 8, -4], [-14, 8, 0, 1, 7, 0, -7, 0], [-21, -7, -21, -15, 7, 0, 15, 2], [0, 1, 22, -1, -3, 2, -4, -2], [-24, -8, 0, -10, 2, 1, -2, 0], [44, -5, 44, 2, 5, 0, -2, 4], [-6, 4, 0, 5, -1, 14, 1, 0], [11, 8, 11, 16, -8, 0, -16, -4], [0, -1, 18, 1, 7, -7, 8, 7], [7, -6, 0, -7, 1, 3, -1, 0], [0, -4, 10, 4, -4, 4, 0, -4], [8, 2, 0, -1, 3, 12, -3, 0], [15, 3, 15, 10, -3, 0, -10, -1], [0, -3, -15, 3, -9, -7, -6, 7], [8, 5, 8, 15, -5, 0, -15, -7], [-9, 4, -9, 8, -4, 0, -8, 7], [0, 4, -7, -4, -8, 10, -12, -10], [7, 4, 7, 0, -4, 0, 0, 5], [18, -11, 18, -9, 11, 0, 9, 3], [0, -2, -1, 2, 15, -7, 17, 7], [23, -3, 0, -1, -2, 8, 2, 0], [18, 7, 0, -4, 11, -3, -11, 0], [0, -4, 23, 4, 3, 4, 7, -4], [0, -1, -22, 1, 3, -3, 4, 3], [0, 7, -39, -7, 0, -2, -7, 2], [-20, 12, -20, 6, -12, 0, -6, 4], [-15, -12, 0, -5, -7, 4, 7, 0], [10, 0, 10, 9, 0, 0, -9, -5], [0, 1, -15, -1, -4, -1, -5, 1], [-9, 1, 0, 1, 0, -10, 0, 0], [0, 5, -10, -5, -4, 6, -9, -6], [-10, -9, -10, -5, 9, 0, 5, 4], [-25, -8, 0, 1, -9, -11, 9, 0], [-7, -17, 0, -6, -11, -1, 11, 0], [0, -6, 19, 6, 5, -3, 11, 3], [5, -9, 0, -12, 3, 3, -3, 0], [0, 4, -12, -4, -1, 5, -5, -5], [3, 5, 0, 17, -12, 1, 12, 0], [18, -6, 18, -17, 6, 0, 17, -4], [7, 3, 7, -16, -3, 0, 16, 2], [0, 11, 9, -11, 0, -1, -11, 1], [0, 9, -14, -9, 16, -3, 7, 3], [-15, -4, 0, 14, -18, -3, 18, 0], [-17, 4, -17, 4, -4, 0, -4, -5], [0, -2, 25, 2, 8, -1, 10, 1], [-9, 12, -9, 7, -12, 0, -7, 8], [0, 10, -18, -10, -3, -6, -13, 6], [-22, -9, 0, -13, 4, -2, -4, 0], [-4, 7, 0, -7, 14, 1, -14, 0], [-6, -10, 0, -4, -6, -4, 6, 0], [10, 0, 10, -5, 0, 0, 5, 10], [0, 0, -18, 0, 6, -11, 6, 11], [-30, -5, -30, 7, 5, 0, -7, -12], [20, 3, 0, 1, 2, -8, -2, 0], [21, 1, 21, -2, -1, 0, 2, 5], [0, 8, -8, -8, 10, 0, 2, 0], [0, -4, -36, 4, 1, 1, 5, -1], [-6, 13, -6, 0, -13, 0, 0, 12], [0, 11, 11, -11, 3, -5, -8, 5], [-24, 2, -24, 3, -2, 0, -3, -7], [0, -2, -3, 2, -11, 3, -9, -3], [-18, -10, 0, 1, -11, 6, 11, 0], [-17, 0, -17, -11, 0, 0, 11, 0], [-11, -4, -11, -17, 4, 0, 17, 12], [-2, -8, -2, -4, 8, 0, 4, -8], [10, 2, 10, 12, -2, 0, -12, 5], [-21, 3, 0, 11, -8, 6, 8, 0], [44, 6, 44, 2, -6, 0, -2, -1], [0, 11, -6, -11, -2, 2, -13, -2], [0, 6, -48, -6, 1, -1, -5, 1], [20, -1, 0, 0, -1, 7, 1, 0], [0, 6, -16, -6, 7, 3, 1, -3], [0, 0, -5, 0, -3, 4, -3, -4], [19, 15, 19, 5, -15, 0, -5, 0], [6, 14, 0, 4, 10, -4, -10, 0], [8, -14, 8, -2, 14, 0, 2, -4], [1, 9, 1, 0, -9, 0, 0, 8], [0, -2, 14, 2, -12, -2, -10, 2], [0, -4, 23, 4, 1, -10, 5, 10], [-2, -6, -2, -26, 6, 0, 26, 2], [4, 9, 0, 22, -13, -5, 13, 0], [-15, 1, -15, -7, -1, 0, 7, 7], [17, -10, 0, -9, -1, -3, 1, 0], [0, -18, -10, 18, -13, 4, 5, -4], [-11, 7, 0, 4, 3, 14, -3, 0], [-34, -5, -34, -13, 5, 0, 13, 9], [-7, 11, 0, -8, 19, -6, -19, 0], [12, 8, 12, 19, -8, 0, -19, -6], [0, -7, -18, 7, -16, -8, -9, 8], [31, 3, 0, -8, 11, -6, -11, 0], [2, 3, 2, 5, -3, 0, -5, 2], [-17, 11, -17, 12, -11, 0, -12, 5], [-33, -2, 0, 6, -8, 9, 8, 0], [-16, -17, -16, -21, 17, 0, 21, 7], [-24, -16, 0, -8, -8, -3, 8, 0], [-22, 6, -22, 5, -6, 0, -5, -13], [0, 5, -58, -5, -4, 0, -9, 0], [0, 15, 9, -15, 0, -1, -15, 1], [12, -16, 0, -15, -1, -5, 1, 0], [13, -10, 0, -1, -9, 19, 9, 0], [0, -3, -23, 3, -2, 7, 1, -7], [1, 5, 0, 9, -4, -14, 4, 0], [3, 10, 0, 10, 0, 6, 0, 0], [0, 11, 2, -11, -5, 3, -16, -3], [0, 5, -18, -5, 4, -9, -1, 9], [20, 5, 20, 13, -5, 0, -13, -5], [-19, -12, 0, -16, 4, 1, -4, 0], [4, 8, 4, 8, -8, 0, -8, -10], [0, 4, 20, -4, 4, -9, 0, 9], [13, 0, 0, 1, -1, 17, 1, 0], [0, 18, -17, -18, 9, -5, -9, 5], [24, -7, 24, -19, 7, 0, 19, -2], [-4, -4, -4, -5, 4, 0, 5, -3], [20, -9, 20, 4, 9, 0, -4, -6], [-33, 0, 0, -3, 3, -11, -3, 0], [0, -6, 11, 6, -18, 4, -12, -4], [10, -1, 10, -10, 1, 0, 10, 16], [3, -2, 3, -3, 2, 0, 3, -3], [37, -4, 0, -12, 8, 11, -8, 0], [-50, 0, -50, -5, 0, 0, 5, -3], [0, -1, -41, 1, 4, -1, 5, 1], [-20, 16, 0, 8, 8, 11, -8, 0], [0, -10, 9, 10, -3, -3, 7, 3], [-21, 14, -21, 18, -14, 0, -18, 3], [28, -23, 28, -19, 23, 0, 19, -9], [0, 11, -26, -11, -13, -1, -24, 1], [0, 7, -25, -7, 3, 0, -4, 0], [0, -1, 0, -6, 1, 0, 6, -9], [-37, 12, 0, 5, 7, 1, -7, 0], [-31, -4, 0, 3, -7, -15, 7, 0], [31, 4, 31, 1, -4, 0, -1, 12], [0, 7, -17, -7, -5, -10, -12, 10], [-46, 4, 0, 13, -9, -1, 9, 0], [0, 9, 18, -9, 13, -3, 4, 3], [-36, 16, 0, 6, 10, 3, -10, 0], [-11, -17, -11, -2, 17, 0, 2, -6], [15, -11, 15, -8, 11, 0, 8, -6], [0, -11, 16, 11, 9, 7, 20, -7], [1, -5, 1, 3, 5, 0, -3, -14], [0, 7, -43, -7, 3, 8, -4, -8], [0, -14, 18, 14, 2, 2, 16, -2], [5, -3, 0, -5, 2, 0, -2, 0], [0, 8, 7, -8, -10, -3, -18, 3], [-3, 6, -3, -6, -6, 0, 6, -6], [8, -23, 0, -4, -19, -3, 19, 0], [43, 13, 0, -2, 15, 8, -15, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_r_l();".
// 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_1008_2_r_l(:prec:=8)
    chi := MakeCharacter_1008_r();
    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_1008_2_r_l();".
// 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_1008_2_r_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 18, 65, 84, 93, -6, 17, 3, 1]>,<11,R![225, 180, 249, 126, 148, 73, 42, 7, 1]>],Snew);
    return Vf;
end function;