
// Make newform 1638.2.p.i in Magma, downloaded from the LMFDB on 10 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_p_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_1638_2_p_i();".
// 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_1638_2_p_i();".
// 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, -8, -8, 4, 3, 2, -2, -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], [-8, -8, 4, 3, 2, -2, -1, 1], [4, 6, -1, -2, 0, 1, 1, 0], [0, -4, -6, 1, 2, 0, -1, -1], [32, 4, -12, -3, 2, 6, 1, -1], [8, -1, -5, -1, 1, 2, 0, -1], [-24, 2, 17, 4, -4, -5, 1, 2]];
    Rf_basisdens := [1, 1, 8, 4, 4, 8, 2, 4];
    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_1638_p();"
function MakeCharacter_1638_p()
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    v := [3, 1, 2];
    // 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_1638_p_Hecke();"
function MakeCharacter_1638_p_Hecke(Kf)
    N := 1638;
    order := 3;
    char_gens := [911, 703, 379];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 1, 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, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 2, 1, -1, 0, 1, 1, 0], [0, 0, -1, 0, 0, 2, -1, 1], [-2, -2, 0, 0, 2, 2, -1, 1], [-1, 0, 1, 0, 0, -2, 0, 1], [-2, 1, -1, -2, 0, 2, -1, 0], [-1, 0, 1, 0, -1, -1, 1, -1], [0, -3, -3, -2, 1, 0, 0, -2], [-1, 1, 3, -1, 2, 0, 1, 0], [0, -1, -1, 0, 2, 2, 0, 3], [0, -1, -1, 8, 1, 2, 0, 2], [-1, -3, -2, -1, 2, 1, -4, 0], [0, -1, -1, 3, 3, 0, 0, 2], [-1, -2, 1, -1, 3, 0, -2, 0], [0, 3, 3, 1, -2, 1, 0, 2], [0, 0, -5, 0, -4, 1, -1, 0], [-4, 0, 4, 0, -4, -4, 2, -2], [5, 1, 2, 0, -3, -3, 3, -3], [0, -4, -4, 3, 1, 4, 0, 1], [0, -1, -1, 0, 0, 1, 0, 0], [-5, 4, 3, -5, 2, 3, 1, 0], [3, 4, -4, 0, 0, 0, -1, 1], [0, -4, -4, 8, 3, 1, 0, 0], [0, 0, 0, 0, -3, 5, 0, 2], [-3, -2, 5, 0, -3, -3, 3, -3], [-7, 1, -4, -7, -4, 1, 0, 0], [0, -4, -4, 3, 3, 5, 0, 4], [0, 4, 4, -1, 0, 2, 0, 6], [0, 3, 3, 1, 3, -4, 0, 2], [3, -5, 3, 3, 7, -1, -4, 0], [-6, 3, 2, -6, -1, 0, 3, 0], [-6, 9, 1, -6, -1, 7, 2, 0], [0, 1, 1, 0, -1, -3, 0, -3], [-5, 1, 0, 0, -1, -1, 7, -7], [0, 10, 10, -2, -10, -2, 0, -2], [0, 3, 3, 4, 2, 2, 0, 7], [4, 1, -4, 0, 3, 3, -4, 4], [-4, 6, 1, -4, 4, 9, -3, 0], [-5, 5, -7, 0, 2, 2, 1, -1], [4, 4, 2, 0, -6, -6, 9, -9], [8, 0, 5, 0, -5, -5, 9, -9], [7, -1, 0, 0, 1, 1, 3, -3], [-11, 0, -5, 0, 5, 5, -1, 1], [8, -5, 3, 8, 7, -1, -4, 0], [-2, -4, 8, -2, 11, -1, -3, 0], [1, 3, -1, 1, 3, 7, -4, 0], [-12, 4, -3, -12, -3, 4, 0, 0], [-13, -4, -3, -13, 7, 6, -10, 0], [-8, 8, -4, -8, -2, 10, -2, 0], [9, -3, 0, 9, -5, -8, 5, 0], [4, 5, 2, 0, -7, -7, -4, 4], [-9, 4, 6, -9, 3, 1, 3, 0], [0, 6, 6, 11, -2, -4, 0, 0], [0, 2, 2, -12, -7, -1, 0, -6], [-4, -2, 12, 0, -10, -10, 6, -6], [-7, 4, 2, -7, -1, 1, 3, 0], [0, 13, 13, -5, -13, -4, 0, -4], [-13, -2, 0, -13, 2, 0, -2, 0], [-5, 0, -11, 0, 11, 11, 4, -4], [-8, -4, -4, 0, 8, 8, -6, 6], [0, 5, 5, 13, -4, -3, 0, -2], [4, 9, 1, 0, -10, -10, 1, -1], [0, 13, 13, -12, -9, -7, 0, -3], [19, -10, -4, 19, 1, -5, -5, 0], [-11, 8, 3, -11, 0, 5, 3, 0], [18, -4, 8, 0, -4, -4, -1, 1], [4, 3, 5, 0, -8, -8, 10, -10], [-1, -8, 7, -1, 12, -3, -5, 0], [-2, 7, -5, -2, -3, 9, -2, 0], [1, 3, -11, 0, 8, 8, -3, 3], [4, -16, 7, 4, 14, -9, -7, 0], [-6, 1, -9, 0, 8, 8, -5, 5], [1, -13, 4, 0, 9, 9, -1, 1], [-19, 3, -5, -19, -3, 5, -2, 0], [12, 1, 11, 0, -12, -12, -1, 1], [0, -4, -4, 3, 3, 9, 0, 8], [7, 3, 2, 0, -5, -5, -2, 2], [0, 2, 2, 6, 0, 5, 0, 7], [-5, -9, 5, -5, 6, -8, -1, 0], [4, 2, 2, 4, -3, -3, 5, 0], [8, 2, 3, 0, -5, -5, -7, 7], [-17, 8, -4, 0, -4, -4, -7, 7], [0, 1, 1, 6, 2, -4, 0, -1], [0, 6, 6, -7, -9, 5, 0, 2], [11, 7, 1, 11, -2, 4, 3, 0], [0, -3, -3, 3, 7, 10, 0, 14], [0, -3, -3, 5, 9, -2, 0, 4], [0, 4, 4, 8, -8, 1, 0, -3], [-12, 3, -3, 0, 0, 0, 10, -10], [-7, 1, 8, -7, 5, -2, 3, 0], [-4, 8, -5, 0, -3, -3, 1, -1], [6, 7, 4, 6, -6, -3, 10, 0], [-8, 2, 4, -8, 6, 4, -2, 0], [1, 12, 13, 1, -1, -2, 14, 0], [0, 4, 4, -19, -5, 9, 0, 8], [0, 3, 3, 5, -6, -4, 0, -7], [0, -16, -16, 5, 11, 1, 0, -4], [0, 11, 11, 1, -6, -5, 0, 0], [2, 2, -2, 2, -8, -4, 6, 0], [12, 1, 2, 0, -3, -3, 4, -4], [-26, 5, -5, 0, 0, 0, 0, 0], [-5, -8, 0, -5, 9, 1, -9, 0], [0, 2, 2, 20, -2, -5, 0, -5], [0, 0, 0, -3, -4, 12, 0, 8], [0, -6, -6, 6, -3, 11, 0, 2], [-4, -10, -11, -4, 5, 6, -16, 0], [15, -16, -9, 15, -1, -8, -8, 0], [0, 0, 0, -10, -14, -1, 0, -15], [10, -4, 6, 10, 5, -5, 1, 0], [-12, -2, -13, 0, 15, 15, -2, 2], [7, -7, -2, 0, 9, 9, -5, 5], [0, -4, -4, 8, 14, -12, 0, -2], [0, -5, -5, 1, -7, -2, 0, -14], [0, -6, -6, 25, 1, 5, 0, 0], [15, -3, 7, 15, 10, 0, -3, 0], [0, -13, -13, -6, 0, 6, 0, -7], [1, 0, -3, 0, 3, 3, -10, 10], [0, -2, -2, 13, 3, -4, 0, -3], [0, -19, -19, 12, 12, 1, 0, -6], [11, 7, 1, 0, -8, -8, 5, -5], [2, -3, -14, 2, -8, 3, -6, 0], [-24, 9, 3, -24, 0, 6, 3, 0], [-13, -6, -11, -13, -7, -2, -4, 0], [0, 7, 7, 27, 3, 2, 0, 12], [33, 5, 1, 0, -6, -6, -2, 2], [1, -9, -5, 0, 14, 14, -1, 1], [-7, 3, -7, 0, 4, 4, -3, 3], [7, -6, 3, 0, 3, 3, -15, 15], [-13, -6, -11, -13, -10, -5, -1, 0], [-3, 4, 10, 0, -14, -14, 15, -15], [13, 4, -6, 13, 2, 12, -8, 0], [0, -20, -20, 0, 9, 7, 0, -4], [4, 6, 6, 4, -5, -5, 11, 0], [-5, 5, -12, 0, 7, 7, 10, -10], [18, 13, 23, 18, 8, -2, 15, 0], [18, -3, 3, 18, -7, -13, 10, 0], [-19, 9, 7, -19, -7, -5, 14, 0], [0, -4, -4, -10, 9, -11, 0, -6], [-31, -5, 2, 0, 3, 3, -2, 2], [-1, 14, 5, 0, -19, -19, 5, -5], [0, 15, 15, -24, -15, -4, 0, -4], [-19, 9, -7, -19, -9, 7, 2, 0], [-26, -2, -4, 0, 6, 6, 6, -6], [-13, 11, 8, 0, -19, -19, -7, 7], [0, 29, 29, -10, -17, -12, 0, 0], [13, -2, -2, 0, 4, 4, -8, 8], [0, -5, -5, 15, 18, 10, 0, 23], [27, -22, -3, 27, 4, -15, -7, 0], [3, 5, 9, 3, 7, 3, 2, 0], [13, 1, 21, 0, -22, -22, 11, -11], [-1, -10, -16, -1, -5, 1, -11, 0], [3, -20, 10, 0, 10, 10, 2, -2], [0, 6, 6, -29, -8, 10, 0, 8], [-29, 3, 2, 0, -5, -5, 4, -4], [13, 9, 22, 0, -31, -31, 6, -6], [19, 8, -18, 0, 10, 10, -5, 5], [-6, 7, -7, 0, 0, 0, -8, 8], [-17, -5, -6, 0, 11, 11, -15, 15], [0, -15, -15, 2, 12, 9, 0, 6], [0, 12, 12, 10, -9, -1, 0, 2], [0, -6, -6, 9, 15, 8, 0, 17], [17, -17, 9, 0, 8, 8, -1, 1], [14, -4, 9, 0, -5, -5, -4, 4], [-24, 7, 5, 0, -12, -12, 1, -1], [0, -3, -3, 9, 0, -9, 0, -12], [27, -7, -3, 0, 10, 10, -3, 3], [-36, 10, 9, -36, 12, 13, -3, 0], [4, 17, -9, 0, -8, -8, -1, 1], [24, 4, 4, 24, -11, -11, 15, 0], [0, -5, -5, -15, -2, 2, 0, -5], [0, -8, -8, 0, 13, -6, 0, -1], [19, 9, -6, 0, -3, -3, 8, -8], [-6, 19, 0, -6, -2, 17, 2, 0], [0, -3, -3, 34, 6, 10, 0, 13], [0, -2, -2, -4, -12, 22, 0, 8], [-5, -5, -6, -5, -4, -3, -2, 0], [0, 22, 22, 0, -25, -3, 0, -6], [-3, 3, -5, -3, -1, 7, -4, 0], [0, -14, -14, -21, 0, 5, 0, -9], [-14, -17, -15, -14, 8, 6, -23, 0], [-1, 9, -21, 0, 12, 12, -7, 7], [-1, -2, 1, 0, 1, 1, 2, -2], [6, -29, -2, 6, 0, -27, -2, 0], [4, 5, 0, 0, -5, -5, 11, -11], [-2, 5, 15, -2, -3, -13, 18, 0], [0, -15, -15, -31, 8, 2, 0, -5], [0, -7, -7, -2, 9, -8, 0, -6], [0, 19, 19, -21, -17, -2, 0, 0], [35, 7, -3, 35, -15, -5, 12, 0], [10, 3, 6, 0, -9, -9, -6, 6], [0, 2, 2, -9, -8, 8, 0, 2], [-15, 14, -15, -15, -9, 20, -6, 0], [0, 17, 17, -13, -4, -12, 0, 1], [10, 7, -12, 10, -1, 18, -11, 0], [-13, -5, -13, 0, 18, 18, -15, 15], [0, 13, 13, 14, -13, -5, 0, -5], [-18, -12, -8, 0, 20, 20, 2, -2], [-2, -9, 0, 0, 9, 9, 8, -8], [0, -25, 2, 0, 8, -19, -6, 0], [-16, 8, -8, -16, -5, 11, -3, 0], [0, 4, 4, 32, 0, -3, 0, 1], [0, 5, 5, -6, 16, -24, 0, -3], [0, 18, 18, -17, -7, -5, 0, 6], [-16, -8, -20, -16, -3, 9, -17, 0], [3, -17, -4, 0, 21, 21, -11, 11], [-35, -12, 10, -35, 6, -16, 4, 0], [-7, -20, 2, 0, 18, 18, -7, 7], [18, 5, 2, 18, -13, -10, 15, 0], [43, -11, 6, 43, 7, -10, -1, 0], [0, -8, -8, -9, -10, 11, 0, -7], [-16, -8, 19, 0, -11, -11, 9, -9], [-14, 16, 17, -14, 8, 7, 9, 0], [-3, -12, 8, 0, 4, 4, -2, 2], [-17, -15, 4, -17, 1, -18, 3, 0], [-2, 11, 13, -2, 8, 6, 5, 0], [0, 7, 7, -8, -16, -11, 0, -20], [-29, -13, 3, -29, 16, 0, -13, 0], [0, 6, 6, -30, 8, 2, 0, 16], [-10, 26, -10, 0, -16, -16, 20, -20], [0, 19, 19, 9, -23, -3, 0, -7], [27, 0, 0, 0, 0, 0, 13, -13], [0, 13, 6, 0, 1, 8, 5, 0], [-4, 19, -18, 0, -1, -1, -5, 5], [-7, -21, 17, -7, 29, -9, -12, 0], [6, -11, -6, 0, 17, 17, -2, 2], [0, 20, 20, -54, -14, -14, 0, -8], [34, 2, -14, 34, -19, -3, 5, 0], [22, -4, -11, 0, 15, 15, 4, -4], [-4, 9, 8, -4, -1, 0, 9, 0], [-8, 11, 2, 0, -13, -13, -1, 1], [39, 24, 8, 0, -32, -32, 12, -12], [-15, 10, 8, -15, -13, -11, 21, 0], [0, 13, 13, -22, 3, -12, 0, 4], [-10, -18, 6, 0, 12, 12, -9, 9], [0, -2, -2, 25, -6, -7, 0, -15], [18, -13, 7, 0, 6, 6, 6, -6], [0, 14, 14, 10, -16, -6, 0, -8], [0, 17, 17, -36, -11, 2, 0, 8], [8, 9, -6, 8, 4, 19, -10, 0], [-7, -16, -19, 0, 35, 35, -7, 7], [0, 1, 1, 4, 6, 9, 0, 16], [-8, -18, 16, -8, 6, -28, 10, 0], [0, -15, -15, 18, 5, -12, 0, -22], [0, 0, 0, -46, 5, -2, 0, 3], [-5, -14, -7, -5, 8, 1, -15, 0], [0, -1, -1, 12, -18, 2, 0, -17], [4, 3, -3, 0, 0, 0, -6, 6], [8, 7, -13, 8, -4, 16, -9, 0], [-14, 9, 17, -14, 13, 5, 4, 0], [-4, -4, 11, -4, 11, -4, 0, 0], [0, -1, -1, -32, 0, 6, 0, 5], [0, 17, 17, 9, -39, 2, 0, -20], [-43, 28, 5, -43, -5, 18, 10, 0], [0, 20, 20, -33, -20, 1, 0, 1], [5, -12, 9, 0, 3, 3, 5, -5], [41, 8, -7, 41, -21, -6, 14, 0], [20, -8, -17, 0, 25, 25, -5, 5], [16, 2, 3, 0, -5, -5, 2, -2], [10, -19, -6, 0, 25, 25, -22, 22], [0, 24, 24, 2, -2, 1, 0, 23], [8, -12, -10, 8, -5, -7, -5, 0], [0, -24, -24, 11, 6, 7, 0, -11], [26, 6, -10, 26, -21, -5, 11, 0], [-20, 5, -4, -20, 12, 21, -16, 0], [19, 8, -2, 0, -6, -6, 11, -11], [0, 30, 30, 8, -19, -10, 0, 1], [-1, -10, 0, 0, 10, 10, 7, -7], [0, 6, 6, -28, -20, -2, 0, -16], [36, 5, 15, 36, 5, -5, 10, 0], [-20, 7, -14, -20, -19, 2, 5, 0], [0, -8, -8, 15, 3, 8, 0, 3], [-18, -10, 4, 0, 6, 6, 2, -2], [16, 22, -9, 0, -13, -13, -3, 3], [0, 15, 15, -17, -10, 3, 0, 8], [0, 8, 8, 37, -19, 3, 0, -8], [24, -2, 19, 0, -17, -17, 7, -7], [0, 22, 22, 28, 1, -7, 0, 16], [-12, 4, 25, 0, -29, -29, 4, -4], [-25, 15, 4, 0, -19, -19, 14, -14], [-15, 16, 3, 0, -19, -19, -16, 16], [-21, -8, 1, 0, 7, 7, 0, 0], [-52, -15, -7, 0, 22, 22, -12, 12], [17, -3, 27, 17, 10, -20, 17, 0], [4, -19, -5, 4, 9, -5, -14, 0], [4, -21, 8, 4, -1, -30, 9, 0], [0, 8, 8, -2, 3, 9, 0, 20], [-14, -9, 2, 0, 7, 7, -25, 25], [-30, -15, -11, -30, 2, -2, -13, 0], [0, -6, -6, -7, -10, 11, 0, -5], [12, 20, -30, 0, 10, 10, -5, 5], [-1, 30, 5, -1, -19, 6, 24, 0], [0, -10, -10, -27, -17, 23, 0, -4], [0, 4, 4, 47, 9, 7, 0, 20], [0, -23, -23, -2, 19, -3, 0, -7], [0, -17, -17, -2, 20, -10, 0, -7], [14, 24, 7, 14, 1, 18, 6, 0], [39, 8, 16, 39, 2, -6, 14, 0], [0, 6, 6, -44, -13, 10, 0, 3], [-5, -22, 10, -5, 24, -8, -14, 0], [5, -5, -25, 0, 30, 30, -18, 18], [-5, -1, -24, -5, -18, 5, -6, 0], [16, 8, 11, 0, -19, -19, 18, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_p_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_p_i();".
// 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_1638_2_p_i(:prec:=8)
    chi := MakeCharacter_1638_p();
    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_1638_2_p_i();".
// 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_1638_2_p_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![289, -510, 1189, 442, 332, 26, 21, 2, 1]>],Snew);
    return Vf;
end function;