
// Make newform 1638.2.x.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_x_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_x_d();".
// 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_x_d();".
// 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 := [324, 0, 540, 0, 205, 0, 26, 0, 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], [-162, 612, 219, 374, 60, 52, 3, 2], [-810, 774, -249, 155, 24, -8, 3, -1], [162, 612, -219, 374, -60, 52, -3, 2], [810, 1206, 249, 155, -24, -8, -3, -1], [0, -954, 0, -665, 0, -112, 0, -5], [114, 0, 97, 0, 20, 0, 1, 0]];
    Rf_basisdens := [1, 1, 432, 432, 432, 432, 432, 24];
    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_x();"
function MakeCharacter_1638_x()
    N := 1638;
    order := 4;
    char_gens := [911, 703, 379];
    v := [4, 2, 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_1638_x_Hecke();"
function MakeCharacter_1638_x_Hecke(Kf)
    N := 1638;
    order := 4;
    char_gens := [911, 703, 379];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 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, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 1, 0, 0, 1, 0], [2, 0, 0, 0, 0, 0, -3, 0], [2, 0, -3, 0, 3, 0, 0, -1], [0, -1, 0, -3, 0, 0, 1, 1], [0, 0, -3, 1, -3, 1, -3, 0], [2, 2, -3, 2, 3, -2, 0, -1], [3, 1, 0, 0, 0, 0, 2, -1], [0, 2, 0, 1, 0, 0, -2, -2], [-2, 0, 6, -2, 0, 0, -2, 0], [0, 2, 3, -1, 3, -1, -5, 0], [2, -2, 0, 0, 0, 2, -2, 0], [2, -2, 0, -2, 0, 2, 0, 2], [-4, -1, 0, 0, -6, 2, 3, 1], [0, -1, 3, 0, 3, 0, -1, 0], [1, -2, 0, 0, -6, 2, -1, 0], [-6, 1, 0, 0, 6, -2, 7, -1], [0, -1, 0, 0, 0, -1, 2, -2], [-4, 3, -6, 3, 6, -3, 0, -4], [-4, 0, 0, 2, 0, 0, -4, 0], [-2, 3, 0, 0, 0, 0, -1, 3], [-3, 3, 6, 0, 0, 0, -6, -3], [4, -3, 0, -3, 0, 3, 0, 4], [-10, -1, -3, -1, 3, 1, 0, 3], [4, 0, 0, 0, 0, 0, 0, 4], [-2, 1, 0, 0, -12, 1, 0, 2], [2, 4, 0, 4, 0, -4, 0, -6], [0, -3, -6, 1, -6, 1, 0, 0], [0, 0, -3, -1, -3, -1, -1, 0], [-3, -1, 0, -1, 0, 0, -2, 1], [0, -2, 0, -4, 0, -4, 4, 0], [-4, 1, 0, 0, -12, 0, 3, 1], [2, -1, 0, 3, 0, 0, 3, 1], [0, 3, -3, 2, -3, 2, -5, 0], [-3, -2, 6, 0, 0, 0, -1, 2], [1, 3, 0, 0, 0, 1, -5, 4], [2, -2, -6, -2, 6, 2, 0, 2], [0, 1, 0, 3, 0, 3, 2, 0], [2, -2, 6, -2, -6, 2, 0, 6], [10, 3, -3, 3, 3, -3, 0, -9], [-1, 1, -6, -2, 0, 0, -2, -1], [4, 1, 0, 0, 0, -2, -3, -1], [0, -1, -3, -1, 3, 1, 0, 5], [-4, 1, -3, 1, 3, -1, 0, 1], [-1, -2, 18, 0, 0, 0, 1, 2], [4, -1, 6, 0, 0, 0, 5, 1], [-1, -1, 0, 0, 6, 0, 2, -1], [0, -5, 0, -1, 0, -1, 2, 0], [-6, -6, 0, 0, 0, 10, 2, 4], [-3, 1, 0, 0, 6, -2, 4, -1], [-6, 5, -3, 5, 3, -5, 0, -1], [-18, -4, 0, -4, 0, 4, 0, 2], [0, 0, -6, 0, 6, 0, 0, -4], [0, 3, 6, -1, 6, -1, -14, 0], [-5, -2, 0, 0, 6, 6, 1, 4], [0, -1, 0, -3, 0, -3, 6, 0], [10, 2, 6, 6, 0, 0, 8, -2], [4, 1, -6, 1, 6, -1, 0, 8], [8, -4, 0, 0, 6, -4, 0, -8], [7, 5, 0, 0, 0, -2, -10, 3], [14, 5, 6, 5, -6, -5, 0, -4], [0, -1, -12, -3, -12, -3, -8, 0], [-14, 2, 0, 0, 6, -4, 16, -2], [3, 6, 0, 0, 6, -10, 1, -4], [0, 1, 3, -2, 3, -2, -1, 0], [-12, 6, 0, 6, 0, -6, 0, -2], [7, -4, 0, 0, 0, 6, -9, 2], [6, 7, 0, 6, 0, 0, -1, -7], [-2, -5, 6, -8, 0, 0, 3, 5], [0, 0, 0, -2, 0, -2, 4, 0], [6, -3, 6, -3, -6, 3, 0, 6], [5, -1, 0, 0, 12, 0, -4, -1], [-13, -4, 0, -3, 0, 0, -9, 4], [0, 2, -6, -2, -6, -2, -6, 0], [-1, 8, 0, 0, 12, -12, 5, -4], [-10, 2, -6, 4, 0, 0, -12, -2], [0, -8, 12, -9, 0, 0, 8, 8], [0, -2, 3, 7, 3, 7, -11, 0], [-3, -3, 0, 0, -6, -2, 8, -5], [-8, 0, 0, 0, 0, 2, 6, 2], [-8, 4, 12, 4, -12, -4, 0, 0], [10, 1, -9, 1, 9, -1, 0, -3], [-4, -8, 6, -8, -6, 8, 0, 6], [5, -3, 0, -7, 0, 0, 8, 3], [3, -1, 0, 0, -18, -2, 0, -3], [11, 3, 0, 3, 0, 0, 8, -3], [-6, -5, 0, -1, 0, 0, -1, 5], [-6, -1, -9, -1, 9, 1, 0, 9], [1, 3, 0, 0, 24, -5, 1, -2], [9, 3, 0, 0, 24, -2, -10, 1], [0, -1, -6, 1, -6, 1, 2, 0], [3, -10, 0, 0, -18, 16, -9, 6], [0, 0, 0, 0, 0, 0, -12, 0], [13, 9, 0, 5, 0, 0, 4, -9], [0, 7, -9, -2, -9, -2, 5, 0], [0, -2, -6, -2, -6, -2, 10, 0], [2, 8, 0, 9, 0, 0, -6, -8], [4, -12, 0, -12, 0, 12, 0, 4], [-4, -7, 12, -8, 0, 0, 3, 7], [6, -13, 9, -13, -9, 13, 0, 9], [0, 0, 6, 8, 6, 8, -2, 0], [0, 2, -6, -4, -6, -4, -2, 0], [13, 5, 6, 2, 0, 0, 8, -5], [0, -5, 18, -6, 0, 0, 5, 5], [2, 3, 0, 0, 12, -4, -1, -1], [-10, -5, 9, -5, -9, 5, 0, -3], [0, -2, -6, 0, -6, 0, 4, 0], [0, -2, -9, 3, -9, 3, -21, 0], [-12, -3, 0, 0, 24, 9, 6, 6], [9, 8, 0, 0, 12, -7, -10, 1], [4, 0, 0, 0, 12, -5, 1, -5], [-14, -1, 0, 7, 0, 0, -13, 1], [0, 3, -6, -5, -6, -5, 2, 0], [0, -5, 24, -3, 0, 0, 5, 5], [6, -1, -6, -1, 6, 1, 0, 6], [-6, -6, -3, -6, 3, 6, 0, 7], [12, -2, -12, -2, 12, 2, 0, 2], [9, 3, 0, 0, -18, -2, -10, 1], [0, 3, -15, -2, -15, -2, -9, 0], [0, -1, 6, -7, 6, -7, 20, 0], [-11, -2, 0, -3, 0, 0, -9, 2], [-3, 3, -12, 8, 0, 0, -6, -3], [0, -6, 0, 0, 0, 0, -8, 0], [3, 1, -42, 4, 0, 0, 2, -1], [-18, 1, 12, 1, -12, -1, 0, -2], [20, -7, 3, -7, -3, 7, 0, 7], [-9, 1, -30, 2, 0, 0, -10, -1], [9, 2, 6, 0, 0, 0, 7, -2], [-14, -1, 0, 0, 6, 6, 9, 5], [0, 6, 18, 2, 18, 2, -10, 0], [20, -4, -12, -4, 12, 4, 0, 0], [-12, 1, 0, 0, 12, -4, 15, -3], [12, -4, 0, -11, 0, 0, 16, 4], [-9, -3, -12, 1, 0, 0, -6, 3], [18, -4, 0, 0, 0, -1, -13, -5], [4, 5, -6, 5, 6, -5, 0, -16], [12, 5, 0, 5, 0, -5, 0, -2], [0, 5, 0, 11, 0, 0, -5, -5], [-2, 10, 0, 0, 6, -2, -6, 8], [0, 12, 6, 8, 6, 8, 2, 0], [-5, -6, 0, 3, 0, 0, 1, 6], [-8, -8, 15, -8, -15, 8, 0, 1], [6, 10, 0, 0, -12, -16, 0, -6], [5, -9, 0, 0, -6, 8, -4, -1], [14, -2, 24, -2, -24, 2, 0, 2], [0, 1, -18, -3, -18, -3, -20, 0], [8, -7, 0, 0, 18, 2, -3, -5], [-3, -1, 0, 0, -30, 10, -6, 9], [2, 6, -9, 6, 9, -6, 0, 3], [0, 2, 3, -7, 3, -7, 3, 0], [0, -11, 0, 1, 0, 1, 16, 0], [0, 2, 12, -2, 12, -2, -16, 0], [18, 7, 15, 7, -15, -7, 0, -15], [-34, -4, -6, -4, 6, 4, 0, 4], [0, 5, -24, 0, 0, 0, -5, -5], [0, 15, 6, 3, 6, 3, -14, 0], [0, 0, -18, -4, 0, 0, 0, 0], [-5, -2, 0, -3, 0, 0, -3, 2], [0, 3, -12, -1, -12, -1, 22, 0], [-16, 10, 0, 0, 0, -7, 13, 3], [0, 12, -12, 4, -12, 4, -4, 0], [23, 4, 0, 0, -12, -11, -16, -7], [9, -1, 0, 0, -24, 11, -19, 10], [34, 7, 0, 7, 0, -7, 0, -4], [0, -4, -12, 4, -12, 4, 8, 0], [-11, -2, -24, 2, 0, 0, -9, 2], [18, 2, -24, 2, 24, -2, 0, -4], [-1, -5, 0, 0, -24, 13, -7, 8], [-17, -2, 0, 0, 12, -2, 21, -4], [0, 4, 0, 0, 0, 0, -4, 0], [-31, 3, 0, 0, 6, -2, 30, 1], [-14, -3, -27, -3, 27, 3, 0, 3], [0, 4, 6, 2, 6, 2, 12, 0], [15, -9, 0, -6, 0, 0, 24, 9], [4, 0, 18, 0, 0, 0, 4, 0], [36, -6, 0, -6, 0, 6, 0, 4], [0, -7, 9, -4, 9, -4, 1, 0], [6, -5, 0, 7, 0, 0, 11, 5], [-12, -14, 6, -14, -6, 14, 0, 16], [26, 9, 6, 9, -6, -9, 0, -14], [16, 7, 24, 10, 0, 0, 9, -7], [16, 7, -18, -6, 0, 0, 9, -7], [22, -8, 18, -8, -18, 8, 0, 16], [0, -1, 0, 9, 0, 9, -6, 0], [-12, -1, 24, 1, 0, 0, -11, 1], [3, -4, -12, 0, 0, 0, 7, 4], [5, -9, 0, 0, -24, 9, -5, 0], [0, 11, 0, 9, 0, 9, 14, 0], [1, -7, 0, 0, 0, 1, 5, -6], [-8, 2, 0, 2, 0, -2, 0, -4], [-5, -16, 0, 0, -24, 19, 2, 3], [0, 4, 6, -6, 6, -6, 26, 0], [18, 12, 9, 12, -9, -12, 0, -11], [15, 5, 0, 0, 6, -10, -10, -5], [0, 3, 6, -13, 6, -13, -18, 0], [-16, 12, 0, 0, 6, -10, 14, 2], [0, 8, 12, -2, 12, -2, -10, 0], [0, -10, -6, -10, 0, 0, 10, 10], [0, -6, -15, 13, -15, 13, 1, 0], [-15, 3, 30, -12, 0, 0, -18, -3], [0, 12, 12, -8, 12, -8, -16, 0], [2, -6, 0, 0, 0, 4, 0, -2], [0, 5, 18, -9, 18, -9, 0, 0], [21, 2, -18, 0, 0, 0, 19, -2], [12, 6, -12, 6, 12, -6, 0, 0], [3, 8, 0, 0, 36, -11, 0, -3], [10, -4, 18, -4, -18, 4, 0, 0], [0, -1, -9, 8, -9, 8, 23, 0], [0, 5, -12, 9, -12, 9, -30, 0], [-6, -2, -6, -2, 6, 2, 0, 4], [-10, -1, 0, 0, 30, -2, 13, -3], [1, 2, -24, -9, 0, 0, -1, -2], [-8, 5, 0, 0, 24, -13, 16, -8], [-4, -17, 12, -17, -12, 17, 0, 14], [0, 5, 15, 10, 15, 10, 15, 0], [-14, -12, 0, 0, -24, 6, 20, -6], [-4, -2, 30, -8, 0, 0, -2, 2], [-18, -4, 30, -4, -30, 4, 0, -2], [-21, 11, 0, 0, 0, -8, 18, 3], [-1, -2, 0, 0, 0, 9, -6, 7], [15, 7, 0, 0, 12, -16, -6, -9], [6, 3, -15, 3, 15, -3, 0, -9], [0, 2, 12, -6, 12, -6, 8, 0], [0, 3, -21, 0, -21, 0, 11, 0], [-36, -6, 12, -6, -12, 6, 0, 10], [-20, 1, 3, 1, -3, -1, 0, 11], [11, 0, -24, 5, 0, 0, 11, 0], [0, 4, 0, -8, 0, -8, -40, 0], [0, -3, -18, -13, -18, -13, -12, 0], [0, 8, 0, 0, 12, -17, 9, -9], [0, -8, 6, -2, 6, -2, 8, 0], [0, 13, -12, -3, -12, -3, 6, 0], [-17, 2, 12, -9, 0, 0, -19, -2], [-14, 3, 0, 0, -36, 5, 6, 8], [-4, -3, -24, 4, 0, 0, -1, 3], [0, 2, -6, -4, -6, -4, 42, 0], [0, -4, -12, 6, -12, 6, 18, 0], [6, -8, 0, 0, 12, -6, 8, -14], [20, 7, 6, 7, -6, -7, 0, -4], [0, -10, -15, -5, -15, -5, 21, 0], [29, -3, 0, 0, 6, 6, -32, 3], [8, -12, 6, -14, 0, 0, 20, 12], [-24, -8, -12, -8, 12, 8, 0, 6], [9, 10, 0, 0, 42, -14, -5, -4], [-24, -3, 0, 0, 18, 14, 13, 11], [-3, -4, 0, 0, -18, -8, 15, -12], [0, 10, -3, 7, -3, 7, -47, 0], [0, -14, 12, -9, 0, 0, 14, 14], [18, -7, 36, -10, 0, 0, 25, 7], [14, 2, -24, 14, 0, 0, 12, -2], [2, 1, 0, 1, 0, -1, 0, 4], [0, -19, -18, -3, -18, -3, 18, 0], [9, 2, 12, 13, 0, 0, 7, -2], [-14, 1, 0, 1, 0, -1, 0, -4], [11, 1, 12, -8, 0, 0, 10, -1], [-2, 4, 12, 4, -12, -4, 0, -18], [16, 2, 12, 11, 0, 0, 14, -2], [0, 5, -12, -5, -12, -5, 10, 0], [2, -3, -6, -3, 6, 3, 0, 4], [25, 9, 18, 6, 0, 0, 16, -9], [0, -7, -27, 4, -27, 4, 7, 0], [16, -3, 0, 0, -36, 10, -23, 7], [0, -5, 18, 3, 18, 3, -40, 0], [6, -1, 0, 0, -24, 8, -13, 7], [-10, 8, -30, 18, 0, 0, -18, -8], [-3, -1, 0, -8, 0, 0, -2, 1], [0, -12, 18, -2, 18, -2, 8, 0], [4, 2, 24, 2, -24, -2, 0, 4], [33, 2, 0, 0, -30, -8, -27, -6], [12, -7, 0, 0, -24, 5, -10, -2], [0, -3, 18, -9, 18, -9, -20, 0], [0, 7, -6, -1, -6, -1, -6, 0], [23, 10, -18, 2, 0, 0, 13, -10], [5, -5, 0, 0, -12, 11, -11, 6], [-2, -2, 0, -13, 0, 0, 0, 2], [21, 7, -18, 24, 0, 0, 14, -7], [20, 14, -12, 14, 12, -14, 0, 0], [0, -7, 0, -11, 0, -11, -24, 0], [18, 7, 0, 3, 0, 0, 11, -7], [0, 7, 6, 5, 6, 5, -2, 0], [17, -3, 6, -4, 0, 0, 20, 3], [4, -4, 0, 0, -12, -5, 5, -9], [0, 4, -18, 0, -18, 0, -16, 0], [-20, 6, 9, 6, -9, -6, 0, 7], [-2, 2, 0, 0, 6, 2, -2, 4], [1, -3, 0, 0, 24, -2, 4, -5], [-30, -10, -3, -10, 3, 10, 0, 25], [6, 6, -21, 6, 21, -6, 0, -21], [0, 14, 3, -3, 3, -3, -1, 0], [-17, 0, 0, 0, -12, 5, 12, 5], [-14, 11, 0, 0, 18, -20, 23, -9], [-38, -6, -12, -6, 12, 6, 0, 20], [0, -3, 30, -3, -30, 3, 0, 8], [0, -6, 0, -12, 0, -12, -20, 0], [-18, -2, 18, -2, -18, 2, 0, 12], [0, 8, -9, 7, -9, 7, -33, 0], [0, -10, 0, 0, 24, -3, 13, -13], [-22, -16, 0, -16, 0, 16, 0, 12], [-8, -4, -18, 6, 0, 0, -4, 4], [0, 11, 0, 3, 0, 3, 6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_x_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_x_d();".
// 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_x_d(:prec:=8)
    chi := MakeCharacter_1638_x();
    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_x_d();".
// 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_x_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, 160, 800, -424, 113, 4, 8, -4, 1]>],Snew);
    return Vf;
end function;