
// Make newform 1638.2.bj.f in Magma, downloaded from the LMFDB on 26 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_bj();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1638_bj_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_bj_f();".
// 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_bj_f();".
// 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, -108, 45, 12, -20, 4, 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], [0, 1, 0, 0, 0, 0, 0, 0], [-27, -45, 42, 10, 10, 2, -1, 1], [27, -27, 30, -10, -10, -2, 1, -1], [162, -153, 18, 52, -26, -4, 8, -5], [243, -225, 54, 26, -22, -2, 13, -7], [-891, 801, -150, -218, 130, 14, -49, 19], [270, -207, 24, 58, -32, -10, 14, -5]];
    Rf_basisdens := [1, 1, 216, 72, 108, 108, 216, 36];
    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_bj();"
function MakeCharacter_1638_bj()
    N := 1638;
    order := 6;
    char_gens := [911, 703, 379];
    v := [6, 6, 5];
    // 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_bj_Hecke();"
function MakeCharacter_1638_bj_Hecke(Kf)
    N := 1638;
    order := 6;
    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, -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, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 1, -1, 0, -1, 2, 0], [0, 0, 0, 0, 0, 0, -1, 0], [-1, 1, 1, -1, 0, 0, 0, 0], [3, -1, 1, -1, 0, -2, 1, 1], [1, -1, -1, -2, -1, -2, 1, 1], [-2, 2, 3, 1, 0, -1, 2, 2], [-1, -1, 0, -2, -2, 1, 3, 1], [-1, 1, -1, 1, 0, 0, -4, 0], [-2, 0, -2, 0, -4, 1, -2, 1], [0, 2, -2, -1, -2, 1, -1, -1], [-1, -1, -3, 2, 2, 1, -1, -1], [0, 0, -5, -1, 1, -1, -2, 0], [3, -3, -2, -3, 0, 2, 1, 5], [3, 1, 4, -1, 0, -3, 3, 4], [-3, -3, -4, -1, -3, 1, -1, -3], [0, 0, 8, -2, 0, -2, 5, 0], [2, 1, -5, -3, -3, -2, 2, 2], [-1, 1, 2, 1, 0, -1, -2, 1], [2, -2, 6, -2, 0, -2, 8, 0], [0, -2, 1, 2, 0, -1, -1, -1], [4, 1, 6, 1, 10, 2, 5, 1], [3, -2, 10, 1, -1, -1, 1, 1], [-9, 1, -1, -2, -4, 2, 5, 1], [2, -2, 6, 2, 0, -4, 12, -4], [-8, 2, -10, -2, 0, 5, 2, -3], [5, -1, 3, -1, 4, 0, 8, 0], [-3, -3, -3, -3, -12, -1, 0, 2], [3, -3, -7, -2, 4, -2, -1, 3], [10, -2, -2, 0, 8, -2, -2, -2], [-1, 4, 5, -4, 0, 1, -3, 3], [-2, 2, 4, 2, 0, -2, 6, 2], [-6, 6, 1, -1, -2, -1, -2, -6], [-2, 2, 1, -1, 0, 1, 7, 2], [6, -6, -3, -6, 0, 3, 3, 9], [6, 2, 4, -2, 0, 0, -2, 2], [6, -4, -3, 1, 1, -1, 2, -4], [-4, 4, 0, -4, 0, 0, 0, 0], [3, -3, -11, -2, -2, -2, -3, 3], [6, 8, 8, 2, 14, 6, 6, 6], [-3, 5, -4, -5, 0, 4, 1, 1], [-7, 7, 2, -1, -8, -1, -2, -7], [-9, 1, 5, 1, 8, 2, -2, -2], [-4, -4, 1, 5, 8, 1, -1, -1], [5, -5, -8, 1, 6, 1, -2, 5], [2, -2, -2, -2, 0, 0, 0, 0], [-1, -3, -2, 8, 4, 5, -5, -5], [-16, -4, -4, 0, -10, 0, -8, -4], [-8, 8, 17, 8, 0, 4, 9, -4], [15, 1, 1, -1, 0, -1, 2, 2], [2, 2, -2, 2, 8, -1, -4, -3], [4, -4, -2, 2, 0, -2, -10, -4], [0, 0, 3, 3, -5, 3, 0, 0], [1, -6, 2, -3, -5, -3, 13, -3], [0, -2, 2, 5, -2, -7, 1, -7], [4, -4, -18, -2, 8, -2, -6, 4], [-5, 7, 6, -5, -2, 2, -2, -2], [-6, 6, -8, 10, 10, 10, -12, -6], [7, -3, -9, -3, 8, -3, -6, 0], [2, 6, 4, 8, 8, -2, -6, -2], [-6, -6, 2, 8, -6, -8, -4, -6], [-7, 9, 0, 9, 4, 2, -9, -7], [-11, 5, 0, -5, 0, 2, 1, 3], [15, 5, -3, -5, 0, 3, 2, 2], [-1, 5, -11, 5, 8, -2, -16, -7], [-6, 2, 2, 0, -2, 0, -14, 2], [2, -8, -5, 8, 0, -1, -1, -7], [7, -7, -4, -7, -6, -7, 5, 7], [-7, 1, 8, 4, 6, 5, -5, -5], [-4, -1, 1, -5, 5, -6, 6, 6], [5, 5, 7, 5, 20, 5, 2, 0], [5, 3, -6, 0, 8, 3, -15, 3], [-5, 5, -9, 4, 24, 4, -9, -5], [-12, 0, -10, 1, 12, 1, -1, -1], [-6, 6, 5, -1, 0, 1, 8, 6], [15, -3, 21, 3, 0, -10, -4, 7], [-4, 4, 8, 4, 0, -4, -5, 4], [-8, 6, -12, 2, 2, 8, -8, -8], [5, -5, 1, 6, 0, -6, 13, -5], [16, -5, -13, -10, 11, 5, -11, 5], [2, -8, -12, -8, -12, -4, -4, 4], [1, 3, 4, -1, -4, 2, -2, -2], [10, -10, 0, -2, 8, -2, 6, 10], [22, -6, -12, -10, 16, 4, -8, 4], [0, -4, 1, 4, 0, 7, -19, -11], [35, 1, -1, -2, 18, 2, -3, 1], [-8, 7, 7, 1, 1, 8, -8, -8], [4, 4, 4, 0, 4, 0, 20, 4], [0, -10, -11, -10, -20, -9, -1, 1], [17, 4, 3, -4, 0, 3, -5, 1], [19, -11, 13, 2, -8, -9, 9, 9], [-3, -9, 4, 13, -6, -13, 7, -9], [10, -2, -2, 0, 8, -2, -2, -2], [-4, 1, 14, 1, -6, 0, 13, -1], [-4, 4, 8, 12, 16, 12, -4, -4], [-12, 8, 0, -1, 4, 7, -7, -7], [-1, 1, 6, -1, 0, 2, -9, -1], [-15, -7, -7, -4, -22, -3, -3, -3], [-13, 5, -5, 5, -16, 5, -10, 0], [17, -7, -3, 7, 0, -3, 2, -4], [8, -4, -9, 3, -4, -1, 1, 1], [13, -13, -15, -8, 6, -8, 3, 13], [-4, 10, 14, 5, 6, 5, 13, 5], [-4, 7, -6, -7, 0, 0, 13, 7], [-13, 7, 1, 7, -12, -3, -6, -10], [-10, 7, -3, -4, 3, 3, -3, -3], [-10, 7, 7, 7, -6, -1, 0, -8], [7, 7, -5, -7, 0, 12, -12, -5], [-8, -6, 2, -11, -14, 5, 21, 5], [1, -1, 18, 3, 22, 3, 8, 1], [-4, -6, -18, 2, 10, -4, 4, 4], [-15, -5, -11, -6, -10, 6, 17, -5], [-17, 7, 2, 7, -20, 0, -5, -7], [17, -1, 4, 5, 8, -5, -11, -1], [8, -8, 6, -10, -8, -10, 12, 8], [-18, 10, 1, -9, -4, 9, -22, 10], [11, 3, 9, 0, 14, 3, 15, 3], [22, -2, -15, -3, 20, 1, -25, 1], [-1, 1, 2, 3, -4, 3, -1, -1], [5, -3, -8, -6, -2, -9, 9, 9], [-9, 6, -10, 2, -3, 4, -28, 4], [-9, 11, 13, -11, 0, 1, -4, 10], [0, -4, 4, 8, -2, -8, -24, -4], [-10, 0, 6, -2, 10, -2, 2, 2], [5, -5, -4, 5, 0, -5, 9, 0], [19, -7, 3, 10, 6, -10, 7, -7], [-10, 10, -13, 1, -20, 1, -12, -10], [9, 7, -5, -7, 0, 18, -24, -11], [-4, 0, -9, 0, -8, 10, -9, 10], [-10, 7, 9, -7, 3, 0, 0, 0], [9, -9, 10, 3, 0, -6, 6, 6], [9, 7, 15, -3, 16, 10, 26, 10], [6, 10, 4, 5, 16, 5, -7, 5], [-30, -2, -2, 0, -16, 0, 10, -2], [0, 2, 0, -11, -2, -9, 9, 9], [2, -2, -2, 0, 0, 0, 8, -2], [-16, 16, 13, -3, 0, 3, 2, 16], [-1, 1, 5, -2, 20, -2, 3, -1], [-22, 12, 2, 3, -10, 9, -11, 9], [0, -4, 34, -4, -8, -2, 38, 2], [4, 0, -5, -7, -4, -7, 7, 7], [12, -12, 6, -18, -6, -18, 18, 12], [14, -4, 6, -4, 20, 4, 10, 8], [13, -11, -11, -4, 2, -7, -7, -7], [12, 0, -8, -8, 6, 8, -20, 0], [20, -12, -15, -12, 16, -4, -3, 8], [-4, 6, -6, -6, 0, 18, -24, -12], [-3, -5, -16, 5, 0, 12, -13, -17], [6, 2, -32, 2, 16, 4, -34, 2], [-15, -5, -5, 0, -10, 0, -33, -5], [6, 9, 9, 11, 15, -2, -2, -2], [36, -7, 0, 7, 0, -6, 5, -1], [-3, -1, 15, 0, -4, -1, 31, -1], [24, 9, 17, 1, 33, 8, 24, 8], [7, -1, 7, 1, 0, -7, 6, 6], [5, -5, -44, -5, -2, -5, -17, 5], [21, 1, -3, -4, 11, 4, 9, 1], [-6, 8, 22, -8, 0, -6, -2, 14], [6, -20, 10, -20, -28, -12, 30, 8], [1, -7, -25, 7, 6, 0, 0, 0], [1, -1, -11, -8, 0, -8, -1, 1], [-2, 12, 10, 12, 20, 14, -2, 2], [-9, 9, -2, 3, -19, 3, -7, -9], [-12, 12, 20, -18, 0, -6, 6, 6], [-4, -8, -22, -8, -24, -12, -14, -4], [-27, 9, -9, -1, -18, 10, -26, 10], [2, -2, 4, -4, -8, -4, 5, 2], [3, 0, 9, 0, 6, -1, 9, -1], [-12, -6, 12, 6, 0, -10, 2, 4], [10, -6, -39, -6, 8, -1, -33, 5], [12, 8, 9, 1, 10, -1, -10, 8], [0, 0, 12, -8, 0, -8, 10, 0], [-13, 3, -10, -13, -5, 13, 1, 3], [25, -3, -3, 3, 0, -5, 10, 2], [9, -9, 9, 0, 29, 0, 9, 9], [-10, 19, 9, -5, -9, 14, -14, -14], [23, -3, 2, -3, 40, 9, 5, 12], [13, -7, 0, -9, 6, 2, 16, 2], [-12, -8, -7, -2, -20, -6, -4, -6], [4, 12, -8, 12, 32, 8, -20, -4], [-7, 8, -9, -8, 0, -1, 19, 9], [4, -8, -22, 8, 0, -8, 30, 0], [-18, 17, 11, 17, -2, -1, -6, -18], [-15, -1, 2, 5, 16, 4, -4, -4], [4, -4, -4, 8, -20, 8, -4, 4], [20, -12, -22, 12, 0, 0, 10, -12], [19, -13, 10, -13, 12, 4, 23, 17], [-2, -10, -2, 16, 12, 6, -6, -6], [4, 20, 17, -3, 12, 3, 26, 20], [-4, 4, 52, -4, 0, -4, 26, -4], [0, 0, 10, 10, 0, -10, 14, 0], [-5, 14, 7, -14, 0, -3, 13, 17], [-8, -10, 22, 6, 18, -4, 4, 4], [-6, 6, -30, 4, -10, 4, -20, -6], [-9, 17, 17, 14, 8, 3, 3, 3], [8, -20, -13, -20, -24, -7, 7, 13], [4, -4, 4, -12, -32, -12, 10, 4], [-25, 18, 11, 18, -14, 3, -7, -15], [15, 5, -18, -5, 0, 0, 23, 5], [11, 5, 13, 8, 8, -8, -12, 5], [10, -10, -45, -5, 4, -5, -15, 10], [2, -6, -4, 0, 4, -6, 6, 6], [-3, -5, -1, 5, 0, 1, -6, -6], [-16, 6, -14, 12, 10, 18, -18, -18], [-18, 6, 14, 8, -12, -2, 14, -2], [18, -10, -22, -10, 16, -6, -12, 4], [-2, 2, 38, 4, 28, 4, 16, -2], [-12, 8, 14, 0, 4, 8, -8, -8], [2, -2, 9, -9, -16, -9, 10, 2], [27, -10, 6, 4, 17, -14, 18, -14], [33, -5, 11, 5, 0, -1, -14, -4], [-3, 15, 25, 11, 12, 4, 24, 4], [7, -5, 8, 13, 1, -13, 1, -5], [16, -4, -5, -1, 6, 1, -36, -4], [1, -3, 33, -3, -4, -5, 36, -2], [8, 8, 1, -8, 0, 19, -31, -11], [3, -3, -25, -4, 35, -4, -9, 3], [-16, -2, -32, 7, 18, 5, -5, -5], [-4, -4, 7, -4, -16, -17, 11, -13], [-8, -4, 12, 10, -12, -14, 18, -14], [10, 6, -16, 6, 32, 12, -22, 6], [-11, -5, -11, -5, -32, 5, -6, 10], [-7, -1, 12, 13, -4, -13, 1, -1], [-10, -13, -5, -2, -23, -11, 5, -11], [37, 1, 11, -1, 0, -11, 12, 12], [35, -22, -26, -19, 13, -3, -11, -3], [-9, 9, -10, 1, 16, 1, -10, -9], [-2, 2, -30, 8, 12, 8, -20, -2], [-16, 18, 44, 18, 4, 10, 26, -8], [16, -12, -14, -10, 4, -2, -6, -2], [11, 13, 3, 26, 24, -13, -33, -13], [-7, 3, 11, -9, 4, -6, 6, 6], [-21, 7, -7, -7, 0, 13, -12, -6], [-9, -11, -8, 11, 0, 10, -23, -21], [-19, 23, 6, 23, 8, 22, -17, -1], [30, -6, -8, -2, 12, 2, -22, -6], [14, -22, 13, 9, 8, -13, 13, 13], [8, -8, -20, 0, 16, 0, -6, 8], [-25, -9, -5, -4, -34, -5, 3, -5], [-21, 11, 13, -3, 10, 8, -8, -8], [-17, 1, 7, 14, -16, -13, -1, -13], [-8, 8, 47, 13, 8, 13, 13, -8], [-14, 0, 2, 22, 14, 22, -22, -22], [39, 11, 7, -4, 25, 4, 23, 11], [-7, 1, 16, -1, 0, -4, -7, 5], [5, -17, -13, 4, -6, -4, 21, -17], [-24, 26, -10, -8, -2, 18, -18, -18], [6, 8, 5, -3, 7, 3, 42, 8], [24, -8, -18, -16, 16, 8, -12, 8], [-20, 10, -6, 4, 10, 14, -14, -14], [17, 1, 19, 0, -18, 1, -1, -1], [-1, -1, -15, -1, -4, -10, -14, -9], [-32, 18, 42, 20, -14, -2, 46, -2], [18, -8, 28, 8, 0, -28, 20, 20], [-13, -15, -6, 9, -14, -9, -31, -15], [-21, 21, 17, 14, 4, 14, -9, -21], [-12, 8, -22, -2, 4, 6, -6, -6], [-16, 16, 2, -16, 0, 8, -2, 8], [13, 7, -3, -10, 10, 10, 37, 7], [12, -12, 9, 12, 0, -3, -15, -9], [9, 9, 27, -1, 18, 10, 46, 10], [15, 3, -14, 3, 36, 4, -17, 1], [52, -4, -15, -6, 48, 2, -20, 2], [3, -9, -15, -6, -3, 6, 31, -9], [-6, 6, 30, 4, -18, 4, 10, -6], [1, -17, -21, -4, -8, 4, 0, -17], [-23, 10, 23, 10, -26, -5, 13, -15], [20, 14, 1, -13, 17, 13, 8, 14], [-5, 5, -1, -10, -20, -10, 2, -5], [11, 7, -25, -7, 0, -2, 36, 9], [5, 2, -7, 2, 14, -3, -9, -5], [-14, 14, -42, -5, 0, 9, -9, -9], [-3, 3, -44, 19, -4, 19, -33, -3], [17, -17, 1, 0, 36, 0, 9, 17], [-6, 0, -26, -5, 6, -5, 5, 5], [3, 5, 16, 11, 4, -11, 29, 5], [-6, 10, 2, 10, 8, 2, -8, -8], [-10, 4, -4, -8, -3, 8, -8, 4], [6, -6, -25, 1, -7, 1, -10, 6], [-30, 18, 34, 20, -12, -2, 30, -2], [20, -12, 6, 10, -8, -2, 2, 2], [-31, -3, 14, 3, 0, -8, -1, 5], [-9, -13, 35, 14, 22, 1, -1, -1], [-2, 0, 12, 0, -4, -16, 12, -16], [-29, 3, -9, -3, 0, 8, -4, -5], [23, 4, 16, -4, 0, 0, -12, 4], [-24, 8, -11, 8, -32, 9, -19, 1], [-48, -8, -9, -1, -28, 1, -9, -8], [11, -11, -4, 1, 21, 1, 3, 11], [-10, -14, -23, -13, -24, -1, -19, -1], [-27, 27, -35, 20, 30, 20, -41, -27], [-8, 8, 2, 15, 0, 23, -23, -23], [24, 10, 5, -5, 17, 5, 18, 10], [-1, 1, 31, 0, -14, 0, 15, -1], [-50, 16, 22, -16, 0, 0, -6, 16], [-35, 9, -3, -9, 0, 3, 6, 6], [-44, 8, 22, 23, -36, -15, 13, -15], [0, -3, 29, -2, -3, -1, 63, -1], [-9, 13, 11, -6, -4, 7, -7, -7], [7, -7, 13, -2, 10, -2, 11, 7], [-13, 9, 2, 9, -8, 5, -7, -4], [1, 11, 27, 13, 12, -2, 30, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_bj_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_bj_f();".
// 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_bj_f(:prec:=8)
    chi := MakeCharacter_1638_bj();
    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_bj_f();".
// 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_bj_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_bj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, 0, 608, 0, 276, 0, 32, 0, 1]>,<11,R![1024, 2304, 2016, 648, -31, -54, 3, 6, 1]>],Snew);
    return Vf;
end function;