
// Make newform 1800.2.d.s in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1800_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1800_d_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_1800_2_d_s();".
// 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_1800_2_d_s();".
// 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, -16, 0, -4, 9, -2, 0, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, -4, -6, 3, 2, 0, 0, -1], [56, -20, -12, -21, 18, 4, 2, -5], [-64, 24, 10, 25, -18, -4, -4, 5], [-36, 10, 8, 15, -10, -2, -2, 3], [36, -8, -12, -15, 10, 4, 2, -3], [44, -10, -9, -18, 12, 4, 3, -4], [-80, 26, 18, 31, -22, -6, -4, 7]];
    Rf_basisdens := [1, 8, 8, 8, 4, 4, 4, 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_1800_d();"
function MakeCharacter_1800_d()
    N := 1800;
    order := 2;
    char_gens := [1351, 901, 1001, 577];
    v := [2, 1, 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_1800_d_Hecke();"
function MakeCharacter_1800_d_Hecke(Kf)
    N := 1800;
    order := 2;
    char_gens := [1351, 901, 1001, 577];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 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, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1, -1, 1], [1, 0, 0, 1, -1, 1, 0, 0], [0, -1, 0, 2, -1, -1, 1, 0], [-1, 0, 0, 1, -1, -1, 2, -2], [1, 2, 3, 1, 1, 1, 0, 2], [-2, -1, 0, -2, 1, -2, 0, -1], [0, -1, -4, -2, 1, 0, -2, 1], [1, 1, 0, 1, -2, -2, -1, 0], [0, 0, 0, -2, -2, 0, 2, 2], [0, -1, 0, 2, -3, -2, 2, 1], [1, 1, 0, -4, -3, 0, 2, 3], [0, 2, 2, 2, 0, 0, 2, 0], [1, 0, 0, -3, 3, 3, 0, 0], [4, -2, -2, 0, -2, 4, -4, 2], [1, -1, -5, 2, -3, 1, 1, -2], [-3, -3, 0, 4, -3, -2, 4, 1], [5, -2, 0, 3, -1, -1, 2, 0], [0, -4, -2, -2, -2, 0, -2, -2], [2, 2, 0, -6, -4, 0, 2, 4], [-4, -2, 0, 6, 0, -2, 0, -2], [0, 2, 0, 0, -6, -4, 0, 2], [2, 0, -1, 4, -4, 2, 2, -2], [0, 3, -2, 0, 3, 0, 0, 3], [2, 2, -6, 0, 2, 2, -2, 4], [-4, -3, 0, 0, 1, 2, 4, 1], [1, -3, 7, 0, -3, 1, -1, -2], [2, 0, -4, 2, -2, 2, 0, 0], [-4, 2, -2, 0, 2, -4, 4, -2], [1, -2, -2, 1, -3, 1, 0, -2], [2, 2, 10, 2, 0, 2, 0, 2], [2, 2, -4, 0, 2, 2, -2, 4], [0, 2, 14, -2, 4, 0, -2, 4], [-3, -1, 0, 1, -8, -4, 5, 4], [6, -7, 0, 8, 7, 3, 3, -4], [-3, 5, 0, -8, -3, 0, -2, 3], [0, 5, 10, 2, 3, 0, 2, 3], [-1, 6, 0, -9, -5, -1, -2, 4], [0, 4, 2, 4, 0, 0, 4, 0], [1, 1, 7, 2, -1, 1, 1, 0], [-7, 0, 0, 3, -3, -3, 0, 0], [0, -4, -3, 2, -6, 0, 2, -6], [8, -4, 0, 4, -4, -2, 6, 2], [-1, 5, 0, -11, -2, 2, -1, 4], [1, 0, 3, -1, 1, 1, -2, 2], [-3, 4, 11, 0, 4, -3, 3, 1], [6, -1, 0, 2, -7, -4, 4, 3], [-3, -5, -17, -6, 1, -3, -3, -2], [7, -2, -2, 1, -3, 7, -6, 4], [-9, -2, 0, -3, -3, 1, 6, 4], [5, 6, 0, -10, -4, 0, -2, 4], [0, -2, 4, -4, 2, 0, -4, 2], [0, 2, 4, 4, -2, 0, 4, -2], [6, 1, 6, 0, 1, 6, -6, 7], [-2, -1, -16, 4, -5, -2, 6, -7], [0, -2, 0, 4, 6, 2, -2, -4], [-12, 1, 0, -2, 9, 5, -5, -4], [-6, -3, 0, -2, -1, 2, 6, 3], [-7, 3, 0, 0, 3, 0, -6, -3], [14, -2, 0, 6, 4, 0, -2, -4], [1, 3, 0, -4, -1, 0, -2, 1], [10, 2, 0, -6, -4, 0, 2, 4], [2, 4, 7, 6, -2, 2, 4, 0], [-8, -2, 0, 4, 2, 0, 0, -2], [2, 0, 16, -2, 2, 2, -4, 4], [-2, 0, 7, -2, 2, -2, 0, 0], [8, 6, 0, -2, -8, -6, -4, 2], [-4, -4, 12, -4, 0, -4, 0, -4], [-9, -2, 0, -3, 1, -9, 6, -8], [11, -4, 0, 5, -5, -3, 6, 2], [3, 2, -9, 0, 2, 3, -3, 5], [-6, 3, 0, -6, 7, 5, -5, -2], [5, 0, 3, 7, -7, 5, 2, -2], [-2, 8, 4, -2, 10, -2, 0, 8], [-2, -5, 14, -2, -3, -2, 0, -5], [18, -1, 0, 6, 7, 1, -5, -6], [2, -1, 0, 8, 7, 0, -6, -7], [3, -4, 0, 14, 2, -4, -2, -6], [-1, 2, 2, -5, 7, -1, -4, 6], [0, 4, 8, -6, 10, 0, -6, 10], [-17, -2, 0, 1, 5, 3, 0, -2], [2, 0, -5, -2, 2, 2, -4, 4], [11, 3, 0, -9, -2, 2, 1, 4], [6, -2, 0, -2, 8, 6, 0, -2], [6, -2, 0, 10, 4, -2, -4, -6], [2, -4, -14, 0, -4, 2, -2, -2], [2, 1, 6, 2, -1, 2, 0, 1], [-2, -2, 10, -4, 2, -2, -2, 0], [-18, -3, 0, 4, 5, 2, 0, -3], [-10, 12, 0, -14, -2, 0, -10, 2], [7, 4, 13, 4, 0, 7, -3, 7], [-2, 8, -8, 2, 6, -2, 4, 4], [5, 2, -3, 9, -7, 5, 4, -2], [-8, 3, -6, 2, 1, -8, 10, -7], [8, -2, -18, -2, 0, 8, -10, 8], [0, -9, 0, 8, 15, 8, 2, -7], [5, 3, 0, 4, -5, -6, -4, -1], [-7, 3, -9, -4, 7, -7, 3, 0], [-2, 10, 0, -8, -2, -2, -10, 0], [-20, 6, 0, -12, -2, 2, -2, 4], [2, 5, 0, -8, 1, 2, -4, 1], [-6, -5, 0, 10, 17, 6, -6, -11], [7, -6, -15, 3, -9, 7, -4, -2], [-6, -4, -11, 0, -4, -6, 6, -10], [-10, 3, 0, -2, 17, 8, -12, -9], [-5, -4, -6, -3, -1, -5, 2, -6], [14, 10, 0, -10, 12, 6, -16, -6], [3, 0, 0, -2, -10, -4, 6, 6], [2, 0, -14, -4, 4, 2, -6, 6], [-14, 4, 0, -8, -8, -2, 2, 6], [3, -6, 6, -3, -3, 3, -6, 0], [5, 4, 7, 7, -3, 5, 2, 2], [1, 7, 0, -7, 8, 4, -11, -4], [14, 2, 0, -10, 12, 10, -4, -2], [-10, 2, 0, -4, -14, -6, 6, 8], [4, -8, 12, -4, -4, 4, -8, 0], [-14, -2, 0, 6, 20, 8, -10, -12], [7, -2, -6, -1, -1, 7, -8, 6], [-4, -6, -4, -2, -4, -4, 2, -8], [0, 8, 22, 10, -2, 0, 10, -2], [15, 4, 0, -5, -3, -1, -2, 2], [-16, -2, 0, -4, -6, 0, 8, 6], [2, 0, 4, 6, -6, 2, 4, -4], [4, 7, 24, 6, 1, 4, 2, 5], [7, 3, -19, 4, -1, 7, -3, 6], [-19, 0, 0, 3, 5, 1, -4, -4], [-3, -8, -7, -4, -4, -3, -1, -7], [20, 4, 0, -4, 0, 0, -4, 0], [-12, 0, 4, -4, 4, -12, 8, -8], [0, 6, -4, 4, 2, 0, 4, 2], [-4, -4, 0, 2, 2, 2, 4, 0], [-24, -5, 0, 4, 5, 3, 3, -2], [-12, 0, 0, 8, -8, -8, 0, 0], [-3, 6, 0, -8, 6, 4, -8, -2], [9, 2, 0, -3, -19, -9, 8, 10], [15, 3, 0, -8, -5, 0, 2, 5], [-6, 6, 0, -2, 0, -2, -8, -2], [8, 5, 0, -2, -9, -6, -2, 3], [-9, 6, 23, 3, 3, -9, 12, -6], [6, -4, -22, 8, -12, 6, 2, -6], [-3, -4, -13, 4, -8, -3, 7, -11], [20, 3, 0, -4, -21, -10, 8, 11], [-8, 2, 24, -2, 4, -8, 6, -4], [-3, -6, 0, -5, -1, 5, 12, 6], [-12, -3, 0, 4, -13, -7, 9, 6], [-4, 2, -24, -8, 10, -4, -4, 6], [-6, 0, 12, -10, 10, -6, -4, 4], [2, -8, 8, 2, -10, 2, 0, -8], [4, 1, 0, -12, 7, 9, 1, 2], [-26, 10, 0, -18, -4, 2, -4, 6], [-3, 3, 0, -4, 19, 10, -12, -9], [-16, 0, 18, -6, 6, -16, 10, -10], [8, 0, 0, -4, 4, 4, 0, 0], [-10, -4, 0, 10, -2, -4, 2, -2], [9, -1, 0, 7, 18, 6, -11, -12], [8, 1, 0, -4, 5, 4, -2, -1], [6, 8, -3, -2, 10, 6, -8, 16], [-14, -8, 2, -4, -4, -14, 10, -18], [12, 5, 0, -14, -1, 4, 0, 5], [4, 4, 4, 8, -4, 4, 4, 0], [-2, -8, 6, 0, -8, -2, 2, -10], [0, -4, -38, -4, 0, 0, -4, 0], [12, -6, -6, 4, -10, 12, -8, 2], [4, -6, -24, 0, -6, 4, -4, -2], [-25, -7, 0, 7, 8, 4, 3, -4], [-20, -4, 0, 8, 20, 8, -8, -12], [13, 8, 0, -16, 16, 12, -12, -4], [14, 4, 0, -14, 6, 8, -2, 2], [-3, 6, 8, 5, 1, -3, 8, -2], [-9, 7, -7, 2, 5, -9, 11, -4], [22, 6, 0, -22, 16, 16, -6, 0], [-18, 8, 17, 2, 6, -18, 20, -12], [-23, 3, 0, 1, -4, -4, -3, 0], [-10, 6, 0, -18, 0, 6, 0, 6], [1, -2, -51, -3, 1, 1, -4, 2], [-8, 9, -4, -6, 15, -8, 2, 7], [15, -10, -7, 4, -14, 15, -11, 1], [9, -5, 11, -6, 1, 9, -15, 10], [-3, -12, -21, -4, -8, -3, -1, -11], [-14, -8, -4, -10, 2, -14, 4, -12], [-26, -7, 0, 10, 1, -1, 5, -2], [2, 0, 14, -2, 2, 2, -4, 4], [-4, 4, 34, -10, 14, -4, -6, 10], [6, 12, 34, 4, 8, 6, -2, 14], [-6, -1, 0, -6, -13, -3, 11, 10], [-9, -5, 0, -8, 19, 16, 2, -3], [-15, 10, 0, -20, -10, 0, 0, 10], [-7, -4, 0, 3, 9, 5, 0, -4], [-8, -4, -1, -14, 10, -8, -6, 2], [-6, -11, 0, 22, -5, -8, 8, -3], [9, 0, 17, 11, -11, 9, 2, -2], [-12, 7, 4, -10, 17, -12, 2, 5], [8, 3, 0, -6, 5, 4, -4, -1], [0, -4, -18, 0, -4, 0, 0, -4], [-2, -8, 0, 6, 6, 4, 6, -2], [-4, -12, 0, 20, -8, -8, 12, 0], [2, 2, 6, -2, 4, 2, -4, 6], [8, -9, -20, -4, -5, 8, -12, 3], [8, 9, -6, 4, 5, 8, -4, 13], [-16, 6, 0, -4, -18, -10, 2, 8], [2, 12, 0, -4, -20, -14, -6, 6], [10, -16, 0, 14, 14, 8, 10, -6], [8, 0, 4, 8, -8, 8, 0, 0], [-19, -2, 0, -7, -15, -3, 14, 12], [-27, -1, 0, 7, 10, 2, -7, -8], [6, 5, 0, 4, -19, -14, 0, 5], [-2, 0, 0, -2, -14, -6, 8, 8], [-3, -4, 29, -5, 1, -3, -2, -2], [-2, 12, 23, 12, 0, -2, 14, -2], [2, -10, -14, -12, 2, 2, -14, 4], [5, 16, 41, 16, 0, 5, 11, 5], [16, -14, 0, 16, 18, 8, 4, -10], [-2, -16, 0, 34, 18, 0, -2, -18], [-10, -16, 0, 22, -10, -8, 18, 2], [5, -2, -13, 12, -14, 5, 7, -9], [-12, 5, 0, -16, -19, -4, 10, 15], [0, 5, 38, 10, -5, 0, 10, -5], [17, 0, 0, 1, -13, -7, 6, 6], [-1, -7, 9, 0, -7, -1, 1, -8], [7, 17, 0, -11, -14, -10, -13, 4], [34, 6, 0, -12, -14, -4, 4, 10], [2, -10, -22, -4, -6, 2, -6, -4], [-12, 13, 0, -20, -11, -2, -4, 9], [13, 13, -13, 6, 7, 13, -7, 20], [-3, 4, 24, -9, 13, -3, -6, 10], [4, 10, 0, -8, -10, -6, -6, 4], [1, 10, 27, -4, 14, 1, -5, 15], [9, -4, -20, -7, 3, 9, -16, 12], [24, 9, 0, 2, -19, -15, -5, 4], [-6, 4, -16, -2, 6, -6, 4, 0], [-3, 15, 0, -23, 8, 8, -15, 0], [-10, 8, 0, 4, -12, -12, -8, 0], [24, -6, 0, 4, 14, 8, 0, -6], [4, 0, -20, 16, -16, 4, 12, -12], [2, 4, 0, -18, 2, 8, 2, 6], [7, 12, 0, -29, 1, 9, -4, 8], [8, -14, -22, -4, -10, 8, -12, -2], [25, 10, 0, -13, -21, -9, 2, 12], [10, -9, 0, 8, 11, 6, 4, -5], [-7, 18, -9, 1, 17, -7, 8, 10], [-1, -6, -17, -16, 10, -1, -15, 9], [-9, 1, -3, 6, -5, -9, 15, -14], [0, 0, -10, 4, -4, 0, 4, -4], [42, 0, 0, 2, -6, -4, 2, 2], [13, 2, 3, 12, -10, 13, -1, 3], [13, 16, 32, 9, 7, 13, -4, 20], [-7, -2, 5, 9, -11, -7, 16, -18], [8, 4, 34, 2, 2, 8, -6, 10], [10, 8, 0, -12, 24, 14, -18, -10], [16, -7, 0, 4, -19, -8, 18, 11], [2, 4, 56, 6, -2, 2, 4, 0], [21, 8, 0, -14, -22, -8, 6, 14], [-19, 18, 0, -15, -19, -11, -10, 8], [-10, 0, -12, -6, 6, -10, 4, -4], [-5, -3, -11, -18, 15, -5, -13, 10], [-16, 2, 0, -8, 10, 8, -4, -2], [-5, 2, 0, 7, 11, 1, -12, -10], [-4, 8, 30, -4, 12, -4, 0, 8], [3, 0, -9, 20, -20, 3, 17, -17], [16, -17, 0, 22, 21, 8, 4, -13], [-12, 10, 4, 6, 4, -12, 18, -8], [12, 5, 0, -6, -15, -7, 3, 8], [14, 12, 14, 18, -6, 14, 4, 8], [2, 8, 36, -6, 14, 2, -8, 16], [12, -12, -30, -10, -2, 12, -22, 10], [18, 10, 0, -34, 4, 14, 0, 10], [-32, 0, 0, 4, 28, 12, -16, -16], [4, -8, 0, 12, 16, 6, -2, -10], [-11, 3, 11, -8, 11, -11, 3, 0], [24, 6, 0, -12, 2, 4, -4, 2], [6, 16, 6, 12, 4, 6, 6, 10], [-1, -9, 0, 19, -10, -10, 9, 0], [10, 8, 13, 8, 0, 10, -2, 10], [-3, -6, -11, -12, 6, -3, -9, 3], [28, -2, 0, -2, 16, 10, -4, -6], [11, 5, -3, 12, -7, 11, 1, 4], [-9, 10, 0, -4, 6, 0, -16, -6], [-4, -14, -32, -12, -2, -4, -8, -6], [12, -5, 2, -2, -3, 12, -14, 9], [-10, 18, 0, -34, -16, 0, -2, 16], [4, 3, 44, 12, -9, 4, 8, -5], [9, 9, 15, 0, 9, 9, -9, 18], [27, 11, 0, -28, -13, 2, 4, 15], [-18, 16, 0, -10, -26, -16, -6, 10], [-8, 4, -49, 0, 4, -8, 8, -4], [7, -20, 0, 19, 1, 1, 20, 0], [44, -4, 0, 10, -6, -6, 4, 0], [22, 9, 0, -20, -11, 0, 2, 11], [16, 12, 6, 10, 2, 16, -6, 18], [6, -14, 0, 26, 8, -2, 4, -10], [-8, -10, -16, 0, -10, -8, 8, -18], [-6, 2, -56, 6, -4, -6, 12, -10], [4, 9, 0, -12, 15, 9, -15, -6], [-18, 5, -10, -2, 7, -18, 16, -11], [-9, -9, 0, 11, -2, -2, 9, 0], [-33, -6, 0, 23, 3, -7, -4, -10], [11, 8, 16, -5, 13, 11, -16, 24], [12, -12, 0, 20, 8, 0, 4, -8], [12, -4, -10, 12, -16, 12, 0, -4], [6, -12, 0, 22, 6, -2, 4, -8], [-27, 1, 0, 3, -12, -8, 3, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1800_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1800_2_d_s();".
// 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_1800_2_d_s(:prec:=8)
    chi := MakeCharacter_1800_d();
    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_1800_2_d_s();".
// 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_1800_2_d_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1800_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 0, 156, 0, 134, 0, 28, 0, 1]>,<11,R![1600, 0, 1344, 0, 336, 0, 32, 0, 1]>,<13,R![9, -32, -22, 0, 1]>,<37,R![64, -128, -64, 0, 1]>,<41,R![328, -56, -64, 0, 1]>,<53,R![-152, 680, -120, -4, 1]>],Snew);
    return Vf;
end function;