
// Make newform 2736.2.s.bc in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_s_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_2736_2_s_bc();".
// 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_2736_2_s_bc();".
// 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 := [9, 12, 28, -4, 21, 0, 8, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [3206, -132, -272, -4652, -152, -84, -199, -16], [11973, 1254, 2584, 1764, 1444, 798, -231, 152], [-5760, 16828, -12640, 13440, -1323, 6080, -1760, 754], [11088, -56579, 24332, -25872, 15594, -11704, 3388, -815], [-9252, 4436, -20303, 21588, -6978, 9766, -2827, 1052], [-13760, -4587, -9452, -8909, -5282, -2919, 510, -556], [-5004, 15574, -10981, 11676, -2767, 5282, -1529, 602]];
    Rf_basisdens := [1, 4243, 4243, 12729, 12729, 12729, 4243, 4243];
    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_2736_s();"
function MakeCharacter_2736_s()
    N := 2736;
    order := 3;
    char_gens := [1711, 2053, 1217, 1009];
    v := [3, 3, 3, 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_2736_s_Hecke();"
function MakeCharacter_2736_s_Hecke(Kf)
    N := 2736;
    order := 3;
    char_gens := [1711, 2053, 1217, 1009];
    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], [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, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0], [-1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1], [0, 0, 0, 0, 2, 0, 2, 0], [-1, 0, 0, -1, 0, 1, -1, -1], [0, 0, 0, 1, 2, -1, 0, 2], [0, 0, 0, 0, 0, 0, 0, -2], [0, -1, 1, 0, 0, 0, 1, 0], [1, -2, 2, 0, 0, 0, 2, 0], [0, 0, -2, 0, 2, 0, 2, -2], [0, -1, 1, 0, 3, -1, 3, 1], [0, 0, 0, 2, -2, -2, 0, 0], [0, 0, 0, -5, 2, 3, 0, 0], [-3, -1, 2, -3, 0, -1, 0, 2], [0, 0, 0, 2, 0, 3, 0, -1], [0, 0, 0, 1, -1, -2, 0, -2], [2, 2, 2, 2, -2, 2, -2, 2], [-1, -4, 0, -1, 0, -4, 0, 0], [6, 3, 1, 6, -1, 3, -1, 1], [-2, -2, -2, 0, 0, 0, 2, 0], [0, 0, 0, 5, -4, 1, 0, -2], [7, 1, 1, 7, -2, 1, -2, 1], [0, 0, 0, 4, -2, 4, 0, 0], [-3, -4, 0, 0, 0, 0, -1, 0], [4, 0, -2, 0, 0, 0, -4, 0], [5, -1, -1, 5, 2, -1, 2, -1], [3, -1, 0, 0, 0, 0, 2, 0], [0, 0, 0, 6, 0, 6, 0, -2], [6, -2, 2, 6, -2, -2, -2, 2], [0, 0, 0, -4, 4, -4, 0, 2], [0, 0, 0, -4, 3, -1, 0, 1], [-9, -1, -2, -9, -2, -1, -2, -2], [0, 4, 4, 0, 0, 0, 0, 0], [1, 0, 0, 1, 6, 0, 6, 0], [-1, 0, 4, 0, 0, 0, -1, 0], [0, 0, 0, 13, -2, 3, 0, -2], [4, -4, 2, 4, 0, -4, 0, 2], [-7, 3, 2, 0, 0, 0, 4, 0], [0, 0, 0, -3, -2, 3, 0, -5], [13, -1, 0, 0, 0, 0, 2, 0], [0, -5, 3, 0, -2, -5, -2, 3], [-9, -1, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, -1, 0, 1], [6, -5, 1, 6, 1, -5, 1, 1], [6, -5, -3, 6, -1, -5, -1, -3], [7, 1, 0, 0, 0, 0, 6, 0], [12, 3, -1, 0, 0, 0, -4, 0], [12, -4, 0, 12, -6, -4, -6, 0], [5, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, -6, 0, 2], [0, 0, 0, 4, 4, -4, 0, -4], [0, 0, 0, 9, 2, -3, 0, 4], [10, 2, -2, 10, 6, 2, 6, -2], [1, 1, 4, 1, 2, 1, 2, 4], [-6, -2, -6, -6, 0, -2, 0, -6], [-3, -1, 3, 0, 0, 0, 2, 0], [0, 0, 0, -13, 0, -1, 0, 0], [2, 6, -2, 2, -8, 6, -8, -2], [16, 8, -4, 0, 0, 0, -4, 0], [15, 7, 1, 15, -2, 7, -2, 1], [1, -1, 6, 0, 0, 0, -2, 0], [0, 0, 0, 9, -6, -1, 0, 3], [0, 0, 0, 15, 2, -1, 0, 6], [-14, -1, 1, 0, 0, 0, 5, 0], [6, -1, -1, 6, -10, -1, -10, -1], [-11, 3, 4, -11, 4, 3, 4, 4], [-2, -5, -1, 0, 0, 0, 4, 0], [-9, 3, 6, 0, 0, 0, 2, 0], [4, 4, -4, 4, -4, 4, -4, -4], [0, 0, 0, -6, -1, 3, 0, 1], [-1, 1, 1, 0, 0, 0, 10, 0], [-2, -5, -7, 0, 0, 0, -3, 0], [-7, -5, 0, -7, 2, -5, 2, 0], [0, 0, 0, -7, 4, -7, 0, 2], [4, 3, -5, 4, 0, 3, 0, -5], [-3, 1, -6, -3, -4, 1, -4, -6], [0, 0, 0, 1, -6, -9, 0, 7], [21, -5, 0, 0, 0, 0, -2, 0], [11, -3, -3, 11, -6, -3, -6, -3], [0, 0, 0, 6, -2, 10, 0, 4], [0, 0, 0, -4, 6, -3, 0, 5], [7, 2, 2, 7, -9, 2, -9, 2], [0, 0, 0, 4, -8, 8, 0, -2], [4, -4, -2, 0, 0, 0, -4, 0], [4, 3, -1, 0, 0, 0, 6, 0], [13, -3, 0, 13, 6, -3, 6, 0], [-11, 10, -2, 0, 0, 0, 5, 0], [14, 2, 0, 0, 0, 0, 6, 0], [0, 0, 0, 14, -10, -2, 0, -2], [4, -4, 0, 0, 0, 0, 8, 0], [14, -10, 6, 14, -6, -10, -6, 6], [1, 2, 6, 1, -1, 2, -1, 6], [0, 0, 0, 4, 0, -8, 0, -2], [0, 0, 0, -28, 0, 4, 0, 0], [29, 1, -2, 0, 0, 0, 2, 0], [0, 0, 0, -15, -3, 4, 0, 0], [0, 0, 0, -7, 2, -4, 0, -8], [0, 0, 0, -6, 1, -1, 0, 1], [0, 0, 0, 8, 4, 0, 0, -6], [0, 4, 2, 0, 0, 0, 0, 0], [4, -4, 6, 0, 0, 0, -2, 0], [5, 4, -8, 0, 0, 0, -3, 0], [5, 3, -1, 0, 0, 0, 2, 0], [-15, 3, -2, -15, 10, 3, 10, -2], [0, 0, 0, -17, -4, -5, 0, -2], [0, 0, 0, -11, 2, -1, 0, 4], [2, -1, -9, 0, 0, 0, -10, 0], [-2, 5, -1, 0, 0, 0, 1, 0], [3, 5, -7, 3, -6, 5, -6, -7], [0, 0, 0, -13, 2, 7, 0, -6], [-23, 4, -4, 0, 0, 0, 5, 0], [0, 0, 0, -5, -1, -2, 0, -2], [-8, 8, -6, -8, -2, 8, -2, -6], [-11, -2, 6, -11, -1, -2, -1, 6], [20, 8, -2, 0, 0, 0, 0, 0], [-8, -8, -4, 0, 0, 0, 6, 0], [0, 0, 0, -7, 0, -9, 0, 4], [0, 0, 0, -2, -8, 4, 0, 0], [-20, 5, -3, 0, 0, 0, -2, 0], [8, 0, 4, 0, 0, 0, -8, 0], [9, 15, 0, 0, 0, 0, -6, 0], [6, -2, -6, 0, 0, 0, -8, 0], [7, 7, -8, 7, 4, 7, 4, -8], [0, 0, 0, -4, 4, -13, 0, -1], [27, -3, -2, 27, 0, -3, 0, -2], [2, -5, 1, 2, 3, -5, 3, 1], [-15, 3, -9, 0, 0, 0, -6, 0], [-11, -12, 4, -11, 9, -12, 9, 4], [-1, -3, 8, -1, -6, -3, -6, 8], [0, 0, 0, -17, -5, 6, 0, 6], [-10, 7, -1, -10, 12, 7, 12, -1], [-23, 5, -4, 0, 0, 0, 6, 0], [0, 0, 0, 21, -4, -6, 0, 6], [0, 0, 0, -4, 10, 4, 0, -6], [-29, 2, 2, 0, 0, 0, -7, 0], [-4, 12, 0, 0, 0, 0, -2, 0], [-13, -1, 0, 0, 0, 0, 12, 0], [0, 0, 0, -2, -8, 2, 0, -6], [0, 0, 0, -40, 8, 8, 0, 2], [0, 0, 0, 22, 0, 2, 0, 6], [0, 0, 0, -24, -4, -4, 0, 2], [2, 1, 1, 0, 0, 0, 8, 0], [-25, 5, 0, -25, 10, 5, 10, 0], [7, 2, 2, 7, -2, 2, -2, 2], [24, -8, 0, 24, 0, -8, 0, 0], [0, 0, 0, -3, -13, 10, 0, 2], [-18, 10, 8, 0, 0, 0, 10, 0], [23, 14, -2, 23, -2, 14, -2, -2], [-15, 5, 8, 0, 0, 0, 16, 0], [0, 0, 0, 22, 1, 3, 0, -3], [0, 0, 0, 5, -12, 11, 0, -6], [0, 4, 0, 0, 8, 4, 8, 0], [-6, -10, 0, 0, 0, 0, -2, 0], [-3, -14, 6, 0, 0, 0, -7, 0], [7, -5, 4, 7, -4, -5, -4, 4], [0, 0, 0, -18, 2, 7, 0, -9], [0, 0, 0, 12, -8, 4, 0, -8], [14, 6, -6, 14, 2, 6, 2, -6], [7, -13, -8, 7, 4, -13, 4, -8], [16, -5, -3, 16, -11, -5, -11, -3], [-14, -10, 4, -14, 10, -10, 10, 4], [32, 0, 6, 0, 0, 0, -2, 0], [5, -9, -8, 5, -4, -9, -4, -8], [12, -5, -7, 12, 9, -5, 9, -7], [0, 0, 0, -5, 14, 4, 0, 4], [22, -8, -4, 22, -8, -8, -8, -4], [0, 0, 0, 8, -2, 0, 0, 10], [5, -1, -6, 0, 0, 0, -4, 0], [18, -3, 1, 18, -4, -3, -4, 1], [-20, -16, 2, -20, 0, -16, 0, 2], [18, 5, -7, 0, 0, 0, -14, 0], [0, 0, 0, 23, -13, -14, 0, -2], [0, 0, 0, 0, 0, -8, 0, 4], [0, 0, 0, -22, -3, 3, 0, -7], [-28, -4, -4, -28, 0, -4, 0, -4], [-5, -6, 10, 0, 0, 0, 7, 0], [15, 2, 10, 15, -6, 2, -6, 10], [0, 0, 0, -21, 6, 7, 0, -11], [-18, -18, 6, 0, 0, 0, 6, 0], [0, 0, 0, -37, 6, -2, 0, -2], [-9, -13, 6, -9, 4, -13, 4, 6], [24, 4, 2, 0, 0, 0, 0, 0], [-12, 4, 4, 0, 0, 0, -16, 0], [0, 0, 0, 1, 2, -5, 0, -9], [45, 0, 4, 45, 1, 0, 1, 4], [-1, 1, 5, 0, 0, 0, -18, 0], [11, 1, 4, 0, 0, 0, -12, 0], [0, 0, 0, 27, -8, 2, 0, -6], [0, 0, 0, 30, 2, 2, 0, 12], [10, -5, 13, 0, 0, 0, -7, 0], [8, 8, -2, 8, -18, 8, -18, -2], [0, 0, 0, 21, -8, 15, 0, -6], [0, 0, 0, -40, 2, -8, 0, 2], [0, 0, 0, -27, 18, -5, 0, 7], [-13, 0, 4, -13, -2, 0, -2, 4], [-40, 0, 6, 0, 0, 0, 4, 0], [36, -4, 0, 0, 0, 0, 8, 0], [0, 0, 0, 11, -8, 3, 0, 0], [0, 0, 0, 9, -6, 5, 0, -1], [8, -11, -3, 8, 16, -11, 16, -3], [4, -3, -3, 4, 6, -3, 6, -3], [-5, 17, -10, -5, 4, 17, 4, -10], [0, 0, 0, -8, 10, 8, 0, 10], [0, 0, 0, 8, 0, 0, 0, -4], [0, 0, 0, -30, 6, 6, 0, 0], [-16, 8, 0, -16, -12, 8, -12, 0], [-23, 8, 0, 0, 0, 0, 5, 0], [22, 1, -7, 22, 14, 1, 14, -7], [0, 0, 0, 0, -6, 0, 0, 0], [-30, -9, 9, 0, 0, 0, 3, 0], [0, 0, 0, -11, -2, -5, 0, -16], [-21, 5, -6, 0, 0, 0, 0, 0], [0, 0, 0, -27, 0, 10, 0, -2], [-36, 4, -8, -36, 0, 4, 0, -8], [-5, -1, 0, 0, 0, 0, 18, 0], [4, -8, -6, 0, 0, 0, 4, 0], [21, -3, -4, 21, -2, -3, -2, -4], [0, 0, 0, 42, 4, 1, 0, 1], [19, 0, 4, 0, 0, 0, -15, 0], [25, -11, 8, 25, 2, -11, 2, 8], [-11, -10, -14, -11, -3, -10, -3, -14], [-6, 14, -8, -6, 10, 14, 10, -8], [0, 0, 0, 0, -8, -11, 0, 1], [8, -8, -4, 0, 0, 0, 0, 0], [9, -9, 2, 9, 14, -9, 14, 2], [2, -2, 10, 2, 16, -2, 16, 10], [10, -2, 16, 0, 0, 0, 14, 0], [0, 0, 0, 21, -2, 4, 0, 0], [0, 0, 0, 13, -9, 2, 0, -2], [-1, -13, 1, 0, 0, 0, 6, 0], [8, 8, 6, 0, 0, 0, 8, 0], [16, -4, 0, 0, 0, 0, 8, 0], [-49, -3, -4, -49, 4, -3, 4, -4], [0, 7, 7, 0, 0, 0, -6, 0], [-4, 4, 8, 0, 0, 0, -4, 0], [10, -2, -4, 10, -18, -2, -18, -4], [0, 0, 0, -5, 2, 9, 0, -8], [1, 15, 0, 1, 6, 15, 6, 0], [-30, 10, -2, 0, 0, 0, 0, 0], [0, 0, 0, -28, 13, -13, 0, 9], [0, 0, 0, -19, 26, -2, 0, 10], [9, 5, 4, 9, -16, 5, -16, 4], [38, 6, 2, 0, 0, 0, -2, 0], [0, 0, 0, 4, -8, 16, 0, -16], [0, 0, 0, -25, 12, -19, 0, 10], [35, 8, -8, 35, -13, 8, -13, -8], [0, 0, 0, -11, 6, -17, 0, 8], [23, -10, -2, 0, 0, 0, -2, 0], [25, 5, 0, 25, 4, 5, 4, 0], [10, -6, -2, 0, 0, 0, 18, 0], [0, 0, 0, 25, 10, 7, 0, -1], [-1, 7, -2, -1, 10, 7, 10, -2], [0, 0, 0, 0, 16, 4, 0, -10], [0, 0, 0, 4, 4, 11, 0, 11], [-5, -6, 2, 0, 0, 0, 21, 0], [-8, 0, 12, -8, 20, 0, 20, 12], [0, 0, 0, -12, -6, 13, 0, -15], [0, 0, 0, -23, -5, 0, 0, -8], [23, -3, 6, 23, -14, -3, -14, 6], [-36, 5, 1, -36, -4, 5, -4, 1], [15, 6, 6, 15, 6, 6, 6, 6], [0, 0, 0, 11, -2, -9, 0, -8], [-55, -2, 6, 0, 0, 0, 7, 0], [12, 4, 12, 0, 0, 0, 10, 0], [0, 0, -14, 0, 0, 0, -10, 0], [0, 0, 0, 9, 5, 0, 0, 8], [0, 0, 0, 8, -2, 0, 0, -10], [-12, 9, -3, 0, 0, 0, -4, 0], [-11, -11, 3, 0, 0, 0, 2, 0], [-6, 8, -8, -6, -16, 8, -16, -8], [-28, -4, 8, 0, 0, 0, 8, 0], [0, 0, 0, -34, 2, -3, 0, -15], [-40, -11, 7, -40, 11, -11, 11, 7], [8, -16, -12, 0, 0, 0, 16, 0], [10, 5, 13, 10, -12, 5, -12, 13], [0, 0, 0, -3, -4, -2, 0, -18], [0, 0, 0, 10, 18, -2, 0, 12], [-31, -1, 10, 0, 0, 0, 12, 0], [-20, 16, 8, 0, 0, 0, 0, 0], [0, 0, 0, 35, -2, 1, 0, -4], [-41, -4, 4, 0, 0, 0, 2, 0], [5, -3, -13, 5, 2, -3, 2, -13], [0, 0, 0, 12, 0, -8, 0, 6], [-37, -2, 2, 0, 0, 0, -4, 0], [0, 0, 0, -31, 2, 9, 0, -10], [22, -3, 11, 22, -3, -3, -3, 11], [4, -4, -12, 0, 0, 0, -14, 0], [4, 12, -6, 0, 0, 0, 10, 0], [15, 17, -8, 0, 0, 0, -18, 0], [0, 0, 0, 20, 8, 4, 0, 10], [14, -14, 8, 0, 0, 0, 2, 0], [9, -20, -4, 9, 2, -20, 2, -4], [-7, -7, 16, 0, 0, 0, 6, 0], [0, 0, 0, 42, -8, 2, 0, -6], [17, 9, 6, 17, 4, 9, 4, 6], [-9, -7, 4, -9, 8, -7, 8, 4], [-35, 6, 6, 0, 0, 0, -1, 0], [5, -17, 3, 5, -2, -17, -2, 3], [28, 4, -8, 28, -24, 4, -24, -8], [0, 0, 0, -41, -19, 4, 0, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_2_s_bc();".
// 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_2736_2_s_bc(:prec:=8)
    chi := MakeCharacter_2736_s();
    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_2736_2_s_bc();".
// 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_2736_2_s_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![144, -240, 352, -176, 108, -24, 20, -4, 1]>,<7,R![3, 2, -8, -2, 1]>,<11,R![-4, -12, -4, 4, 1]>,<13,R![32761, 1448, 5494, -240, 719, -16, 30, 0, 1]>],Snew);
    return Vf;
end function;