
// Make newform 2736.2.bm.r in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_bm();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_bm_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_bm_r();".
// 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_bm_r();".
// 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 := [19, 116, 300, 114, -5, -30, 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], [154035, -278981, 80360, 50938, -15007, 6647, -2059, 44], [-308070, 250249, 146993, -101876, 30014, -13294, 4118, -88], [-1214655, 131509, 373886, 251431, -114745, 21212, -15922, 3500], [-420924, -309125, -315175, -2846, 111605, -10351, 8669, -3772], [414680, 1282798, 830669, -28670, -142319, -7253, -10484, 5106], [-78329, -273328, -124940, 254, 29375, -178, 2264, -1002], [-2690259, -9255799, -3586874, 99677, 958120, -5933, 65584, -32081]];
    Rf_basisdens := [1, 307713, 307713, 307713, 307713, 307713, 43959, 307713];
    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_bm();"
function MakeCharacter_2736_bm()
    N := 2736;
    order := 6;
    char_gens := [1711, 2053, 1217, 1009];
    v := [3, 6, 6, 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_bm_Hecke();"
function MakeCharacter_2736_bm_Hecke(Kf)
    N := 2736;
    order := 6;
    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, 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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 1, -1], [-1, 0, 1, 0, -1, 0, -2, 0], [-1, 0, 0, 0, 0, 1, -1, 0], [0, -1, -1, 0, 0, 1, 1, 0], [-1, 0, -2, 1, 1, 0, 1, -1], [0, -1, -1, 0, 1, 1, 1, -1], [-2, -1, -1, 2, 0, 1, 1, -1], [2, 2, 0, 0, 0, -2, 0, 0], [-4, -2, -1, 1, -1, 1, 1, 0], [-1, 0, 2, 0, -2, 0, -2, 0], [1, 2, 0, -1, 1, 0, -1, -1], [-2, 1, 0, -1, -1, 0, 1, -1], [-1, 0, -1, -2, 0, 0, -1, 1], [1, 0, 1, 0, 0, 0, 0, 0], [-2, -1, 0, -2, 0, 2, -3, 2], [-2, 1, 1, 0, -2, 1, -1, 0], [-1, 0, 1, 0, -2, 0, -5, 4], [1, 0, 2, 1, -1, 0, 1, -1], [-1, 0, 0, -2, 0, 0, -3, 2], [2, -1, -2, 3, 1, 2, 7, -3], [4, 0, -1, -1, 1, 2, 8, 2], [-10, 0, 2, 2, 0, 0, -5, -1], [5, -1, 0, 0, -1, 0, -5, 0], [-1, -2, 3, 0, -6, -2, -5, 3], [-3, 0, 1, 0, 1, 0, 0, 0], [0, -4, -2, 4, -2, 2, 2, 0], [-1, 1, 0, 0, -1, 0, -1, 0], [6, 0, -1, 0, 1, 0, 12, 0], [0, 0, 0, 0, 0, 0, 2, 0], [-3, 0, 0, 1, -3, 0, -1, 1], [0, 0, 0, 0, 0, 0, -6, 0], [12, -1, -1, -2, 0, 1, 6, 1], [1, -2, -2, -2, 1, 4, 2, 2], [-8, 0, 0, 2, 0, 0, 0, 0], [-9, 0, -4, 2, 2, 0, -5, -2], [3, 0, 1, 0, -1, -2, 4, 0], [0, 1, -1, 0, 2, 1, 19, -3], [0, -4, 0, 0, 0, 0, 4, 0], [-3, -2, 2, 2, 2, 1, 1, 0], [0, 1, -1, 4, 0, -1, 2, -2], [-1, 0, -5, 1, 5, 1, 1, -2], [-2, 3, 0, -4, 3, 0, 6, -4], [6, 0, 0, -1, 0, 0, 0, 0], [0, 3, -1, 6, 0, -3, 2, -3], [-4, -3, 2, 3, -1, 6, 1, -3], [-10, 1, 2, -1, -1, -2, -13, 1], [9, -2, -1, -3, -1, 1, 1, 0], [8, 2, 1, -4, 1, -1, -1, 0], [7, 0, 2, -3, -1, 0, 3, 3], [-3, 0, 2, 0, -2, -3, -9, 0], [12, 4, -2, 0, 0, -4, 5, 0], [-8, 0, 4, 0, 0, 0, -6, 0], [-1, -2, 1, 0, 0, 2, 0, 0], [-1, 2, 0, 1, -1, 0, -3, 1], [-5, -2, 0, 0, 1, 0, 8, 0], [-6, 4, 0, -2, 0, 0, 4, -2], [1, 6, 1, -6, 1, -3, -3, 0], [2, 0, 2, 6, 0, 0, 3, -3], [-4, 0, 0, 0, -4, 0, 0, 0], [-2, 0, 2, -2, -2, -4, -12, 4], [4, 1, 0, 1, 0, -2, 4, -1], [9, 0, -1, -1, 1, -3, 13, 2], [-2, 1, 1, 0, -2, 1, -12, 8], [5, 4, 1, -8, 0, -4, -4, 4], [-4, -6, -1, 1, -1, 3, 3, 0], [0, -5, 0, 2, -1, 0, 2, 2], [2, -3, 0, 0, -4, 0, -3, 0], [6, -6, -1, -6, -1, 3, 3, 0], [0, -4, 1, 0, 1, 2, 2, 0], [6, 4, 0, -4, 6, 0, 0, -4], [20, -3, -3, 2, 0, 3, 14, -1], [-3, 0, 1, 4, -1, 3, 5, -8], [-12, 6, 1, 1, 1, -3, -3, 0], [-13, -1, -10, 7, 5, 2, 0, -7], [0, 4, -4, 0, 8, 4, 5, -1], [2, -1, -6, -2, 3, 2, 4, 2], [4, 4, 0, 1, -2, 0, -11, 1], [12, 1, -1, 0, 0, -1, 6, 0], [-3, 0, -5, 1, 5, -3, -7, -2], [-7, -7, 0, 2, -7, 0, 5, 2], [3, 0, -3, 0, 6, 0, -9, -3], [-10, -1, 1, 8, 0, 1, -1, -4], [2, 2, -2, -4, 1, -4, -3, 4], [-22, 2, 0, -4, 0, -2, -14, 2], [-4, 0, -4, 4, 4, 4, 4, -8], [-22, -6, -1, 4, -1, 3, 3, 0], [9, 2, -2, 4, 1, -4, 12, -4], [-5, 0, 1, -2, -1, -2, -16, 4], [-2, 0, 7, -3, -7, -4, -14, 6], [7, 6, 1, 2, 0, -6, 1, -1], [-14, 8, 2, 2, 2, -4, -4, 0], [-3, 2, 0, -5, -3, 0, 3, -5], [-4, -6, 0, 4, -3, 0, 3, 4], [0, 4, -4, 4, 0, -4, 2, -2], [-20, 2, -10, 8, 0, -2, -2, -4], [-6, 0, 5, -2, -5, 0, -16, 4], [3, -2, -1, 0, 2, -2, 5, -8], [-4, 0, 4, 0, -8, 0, -9, 4], [2, -1, -1, 0, 2, -1, -4, -5], [0, 2, -2, 0, 4, 2, 8, -6], [24, -4, 4, 0, 4, 2, 2, 0], [12, 0, -1, 3, 1, 0, 30, -6], [17, 0, 1, -2, -1, 2, 32, 4], [11, 6, -1, -2, -1, -3, -3, 0], [3, -5, 0, 5, -3, 0, -6, 5], [-2, -2, 4, 0, -8, -2, -11, -3], [4, 1, -5, 0, 10, 1, 17, -1], [2, 0, -1, 0, 1, 3, 7, 0], [-14, -10, -1, -3, -1, 5, 5, 0], [-7, -1, -2, 0, 1, 2, -5, 0], [7, -4, -3, 0, 6, -4, 12, 2], [-7, 0, 3, 4, -3, 6, 0, -8], [-15, 0, -5, 12, 0, 0, 1, -6], [-9, -2, 0, 1, -3, 0, 7, 1], [10, -2, 0, 2, 3, 0, -7, 2], [-10, 0, 0, -4, 0, -2, -30, 8], [20, -8, -3, -3, -3, 4, 4, 0], [1, -3, 2, 0, -4, -3, -14, -4], [-8, 4, 0, 8, 0, -4, -2, -4], [-10, 0, -7, 2, 7, 3, -13, -4], [-2, 0, -2, 0, 2, 8, 4, 0], [27, 2, 1, -1, 1, -1, -1, 0], [-10, 0, 10, -4, -10, -8, -36, 8], [-26, 0, -4, 3, 2, 0, -21, -3], [6, -5, -1, 0, 2, -5, 5, 0], [-4, -1, 0, -2, 2, 0, 9, -2], [-20, -1, 0, 3, -3, 0, 15, 3], [-5, -6, -3, 4, -3, 3, 3, 0], [0, 2, 0, -4, 5, 0, 7, -4], [-3, 3, -6, 9, 3, -6, 6, -9], [-3, -2, 5, 0, -10, -2, -19, -6], [-8, 5, 10, -2, -5, -10, -20, 2], [-2, -12, -1, 0, -1, 6, 6, 0], [0, -2, 2, 0, -4, -2, -9, 8], [21, -8, -7, 2, 0, 8, 19, -1], [-3, -4, 3, -8, 3, 2, 2, 0], [4, 0, -3, 3, 3, 2, 16, -6], [-6, 4, 2, -5, 2, -2, -2, 0], [2, 2, -4, 0, 8, 2, 12, -6], [4, -4, 0, 0, 0, -4, -10, 6], [18, -2, 8, -4, 0, 2, 4, 2], [0, -2, 2, 0, -4, -2, -4, 8], [4, 0, -1, 4, 1, -5, 11, -8], [-13, 5, 2, 9, -1, -10, -10, -9], [-15, 2, -4, -2, 2, -4, -17, 2], [16, -8, 0, -4, -2, 0, -6, -4], [11, -4, -7, 4, 0, 4, 13, -2], [22, 8, 3, -7, 3, -4, -4, 0], [11, 8, 0, -2, 4, 0, -13, -2], [20, -8, 0, 3, 0, 4, 4, 0], [-30, 1, -5, -2, 0, -1, -14, 1], [-3, 5, -2, 0, 4, 5, 0, 2], [6, -2, 0, 0, 0, 4, 8, 0], [-10, -12, 1, 5, 1, 6, 6, 0], [-15, 0, -3, 4, 3, -2, -24, -8], [-18, -2, -4, 1, 2, 4, -13, -1], [4, 1, -5, 0, 10, 1, 11, 4], [32, -2, 10, -8, 0, 2, 8, 4], [-11, -10, 0, 1, 1, 0, 21, 1], [-4, 2, 0, -7, 2, 0, 11, -7], [-2, -11, 0, 3, -9, 0, 1, 3], [5, -2, -6, 5, 3, 4, 15, -5], [2, 0, -9, 5, 9, 4, 18, -10], [-6, -7, -8, 10, 4, 14, 15, -10], [22, 5, -2, -1, 1, -10, 17, 1], [2, 0, -2, 0, 4, 0, 25, -6], [6, 2, 0, -2, 0, 0, -6, -2], [-1, -2, 3, 0, -6, -2, 27, 1], [3, 14, 4, -4, 4, -7, -7, 0], [-14, 1, 0, 6, -3, 0, 4, 6], [18, -2, 0, 0, 0, 0, -16, 0], [-14, -10, -1, 10, -1, 5, 5, 0], [1, 2, -3, 0, 6, 2, 21, -2], [2, 2, -4, 0, 8, 2, 8, -6], [34, 1, -1, -2, 0, -1, 16, 1], [-4, 4, -12, -4, 6, -8, -6, 4], [-5, 8, 5, -4, 5, -4, -4, 0], [15, 8, 8, -8, -4, -16, -5, 8], [13, 5, 8, -14, 0, -5, -7, 7], [12, -8, -7, 7, -7, 4, 4, 0], [8, 0, 6, 8, 0, 0, 5, -4], [4, 0, 0, -1, 4, 0, 1, -1], [0, 0, 12, -4, -12, -6, -14, 8], [26, -8, -6, 2, -6, 4, 4, 0], [0, -9, -3, 0, 0, 9, 6, 0], [12, 0, -2, 0, 1, 0, 13, 0], [-5, 0, 11, -2, -11, -7, -21, 4], [3, 0, 2, -8, -2, -3, -13, 16], [-14, 0, -4, 16, 0, 0, 3, -8], [-23, 8, 5, 2, 0, -8, -17, -1], [-2, 14, 7, -7, 7, -7, -7, 0], [-5, 6, 0, 1, 13, 0, 11, 1], [27, -7, -2, -8, 0, 7, 14, 4], [-17, 4, 3, -2, 0, -4, -13, 1], [-2, -1, 3, 0, -6, -1, -26, 6], [3, -4, -12, 2, 6, 8, 15, -2], [8, 16, 6, -6, 6, -8, -8, 0], [-16, 0, 0, 0, 0, 10, -22, 0], [-12, -4, 4, 6, 0, 4, -3, -3], [5, -1, -4, 0, 8, -1, -3, 7], [0, -5, 0, 0, 1, 0, 6, 0], [8, -7, 0, 2, -3, 0, -6, 2], [-4, 7, 0, 2, 14, 0, 9, 2], [-7, 2, 5, 0, -10, 2, -13, -3], [0, 2, -2, 4, 0, -2, 2, -2], [3, -2, -1, 0, 2, -2, 7, -9], [-36, 2, -8, 6, 4, -4, -28, -6], [-25, 0, -7, 12, -7, 0, 0, 0], [20, -3, -6, 2, 3, 6, 28, -2], [1, 2, -3, 0, 6, 2, 25, -11], [-16, 0, 3, -3, -3, 9, -29, 6], [-8, 3, 5, 0, -10, 3, 9, 3], [-17, 2, -6, 12, -6, -1, -1, 0], [10, 6, -6, -4, 0, -6, 3, 2], [-4, -16, 0, 6, -4, 0, 10, 6], [6, 0, 11, -4, -11, -6, -2, 8], [-6, -4, -2, -4, -2, 2, 2, 0], [-5, 2, -6, -4, 3, -4, -8, 4], [-12, 3, -13, -4, 0, -3, -3, 2], [3, 0, -5, -6, -5, 0, 0, 0], [-23, 0, 0, 0, 1, 0, 24, 0], [0, 2, 0, 2, 1, 0, -3, 2], [15, -2, 2, -11, -1, 4, 5, 11], [4, -5, 1, 0, -2, -5, -5, 4], [-2, 0, 0, -4, 0, -4, -16, 8], [13, -5, -14, 3, 7, 10, 28, -3], [8, -2, 16, 0, -8, 4, 2, 0], [0, 0, -5, -1, 5, 6, 4, 2], [2, 2, -4, 0, 8, 2, 37, -12], [-1, 2, -1, 0, 2, 2, 11, -8], [-11, -10, 2, 5, 2, 5, 5, 0], [14, 0, -16, 4, 16, 10, 46, -8], [10, 0, -2, -2, 2, 2, 18, 4], [2, -9, 0, 10, -10, 0, -13, 10], [16, -6, 3, 4, 3, 3, 3, 0], [20, -8, 6, -6, 6, 4, 4, 0], [-1, 2, 0, 5, -7, 0, -13, 5], [-2, 3, -1, 0, 2, 3, 33, -15], [7, 1, 10, -1, -5, -2, 0, 1], [4, 0, -6, 8, 6, 2, 26, -16], [-2, -9, -3, -6, 0, 9, 2, 3], [-28, 6, -12, 8, 0, -6, -7, -4], [-18, -4, 0, 3, 0, 0, 19, 3], [-18, 0, 9, 3, -9, 12, -18, -6], [18, 4, 2, 0, 0, -4, 6, 0], [-7, 9, -2, 0, 4, 9, -26, -4], [-42, -2, -2, -2, 1, 4, -41, 2], [-4, -3, 11, 6, 0, 3, -3, -3], [-15, -4, 6, -2, 6, 2, 2, 0], [-4, 4, 0, -9, 0, -8, -17, 9], [20, 0, 7, -3, -7, -2, 32, 6], [20, -1, 9, -4, 0, 1, 4, 2], [-17, 0, 2, -20, -1, 0, -38, 20], [-14, 8, 2, 4, 0, -8, -10, -2], [-6, 5, 1, 0, -2, 5, 19, -4], [7, -4, 7, 6, 7, 2, 2, 0], [-16, -8, 0, 4, -4, 0, 16, 4], [-2, -7, 9, 0, -18, -7, -19, 10], [-5, 2, 3, 0, -6, 2, -41, 2], [-17, -5, -2, 3, 1, 10, -8, -3], [-20, -1, 2, 8, -1, 2, -12, -8], [9, 2, 0, -4, -8, 0, -15, -4], [-3, -4, 7, 0, -14, -4, -10, 0], [9, 0, -7, -4, -7, 0, 0, 0], [10, 0, 9, -5, -9, 2, 12, 10], [2, 0, 1, 9, 1, 0, 0, 0], [3, 0, -3, 0, 6, 0, 5, 8], [1, -2, 1, 0, -2, -2, -41, -3], [-2, 0, -11, 4, 11, 1, 5, -8], [-11, -2, -6, 9, -6, 1, 1, 0], [-6, 6, -8, 14, 4, -12, 6, -14], [24, 0, 0, 4, 0, -12, 44, -8], [24, -7, -7, -8, 0, 7, 15, 4], [-10, 15, 0, -1, 9, 0, 5, -1], [-12, 0, 2, 4, -2, -6, -22, -8], [-6, -7, 0, 12, -13, 0, -12, 12], [16, 14, 4, -16, 0, -14, -9, 8], [-29, -6, 1, 6, 0, 6, -9, -3], [-25, 10, 6, -16, 6, -5, -5, 0], [22, 0, 4, 0, -4, -4, 40, 0], [26, -7, -15, -6, 0, 7, 21, 3], [-5, 0, -6, 0, 6, 12, 2, 0], [6, 7, 0, -3, -6, 0, -16, -3], [-8, 6, 10, -16, 0, -6, -20, 8], [29, -4, -2, 8, -2, 2, 2, 0], [-47, 2, -1, -12, 0, -2, -30, 6], [-2, 3, 0, 5, 3, 0, -3, 5], [-8, 0, -5, 5, 5, 2, -4, -10], [-8, -16, -3, -3, -3, 8, 8, 0], [3, 0, 13, -9, -13, -3, -15, 18], [8, 8, -12, 0, 0, -8, 6, 0], [-30, 0, 3, -5, 3, 0, 0, 0], [-7, 2, 0, 2, -12, 0, -9, 2], [12, -16, -5, 16, -5, 8, 8, 0], [4, 2, -6, 0, 12, 2, 10, -4], [-2, -2, 16, 3, -8, 4, -5, -3], [6, -1, -8, 10, 4, 2, 21, -10], [9, 0, 19, -8, -19, -20, -18, 16], [-11, -1, -22, 6, 11, 2, 7, -6], [-4, -18, 0, 10, -22, 0, -10, 10], [-23, 4, -5, 0, 0, -4, -11, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_bm_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_2_bm_r();".
// 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_bm_r(:prec:=8)
    chi := MakeCharacter_2736_bm();
    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_bm_r();".
// 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_bm_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_bm();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![324, -648, 1044, -576, 286, -44, 18, -2, 1]>,<7,R![9, 0, 1056, 0, 418, 0, 40, 0, 1]>,<11,R![2916, 0, 2376, 0, 520, 0, 40, 0, 1]>,<23,R![12996, 2736, -4140, -912, 1378, -228, -26, 6, 1]>,<31,R![-45, -282, 10, 14, 1]>],Snew);
    return Vf;
end function;