
// Make newform 729.2.e.k in Magma, downloaded from the LMFDB on 21 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_729_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_729_e_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_729_2_e_k();".
// 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_729_2_e_k();".
// 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 := [1, 0, 132, 0, 306, 0, 266, 0, 105, 0, 18, 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, 0, 0, 0], [8, 31, 63, -21, 58, -106, 15, -72, 1, -16, 0, -1], [1, 16, 163, 63, 285, 58, 160, 15, 33, 1, 2, 0], [11, 126, -159, 301, -285, 204, -160, 47, -33, 3, -2, 0], [-3, 0, 222, 0, 343, 0, 175, 0, 34, 0, 2, 0], [3, 59, -411, 58, -702, 15, -394, 1, -82, 0, -5, 0], [3, -59, -411, -58, -702, -15, -394, -1, -82, 0, -5, 0], [5, -90, 474, -37, 760, 91, 409, 71, 83, 16, 5, 1], [7, 78, 356, -158, 583, -389, 321, -232, 66, -49, 4, -3], [-2, -42, 0, -335, 0, -477, 0, -249, 0, -50, 0, -3], [13, -22, 549, 495, 881, 866, 482, 481, 99, 99, 6, 6], [11, -78, 356, 158, 583, 389, 321, 232, 66, 49, 4, 3]];
    Rf_basisdens := [1, 8, 8, 8, 4, 8, 8, 8, 4, 4, 8, 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_729_e();"
function MakeCharacter_729_e()
    N := 729;
    order := 9;
    char_gens := [2];
    v := [4];
    // 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_729_e_Hecke();"
function MakeCharacter_729_e_Hecke( : Kf := false)
    N := 729;
    order := 9;
    char_gens := [2];
    char_values := [[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, -1, 0, -1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, -1, 1, 1, 0, 0, 1, -1, 1], [1, 0, -1, -1, -1, 0, 3, 2, 1, 0, 0, -1], [2, -2, 0, 1, 1, -2, 1, -2, 1, 0, 0, 0], [0, -1, -1, 1, 0, -1, 0, -1, 1, 2, -1, 1], [1, -2, -1, -1, -1, 2, 1, 1, 1, 2, -1, 1], [2, -1, -1, 0, 0, -1, 2, 2, 2, 2, 1, -2], [-1, -2, 0, 1, 1, -2, -1, -1, -1, 2, 0, 1], [-1, -3, 2, -2, -1, 3, -1, 2, -1, -2, -2, 0], [0, -2, 0, 1, 0, 1, 2, -1, 2, 0, 1, 0], [0, 5, 2, 0, 0, -4, -1, -1, -1, -2, 1, -2], [3, -1, 1, 2, 1, 3, 1, 1, -1, 2, -2, 1], [6, 0, 0, 2, 2, -6, 0, -2, 2, 2, 2, -2], [3, 0, 2, 2, 2, 0, -6, 1, -2, 0, 0, 1], [-1, -2, -1, 0, 0, -3, 1, -2, -1, 0, 1, -2], [-1, 0, -3, -1, -1, 1, 4, 3, 0, 4, -1, 1], [1, 0, 1, 0, 1, 3, 2, 0, 0, 3, 0, 2], [-4, -5, -1, 0, -2, -2, 3, -3, 1, -4, 0, 2], [2, -6, 0, 2, 2, 2, 0, 0, 2, 2, -2, 0], [-3, -3, 3, 3, 0, 0, -3, -3, -3, 2, 0, 3], [2, -6, -2, 2, -1, 3, 3, 3, 3, -2, 1, 0], [4, 0, -2, 2, 1, 1, 1, -1, 1, 2, 1, 0], [4, -1, 0, -1, 0, -7, 7, -1, 4, -4, 4, -4], [3, -6, 0, 2, 2, -3, 3, -2, 2, 2, -1, 1], [1, 0, 0, 1, 0, 8, -2, 1, -1, 8, 0, 1], [-1, 2, 1, 3, 1, -1, -2, 3, 0, -2, 1, -3], [-8, -4, 1, 0, -3, 3, 5, -4, 1, 0, -1, 2], [-2, 2, 2, 0, 0, 5, -1, 2, -4, 0, -2, -2], [-4, 7, 0, 1, 3, -3, -3, -2, 0, -3, 3, -1], [-1, -4, -1, -3, 0, -4, 5, 2, -1, 2, -2, 1], [-1, -4, 0, 3, -1, 0, 1, -3, -1, 4, 4, -1], [1, -9, -2, 2, -2, 1, 4, 2, 4, -1, 1, 0], [3, 4, 0, 4, 3, 1, -7, -4, -1, 0, 1, 3], [-3, 4, 1, -3, -3, 7, -1, 6, -1, -3, 3, 0], [0, 6, -4, -3, 1, 0, -1, 8, 1, 2, 2, -2], [0, 0, -5, -2, 0, 0, 1, -1, 0, 1, 0, 2], [1, 5, -1, 0, 6, -1, -4, 5, 2, 0, 1, 1], [-1, 2, 6, 3, 1, -1, -4, 8, 0, -4, 1, -3], [-3, -8, -2, 0, 2, 3, 2, 7, 2, 8, -6, 6], [5, 6, 3, 2, 2, -4, -6, -6, -4, 2, 4, -3], [2, -4, -1, 3, 0, -1, 2, 2, 2, -1, 1, -2], [-2, 9, 4, -4, -2, 7, -2, 5, -2, -10, 1, 0], [5, -3, 0, 3, 4, -3, -10, -3, -1, 9, -1, 4], [-4, -2, 4, 5, 0, 2, -6, -5, -4, -4, 0, 4], [-6, -1, -4, -3, -3, -1, 5, 5, 5, -2, -5, 4], [3, 0, -5, 1, 3, -3, 10, 5, 0, 10, 3, -1], [4, 0, -4, -7, -4, 0, 9, 8, 7, 0, 0, 2], [0, 7, 7, 3, -4, 0, -10, -5, -4, 2, 0, 0], [3, 2, 5, -2, -3, 2, 6, -4, -5, -7, 2, 1], [-5, -2, -1, -4, -4, 2, 7, 7, 4, -4, -4, 1], [5, 4, 0, -5, 1, -2, -7, 5, -1, 2, -6, 1], [2, 3, 1, -1, -1, 3, 0, 0, 0, -5, 4, 0], [5, -5, -2, 5, 0, -1, 3, 4, 5, 8, 3, -5], [0, -10, -2, -1, -2, -3, 13, -2, 2, -10, 1, 1], [7, 0, 0, -2, 0, -4, 16, 7, 2, -4, 0, 1], [-6, -9, 3, 0, -3, -9, -6, -9, 3, 0, -3, 6], [14, -6, -3, 0, 0, -15, -3, -6, 3, 0, 3, 0], [-5, 0, -1, -1, -1, -6, -6, -4, 1, -6, 0, 5], [-3, 4, -2, 0, 2, 3, 2, -2, 2, -13, 6, -6], [12, 11, -7, -5, -3, -7, 9, 2, 7, 8, 5, -2], [-7, 9, 0, -7, -3, 2, 3, 7, -7, -9, -4, -3], [-5, -6, 3, 6, 6, 3, -9, -9, -3, -8, 3, -3], [-9, -3, -5, 1, -10, 5, 3, -1, 5, 4, -1, 11], [6, 3, 0, -1, -1, -6, -3, -8, -1, -1, -1, 1], [-9, 0, 5, 5, 5, -3, -21, -14, -5, -3, 0, 4], [-11, 0, 2, 5, 2, 0, -9, -13, -5, 0, 0, 11], [6, 8, 2, -5, 3, 2, 0, 8, -2, -4, 5, -8], [8, -5, 0, -2, -2, -8, -1, 2, 5, 8, 0, -2], [-4, 3, 4, -4, -4, 0, 0, 0, 0, 4, -8, 0], [6, -6, 0, 0, 6, -1, -6, -7, -6, 12, -5, 0], [0, 15, -1, -10, 0, 5, -4, 0, 0, 2, -1, 0], [-15, 3, -4, -3, 1, 15, -7, 8, 1, -4, -7, 7], [9, -3, -2, 7, 6, -6, -2, -10, 0, -2, 6, -7], [4, -4, -4, 0, 0, -4, 8, -4, 2, 0, 4, -2], [5, 1, 0, 1, 6, -6, -6, 7, 0, -6, 6, -1], [-11, -5, 8, 5, -3, 11, -3, -8, -3, -8, -3, 3], [-1, 5, -1, -6, 0, 5, -4, 5, -1, -10, -2, 1], [-1, 8, 0, 3, 2, 15, -5, -3, -4, -5, 1, 2], [-15, -14, 0, 4, 3, 1, 8, -4, -4, 3, 1, 3], [-12, 7, 7, -3, -3, -11, -13, -18, -7, -6, 3, 0], [6, 0, 3, 5, 2, -6, 6, -5, 2, -7, 11, -11], [19, -1, -2, 0, 9, 0, 5, -1, 1, 0, 2, -1], [-8, -7, 5, 0, -3, 8, 5, -7, -1, 0, -5, 4], [-2, 0, -5, 3, -5, 0, 17, 3, -3, 0, 0, 2], [9, 7, -2, -6, -4, -9, -10, -14, -4, 2, -9, 9], [-6, -1, -2, 5, 5, -7, -7, -7, 3, -4, -3, 2], [-7, 12, 1, -1, -1, -6, 0, 0, 0, -2, 1, 0], [-4, 0, 2, -2, -1, -12, -1, -10, -1, -2, 10, 0], [-9, 4, 0, -5, 3, 4, -1, 5, -7, -6, -8, 3], [2, 10, 1, -1, 0, 11, -12, 4, 2, -4, 3, -2], [-8, 0, 0, 7, 0, 2, 7, -8, -7, 2, 0, 4], [7, 5, 8, 0, 3, 2, -19, 5, -7, 0, -8, 1], [5, -13, 3, -3, -8, 8, 2, 8, 0, 2, -8, 3], [-13, 7, -1, -3, -2, -5, -15, -3, 1, 2, 3, -1], [-13, 0, -3, -4, -4, 8, 18, 18, -4, -10, 4, 3], [-13, -7, 0, 6, 5, 3, 4, -6, -4, -2, 1, 5], [4, -10, 0, 2, 0, -16, 16, -4, 4, -13, 4, -4], [10, -9, 6, 15, 15, -6, -6, -6, 0, 4, 0, -6], [-11, -10, -4, 0, 3, -16, 2, -10, 5, 0, 4, 1], [-11, 0, -2, 0, -2, -12, 8, -9, 0, -12, 0, -7], [-1, -6, 9, 8, 8, -1, 0, 0, -1, -10, 1, -9], [5, -13, 0, 3, 5, -6, -8, -3, 2, 16, -2, 5], [4, 9, 1, -1, -8, -2, 7, -7, 7, 5, 1, 0], [9, 13, 0, 1, 3, -14, -10, -1, 2, -6, -2, 3], [7, -18, 3, -3, -3, 15, 21, 21, 6, 4, -6, -3], [5, 0, 4, -6, 4, -8, 8, 1, 6, -8, 0, -5], [-1, 13, -13, 0, -9, -6, 7, 13, 8, 0, 13, -5], [-13, 11, 3, 3, -2, 2, -22, 2, 0, -22, -2, -3], [16, 0, 11, -1, 11, -15, -3, 5, 1, -15, 0, -10], [0, -1, -1, 1, 3, -19, -3, -16, 1, 2, -1, -2], [-3, -6, 3, 3, 0, 0, -3, 0, -3, -13, 0, 3], [2, 10, 0, -2, -5, -5, 5, 2, 2, -10, 3, -5], [8, -3, -6, 6, 1, -6, 5, -8, 5, 16, -2, 0], [3, 5, 5, -3, -3, 5, -1, -1, -7, -2, 7, -5], [1, 0, -6, 1, -6, 14, 1, 7, -1, 14, 0, -8], [-10, 8, 1, 0, -2, 2, 4, -3, 0, 4, -2, 0], [10, 8, 4, 0, -12, 24, 8, 8, -8, 0, -4, -4], [-7, 10, 3, 7, 3, -3, -12, -2, 0, -12, 3, -7], [11, 28, -3, -2, 1, -11, -23, -5, 1, 0, 3, -3], [-13, 1, 8, 6, 4, 10, -24, 0, -8, 5, -6, 2], [-4, 6, -5, 5, -1, -6, 6, 3, 6, -5, 7, 0], [-12, 3, 0, 0, 3, -4, -3, 8, -3, 6, -14, 0], [-8, -1, 0, 5, 0, 8, -8, -4, -8, 26, -8, 8], [-3, -9, 0, 2, 2, 3, 6, 16, 2, 14, -1, 1], [14, 1, 2, 0, 3, -3, -8, 1, -4, 0, -2, -2], [10, 0, 2, 8, 2, 18, -12, 8, -8, 18, 0, -1], [-4, -3, 6, 11, 11, -4, 0, 0, -4, -10, 4, -6], [-4, -12, 3, -3, -11, 15, 8, 16, 8, -8, -11, 0], [-15, -20, 0, -5, -3, -2, 23, 5, 5, 0, -2, -3], [6, -4, 8, 0, 0, 11, 5, 5, -7, -2, 7, -8], [-3, 7, 7, 6, 9, 4, -25, 0, -7, 12, -6, -3], [9, -21, -2, 1, -12, 12, 22, 17, 0, 22, -12, -1], [-4, 15, 3, 0, 0, 15, -6, 15, 0, 0, -3, 3], [-7, -3, 6, -10, -10, 10, -17, -6, 0, -17, -10, 10], [-12, 2, 5, 4, -6, 8, -12, -7, -5, 5, -4, 10], [-15, 8, -5, -7, -7, -4, 8, 8, 3, -10, -3, 5], [17, 3, 0, 9, 1, -12, -10, -9, 8, 6, 8, 1], [0, 3, 7, 1, -4, -1, 0, -13, -7, -2, -1, 5], [9, -6, -3, 0, -3, -6, 6, -6, 0, 0, 3, -3], [-11, 17, -4, 0, -15, 14, 5, 17, -1, 0, 4, -5], [1, -2, -4, 2, 6, -1, 18, 1, 6, -20, 18, -18], [-12, 11, -7, -2, -12, 8, -9, 5, 7, 14, 2, 10], [18, -1, 10, 5, 5, -4, -4, -4, 0, 8, 0, -10], [-1, -1, 5, 6, 0, 2, -7, -4, -1, 8, 4, 1], [2, 3, -10, 10, 11, -12, -1, -4, -1, 10, 4, 0], [3, -15, 5, 9, 4, -3, -1, -19, 4, -4, -7, 7], [-22, 0, -5, -6, -5, -8, 8, -17, 6, -8, 0, 7], [0, 0, -2, -2, 0, 0, -2, -10, 0, -2, 0, 2], [-5, -1, 1, 0, 0, 15, 14, -1, 4, 0, -1, 5], [6, -4, -19, -8, -3, -4, 21, 20, 19, -1, 8, -5], [-17, -2, 8, -10, -10, 11, 1, 1, 1, -25, -1, -8], [11, 11, -1, -12, 0, -10, 11, 2, 11, -22, 10, -11], [-4, 18, 0, -4, 6, -16, -18, 4, -16, -6, -10, 6], [12, 3, -3, -1, 2, -12, 0, 19, 2, 26, 2, -2], [3, 0, 11, 11, 11, 0, -15, -8, -11, 0, 0, -5], [14, -6, 0, 0, 0, -12, -6, -6, 6, 0, 0, 6], [7, 19, 8, 8, 0, -7, -24, -5, 0, 7, 3, -3], [-1, 8, 2, 9, 0, -7, 5, -25, -1, 8, 1, 1], [5, -4, 0, 0, -1, 9, 4, 0, 8, 1, 1, -1], [6, -3, 9, -9, -3, 8, -6, -7, -6, 0, -2, 0], [3, 6, -1, -16, 0, 8, -7, 21, 3, -10, 2, -3], [-22, 26, 1, 0, 4, -4, -20, -3, 0, -20, 4, 0], [-1, 19, 2, 0, -3, -6, -29, 19, -1, 0, -2, 1], [2, -6, -3, -4, -4, 4, 22, -3, 0, 22, -4, 4], [7, -32, 5, 8, 3, -7, 21, -5, 3, 10, -3, 3], [6, -6, -9, 3, 0, 15, -6, 24, 6, 14, -3, -6], [5, -18, 0, 0, 1, 18, -1, 13, -1, -2, -11, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_729_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_729_2_e_k();".
// 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_729_2_e_k(:prec:=12)
    chi := MakeCharacter_729_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_729_2_e_k();".
// 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_729_2_e_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_729_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, -81, 135, 387, 495, 126, 228, 45, 18, 15, 6, 3, 1]>,<5,R![25281, 41499, 34290, 15624, 2547, -1854, -1203, -81, 162, 75, 24, 6, 1]>,<7,R![18496, -65280, 101712, -38936, 5832, -198, -861, 621, -18, -83, 33, -6, 1]>],Snew);
    return Vf;
end function;