
// Make newform 729.2.e.j in Magma, downloaded from the LMFDB on 19 April 2024.
// 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_j();".
// 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_j();".
// 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)
    N := 729;
    order := 9;
    char_gens := [2];
    char_values := [[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 := [[-1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 1], [-1, 0, 1, 1, 0, 1, -2, -1, 0, 0, -1, 1], [2, -2, -1, 1, 0, 0, 1, -1, 1, 2, 0, 0], [-1, 1, 0, -1, -1, 2, 1, 1, -1, -1, 0, -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], [2, -1, -1, 0, 1, -1, 2, 2, 0, 2, 1, 0], [3, 0, 1, -1, 1, -1, -3, 2, 1, -1, 0, -2], [-1, 0, -2, -1, -1, 1, 0, 4, 0, 0, -1, 1], [0, 5, 2, 0, 0, -4, -1, -1, -1, -2, 1, -2], [0, 3, 0, 1, 2, 2, -3, -1, -1, -2, -1, 2], [-2, 0, 2, -2, 0, 4, -2, 4, -2, -4, -2, 0], [3, 1, 2, 2, 0, -3, -4, -2, 0, 4, -1, 1], [1, 2, 1, 0, 0, 3, -1, 2, 1, 0, -1, 2], [3, 1, -3, 0, -1, 0, 3, 2, 3, 4, 0, 1], [2, 6, 0, -1, -1, -2, -3, -3, -1, -3, 2, -2], [4, 3, 0, -2, 0, 4, -5, 2, -1, 2, -2, 0], [-2, 6, 0, -2, -2, -2, 0, 0, -2, -2, 2, 0], [-3, -3, 3, 3, 0, 0, -3, -3, -3, 2, 0, 3], [-3, 0, 3, 1, 3, -2, 1, -6, -1, -2, 0, -1], [1, 0, -1, 1, -1, 4, 1, 2, -1, 4, 0, 1], [-4, 1, 0, 1, 0, 7, -7, 1, -4, 4, -4, 4], [-2, 6, 2, -2, 0, 1, -2, 1, -2, -4, 1, 0], [-7, 1, 1, 0, -1, 7, -2, 7, -1, 2, -1, 1], [1, 1, -1, -2, -3, -4, 5, -4, 1, -2, 2, 1], [8, 4, -1, 0, 3, -3, -5, 4, -1, 0, 1, -2], [-2, 2, 2, 0, 0, 5, -1, 2, -4, 0, -2, -2], [0, 6, 0, -2, 1, -2, -5, 1, 0, 2, 2, -3], [-1, -4, -1, -3, 0, -4, 5, 2, -1, 2, -2, 1], [8, -5, -1, 4, 3, -3, 6, -2, 0, 6, 3, -4], [1, 0, -4, -2, -4, 1, -2, 5, 2, 1, 0, 1], [-1, -3, 1, -1, -4, 4, 6, 4, 0, 6, -4, 1], [0, 1, 0, 0, -3, 7, 2, 0, -1, 0, 3, -3], [4, -6, 1, -1, -4, 4, 3, -1, 3, 2, -2, 0], [-1, 2, 5, 2, 2, 6, -7, 1, -5, 1, -2, 0], [1, 5, -1, 0, 6, -1, -4, 5, 2, 0, 1, 1], [-3, 2, 6, 2, 3, 14, -10, 9, -6, 0, -2, -1], [-5, -8, -2, 2, 2, -2, 0, 5, 0, 3, -6, 0], [-5, -6, -3, -2, -2, 4, 6, 6, 4, -2, -4, 3], [2, -4, -1, 3, 0, -1, 2, 2, 2, -1, 1, -2], [7, 0, 2, -2, 2, -2, 4, 5, 2, -2, 0, 1], [-8, 5, 1, 1, -3, 3, 0, -1, 0, 0, -3, -1], [4, 2, -4, -5, 0, -2, 6, 5, 4, 4, 0, -4], [-6, -1, -4, -3, -3, -1, 5, 5, 5, -2, -5, 4], [-13, -7, 5, 2, -1, 0, -7, -8, -5, -8, -2, 3], [-4, 3, 7, 4, -3, 4, -5, -7, -3, -6, 2, -2], [-2, 7, 4, -4, -7, -2, 3, -4, 3, 0, 0, 0], [5, -11, 0, -1, 2, -1, -2, 1, 5, 11, -3, 2], [5, 2, 1, 4, 4, -2, -7, -7, -4, 4, 4, -1], [-4, -1, -1, -6, -5, 5, -4, 8, 0, -4, -5, 6], [-3, 0, 0, 1, 0, -2, 1, -3, -1, -2, 0, -4], [-5, 5, 2, -5, 0, 1, -3, -4, -5, -8, -3, 5], [-9, 0, 0, 1, -1, -4, 12, -1, 2, -11, 2, -1], [11, -17, -2, 0, 2, -11, 16, -2, 2, 8, -1, 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], [-1, -3, 1, 1, 0, 1, 7, 5, 0, 0, 5, -5], [4, 4, -2, 2, 2, 7, 0, 5, 0, -9, 6, 0], [-13, -8, 0, 2, 5, 5, 1, -2, -7, 2, -3, 5], [-13, 6, -7, -4, -7, 7, 1, 12, 0, 1, -7, 4], [-5, -6, 3, 6, 6, 3, -9, -9, -3, -8, 3, -3], [3, -4, 0, 11, 1, -6, 1, -11, 5, 2, 10, 1], [-8, -3, -1, 1, 0, -2, 1, 7, 1, -7, 1, 0], [-6, 7, 5, 5, 0, 6, -16, -8, 0, -5, -4, 4], [11, 6, -5, -2, 3, -11, 7, 5, 3, 0, 11, -11], [-1, -5, 0, 8, 5, -10, -2, -8, 2, 2, 3, 5], [8, -10, 5, 0, -2, 2, 2, 5, 0, 2, -2, 0], [0, 0, 0, 4, 0, 4, -11, 0, -4, 4, 0, 8], [-1, 0, 6, 6, 6, 6, -13, -7, -6, 6, 0, -5], [0, -15, 1, 10, 0, -5, 4, 0, 0, -2, 1, 0], [4, -3, 1, -1, -4, -11, 3, -16, 3, 8, 7, 0], [-4, 8, 2, -1, -7, 12, -4, 4, -2, 4, 1, 6], [4, -4, -4, 0, 0, -4, 8, -4, 2, 0, 4, -2], [0, 0, 0, -5, 1, 7, 4, 13, 0, -10, 5, -6], [0, -5, 3, -3, -8, 11, 5, 8, 5, -8, -3, 0], [-1, 5, -1, -6, 0, 5, -4, 5, -1, -10, -2, 1], [-4, 7, -4, 1, 3, -3, -6, -14, 0, -6, 3, -1], [-4, 0, 4, -1, -4, 4, -15, 1, 0, -15, -4, 1], [-3, -2, 0, 0, -3, -11, -1, 0, -7, 6, 3, -3], [-2, 0, 2, -2, 3, 4, -5, 4, -5, 5, -11, 0], [-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, 9, -3, 5, 8, -2, -12, 3, 8, 0, 2, -2], [16, 7, 4, -4, 2, 7, -6, -13, -6, 18, -9, 0], [6, 1, 2, -5, -5, 7, 7, 7, -3, 4, 3, -2], [6, 0, 0, 1, 0, 7, -11, 6, -1, 7, 0, -1], [-12, 0, 1, -1, 1, -4, -12, -13, 1, -4, 0, 10], [14, -9, 7, 8, 5, -5, 0, -8, 0, 0, 5, -8], [-2, -10, -1, 1, 0, -11, 12, -4, -2, 4, -3, 2], [-10, -11, 7, 0, -7, 10, 7, -5, -7, -4, -4, 4], [7, 5, 8, 0, 3, 2, -19, 5, -7, 0, -8, 1], [-6, -13, 3, 5, -3, 11, 5, 0, -3, -6, -5, 8], [-5, 12, 0, 1, 3, 4, 1, -1, -1, -16, -2, 3], [13, 0, 3, 4, 4, -8, -18, -18, 4, 10, -4, -3], [-1, 7, -4, 1, 6, -6, 6, -5, 0, 6, 6, -1], [-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], [-1, -3, 0, 2, 2, 1, -6, 12, 2, 18, -7, 7], [1, 6, -9, -8, -8, 1, 0, 0, 1, 10, -1, 9], [14, -11, 2, -2, 3, -3, 6, 1, 0, 6, 3, 2], [-2, 0, -7, -8, -7, 4, 22, 5, 8, 4, 0, 1], [8, -9, -2, 2, -1, 1, 18, -11, 0, 18, -1, -2], [7, -18, 3, -3, -3, 15, 21, 21, 6, 4, -6, -3], [13, -11, -6, 4, 10, -13, 12, -2, 10, 0, 5, -5], [1, -13, 13, 0, 9, 6, -7, -13, -8, 0, -13, 5], [-24, -13, 3, 5, 3, 5, -19, 0, -3, -6, -5, 2], [-31, 9, 1, -11, -12, 31, -8, 14, -12, -6, -10, 10], [-1, 4, 0, 2, -1, 17, 1, -2, -1, -4, 3, -1], [-3, -6, 3, 3, 0, 0, -3, 0, -3, -13, 0, 3], [-7, 5, 2, 3, -2, 2, -7, 5, 0, -7, -2, -3], [6, 0, 5, -1, 5, -8, -9, 1, 1, -8, 0, 2], [3, 5, 5, -3, -3, 5, -1, -1, -7, -2, 7, -5], [-13, 19, 1, -6, -7, 13, -5, 13, -7, -7, 8, -8], [-2, -11, -1, -2, 0, 2, 11, 5, 1, -14, 2, -2], [-10, -8, -4, 0, 12, -24, -8, -8, 8, 0, 4, 4], [-9, -3, 3, 4, 7, 1, -17, 1, -3, 2, -4, -3], [8, 28, -1, 1, 3, -3, -2, -10, -2, 8, 3, 0], [1, 9, 0, -2, -6, -8, 13, 2, 8, -16, 4, -6], [6, 0, 6, 1, 6, 4, -11, 0, -1, 4, 0, -7], [-4, 0, 3, 3, 3, -12, 11, -7, -3, -12, 0, -14], [8, 1, 0, -5, 0, -8, 8, 4, 8, -26, 8, -8], [16, 9, 2, -2, 0, 13, -2, -5, -2, 2, 1, 0], [-14, -1, -2, 0, -3, 3, 8, -1, 4, 0, 2, 2], [8, -9, -8, -2, 6, -8, 10, -10, 6, -9, -1, 1], [4, 3, -6, -11, -11, 4, 0, 0, 4, 10, -4, 6], [-15, 0, 8, 11, 8, 4, -15, -23, -11, 4, 0, 11], [2, 3, -5, 2, 5, -5, -18, -2, 0, -18, 5, -2], [6, -4, 8, 0, 0, 11, 5, 5, -7, -2, 7, -8], [6, -5, 0, -3, 6, 7, -19, 3, -7, 18, -9, 6], [-10, 14, 2, -13, -1, -15, -10, -5, -2, -14, 13, -12], [-4, 15, 3, 0, 0, 15, -6, 15, 0, 0, -3, 3], [-27, -14, 6, 0, -10, 0, -3, -16, -6, -20, 0, 10], [-1, 1, 0, -10, -4, 2, 10, 10, 5, -7, -6, -4], [15, -8, 5, 7, 7, 4, -8, -8, -3, 10, 3, -5], [-14, 5, -8, -8, -9, 9, 12, 5, 0, 12, -9, 8], [-3, 11, 0, 5, 1, -6, -5, -5, -7, -7, 4, 1], [-9, 6, 3, 0, 3, 6, -6, 6, 0, 0, -3, 3], [-11, 17, -4, 0, -15, 14, 5, 17, -1, 0, 4, -5], [3, -2, -6, 6, 4, 2, 2, 5, 2, -17, 18, 0], [-16, 7, 0, -10, 2, 2, 7, 10, -7, -16, -12, 2], [-18, 1, -10, -5, -5, 4, 4, 4, 0, -8, 0, 10], [-1, -1, 5, 6, 0, 2, -7, -4, -1, 8, 4, 1], [-12, 0, 1, 11, 1, 2, -3, -13, -11, 2, 0, 4], [-14, 15, 4, -4, 5, -11, -9, -1, -9, -10, 7, 0], [-14, -5, -6, -5, 1, 14, 3, -2, 1, -15, -7, 7], [2, 2, 2, 2, 2, 12, -4, 10, -2, 4, -2, 0], [5, 1, -1, 0, 0, -15, -14, 1, -4, 0, 1, -5], [-7, 10, 0, 5, 8, -1, -20, -5, -19, 2, -3, 8], [17, 2, -8, 10, 10, -11, -1, -1, -1, 25, 1, 8], [11, 11, -1, -12, 0, -10, 11, 2, 11, -22, 10, -11], [-16, 12, -16, -10, -4, 4, -8, 36, 0, -8, -4, 10], [16, -3, 2, -2, -3, 28, 1, 10, 1, -10, -2, 0], [3, -2, 11, 11, 0, -3, -4, -11, 0, -2, 5, -5], [14, -6, 0, 0, 0, -12, -6, -6, 6, 0, 0, 6], [-3, 19, 0, 0, -8, -10, 8, -7, 8, 4, 3, 0], [-1, 8, 2, 9, 0, -7, 5, -25, -1, 8, 1, 1], [2, -2, 8, 1, 0, 0, -3, -17, 0, -3, 0, -1], [8, 0, 6, -3, 6, 6, -13, 2, 3, 6, 0, -2], [-3, -6, 1, 16, 0, -8, 7, -21, -3, 10, -2, 3], [16, -11, -1, 4, 0, 2, 23, -1, 1, -2, -4, 4], [1, -19, -2, 0, 3, 6, 29, -19, 1, 0, 2, -1], [18, 13, -3, 0, -4, -6, 9, -7, 3, 16, 0, 4], [0, -32, -3, 3, -5, -7, 8, -4, 8, 10, -3, 0], [6, -6, -9, 3, 0, 15, -6, 24, 6, 14, -3, -6], [-18, 0, -1, -1, -1, -5, 9, -17, 1, -5, 0, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_729_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_729_2_e_j();".
// 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_j(: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_j();".
// 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_j( : 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, 324, -144, 360, 144, -69, -72, -18, 12, 15, 6, 1]>,<5,R![25281, 10017, 2808, 3978, 846, -711, 363, 135, -9, -21, 15, -6, 1]>,<7,R![18496, -21216, 79680, -33104, 144, 1296, 921, -522, 162, -2, -12, 3, 1]>],Snew);
    return Vf;
end function;