
// Make newform 729.2.c.a in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_729_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_729_c_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_c_a();".
// 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_c_a();".
// 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, 0, 63, 0, 58, 0, 15, 0, 1, 0, 0, 0], [8, -149, 63, -95, 58, 76, 15, 70, 1, 16, 0, 1], [3, 95, -222, 322, -343, 310, -175, 119, -34, 19, -2, 1], [1, 0, 163, 0, 285, 0, 160, 0, 33, 0, 2, 0], [3, 0, -222, 0, -343, 0, -175, 0, -34, 0, -2, 0], [-5, -28, -474, -79, -760, -121, -409, -73, -83, -16, -5, -1], [-17, 256, -549, 417, -881, 234, -482, 49, -99, 3, -6, 0], [-2, 42, 0, 335, 0, 477, 0, 249, 0, 50, 0, 3], [-5, 0, -474, 0, -760, 0, -409, 0, -83, 0, -5, 0], [-13, 0, -549, 0, -881, 0, -482, 0, -99, 0, -6, 0], [5, 86, 163, 1555, 285, 2416, 160, 1301, 33, 265, 2, 16]];
    Rf_basisdens := [1, 4, 8, 8, 4, 4, 8, 8, 4, 4, 4, 8];
    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_c();"
function MakeCharacter_729_c()
    N := 729;
    order := 3;
    char_gens := [2];
    v := [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_729_c_Hecke();"
function MakeCharacter_729_c_Hecke(Kf)
    N := 729;
    order := 3;
    char_gens := [2];
    char_values := [[0, 0, 0, 0, 0, 0, 0, 0, 1, 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, -1, 1, -1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 1, -1, 0, -1, 0, 0], [0, -1, 1, -1, 0, 0, -1, 1, -1, 0, -1, 0], [1, -1, 1, 0, 0, 0, 1, 1, 0, 0, -1, 1], [1, 0, -1, 1, 1, -1, 0, 0, 1, 0, 0, -1], [-1, 0, 0, 0, 1, 1, 0, 0, 0, -1, -1, 0], [2, 1, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0], [-1, 0, 2, 1, -1, -1, 0, 0, -3, 0, 0, 1], [-5, 1, -1, 2, 0, 0, 2, 1, -6, 0, -1, 2], [0, 0, 2, -2, 0, 2, -2, -1, 4, 2, 0, 0], [-1, 1, 0, 0, -1, 0, 0, 0, 0, 4, 2, 0], [2, 0, 3, -1, 2, 1, -1, -1, 0, 1, 0, -2], [-2, 2, -2, 2, 0, 0, 4, 2, -4, 0, -2, 2], [1, -3, 3, 1, 0, 0, -3, -2, 3, 0, 2, 0], [0, -4, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0], [0, 0, -1, -3, 0, 3, 1, -1, 3, -1, 0, 0], [-5, 0, 0, 2, 0, 0, 3, -1, -4, 0, 1, -1], [0, 0, -3, 1, 0, -1, 5, 2, 1, -5, 0, 0], [2, -2, 0, 0, 2, -2, 0, 0, 0, -4, 0, 0], [2, 3, 0, 0, 3, 3, 0, 0, 0, 3, 0, 0], [-4, 5, -5, -1, 0, 0, 3, -3, -1, 0, 3, -2], [2, 3, -3, -1, 0, 0, -1, -1, 3, 0, 1, -2], [1, 1, 0, 0, 0, -1, 0, 0, 0, -2, -4, 0], [4, -1, 1, -1, 0, 0, 1, 2, 2, 0, -2, 2], [0, -2, 2, 1, 0, 0, 6, 0, 0, 0, 0, 1], [-2, 0, 3, 1, -2, -1, 1, 3, 1, -1, 0, 2], [-6, 1, 0, 0, -2, -3, 0, 0, 0, 3, 1, 0], [2, 1, 0, 0, 2, 0, 0, 0, 0, 1, -4, 0], [2, 0, 1, 0, 2, 0, 4, 1, 0, -4, 0, -2], [2, 4, 0, 0, -1, -3, 0, 0, 0, -2, 2, 0], [1, 0, 4, 1, 1, -1, 0, -4, 3, 0, 0, -1], [-7, 3, -3, -1, 0, 0, 2, -4, -3, 0, 4, -2], [-3, 0, -4, 1, -3, -1, 4, -1, 1, -4, 0, 3], [-3, 0, 1, -1, -3, 1, -1, 3, -9, 1, 0, 3], [-7, 3, -3, 2, 0, 0, -1, -3, -4, 0, 3, 1], [2, 0, 2, -5, 2, 5, -4, -2, 1, 4, 0, -2], [-7, 7, 0, 0, -1, 6, 0, 0, 0, 1, 2, 0], [-2, 0, 0, 6, -2, -6, 4, 3, 6, -4, 0, 2], [6, 0, 0, -6, 0, 0, -5, 0, 6, 0, 0, 2], [-7, 4, 0, 0, -4, -2, 0, 0, 0, 2, 3, 0], [-4, 4, 0, 0, -1, 3, 0, 0, 0, 4, -1, 0], [5, 3, -3, -1, 0, 0, 5, 2, 3, 0, -2, 4], [-4, 0, 3, 1, -4, -1, 6, 1, -5, -6, 0, 4], [-5, 2, 0, 0, 4, 5, 0, 0, 0, 4, 0, 0], [8, -2, 0, 0, 5, 3, 0, 0, 0, 4, -4, 0], [2, 0, -7, -5, 2, 5, 5, 1, 7, -5, 0, -2], [0, -1, 1, 2, 0, 0, -1, 4, -4, 0, -4, -3], [-3, 6, -6, 0, 0, 0, 4, 3, -6, 0, -3, -4], [-2, 0, 11, -5, -2, 5, 0, 3, 1, 0, 0, 2], [5, -3, 0, 0, 4, 4, 0, 0, 0, 2, -1, 0], [-1, 0, -4, 1, -1, -1, 6, 6, -9, -6, 0, 1], [2, 2, -2, -4, 0, 0, 0, 0, 2, 0, 0, 1], [4, 5, 0, 0, -2, 5, 0, 0, 0, 7, -3, 0], [-1, 0, 0, 2, -1, -2, 11, 2, 0, -11, 0, 1], [-15, -5, 5, 1, 0, 0, -9, 0, -15, 0, 0, -2], [-6, -9, 0, 0, -6, -3, 0, 0, 0, -6, 3, 0], [11, -12, 0, 0, 0, 0, 0, 0, 0, -6, 3, 0], [3, 5, -5, 5, 0, 0, -1, 1, 2, 0, -1, 0], [-6, -9, 9, 6, 0, 0, -5, 0, -6, 0, 0, 2], [-5, 0, 8, 7, -5, -7, -6, 3, -5, 6, 0, 5], [3, 0, -9, 7, 3, -7, 0, 4, -9, 0, 0, -3], [8, 3, 0, 0, -3, -6, 0, 0, 0, -9, 3, 0], [1, 0, -4, 5, 1, -5, 2, 10, -9, -2, 0, -1], [-2, -1, 1, -1, 0, 0, 7, -1, -1, 0, 1, -1], [7, 9, -9, 4, 0, 0, 6, -5, 12, 0, 5, 0], [-9, 8, -8, 11, 0, 0, 8, -2, -7, 0, 2, 3], [-5, 0, 5, -2, -5, 2, -3, -3, -5, 3, 0, 5], [2, 0, 5, -5, 2, 5, 3, 0, -3, -3, 0, -2], [-1, -4, 4, 8, 0, 0, 0, 0, -1, 0, 0, 4], [-6, -1, 1, 5, 0, 0, -7, 6, -12, 0, -6, 0], [10, -15, 0, 0, -1, -10, 0, 0, 0, -11, 1, 0], [-4, -12, 12, -7, 0, 0, -16, -3, -1, 0, 3, 1], [-1, 0, 5, -2, -1, 2, -7, 7, -11, 7, 0, 1], [2, -2, 0, 0, 2, 0, 0, 0, 0, 4, 2, 0], [5, 0, 1, 0, 5, 0, -5, 1, 12, 5, 0, -5], [8, -8, 8, -3, 0, 0, -8, 5, 3, 0, -5, -3], [-7, -5, 0, 0, -1, -6, 0, 0, 0, -2, 2, 0], [-2, 0, -8, 4, -2, -4, 3, -1, 12, -3, 0, 2], [-3, 0, 14, 4, -3, -4, -11, -1, -2, 11, 0, 3], [-3, 0, -2, -7, -3, 7, -4, 3, 15, 4, 0, 3], [-2, 2, -2, 11, 0, 0, 4, 5, -7, 0, -5, 2], [9, 10, 0, 0, 1, 9, 0, 0, 0, 12, 1, 0], [-4, 4, 0, 0, -4, -3, 0, 0, 0, 7, -1, 0], [-17, -6, 6, 2, 0, 0, -6, 5, -22, 0, -5, 8], [-3, -3, 3, -9, 0, 0, 13, -6, 3, 0, 6, -4], [14, 2, 0, 0, 3, -5, 0, 0, 0, 2, -2, 0], [11, -7, 7, -1, 0, 0, 0, 0, 11, 0, 0, 1], [-2, -14, 14, -10, 0, 0, -10, 1, -3, 0, -1, 2], [-3, 0, -4, 7, -3, -7, -2, 8, -11, 2, 0, 3], [1, -10, 0, 0, 1, -1, 0, 0, 0, 4, -3, 0], [-18, 1, -1, 4, 0, 0, 3, 0, -18, 0, 0, 7], [11, -2, 0, 0, -1, 3, 0, 0, 0, -8, -7, 0], [-5, 0, 0, 3, -5, -3, -8, -3, 3, 8, 0, 5], [-3, 0, -12, 1, -3, -1, -4, 2, 1, 4, 0, 3], [5, -14, 0, 0, -4, 4, 0, 0, 0, -4, -3, 0], [-5, 0, 7, 4, -5, -4, -9, -1, 0, 9, 0, 5], [-11, 10, 0, 0, 0, 2, 0, 0, 0, -2, -4, 0], [5, -9, 0, 0, 0, -15, 0, 0, 0, -9, 6, 0], [-19, -8, 0, 0, -1, 3, 0, 0, 0, 1, 5, 0], [1, 9, -9, -7, 0, 0, -3, 2, -1, 0, -2, 2], [8, -8, 0, 0, -1, -8, 0, 0, 0, -7, 9, 0], [-5, 0, 13, -2, -5, 2, 3, 2, -6, -3, 0, 5], [8, -3, 3, -1, 0, 0, -7, -7, 15, 0, 7, 1], [-3, 0, -13, -2, -3, 2, 7, 2, -14, -7, 0, 3], [-4, -18, 0, 0, 6, 3, 0, 0, 0, -12, 3, 0], [-1, 5, -5, -5, 0, 0, -3, -4, 3, 0, 4, -10], [0, -7, 0, 0, 5, -9, 0, 0, 0, -15, 8, 0], [-5, 0, 24, 3, -5, -3, -8, 3, -3, 8, 0, 5], [3, -4, 4, -10, 0, 0, -19, -11, 14, 0, 11, -12], [1, 0, -4, 1, 1, -1, 0, -3, 19, 0, 0, -1], [-13, 6, 0, 0, 3, 3, 0, 0, 0, 6, 0, 0], [5, 0, -10, -2, 5, 2, 0, -3, 0, 0, 0, -5], [3, 0, 0, 2, 0, 0, -8, -5, 8, 0, 5, -6], [-13, 4, 0, 0, -7, 3, 0, 0, 0, -2, 5, 0], [12, -8, 8, -8, 0, 0, 6, 6, 6, 0, -6, 7], [-2, 0, -3, 1, -2, -1, 13, 0, -5, -13, 0, 2], [30, 4, 0, 0, 4, -12, 0, 0, 0, 0, -8, 0], [-4, 0, 13, 3, -4, -3, 1, 7, -6, -1, 0, 4], [-29, 2, -2, 3, 0, 0, 10, -2, -27, 0, 2, 1], [6, 0, -9, -8, 6, 8, -7, -4, 4, 7, 0, -6], [11, -1, 1, -7, 0, 0, 3, -6, 17, 0, 6, -5], [3, -4, 4, 14, 0, 0, 8, 3, 0, 0, -3, 0], [13, 1, 0, 0, 0, 5, 0, 0, 0, 1, 8, 0], [7, 11, -11, -1, 0, 0, -5, 2, 5, 0, -2, 2], [15, -4, 0, 0, 2, 3, 0, 0, 0, -3, -4, 0], [3, -16, 16, -1, 0, 0, 2, -2, 5, 0, 2, 6], [11, -11, 0, 0, -4, -11, 0, 0, 0, -7, 6, 0], [-15, 12, -12, 11, 0, 0, 16, -8, -7, 0, 8, 3], [3, 0, 20, -5, 3, 5, -20, 2, 1, 20, 0, -3], [-13, -5, 0, 0, -7, 0, 0, 0, 0, -11, 8, 0], [6, 0, -5, -7, 6, 7, -13, -9, 9, 13, 0, -6], [-13, 0, -7, -2, -13, 2, -1, 1, 4, 1, 0, 13], [-4, 15, 0, 0, -3, 0, 0, 0, 0, -3, 0, 0], [0, 0, 11, 6, 0, -6, -14, -10, -6, 14, 0, 0], [4, 0, -1, -5, 4, 5, -6, 6, 1, 6, 0, -4], [11, -1, 0, 0, 3, 7, 0, 0, 0, 11, -5, 0], [-1, 0, -3, -8, -1, 8, 9, -8, 4, -9, 0, 1], [1, 0, 11, -7, 1, 7, -4, 4, 9, 4, 0, -1], [12, -9, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0], [5, -2, 0, 0, 5, -15, 0, 0, 0, -14, -1, 0], [-4, -8, 8, 18, 0, 0, -5, 2, -6, 0, -2, 6], [-2, 0, -7, 7, -2, -7, -9, 12, -17, 9, 0, 2], [-13, -1, 0, 0, 0, -5, 0, 0, 0, -1, 10, 0], [11, 1, 0, 0, 5, 6, 0, 0, 0, 7, -4, 0], [13, -2, 2, -4, 0, 0, -4, 1, 12, 0, -1, -10], [11, -15, 15, -7, 0, 0, -1, 9, 2, 0, -9, 4], [-4, 23, -23, 7, 0, 0, 15, 5, -9, 0, -5, -1], [2, 0, 2, -2, 2, 2, -4, -2, -8, 4, 0, -2], [-9, 19, 0, 0, -5, 0, 0, 0, 0, 15, 4, 0], [-8, 0, -10, 19, -8, -19, 12, 3, -23, -12, 0, 8], [8, 9, 0, 0, 1, 10, 0, 0, 0, -1, 8, 0], [-1, -11, 0, 0, -1, -12, 0, 0, 0, -11, -10, 0], [-6, 0, -18, 16, -6, -16, 12, 10, -42, -12, 0, 6], [-5, 26, -26, 2, 0, 0, 10, -1, -4, 0, 1, 2], [-2, -3, 3, -5, 0, 0, -3, -11, 9, 0, 11, 0], [8, -6, 0, 0, -6, 0, 0, 0, 0, -6, 6, 0], [-19, 10, -10, 3, 0, 0, 7, 8, -27, 0, -8, 0], [2, -8, 0, 0, 2, 9, 0, 0, 0, -14, -1, 0], [1, 0, 4, -8, 1, 8, -3, -1, 18, 3, 0, -1], [-12, -1, 1, 2, 0, 0, -7, 6, -18, 0, -6, 9], [7, -6, 0, 0, -1, -16, 0, 0, 0, 4, -2, 0], [4, 0, 15, 1, 4, -1, 7, 0, 1, -7, 0, -4], [3, -10, 0, 0, -1, -3, 0, 0, 0, -30, -1, 0], [0, 0, -19, -3, 0, 3, 13, -4, -3, -13, 0, 0], [29, 4, -4, -3, 0, 0, 4, 8, 21, 0, -8, 3], [-1, 6, 0, 0, -9, 3, 0, 0, 0, 18, 3, 0], [-18, 18, -18, 11, 0, 0, 13, 1, -19, 0, -1, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_729_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_729_2_c_a();".
// 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_c_a(:prec:=12)
    chi := MakeCharacter_729_c();
    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_c_a();".
// 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_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_729_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, -54, 324, -126, 360, 207, 381, 144, 99, 24, 15, 3, 1]>],Snew);
    return Vf;
end function;