
// Make newform 729.2.e.a in Magma, downloaded from the LMFDB on 25 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_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_e_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, 0, -1, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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, -1]];
    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, -1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0], [-1, -1, 2, 1, 0, -2], [-1, 0, 1, -1, -1, -2], [-3, 1, -2, 1, 0, -1], [-2, 1, -1, 1, -2, 0], [3, 0, 0, -3, 0, 0], [0, 3, -3, 1, -3, 3], [-3, 1, 0, 2, 2, 2], [-1, -4, -3, 3, 4, 1], [-2, 1, 0, 1, 1, 4], [1, 0, -3, -1, 3, 3], [-2, -3, 1, -3, -2, 0], [-1, 4, -2, -1, 0, 1], [-5, 0, 5, 5, 2, 0], [-6, 0, 0, 0, 3, -3], [-1, -1, -7, 1, 0, 7], [-4, 0, 4, -1, 2, -5], [4, 1, 2, 1, 4, 0], [-3, 3, 6, 3, -9, -9], [0, 3, 3, -2, 3, 3], [5, 2, 4, -4, -2, -5], [-1, 2, 3, -3, -2, 1], [0, -3, -6, 0, -6, -3], [-4, -5, -8, -4, 0, 4], [7, 0, -7, -4, -4, 3], [4, 4, -5, -8, 4, 5], [-3, 0, 0, 0, -3, 3], [-10, 3, 3, 0, -3, 0], [-1, -1, 5, -5, 6, -5], [0, 0, 3, 1, 3, 0], [-6, 13, 0, -7, -7, -4], [-1, -7, -6, 6, 7, 1], [-5, -2, 0, 7, 7, -5], [-5, 9, -5, 9, -5, 0], [5, -8, 4, -1, 0, 1], [-11, -11, 7, 1, 10, -7], [-1, -3, -3, 0, -9, 12], [-1, -1, -1, -2, 3, 1], [3, 7, 7, 4, 0, -4], [-3, -6, 6, 3, 0, 0], [0, -9, 3, 1, 3, -9], [-1, 11, 3, -3, -11, 1], [4, -5, 0, 1, 1, 10], [0, -6, 6, 12, 6, -6], [1, -6, 0, -1, 6, 6], [1, 1, 7, -5, 4, -7], [11, 0, -11, -7, -1, 4], [3, -11, -2, -5, 0, 5], [-2, -8, -10, -8, -2, 0], [3, -9, 0, -3, 9, 9], [6, -8, 0, 2, 2, -16], [-10, 2, 1, -1, -2, 10], [0, 3, 6, -15, 6, 3], [-11, 6, 1, 6, -11, 0], [-2, 0, 2, 5, -13, 3], [-9, 0, 0, 0, -9, 9], [-7, 3, 3, 0, 3, -6], [-4, 0, 4, -13, -1, -17], [12, -2, 16, 4, 0, -4], [-14, 7, -1, 7, -14, 0], [9, 7, 0, -16, -16, 8], [-11, 3, -15, 11, 12, 12], [10, -6, -2, -6, 10, 0], [-1, -5, -2, -1, 0, 1], [4, 0, -4, -4, -25, 0], [-16, 0, 16, 20, -1, 4], [-2, 10, 8, 10, -2, 0], [6, 1, 0, -7, -7, 11], [-4, 20, 4, -4, -20, 4], [8, 17, 9, -9, -17, -8], [0, 0, -9, 15, -9, 0], [5, 19, 4, -1, 0, 1], [-5, -5, -11, 16, -11, 11], [-1, -18, -18, 0, 3, 15], [-10, -10, 5, 13, -3, -5], [-9, 4, -11, -2, 0, 2], [0, -15, 6, -8, 6, -15], [-6, 4, 0, 2, 2, -4], [7, -8, 0, 1, 1, 4], [-14, 0, -14, 0, -14, 0], [-16, 4, -5, 11, 0, -11], [-3, -9, -9, 0, 24, -15], [-19, 15, 15, 0, -9, -6], [2, 0, -2, 14, 2, 16], [12, 16, 7, -5, 0, 5], [-18, -21, 12, 18, 9, 9], [2, -7, 7, -7, 7, -2], [-10, -13, 6, -6, 13, 10], [-17, 22, 0, -5, -5, -5], [0, 12, -3, 21, -3, 12], [-2, 0, 2, -13, 14, -15], [8, -9, -9, 0, 12, -3], [8, 8, 8, -2, -6, -8], [-2, -5, 2, -5, -2, 0], [24, 3, -3, -24, 0, 0], [-15, 4, 0, 11, 11, -4], [0, -3, 3, -27, 3, -3], [-2, -15, 3, 2, 12, 12], [17, 3, 3, 0, 15, -18], [2, 0, -2, 5, -16, 7], [15, -6, 15, -15, -9, -9], [-15, 4, 0, 11, 11, 14], [8, 2, -15, 15, -2, -8], [13, -11, 0, -2, -2, -23], [-2, 12, 3, 2, -15, -15], [10, 0, -10, -22, 17, -12], [-6, -9, -9, 0, 21, -12], [-1, -1, -1, 7, -6, 1], [-7, 0, 7, 2, -19, -5], [4, 16, -10, 16, 4, 0], [0, -6, 3, -11, 3, -6], [-12, 1, 0, 11, 11, -7], [-7, -13, -23, 23, 13, 7], [-8, 15, 12, 8, -27, -27], [-2, 0, 2, 14, 5, 12], [10, 10, 13, 7, -17, -13], [-3, 9, 9, 0, -3, -6], [-19, -19, 23, 4, 15, -23], [6, 1, -11, -17, 0, 17], [10, -20, 2, -20, 10, 0], [-7, -16, -11, 11, 16, 7], [-1, -1, 18, -18, 1, 1], [0, -12, 3, -9, 3, -12], [-13, 13, 19, 32, 0, -32], [-15, 0, 0, 0, -24, 24], [11, 0, -11, 2, -1, 13], [-21, 12, -12, 21, 0, 0], [8, 14, -2, 2, -14, -8], [-14, 16, 0, -2, -2, 13], [19, -15, 0, -19, 15, 15], [-14, 0, -23, 0, -14, 0], [10, 10, 25, -23, 13, -25], [-25, -15, -15, 0, 21, -6], [-10, -10, -25, 19, -9, 25], [-5, -11, -10, -11, -5, 0], [0, 33, -6, 0, -27, -27], [19, 1, 0, -20, -20, -8], [19, -15, 16, -15, 19, 0], [18, 9, 9, 0, 0, -9], [17, 0, 0, 0, 3, -3], [9, 13, 16, 7, 0, -7], [1, 4, 17, 4, 1, 0], [-9, -3, -15, 9, 18, 18], [0, 6, 18, -8, 18, 6], [-10, -4, -21, 21, 4, 10], [11, -17, 7, -4, 0, 4], [1, 0, -1, -4, 17, -3], [10, 10, -29, 4, -14, 29], [-3, 9, 9, 0, -21, 12], [10, -5, -10, -5, 10, 0], [3, 18, -18, -3, 0, 0], [0, 24, 0, 1, 0, 24], [-9, -2, 0, 11, 11, -1], [-4, 4, -8, -4, 0, 4], [-14, 0, 14, 5, 2, -9], [11, -6, -6, 0, 12, -6], [0, -23, -11, -11, 0, 11], [0, -9, -6, -8, -6, -9], [21, -5, 0, -16, -16, -22], [-19, -10, -9, 9, 10, 19], [0, 18, 0, -3, 0, 18], [13, 13, -23, -17, 4, 23], [-24, 18, 18, 0, -15, -3], [-1, -1, -16, 19, -18, 16], [0, -14, -2, -2, 0, 2], [0, 0, 0, -2, 0, 0], [5, 5, 28, -28, -5, -5]];
    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_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_e_a(:prec:=6)
    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_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_e_a( : 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, 27, 36, 30, 18, 6, 1]>,<5,R![9, -27, 9, 24, 18, 3, 1]>,<7,R![289, 459, 342, 152, 45, 9, 1]>],Snew);
    return Vf;
end function;