
// Make newform 729.2.e.h 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_h();".
// 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_h();".
// 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 := [[0, -1, -1, 1, 1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 2, -1, 1, -2], [-1, 0, 1, 2, -1, 1], [1, 1, 3, 2, 0, -2], [1, -2, -1, -2, 1, 0], [3, 0, 0, -3, 0, 0], [0, 0, 3, 1, 3, 0], [1, 2, 0, -3, -3, 2], [3, -4, 2, -2, 4, -3], [1, 1, 0, -2, -2, 4], [1, -3, 3, -1, 0, 0], [5, -2, 1, -2, 5, 0], [-1, 4, 1, 2, 0, -2], [5, 0, -5, 0, 2, 5], [-6, -3, -3, 0, 0, 3], [0, 0, -7, -1, 1, 7], [-1, 0, 1, 5, 2, 4], [-5, 4, 2, 4, -5, 0], [-3, 6, -9, 3, 3, 3], [0, -6, 3, -2, 3, -6], [-4, 2, 1, -1, -2, 4], [-3, 2, -4, 4, -2, 3], [0, 9, -3, 0, -3, 9], [-4, -5, 4, 8, 0, -8], [-4, 0, 4, -3, -4, -7], [4, 4, -5, 4, -8, 5], [-3, 3, 3, 0, 0, -3], [-10, 0, 0, 0, 3, -3], [6, 6, 5, -1, -5, -5], [0, -3, 0, 1, 0, -3], [13, -7, 0, -6, -6, -4], [6, -7, 5, -5, 7, -6], [-2, 7, 0, -5, -5, -5], [-4, -5, -5, -5, -4, 0], [-1, -8, -5, -4, 0, 4], [10, 10, 7, -11, 1, -7], [-1, 12, 12, 0, -3, -9], [3, 3, -1, -1, -2, 1], [4, 7, -3, -7, 0, 7], [-3, 6, 0, 3, -6, -6], [0, 6, -9, 1, -9, 6], [-3, 11, -4, 4, -11, 3], [-5, 1, 0, 4, 4, 10], [0, 0, -6, 12, -6, 0], [1, 0, 6, -1, -6, -6], [4, 4, 7, 1, -5, -7], [-7, 0, 7, -4, -1, -11], [-5, -11, -3, 2, 0, -2], [10, -2, -10, -2, 10, 0], [3, 0, 9, -3, -9, -9], [-8, 2, 0, 6, 6, -16], [-1, 2, -11, 11, -2, 1], [0, -9, 3, -15, 3, -9], [5, -11, 1, -11, 5, 0], [5, 0, -5, -3, -13, 2], [-9, 9, 9, 0, 0, -9], [-7, -6, -6, 0, 3, 3], [-13, 0, 13, 17, -1, 4], [4, -2, -12, -16, 0, 16], [7, -14, -1, -14, 7, 0], [7, -16, 0, 9, 9, 8], [-11, -15, 12, 11, 3, 3], [-4, 10, -2, 10, -4, 0], [-1, -5, 1, 2, 0, -2], [-4, 0, 4, 0, -25, -4], [20, 0, -20, -4, -1, 16], [-8, -2, 8, -2, -8, 0], [1, -7, 0, 6, 6, 11], [-4, 20, -8, 8, -20, 4], [-9, 17, -1, 1, -17, 9], [0, 9, 0, 15, 0, 9], [-1, 19, -5, -4, 0, 4], [-11, -11, -11, -5, 16, 11], [-1, 15, 15, 0, -18, 3], [-3, -3, 5, -10, 13, -5], [-2, 4, 9, 11, 0, -11], [0, 9, -15, -8, -15, 9], [4, 2, 0, -6, -6, -4], [-8, 1, 0, 7, 7, 4], [14, -14, -14, -14, 14, 0], [11, 4, 16, 5, 0, -5], [-3, -15, -15, 0, -9, 24], [-19, -6, -6, 0, 15, -9], [14, 0, -14, -16, 2, -2], [-5, 16, -12, -7, 0, 7], [-18, 12, 9, 18, -21, -21], [-7, -7, -5, 5, 7, 7], [-6, -13, -16, 16, 13, 6], [22, -5, 0, -17, -17, -5], [0, -9, 12, 21, 12, -9], [-13, 0, 13, 15, 14, 2], [8, -3, -3, 0, -9, 12], [-6, -6, 8, 8, -2, -8], [7, -2, 2, -2, 7, 0], [24, -3, 0, -24, 3, 3], [4, 11, 0, -15, -15, -4], [0, 0, -3, -27, -3, 0], [-2, 3, 12, 2, -15, -15], [17, -18, -18, 0, 3, 15], [5, 0, -5, -7, -16, -2], [15, 15, -9, -15, -6, -6], [4, 11, 0, -15, -15, 14], [15, 2, 23, -23, -2, -15], [-11, -2, 0, 13, 13, -23], [-2, 3, -15, 2, 12, 12], [-22, 0, 22, 12, 17, -10], [-6, -12, -12, 0, -9, 21], [-6, -6, -1, -1, 7, 1], [2, 0, -2, 5, -19, 7], [-20, 4, -10, 4, -20, 0], [0, 3, -6, -11, -6, 3], [1, 11, 0, -12, -12, -7], [23, -13, 16, -16, 13, -23], [-8, 12, -27, 8, 15, 15], [14, 0, -14, -12, 5, 2], [-17, -17, 13, 10, 7, -13], [-3, -6, -6, 0, 9, -3], [15, 15, 23, -19, 4, -23], [-17, 1, -6, 11, 0, -11], [10, 10, 2, 10, 10, 0], [11, -16, 4, -4, 16, -11], [-18, -1, -19, 19, 1, 18], [0, 9, -12, -9, -12, 9], [32, 13, 13, -19, 0, 19], [-15, 24, 24, 0, 0, -24], [2, 0, -2, -13, -1, -11], [-21, -12, 0, 21, 12, 12], [2, 14, 10, -10, -14, -2], [16, -2, 0, -14, -14, 13], [19, 0, 15, -19, -15, -15], [14, -14, -23, -14, 14, 0], [13, 13, 25, 10, -23, -25], [-25, -6, -6, 0, -15, 21], [-9, -9, -25, -10, 19, 25], [16, -5, -10, -5, 16, 0], [0, -6, -27, 0, 33, 33], [1, -20, 0, 19, 19, -8], [-4, 19, 16, 19, -4, 0], [18, -9, -9, 0, 9, 0], [17, -3, -3, 0, 0, 3], [7, 13, -9, -16, 0, 16], [-5, 1, 17, 1, -5, 0], [-9, -15, 18, 9, -3, -3], [0, -24, 6, -8, 6, -24], [21, -4, 11, -11, 4, -21], [-4, -17, -11, -7, 0, 7], [-4, 0, 4, 3, 17, -1], [-14, -14, -29, 10, 4, 29], [-3, 12, 12, 0, 9, -21], [-5, 10, -10, 10, -5, 0], [3, -18, 0, -3, 18, 18], [0, -24, 24, 1, 24, -24], [-2, 11, 0, -9, -9, -1], [-4, 4, 4, 8, 0, -8], [5, 0, -5, 9, 2, 14], [11, -6, -6, 0, -6, 12], [-11, -23, 0, 11, 0, -11], [0, 15, -9, -8, -9, 15], [-5, -16, 0, 21, 21, -22], [9, -10, -10, 10, 10, -9], [0, -18, 18, -3, 18, -18], [4, 4, -23, 13, -17, 23], [-24, -3, -3, 0, 18, -15], [-18, -18, -16, -1, 19, 16], [-2, -14, 0, 2, 0, -2], [0, 0, 0, -2, 0, 0], [-28, 5, -23, 23, -5, 28]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : 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, 0, 9, 3, 0, -3, 1]>,<5,R![9, 0, 36, -30, 0, 3, 1]>,<7,R![289, -153, 36, -10, 0, 0, 1]>],Snew);
    return Vf;
end function;