
// Make newform 1323.2.f.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1323_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1323_f_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_1323_2_f_d();".
// 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_1323_2_f_d();".
// 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;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1], [0, 1, 1, 0, -1, 0], [0, 0, 0, 0, 1, -1], [0, 1, 0, 0, -1, -1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1];
    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_1323_f();"
function MakeCharacter_1323_f()
    N := 1323;
    order := 3;
    char_gens := [785, 1081];
    v := [1, 3];
    // 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_1323_f_Hecke();"
function MakeCharacter_1323_f_Hecke(Kf)
    N := 1323;
    order := 3;
    char_gens := [785, 1081];
    char_values := [[-1, 1, 0, 0, 0, 0], [1, 0, 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 := [[0, 1, 0, -1, -1, 1], [0, 0, 0, 0, 0, 0], [-1, 1, -1, 1, 0, -1], [0, 0, 0, 0, 0, 0], [0, 2, 1, -1, -1, 1], [-1, 1, 4, -4, 0, 2], [2, 0, 0, -1, 0, 0], [1, 0, 0, 1, 2, 0], [4, -4, -2, 2, 0, 1], [0, 3, -4, 5, 5, -5], [-1, 1, -3, 3, 0, 3], [-1, 0, 0, 6, 3, 0], [0, 0, 1, -1, 0, -1], [0, 1, 1, 1, 1, -1], [0, -1, -3, -2, -2, 2], [-2, 0, 0, 1, 3, 0], [1, -1, 0, 0, 0, 5], [0, 2, 0, 3, 3, -3], [4, -4, 3, -3, 0, 0], [-3, 0, 0, 3, -3, 0], [7, 0, 0, -5, -4, 0], [0, 7, -3, 0, 0, 0], [0, 6, 1, 4, 4, -4], [4, 0, 0, 4, -3, 0], [0, -1, 8, -7, -7, 7], [0, 7, -1, -2, -2, 2], [2, -2, 0, 0, 0, -3], [-1, 0, 0, -4, 0, 0], [5, 0, 0, -3, 0, 0], [12, -12, -4, 4, 0, -2], [-4, 0, 0, -8, -13, 0], [5, -5, -10, 10, 0, -7], [0, 5, 7, -7, -7, 7], [5, -5, -5, 5, 0, -10], [-2, 2, 7, -7, 0, 7], [0, 1, 12, -9, -9, 9], [-1, 1, 5, -5, 0, 10], [2, 0, 0, -3, 0, 0], [3, -3, 4, -4, 0, 14], [0, -13, 3, 7, 7, -7], [7, 0, 0, 1, 0, 0], [7, 0, 0, 6, 3, 0], [0, 1, 9, -1, -1, 1], [-5, 5, 0, 0, 0, 3], [-15, 0, 0, 3, -3, 0], [-2, 0, 0, 3, 3, 0], [1, -1, 2, -2, 0, -2], [0, 5, -6, 6, 6, -6], [0, 7, 2, 1, 1, -1], [5, -5, 0, 0, 0, -12], [-7, 0, 0, -4, -9, 0], [3, -3, 2, -2, 0, -8], [0, -10, 0, -3, -3, 3], [-6, 0, 0, 15, 6, 0], [-16, 16, 2, -2, 0, -4], [0, -1, -17, 14, 14, -14], [-7, 0, 0, -7, -9, 0], [-8, 0, 0, 0, 3, 0], [0, -2, -3, -6, -6, 6], [0, 18, -7, 8, 8, -8], [5, -5, -3, 3, 0, -15], [-3, 3, -2, 2, 0, -7], [-2, 0, 0, -1, -8, 0], [16, -16, 3, -3, 0, -1], [0, -1, 10, -17, -17, 17], [0, -8, -3, 11, 11, -11], [0, -14, -1, -4, -4, 4], [-5, 5, 12, -12, 0, -3], [5, -5, -15, 15, 0, -8], [0, 8, 8, -10, -10, 10], [0, -8, 2, -5, -5, 5], [2, 0, 0, 11, 6, 0], [0, 5, -5, -5, -5, 5], [-5, 5, -5, 5, 0, -1], [-1, 0, 0, -5, 5, 0], [11, -11, 11, -11, 0, -1], [0, -7, 13, -4, -4, 4], [19, 0, 0, 6, 3, 0], [4, -4, 7, -7, 0, -11], [-4, 4, 18, -18, 0, 12], [12, -12, -8, 8, 0, -13], [0, 13, 15, -6, -6, 6], [-5, 0, 0, 19, 15, 0], [-17, 0, 0, -4, 10, 0], [0, 11, -8, -11, -11, 11], [0, -2, -6, -4, -4, 4], [0, 0, 0, -3, -6, 0], [0, 19, 15, -12, -12, 12], [0, -6, 7, -2, -2, 2], [4, -4, 6, -6, 0, 3], [-23, 0, 0, -8, 3, 0], [0, 25, -7, 4, 4, -4], [-7, 0, 0, 17, 1, 0], [-9, 9, -7, 7, 0, 13], [-2, 2, -18, 18, 0, -6], [-12, 0, 0, 18, 9, 0], [8, -8, 2, -2, 0, 11], [-13, 0, 0, -4, -6, 0], [-11, 0, 0, -3, -15, 0], [11, 0, 0, 8, 10, 0], [0, -8, -20, 1, 1, -1], [1, 0, 0, -17, -21, 0], [10, -10, -21, 21, 0, -13], [0, -17, 6, -10, -10, 10], [13, -13, -14, 14, 0, -13], [16, 0, 0, 9, 9, 0], [0, -18, 13, 7, 7, -7], [19, 0, 0, 13, 6, 0], [14, -14, -12, 12, 0, -14], [0, -22, -1, 5, 5, -5], [17, -17, 8, -8, 0, 22], [-1, 0, 0, -18, 6, 0], [24, -24, 11, -11, 0, -5], [0, -22, 0, 15, 15, -15], [2, 0, 0, 16, -1, 0], [0, 14, 10, -13, -13, 13], [-19, 19, -2, 2, 0, -13], [-7, 0, 0, -1, 18, 0], [-8, 8, -23, 23, 0, -2], [0, -3, 5, -28, -28, 28], [5, -5, -22, 22, 0, -5], [0, -14, 3, -9, -9, 9], [0, -1, -24, 1, 1, -1], [-32, 0, 0, -2, 0, 0], [0, -4, 15, -9, -9, 9], [-6, 0, 0, 6, 12, 0], [0, -2, -12, 6, 6, -6], [-11, 0, 0, -5, -21, 0], [0, -13, -19, 17, 17, -17], [11, -11, -12, 12, 0, 6], [-31, 0, 0, 3, 3, 0], [-6, 6, 23, -23, 0, 10], [-11, 11, -11, 11, 0, -16], [14, 0, 0, -5, 8, 0], [-10, 10, -16, 16, 0, 2], [20, -20, 12, -12, 0, 9], [-1, 0, 0, -4, 12, 0], [-19, 19, -9, 9, 0, -15], [-24, 24, 19, -19, 0, 2], [-8, 0, 0, -5, -12, 0], [4, 0, 0, 15, -9, 0], [0, -5, -3, -22, -22, 22], [-32, 32, -12, 12, 0, 0], [-3, 0, 0, -21, -3, 0], [-5, 0, 0, -11, -10, 0], [0, -9, 10, -8, -8, 8], [0, 11, -15, 0, 0, 0], [0, 4, 14, -17, -17, 17], [-31, 31, 0, 0, 0, -12], [2, 0, 0, -1, -12, 0], [-2, 2, -9, 9, 0, 0], [33, 0, 0, 6, 21, 0], [5, 0, 0, -18, 3, 0], [-4, 4, 26, -26, 0, 17], [0, -17, 6, 18, 18, -18], [0, 9, -22, 11, 11, -11], [8, 0, 0, 12, 3, 0], [0, -25, -12, 19, 19, -19], [-2, 0, 0, -1, 16, 0], [31, -31, 15, -15, 0, -1], [0, -8, -12, 17, 17, -17], [-18, 0, 0, 12, 24, 0], [22, -22, 11, -11, 0, 28], [-26, 0, 0, 1, -18, 0], [17, -17, 24, -24, 0, 16], [0, -32, -4, 13, 13, -13], [35, 0, 0, 15, 0, 0], [0, 2, 24, -12, -12, 12], [14, 0, 0, -11, 14, 0], [-13, 13, -19, 19, 0, -4], [0, 41, -2, -1, -1, 1], [-28, 28, -3, 3, 0, 9], [7, -7, -8, 8, 0, -8], [0, 19, -4, 8, 8, -8], [5, -5, -6, 6, 0, 6], [-18, 18, 31, -31, 0, 11], [0, 7, 7, 7, 7, -7], [19, 0, 0, -20, -21, 0], [16, 0, 0, 11, 31, 0], [0, -37, -15, 3, 3, -3], [0, -2, 26, -7, -7, 7], [0, 9, -29, 4, 4, -4], [-8, 8, -16, 16, 0, -14], [19, 0, 0, -29, -18, 0], [-10, 10, -6, 6, 0, 1], [0, 19, 23, -2, -2, 2], [-7, 0, 0, -18, -27, 0], [0, 14, -5, 22, 22, -22], [16, -16, 21, -21, 0, -21], [-1, 0, 0, -31, -12, 0], [25, 0, 0, -17, -1, 0], [0, -15, 23, -25, -25, 25], [-19, 0, 0, -15, -27, 0], [0, 20, 3, -5, -5, 5], [-17, 0, 0, 28, 27, 0], [-32, 32, 12, -12, 0, -1], [13, -13, -11, 11, 0, 20], [0, -5, -21, 36, 36, -36], [0, -36, 1, -8, -8, 8], [-11, 0, 0, -14, 6, 0], [-25, 25, 30, -30, 0, 16], [0, 5, 32, -10, -10, 10], [-43, 43, 0, 0, 0, 3], [0, -16, 15, -21, -21, 21], [6, 0, 0, -6, 27, 0], [-40, 0, 0, 8, 3, 0], [-26, 0, 0, -6, -30, 0], [16, -16, 4, -4, 0, -17], [0, 12, 8, -10, -10, 10], [14, -14, 27, -27, 0, 9], [14, 0, 0, -36, -21, 0], [21, -21, -5, 5, 0, 8], [0, -20, -2, -14, -14, 14], [0, -10, 27, -11, -11, 11], [-2, 2, -3, 3, 0, -34], [0, 17, -6, 27, 27, -27], [28, -28, -32, 32, 0, -19], [0, 25, 17, 7, 7, -7], [-40, 0, 0, 17, 12, 0], [21, -21, -4, 4, 0, 7], [-11, 11, 0, 0, 0, 27], [-22, 22, -8, 8, 0, -4], [16, -16, 4, -4, 0, -20], [-1, 0, 0, -10, 7, 0], [-6, 6, -8, 8, 0, -10], [0, -20, -29, -2, -2, 2], [0, -28, 27, -11, -11, 11], [-20, 0, 0, 1, 3, 0], [0, -34, -23, 1, 1, -1], [0, -5, -9, 23, 23, -23], [-11, 11, -11, 11, 0, 17], [22, 0, 0, -2, -10, 0], [-20, 20, -10, 10, 0, -2], [-7, 7, -3, 3, 0, 4], [0, 2, 2, 20, 20, -20], [0, 19, -31, 43, 43, -43], [50, -50, -10, 10, 0, -2], [2, 0, 0, 11, 15, 0], [33, -33, -13, 13, 0, -17], [-3, 0, 0, -3, 6, 0], [0, 14, 13, -7, -7, 7], [25, 0, 0, -5, 5, 0], [-16, 16, -19, 19, 0, -23], [-7, 0, 0, 27, -9, 0], [6, -6, 22, -22, 0, 35], [0, -10, -15, -2, -2, 2], [7, 0, 0, 11, 1, 0], [-2, 2, -3, 3, 0, 29], [-17, 17, 43, -43, 0, 14], [3, 0, 0, -12, -18, 0], [16, -16, -1, 1, 0, -17], [10, 0, 0, 34, 12, 0], [38, -38, 12, -12, 0, 1], [0, 2, -7, -4, -4, 4], [0, 19, 5, -38, -38, 38], [-4, 0, 0, -19, -30, 0], [5, 0, 0, 3, 18, 0], [0, 41, -21, 18, 18, -18], [-3, 0, 0, -12, -21, 0], [-17, 0, 0, -32, -28, 0], [0, -26, 14, -19, -19, 19], [0, 18, 5, -10, -10, 10], [-16, 16, -15, 15, 0, -21], [7, 0, 0, -34, -14, 0], [16, -16, -24, 24, 0, -16], [0, 2, -30, 24, 24, -24], [3, 0, 0, 9, 12, 0], [0, 15, 1, 31, 31, -31], [16, -16, 21, -21, 0, -15], [-46, 46, -6, 6, 0, 1], [-28, 28, -13, 13, 0, 16], [11, 0, 0, -15, -6, 0], [0, 5, 6, -36, -36, 36], [28, -28, -33, 33, 0, -12], [-25, 25, -24, 24, 0, -27], [31, 0, 0, -12, -18, 0], [10, -10, 10, -10, 0, 19], [0, 1, 3, -24, -24, 24], [2, 0, 0, -18, -12, 0], [0, 35, -15, 7, 7, -7], [-17, 17, -24, 24, 0, -4], [-26, 26, -9, 9, 0, -27], [0, 12, -20, 13, 13, -13], [0, -1, 32, 11, 11, -11], [-10, 10, 15, -15, 0, 12], [-7, 0, 0, -25, -6, 0], [-7, 0, 0, 9, -12, 0], [-3, 3, 32, -32, 0, 31], [0, -7, 18, 9, 9, -9], [-48, 0, 0, -9, 12, 0], [0, -52, 10, 23, 23, -23], [26, 0, 0, -16, -12, 0], [13, -13, -5, 5, 0, 34], [-3, 3, 13, -13, 0, 8], [0, 22, -26, 37, 37, -37], [37, -37, -3, 3, 0, 17], [0, 23, 6, 9, 9, -9], [0, 30, 16, 10, 10, -10], [31, 0, 0, -26, -9, 0], [0, -1, -28, 8, 8, -8], [-43, 43, 12, -12, 0, 0], [26, 0, 0, 17, 51, 0], [5, 0, 0, -1, -5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1323_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1323_2_f_d();".
// 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_1323_2_f_d(:prec:=6)
    chi := MakeCharacter_1323_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1323_2_f_d();".
// 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_1323_2_f_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1323_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 0, 9, -6, 9, -3, 1]>,<5,R![9, 0, 9, 6, 9, 3, 1]>],Snew);
    return Vf;
end function;