
// Make newform 666.2.x.c in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_666_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_666_x_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_666_2_x_c();".
// 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_666_2_x_c();".
// 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_666_x();"
function MakeCharacter_666_x()
    N := 666;
    order := 9;
    char_gens := [371, 631];
    v := [9, 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_666_x_Hecke();"
function MakeCharacter_666_x_Hecke(Kf)
    N := 666;
    order := 9;
    char_gens := [371, 631];
    char_values := [[1, 0, 0, 0, 0, 0], [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, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0], [-1, -1, -2, 1, 0, 2], [1, 1, -1, 0, -1, 1], [0, -1, 1, 3, 1, -1], [-1, -1, 0, 2, 2, 3], [0, 0, 0, 0, 3, 0], [0, 3, -1, 3, 0, 0], [5, -2, 3, -5, -1, -1], [0, 4, -5, 0, -5, 4], [-3, 1, 1, 0, 0, -1], [-1, -2, -4, 5, 2, 2], [-4, -2, 0, 6, 6, -2], [2, 1, 1, 0, 5, -6], [1, 0, -4, -1, 4, 4], [5, -3, 1, -4, 0, 4], [-3, 1, 5, 8, 0, -8], [0, 4, 0, -4, -4, 4], [-1, -1, -7, 1, 0, 7], [4, -6, -8, -6, 4, 0], [-6, -1, -1, 0, -2, 3], [6, 6, 1, -2, -4, -1], [1, 0, -1, -4, 3, -3], [8, 1, 3, -5, 0, 5], [14, 0, 2, -14, -2, -2], [-4, -3, 7, 4, -4, -4], [0, -11, 3, 0, 3, -11], [0, -3, 0, 3, 3, 5], [0, -5, 8, -8, 5, 0], [-2, 7, 2, -2, -7, 2], [-3, 0, 4, 7, 0, -7], [-8, 0, 8, 4, -8, -4], [0, -3, 0, 3, 0, -3], [-10, 0, 10, 6, 4, -4], [5, 5, 5, 0, -5, 0], [-5, -1, 5, -1, -5, 0], [0, 0, 0, -1, -16, -1], [5, -1, 1, -4, 0, 4], [1, 9, 3, 9, 1, 0], [1, 8, 5, -5, -8, -1], [4, 0, 0, 0, 7, -7], [8, -1, 0, -7, -7, 10], [8, 5, 5, 0, -10, 5], [0, 11, -6, 6, -6, 11], [-6, -7, 5, -5, 7, 6], [3, -1, 3, -3, -2, -2], [0, -4, 10, -5, 10, -4], [-10, 5, 5, 0, -4, -1], [-9, -9, -5, 7, 2, 5], [-1, -1, 14, 2, -1, -14], [0, 6, -6, 5, -6, 6], [-5, 0, 5, -6, -3, -11], [9, 0, -4, 0, 9, 0], [0, -1, 6, 7, 6, -1], [8, 5, 13, -13, -5, -8], [-5, -9, 0, 14, 14, -2], [13, 12, -10, -13, -2, -2], [-8, 1, -6, 6, -1, 8], [6, 16, -7, 16, 6, 0], [3, 14, -7, -10, 0, 10], [-9, 8, 0, 1, 1, -6], [0, -2, 1, -2, 0, 0], [0, -1, -1, 10, -1, -1], [-8, -4, -8, 0, 0, 0], [-2, 0, 2, 1, -14, -1], [-11, -5, -7, 4, 0, -4], [4, 10, -7, 7, -10, -4], [-4, -3, 0, 7, 7, -7], [11, 7, -6, -11, -1, -1], [3, -3, 10, 7, 0, -7], [-13, 0, 13, -3, 2, -16], [0, 7, -3, -4, -3, 7], [16, -23, -13, -23, 16, 0], [13, -5, -7, -5, 13, 0], [7, 0, -7, -2, -11, 5], [-20, 4, 0, 16, 16, -7], [-7, 3, -9, 3, -7, 0], [-6, 7, 6, 6, -13, -13], [-21, -6, -6, 0, 0, 6], [-8, -1, -5, 5, 1, 8], [17, 8, 2, -2, -8, -17], [-7, 1, -7, 7, 6, 6], [6, 6, 0, -12, -12, -2], [0, 21, -14, -1, -14, 21], [14, 14, 13, -20, 6, -13], [13, 2, 2, 0, 9, -11], [5, 5, 9, -9, 4, -9], [1, 5, 0, -6, -6, 23], [-7, 0, 7, 11, -5, 4], [-14, 7, 19, 7, -14, 0], [14, -15, -1, -14, 16, 16], [-20, 5, -12, 12, -5, 20], [20, -17, -17, 0, 2, 15], [2, -16, 19, -2, -3, -3], [4, 12, 7, 12, 4, 0], [-3, 28, -1, 2, 0, -2], [-3, 0, 3, 8, -14, 5], [-2, -9, 13, -9, -2, 0], [-19, -19, -8, 12, 7, 8], [-4, -3, 14, 4, -11, -11], [0, -16, 5, -27, 5, -16], [1, -9, -14, 14, 9, -1], [0, 8, -1, -2, -1, 8], [-3, -2, -1, 3, 3, 3], [-6, -1, 2, 8, 0, -8], [-6, 7, -7, -1, 0, 1], [5, 5, -11, 0, -5, 11], [-7, -7, -7, 0, -19, 26], [-3, -3, -7, -5, 8, 7], [-3, 0, 3, -8, 18, -11], [3, 1, -19, -22, 0, 22], [-3, 0, -19, -16, 0, 16], [9, 7, -6, 6, -7, -9], [-9, 5, 15, 9, -20, -20], [2, 23, -13, 13, -23, -2], [11, -15, 11, -11, 15, -11], [-14, 9, 15, 14, -24, -24], [-1, 6, -20, 6, -1, 0], [7, 0, 0, -7, -7, -20], [-7, -7, 24, 4, 3, -24], [1, 1, -17, -4, 3, 17], [23, 23, 7, -8, -15, -7], [0, -12, 12, -13, 12, -12], [4, 0, -4, -5, -33, -1], [13, 15, 33, -33, -15, -13], [3, -7, -12, 12, 7, -3], [-9, -1, -1, 0, -7, 8], [-4, 0, 13, 17, 0, -17], [-9, -16, 0, 25, 25, -19], [7, 7, 15, -9, 2, -15], [-2, 15, 15, 0, 3, -18], [-4, -3, 14, -3, -4, 0], [0, 23, -16, -5, -16, 23], [-9, 0, 9, -10, -13, -19], [-16, 11, -5, 11, 0, -11], [0, 6, 17, 11, 17, 6], [-9, -2, 0, 11, 11, 25], [-16, -2, -23, 16, 25, 25], [-3, 0, 3, 7, 9, 4], [12, 12, 26, -4, -8, -26], [23, -25, -6, -25, 23, 0], [-1, -1, 5, -10, 11, -5], [-17, 0, 17, 16, 21, -1], [-2, 9, 0, -7, -7, 9], [3, 20, 16, 13, 0, -13], [23, -4, 22, -22, 4, -23], [-12, -13, -21, 21, 13, 12], [1, -26, -26, 0, 12, 14], [0, 18, 2, 12, 2, 18], [13, 16, 12, -12, -16, -13], [0, 23, 8, 2, 8, 23], [-10, -10, -2, -9, 19, 2], [0, 0, -11, -7, 7, 11], [-10, -20, -20, 0, -5, 25], [-13, -8, -9, -8, -13, 0], [-3, 24, -17, 3, -7, -7], [-1, 21, 21, 0, -11, -10], [22, -9, 0, -13, -13, -21], [17, 25, -6, 6, -25, -17], [-23, 15, -16, 7, 0, -7], [-13, -22, 6, 19, 0, -19], [-12, 0, 12, 33, -19, 21], [19, 19, 12, -9, -10, -12], [-23, 0, 23, 32, 5, 9], [4, -1, -23, -27, 0, 27], [18, 8, -10, -28, 0, 28], [0, -6, 4, 22, 4, -6], [-20, -9, -19, 19, 9, 20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_666_x_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_666_2_x_c();".
// 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_666_2_x_c(:prec:=6)
    chi := MakeCharacter_666_x();
    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_666_2_x_c();".
// 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_666_2_x_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_666_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -3, 69, -8, -6, 3, 1]>],Snew);
    return Vf;
end function;