
// Make newform 567.2.i.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_567_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_567_i_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_567_2_i_f();".
// 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_567_2_i_f();".
// 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 := [81, 0, -198, 0, 403, 0, -180, 0, 59, 0, -9, 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], [0, 76298, 0, 1782, 0, -3627, 0, 3481, 0, -531, 0, 81], [-60864, 0, 90554, 0, -27234, 0, 7472, 0, -767, 0, 117, 0], [240966, 0, -490451, 0, 202311, 0, -71803, 0, 10953, 0, -1298, 0], [-186237, 0, 142112, 0, -98847, 0, 28501, 0, -5466, 0, 461, 0], [0, -438921, 0, 490451, 0, -202311, 0, 71803, 0, -10953, 0, 1298], [0, 154944, 0, 490451, 0, -202311, 0, 71803, 0, -10953, 0, 1298], [-68439, 0, -6336, 0, 12896, 0, -9933, 0, 1888, 0, -288, 0], [0, -90434, 0, -6336, 0, 12896, 0, -9933, 0, 1888, 0, -288], [-271656, 0, 552916, 0, -273351, 0, 80948, 0, -12348, 0, 1138, 0], [0, -1055934, 0, 2149199, 0, -1022364, 0, 314647, 0, -47997, 0, 4712], [0, 172044, 0, -1763849, 0, 809244, 0, -287212, 0, 43812, 0, -5192]];
    Rf_basisdens := [1, 21995, 21995, 197955, 65985, 197955, 197955, 21995, 21995, 65985, 197955, 197955];
    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_567_i();"
function MakeCharacter_567_i()
    N := 567;
    order := 6;
    char_gens := [407, 325];
    v := [5, 5];
    // 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_567_i_Hecke();"
function MakeCharacter_567_i_Hecke(Kf)
    N := 567;
    order := 6;
    char_gens := [407, 325];
    char_values := [[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [2, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 0, 0], [0, 1, 0, 0, 0, -2, -1, 0, 0, 0, 1, 1], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [-1, 0, 1, 2, 0, 0, 0, -2, 0, 3, 0, 0], [0, 0, -2, 2, -1, 0, 0, 1, 0, 1, 0, 0], [0, -1, 0, 0, 0, 1, 1, 0, 1, 0, -2, -2], [-1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 1, 0, 0, 0, 0, 0, -2], [0, -2, 0, 0, 0, 2, 1, 0, 0, 0, -1, -2], [0, -2, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1], [-1, 0, 2, 2, 0, 0, 0, -2, 0, 2, 0, 0], [-7, 0, 2, 0, 4, 0, 0, -1, 0, -2, 0, 0], [0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, -1], [-3, 0, 3, -3, 3, 0, 0, -2, 0, -2, 0, 0], [5, 0, 2, 0, 4, 0, 0, -1, 0, -2, 0, 0], [0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2], [0, -2, 0, 0, 0, 0, 0, 0, 1, 0, -2, -4], [4, 0, -2, 4, -2, 0, 0, 2, 0, 2, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0, -2, 0, 4, 0], [1, 0, 2, 1, 2, 0, 0, -2, 0, -2, 0, 0], [0, 5, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0], [2, 0, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 2, -2, 0, 1, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, 2, 0, 4, 0, -2, 3], [0, 2, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0], [-2, 0, 0, 1, 1, 0, 0, 3, 0, -2, 0, 0], [0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1], [0, 0, 2, -10, 1, 0, 0, -1, 0, -3, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 10, 0, 0, 0, 0, 0, -3, 0, 0], [0, -4, 0, 0, 0, 1, 1, 0, -1, 0, 2, -8], [0, 2, 0, 0, 0, -2, 0, 0, -1, 0, -1, 1], [0, 4, 0, 0, 0, -6, -3, 0, 0, 0, -2, 4], [-3, 0, 3, 6, 0, 0, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 3, -1], [-5, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -2, 0, -4, 0, 4, -4], [-5, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, -4, -1, 0, 0, 1, 0, 7, 0, 0], [-3, 0, 7, -3, 7, 0, 0, -3, 0, -3, 0, 0], [0, 3, 0, 0, 0, -1, -1, 0, 0, 0, 0, 6], [2, 0, -4, 2, -4, 0, 0, 3, 0, 3, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 11, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0], [-10, 0, 0, 5, -4, 0, 0, 4, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, -2, 0, -2, -3], [0, 0, 0, 0, 0, 0, 2, 0, -6, 0, 3, 0], [0, -1, 0, 0, 0, -3, 3, 0, 4, 0, 0, 0], [0, -1, 0, 0, 0, 0, 6, 0, 2, 0, -1, 1], [22, 0, 0, -11, 0, 0, 0, -2, 0, 1, 0, 0], [0, 0, -2, -5, -1, 0, 0, 1, 0, -7, 0, 0], [0, 1, 0, 0, 0, -2, -1, 0, 0, 0, -6, 1], [-8, 0, 1, 16, 0, 0, 0, -1, 0, 1, 0, 0], [0, -1, 0, 0, 0, 1, 1, 0, -4, 0, 8, -2], [5, 0, -3, -10, 0, 0, 0, 1, 0, 1, 0, 0], [0, 4, 0, 0, 0, 2, 0, 0, -4, 0, -4, 2], [3, 0, -4, -6, 0, 0, 0, 6, 0, -8, 0, 0], [0, 0, 0, 0, 0, -1, -2, 0, 1, 0, -1, 8], [-2, 0, 3, 0, 6, 0, 0, -2, 0, -3, 0, 0], [0, 0, -2, 8, -1, 0, 0, 1, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4], [8, 0, -11, 8, -11, 0, 0, 7, 0, 7, 0, 0], [0, -6, 0, 0, 0, 5, 5, 0, 0, 0, 0, -12], [0, -7, 0, 0, 0, -4, 4, 0, 1, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, -6, 0, 2, 0, 0], [4, 0, -2, -4, 2, 0, 0, 10, 0, -10, 0, 0], [-2, 0, 1, 0, 2, 0, 0, 2, 0, -1, 0, 0], [0, -3, 0, 0, 0, 0, -7, 0, 4, 0, -2, 3], [0, -7, 0, 0, 0, 1, -1, 0, -4, 0, 0, 0], [16, 0, 0, -8, -3, 0, 0, 7, 0, -2, 0, 0], [0, -5, 0, 0, 0, -6, -3, 0, 0, 0, 5, -5], [7, 0, -2, -14, 0, 0, 0, -1, 0, 4, 0, 0], [0, -4, 0, 0, 0, -2, -2, 0, 1, 0, -2, -8], [1, 0, 0, -1, 0, 0, 0, 3, 0, -3, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, -8, -5], [10, 0, 0, -20, 0, 0, 0, -1, 0, 2, 0, 0], [-3, 0, -4, 6, 0, 0, 0, 0, 0, 4, 0, 0], [0, 0, 0, 0, 0, 2, 4, 0, -5, 0, 5, -1], [0, 0, 0, 0, 0, -2, -4, 0, 3, 0, -3, -1], [19, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, -5, 0, 2, 0, -1, 5], [0, 0, 0, -14, 0, 0, 0, 0, 0, -6, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, -5, 0, 10, 2], [0, -6, 0, 0, 0, -7, -7, 0, -1, 0, 2, -12], [-4, 0, 1, 4, -1, 0, 0, 6, 0, -6, 0, 0], [0, 5, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-4, 0, -1, 4, 1, 0, 0, -2, 0, 2, 0, 0], [0, -6, 0, 0, 0, 0, 0, 0, 1, 0, 1, -3], [0, -9, 0, 0, 0, 0, -4, 0, -6, 0, 3, 9], [0, 5, 0, 0, 0, 0, 6, 0, -2, 0, 1, -5], [2, 0, 0, -1, -12, 0, 0, -6, 0, 9, 0, 0], [0, 0, -6, 16, -3, 0, 0, 3, 0, -3, 0, 0], [4, 0, -6, -4, 6, 0, 0, 3, 0, -3, 0, 0], [0, 7, 0, 0, 0, 2, 1, 0, 0, 0, 8, 7], [0, -8, 0, 0, 0, 5, 0, 0, 0, 0, 0, -4], [0, 0, 0, 0, 0, 1, 2, 0, 8, 0, -8, 5], [14, 0, -1, 0, -2, 0, 0, -1, 0, 1, 0, 0], [-8, 0, 15, -8, 15, 0, 0, -9, 0, -9, 0, 0], [0, -2, 0, 0, 0, 0, -8, 0, 0, 0, 0, 2], [0, 4, 0, 0, 0, 3, 3, 0, 1, 0, -2, 8], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 3, 14], [-7, 0, 7, -7, 7, 0, 0, -5, 0, -5, 0, 0], [6, 0, 5, 6, 5, 0, 0, 2, 0, 2, 0, 0], [10, 0, -6, -10, 6, 0, 0, 9, 0, -9, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0], [-4, 0, 0, 2, -1, 0, 0, 13, 0, -6, 0, 0], [-10, 0, -3, 0, -6, 0, 0, 0, 0, 3, 0, 0], [0, 2, 0, 0, 0, -1, 1, 0, 3, 0, 0, 0], [-14, 0, 0, 7, 7, 0, 0, -3, 0, -2, 0, 0], [0, -10, 0, 0, 0, 0, -3, 0, -4, 0, 2, 10], [0, 1, 0, 0, 0, 8, 4, 0, 0, 0, 3, 1], [0, 19, 0, 0, 0, -2, -1, 0, 0, 0, 6, 19], [12, 0, 1, -24, 0, 0, 0, 6, 0, -13, 0, 0], [-16, 0, 4, 16, -4, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, -2, 0, 0, 3, 0, 3, -2], [0, -5, 0, 0, 0, -8, -4, 0, 0, 0, 3, -5], [1, 0, 11, -2, 0, 0, 0, -4, 0, -3, 0, 0], [0, 0, 0, 0, 0, -5, -10, 0, -9, 0, 9, -10], [19, 0, -3, 0, -6, 0, 0, 1, 0, 3, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 2, 0, -4, -4], [-8, 0, -5, -8, -5, 0, 0, 5, 0, 5, 0, 0], [-5, 0, 8, -5, 8, 0, 0, -5, 0, -5, 0, 0], [10, 0, -6, -10, 6, 0, 0, -9, 0, 9, 0, 0], [0, -13, 0, 0, 0, 1, -1, 0, -9, 0, 0, 0], [-28, 0, -4, 28, 4, 0, 0, -2, 0, 2, 0, 0], [25, 0, -8, 0, -16, 0, 0, 8, 0, 8, 0, 0], [0, 3, 0, 0, 0, 0, 5, 0, 4, 0, -2, -3], [12, 0, 0, -6, -8, 0, 0, 10, 0, -1, 0, 0], [0, -1, 0, 0, 0, 0, 3, 0, 2, 0, -1, 1], [-5, 0, 4, 10, 0, 0, 0, 2, 0, -8, 0, 0], [0, -4, 0, 0, 0, -2, -2, 0, 2, 0, -4, -8], [0, -14, 0, 0, 0, -2, -1, 0, 0, 0, 6, -14], [3, 0, 3, -6, 0, 0, 0, 6, 0, -15, 0, 0], [0, 0, 0, 0, 0, 4, 8, 0, 5, 0, -5, -13], [2, 0, -1, 0, -2, 0, 0, 2, 0, 1, 0, 0], [0, 0, 0, 0, 0, -1, -2, 0, 8, 0, -8, -4], [0, 0, -9, 0, -9, 0, 0, 13, 0, 13, 0, 0], [0, 4, 0, 0, 0, 0, -5, 0, -4, 0, 2, -4], [-15, 0, -8, -15, -8, 0, 0, -3, 0, -3, 0, 0], [0, 3, 0, 0, 0, 8, 8, 0, 5, 0, -10, 6], [7, 0, -4, 7, -4, 0, 0, 1, 0, 1, 0, 0], [0, 2, 0, 0, 0, -1, 1, 0, 13, 0, 0, 0], [-2, 0, -2, 2, 2, 0, 0, -2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 3, 0, 0, -3, 0, -3, 0], [-23, 0, -2, 0, -4, 0, 0, 11, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0], [0, 0, -4, 8, -2, 0, 0, 2, 0, -6, 0, 0], [0, -17, 0, 0, 0, -6, -3, 0, 0, 0, 4, -17], [0, 0, -8, 11, -4, 0, 0, 4, 0, -2, 0, 0], [0, -22, 0, 0, 0, -8, 0, 0, 4, 0, 4, -11], [-2, 0, -9, 4, 0, 0, 0, -4, 0, 17, 0, 0], [0, -2, 0, 0, 0, -4, 0, 0, -3, 0, -3, -1], [0, 0, 0, 0, 0, 5, 10, 0, -5, 0, 5, -1], [0, 0, 0, 0, 0, -3, -6, 0, 10, 0, -10, 8], [0, 0, 6, -2, 3, 0, 0, -3, 0, 6, 0, 0], [0, -5, 0, 0, 0, 9, 9, 0, 0, 0, 0, -10], [0, 0, 0, 0, 0, 2, 4, 0, 4, 0, -4, -13], [0, 9, 0, 0, 0, -4, -4, 0, -5, 0, 10, 18], [2, 0, 1, -2, -1, 0, 0, -9, 0, 9, 0, 0], [40, 0, 0, -20, -1, 0, 0, -1, 0, 1, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_567_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_567_2_i_f();".
// 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_567_2_i_f(:prec:=12)
    chi := MakeCharacter_567_i();
    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_567_2_i_f();".
// 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_567_2_i_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_567_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 0, 25, 0, 10, 0, 1]>,<11,R![50625, 0, -48150, 0, 37471, 0, -7468, 0, 1155, 0, -37, 0, 1]>,<13,R![675, 135, -171, -36, 51, -12, 1]>],Snew);
    return Vf;
end function;