
// Make newform 6027.2.a.w in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6027_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6027_2_a_w();".
// 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_6027_2_a_w();".
// 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 := [-2, 6, 4, -6, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [4, -3, -3, 1, 0], [2, 2, -5, -2, 1], [-1, -6, 3, 3, -1]];
    Rf_basisdens := [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_6027_a();"
function MakeCharacter_6027_a()
    N := 6027;
    order := 1;
    char_gens := [4019, 493, 2794];
    v := [1, 1, 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;

function MakeCharacter_6027_a_Hecke(Kf)
    return MakeCharacter_6027_a();
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, 0], [-1, 0, 0, 0, 0], [-1, 0, 0, -1, 0], [0, 0, 0, 0, 0], [1, 1, 1, 0, 1], [-1, 0, -1, 1, 0], [-2, 0, 0, 1, 0], [-3, 0, 0, -2, -1], [1, 1, 0, 0, -2], [0, 2, -1, -1, 1], [-1, 0, 0, 2, 1], [-2, 1, 1, 1, -1], [1, 0, 0, 0, 0], [-2, 2, 2, -3, -2], [3, -1, 1, 2, 0], [-3, 2, 1, -1, -2], [-2, 2, 0, 1, 0], [2, 1, 5, -2, 0], [-3, -1, -1, -2, -2], [2, -3, -3, 2, 2], [3, -2, 0, 0, 1], [4, -2, 0, 4, 1], [-8, -2, 2, 0, 2], [-1, -3, -1, 0, -1], [2, -2, 1, -1, -5], [-3, 3, -1, -2, -1], [-1, 6, 1, -4, -2], [4, 1, 0, 2, -1], [-1, 3, 1, -1, 5], [3, 1, -1, -4, 1], [-3, 3, -3, 2, 1], [-1, 1, 1, 3, 3], [-1, 1, 4, -1, 2], [-8, 2, 3, -2, -1], [-3, 0, 2, -4, -5], [-5, 7, 1, -4, -1], [7, 4, 6, 4, 5], [-1, -3, -1, -1, 5], [-3, -3, -7, 1, 0], [2, 1, -1, 6, 1], [3, -2, -4, 2, -3], [1, -4, -2, 0, 3], [1, -2, -4, 4, 7], [-4, -6, -4, 1, -2], [-8, -4, -4, -2, -2], [-6, -3, -5, 2, -2], [-6, 10, 2, -4, -3], [8, 2, -1, 2, 7], [-9, 7, 3, -4, -6], [15, 1, -2, -1, 0], [7, -4, -2, 6, -3], [7, -5, 1, 6, 1], [-6, 6, -4, -2, 4], [3, -5, 3, 2, 1], [-4, 2, 6, 4, 0], [5, -5, 3, 2, -3], [-11, 10, -2, -4, -4], [5, -7, 0, -6, -2], [-10, -7, -3, 4, 1], [10, -1, 2, 3, 5], [-2, -11, -7, 6, 6], [-2, 2, 4, 3, 4], [-2, -1, -2, -2, -1], [0, 0, 0, 8, -1], [10, -7, 0, 7, 3], [2, -11, -8, 7, 9], [3, -7, -5, -7, 0], [-7, 8, 4, 5, -4], [15, -6, 2, -8, -3], [-9, 10, -2, 2, 1], [-11, -5, -3, 0, -1], [5, 0, 1, 4, 8], [-14, 3, 0, 6, -1], [-11, 4, 2, 1, -2], [-8, -6, -2, -7, -8], [12, -10, -8, 4, 7], [3, -8, 2, 0, -5], [11, -6, -2, 2, 5], [-1, 5, 1, 4, -3], [-1, 5, 7, 6, 1], [2, 0, -8, 1, 6], [-8, 10, 2, 5, 6], [5, -8, -4, 8, -5], [-8, -6, -10, 6, 2], [-11, -2, 0, 4, 1], [3, 14, 4, 0, 1], [4, 1, 7, -6, -8], [-14, 2, 8, 0, -6], [5, -6, 0, -2, 0], [-7, -1, 5, -10, 2], [-9, 1, -1, -9, -1], [5, -1, -1, 4, 9], [7, -13, 3, 10, -1], [6, -3, -1, 2, 6], [21, -7, -3, 4, 5], [7, -3, -1, 1, -4], [-11, -7, -9, 3, -3], [-9, 5, 11, -10, -7], [9, -10, -4, 10, -1], [-8, -9, -3, 1, 3], [0, -8, -4, -9, 5], [-11, -6, 4, -6, 7], [-14, 4, 0, -5, -5], [14, -5, -9, 12, 14], [3, 3, 13, 0, 3], [-9, 6, 4, 12, -3], [4, 8, 10, -8, -8], [2, 10, -2, -6, 6], [-1, 7, 4, -8, -6], [12, -4, -1, 1, 3], [-11, -2, 6, -8, 3], [-11, -8, -2, 10, -2], [0, 2, 4, 10, 0], [-23, 2, 3, -8, 0], [2, 14, 4, 0, 6], [-15, 6, 10, -4, -9], [-2, 3, 5, -2, 0], [11, 3, -3, 1, -13], [-2, 3, 11, -14, -6], [19, -11, 1, -4, -5], [1, -5, -7, -2, 7], [6, -14, -4, -5, 4], [11, -7, 1, 4, -7], [3, -15, -5, 2, 5], [-1, -1, 1, 6, 1], [-11, 3, -7, -4, -9], [5, -9, -7, 2, 7], [-1, 9, 5, -12, -9], [10, -2, 2, 0, -6], [6, 0, 0, 4, -6], [-6, 0, -6, -3, -2], [-13, -6, -2, -2, 1], [9, -9, 11, 4, 0], [3, -11, -7, 8, 5], [-22, -3, 3, -7, 1], [-2, -4, -4, 8, -2], [13, -1, 1, -2, 9], [-7, -9, -2, -10, -4], [-4, -10, -4, -10, -10], [-8, 9, -10, 7, -1], [10, 1, -6, -4, -7], [-8, 1, -3, -8, -6], [16, -12, 2, 4, 8], [-26, 18, 0, -12, -2], [15, -9, 5, 2, -9], [0, 6, 2, 3, -7], [5, -1, -11, -2, -3], [2, 3, -1, -5, 1], [-4, 8, -5, 2, 9], [0, 15, 2, -2, 1], [-16, 2, 2, -10, -10], [28, -5, -1, 5, 9], [-19, 3, 5, -3, 4], [-24, -6, -4, -11, -1], [0, -4, -12, -2, 2], [-22, 17, -1, 6, 6], [6, 8, -2, -5, -9], [4, -12, -12, -9, -4], [19, -4, 12, -8, -7], [4, -3, 5, 4, -7], [25, 14, 9, -2, 4], [9, 2, 2, 18, 5], [-20, 2, -14, 9, 3], [23, 3, 7, -4, 0], [-19, 6, 12, -6, -5], [5, -1, -3, -11, 3], [-14, 0, -8, -9, -11], [-29, 3, -4, -19, -14], [15, -2, -8, 2, 14], [-8, -2, -2, -10, -12], [0, 18, 10, -6, -4], [-2, 1, -3, 10, 10], [21, 7, 7, 2, 5], [-27, -4, -8, 11, 6], [-15, 11, 9, -8, -17], [-11, 13, 11, -3, 1], [12, 4, 10, -3, -1], [-26, 9, -1, -4, -2], [13, -7, 11, -6, 1], [-6, 3, -10, 7, 13], [-13, -5, -1, -8, -9], [-9, 1, -3, 3, -9], [-18, 10, 0, 12, 4], [-22, -5, 5, 6, 7], [11, -6, -3, 0, 12], [-22, 4, 6, -8, -4], [-20, -16, -10, 8, 2], [-25, 8, -1, -20, -4], [-3, -11, 3, -1, -7], [-9, 11, -5, 20, 7], [-3, -16, 4, 2, 5], [-14, -19, -11, 12, 4], [36, 0, -2, 1, -12], [33, -21, -3, 16, 11], [25, -8, -1, 12, -4], [-34, 10, 2, -10, 0], [15, -13, 1, 15, 5], [-12, 6, 2, -5, 18], [-9, 13, 21, -23, -17], [-10, -24, -12, 12, 6], [-19, 9, 17, -10, -7], [14, -5, -8, 18, 13], [4, 17, 9, 2, -4], [5, -1, -16, -11, 2], [19, -1, -7, -8, 7], [-17, -3, -13, -12, -11], [-23, 1, 4, -12, -6], [6, 7, 1, -4, -12], [22, -3, 7, -6, 4], [-26, -13, -7, -4, 2], [-23, 5, 1, -11, -1], [-3, 9, 3, -6, 13], [-31, 11, 1, -17, 1], [29, -7, -11, 18, 11], [14, 4, -8, -4, 5], [-21, 15, 11, -10, -7], [0, -16, -6, -2, 13], [-2, 0, -16, 8, 8], [-8, 8, 10, -14, -22], [1, -8, -16, 2, 3], [7, 1, -7, 11, 17], [-6, -1, -20, 2, -3], [-16, -1, -4, -19, -15], [-17, 15, 7, -17, -8], [-19, -19, -1, 12, -3], [0, 26, 8, -8, -18], [-19, 12, -6, 9, 22], [6, 14, -9, -8, -7], [13, 1, -11, 20, -1], [21, -5, 14, -18, -10], [2, -11, 1, 16, 3], [-24, 11, 11, -18, -8], [6, -18, -4, 9, 4], [16, 10, 10, 10, 6], [3, 1, 9, 16, -3], [23, -7, 3, 0, 7], [-34, 20, -1, 5, 3], [3, 2, 1, 7, 2], [10, -18, -5, -12, 1], [24, -4, -2, -8, 4], [-13, -13, -7, 10, 3], [-26, 18, 10, -20, -12], [-7, 7, 11, -4, -5], [-41, 10, 6, -21, -10], [-5, 2, 6, -4, -6], [-15, 17, 11, 2, 5], [-18, 8, -1, -10, -5], [-13, 3, 13, -15, 6], [27, 1, 3, 22, 5], [-23, 8, 13, -10, -6], [-8, 9, 5, 0, -13], [-9, 12, 26, -20, -18], [-19, 4, -13, 18, 12], [4, -5, 13, -8, -6], [5, 5, 9, -6, 3], [21, -13, 10, 8, 8], [7, 9, 1, -9, 3], [-2, -20, -14, 0, 8], [19, -33, 1, 11, 7], [-14, 0, 5, -5, -9], [3, 1, -13, 15, 7], [0, 12, -16, -6, -2], [-10, 1, 6, -13, -1], [-28, -8, -7, 5, 1], [-33, -8, -4, 11, 2], [-31, -8, 7, -14, 6], [13, -14, 6, 12, 3], [17, -12, -6, -8, 6], [-21, 3, 13, -4, 3], [-44, 8, 3, -13, -11], [-4, 10, -3, 17, 19], [2, 24, 8, 12, 2], [9, -2, -10, -8, 11], [9, 11, 11, -11, -13], [35, 4, 10, -4, -3], [1, 4, -16, -2, -5], [8, -1, 1, 14, 4], [-5, -9, -21, 2, 3], [14, 0, 20, -3, -2], [18, 18, 14, 7, 7], [14, 8, 8, -4, 10], [-14, 8, 18, -15, -23], [5, 25, 7, -9, 11], [-32, 5, -1, 10, 6], [-25, 17, 17, -10, -13], [0, 3, -7, 2, 0], [-39, 21, 1, -3, 3], [-17, -2, -2, 6, -11], [-16, 10, 0, 22, 4], [-18, 26, 2, -3, -2], [0, -20, -7, 15, 9], [23, -17, -15, 22, 19], [42, 13, 5, 8, 6], [0, -14, -6, 3, 12], [5, 3, -5, -6, 19], [-4, -2, 4, -21, -18], [1, 21, 13, -20, -11], [-42, 7, -3, -16, -9], [-6, 14, 8, 3, 3], [2, 9, 15, -2, -10], [-11, -9, 3, -6, 15], [-1, 13, 8, -5, -14], [15, 5, -3, -8, -21], [3, -10, -16, 6, 1], [8, 12, 0, -2, 0], [-8, -20, -16, 6, 2], [12, 2, 3, 16, 7], [-41, 19, 9, -8, -9], [-9, -6, -6, -4, 13], [41, -3, 1, 21, 9], [23, 3, -7, 7, -16], [-33, -5, 4, -5, 6], [38, -11, 9, 10, 6], [37, -7, -11, -5, 7], [12, 2, -1, -12, 13], [51, -8, 0, 14, 13], [-3, -21, -11, 10, 25], [-28, 9, 2, -24, -9], [14, -7, -12, 9, 5], [-5, -10, 4, -22, -15], [-26, 10, 3, 8, 3], [-5, 24, 0, -5, 2], [-17, 11, 13, 2, 1], [-10, 18, 8, 10, 2], [11, -4, 2, -4, -29], [-37, 2, -13, -21, -14], [-14, -4, -18, 23, 17], [-13, 19, 17, -18, -3], [22, -22, -16, 18, 24], [3, -10, -14, -4, -1], [14, 0, -12, 13, -6], [-22, 15, -13, -6, 2], [17, 0, 1, 14, -8], [14, -2, 0, 24, 9], [33, 1, 9, -13, -23], [35, 5, 23, -16, -8], [7, -11, 1, 15, -19], [7, 6, 0, 17, -4], [-9, 9, -4, -3, -18], [35, -22, -2, -6, 1], [23, -7, -13, -5, 11], [9, 36, 20, -16, -17], [-26, -2, 6, -10, -18], [-30, -14, -4, -4, 2], [-15, 21, 11, 18, -1], [18, -16, -3, 16, 15], [-7, -6, -16, -2, -17], [3, 3, 3, 10, -3], [15, 1, -15, 2, -4], [-29, 19, 7, -20, 1], [-40, -3, -9, -2, 0], [-44, 5, 3, -9, -1], [6, -7, 6, 7, -13], [37, -16, -6, 16, 3], [-39, 1, 7, 1, 7], [0, -16, -22, 13, 2], [34, 18, 10, 10, 6], [-12, -18, 4, 0, -4], [-17, 9, 11, -24, 5], [-26, -2, 13, 4, 15], [11, -2, 0, -14, -14], [-31, -11, -9, 2, -3], [13, -6, 2, -14, 11], [-8, 29, -2, -10, 9], [-27, 24, -10, 0, 5], [10, 4, -16, 22, 6], [-35, -3, 1, 5, 3], [9, 21, -3, 19, -7], [2, -3, 11, -26, -15], [-30, 11, 3, -18, -18], [-26, 18, 0, -6, -2], [-29, 16, 16, -30, -27], [17, -20, 4, -12, -5], [13, -33, -4, 18, 18], [-29, 13, 5, -18, -19], [-2, 0, 4, -19, -26], [17, -5, -2, -4, 6], [4, -44, -12, -2, 10], [-11, 17, 11, -19, -29], [-25, 2, 1, -11, 10], [3, -7, -27, 3, 1], [9, -15, -15, 18, 9], [-16, -34, -6, 13, 4], [-6, 6, -20, -6, 8], [-27, -22, -8, -12, -9], [11, 5, 7, 15, -9], [-22, 24, 4, -24, 5], [-12, -19, 0, -21, 7], [-22, 18, 8, -22, -24], [-16, -6, -16, 21, 3], [0, -6, 4, 17, 22], [21, -1, -19, 13, 7], [-22, -7, 11, 6, -6], [4, 1, 24, 0, -3], [8, -15, -20, -12, -3], [-13, 1, 1, 6, 7], [-6, -10, 0, 6, 4], [-26, -11, -11, -1, 7], [0, -8, -20, 25, 18], [-20, -11, -9, -30, -8], [-19, 23, 11, 12, 3], [-9, -9, -10, -1, -6], [15, 5, 11, 16, -5], [-8, 12, 4, -4, 20], [-17, 18, 14, -17, 4], [51, -3, 3, 22, 13], [28, 22, 20, -18, 6], [2, -7, -1, -12, -38], [6, -8, 4, 6, -10], [-23, 9, -7, 21, 9], [-8, 7, 23, -2, 14], [12, 2, -22, 10, 8], [-3, 6, 26, -14, -11], [-13, 13, 21, -16, -32], [33, -23, -1, 6, -1], [-18, 7, 17, -25, -31], [-44, -16, -13, -8, -15], [-13, -12, -16, 0, 1], [-23, 9, -1, -7, -5], [6, 2, 0, -15, -4], [24, -2, 22, -6, 4], [20, 9, 3, 12, 8], [40, -29, -22, 14, 27], [-12, 22, -6, -2, -1], [44, 7, 3, -6, 0], [21, -26, 4, 8, 0], [43, -18, 2, 22, 11], [30, 14, -1, -17, 1], [-33, -13, -7, -22, -5], [42, -2, 0, 2, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6027_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6027_2_a_w();".
// 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_6027_2_a_w(:prec:=5)
    chi := MakeCharacter_6027_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6027_2_a_w();".
// 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_6027_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -7, 16, -4, -3, 1]>,<5,R![19, 5, -18, -6, 3, 1]>,<13,R![103, 23, -48, -10, 5, 1]>],Snew);
    return Vf;
end function;