
// Make newform 6027.2.a.x 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_x();".
// 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_x();".
// 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, 14, 6, -8, -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], [-3, -2, 1, 0, 0], [2, 7, -4, -3, 1], [-4, 11, -1, -4, 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, 1, 0, 0, -1], [0, 0, 0, 0, 0], [-1, 1, 1, 0, -1], [1, 0, 0, -1, 2], [4, 1, 0, 2, 1], [-1, 1, 1, 1, 0], [-1, -1, 0, 0, -2], [4, -1, -1, 0, 2], [-1, 1, 1, -1, -2], [2, 2, -1, 0, -2], [1, 0, 0, 0, 0], [2, -1, 0, 0, 1], [1, 0, 2, -3, -1], [3, 2, -2, 1, 2], [-2, -3, -2, -2, -1], [0, 0, 0, 5, 1], [5, -6, -2, -3, 3], [6, 0, 0, 3, 3], [-3, 1, -1, 1, -2], [4, -1, -1, -3, 0], [4, -4, -2, 2, 2], [7, -3, 1, 0, 1], [0, 1, -1, 0, 2], [-3, -3, -3, -4, -1], [5, -5, -2, 1, -1], [-6, 8, 1, 3, -4], [-3, 0, -1, 4, -2], [1, 3, 1, 2, -3], [1, -5, -1, -2, 1], [-9, 6, 3, 2, 0], [-1, 2, 2, 2, 3], [-10, -2, -1, -2, -1], [9, 1, -1, 5, 6], [-1, -1, 1, 0, 3], [3, -1, 1, 3, 2], [11, -2, -3, -2, -2], [-7, -1, -2, -3, 6], [4, -1, 1, -6, 1], [-3, 1, -1, -5, -4], [-3, -3, -5, -3, 2], [-3, 1, -1, 3, 0], [-2, 1, -2, 4, 1], [-2, 0, 0, 0, 0], [-10, -2, 0, -7, -3], [14, 3, -1, 1, 2], [-10, 2, 3, 2, -1], [-11, -4, 0, -9, -5], [-1, 4, 0, 0, 1], [-13, -3, -7, -5, 4], [-1, -7, -1, -2, 3], [6, -8, -4, -4, -2], [-1, -1, -1, 6, -1], [0, 2, -4, 0, 6], [-7, 7, -1, 4, -3], [3, 4, 2, 4, 8], [-9, -1, -4, 2, -4], [2, -7, 3, -4, -1], [-2, -7, -3, -13, -3], [-16, -2, 2, -1, -3], [8, -9, 0, 0, -1], [-6, 6, -1, -3, 0], [14, -7, -1, 3, 2], [-4, 1, 1, -5, -7], [6, 1, 1, 1, 7], [-7, 3, 0, -1, -2], [5, 3, 2, -6, -5], [7, -3, 1, -5, 2], [-13, 1, 1, -3, -8], [1, -1, -3, -4, -5], [1, -7, -6, -9, -7], [0, 0, -7, -3, -4], [-3, 1, 8, 0, 1], [0, -7, -6, -2, -7], [6, 9, 3, 5, 2], [3, 7, -5, -1, 2], [5, 3, -1, 1, -2], [-9, 5, -1, -6, 1], [9, -7, -3, 6, 3], [-2, 1, 4, 4, -1], [6, 1, 2, -4, -5], [25, -11, -5, 1, 4], [2, 0, 0, 2, -6], [-5, -5, -5, -9, 2], [5, -3, -1, -11, -2], [20, -2, 2, 1, -3], [-2, -2, -2, -4, 8], [-9, -6, 6, -8, -4], [-3, 10, -2, 9, 1], [-13, -12, -1, -8, 2], [9, 5, 3, 2, 9], [-13, 7, 5, 6, -1], [2, 0, 4, 7, -3], [1, -7, -1, -4, -5], [7, -5, 2, 5, -4], [-5, 10, 3, 8, -8], [-9, 1, -5, -2, 7], [-9, -3, -3, -1, 4], [-16, 2, -1, 2, -6], [-6, 2, 3, 1, -13], [-3, -1, 3, 7, 6], [6, 12, 3, 5, -3], [-6, 4, 4, 7, 3], [23, -1, 1, 0, 3], [-5, 7, 5, 5, -8], [0, -4, -4, 0, 10], [6, -8, -2, -12, 8], [21, -7, 0, -8, 2], [16, -5, -7, 4, 8], [-11, 13, 7, 7, 6], [17, -4, 2, 10, 0], [-6, -8, -2, 2, 8], [-7, 3, 6, -1, -9], [10, -8, -2, -6, 4], [15, 7, 1, -1, -4], [12, -4, 0, 3, -3], [7, -4, 3, 0, 14], [-6, -12, 0, -3, -3], [15, 3, -3, -4, -11], [7, -13, -5, -10, 5], [-12, 13, 0, 2, 3], [19, -3, -1, -4, -1], [1, 5, 5, -10, -1], [1, 1, 5, 0, 1], [-11, -7, -7, -8, 1], [-7, 7, -3, -6, 9], [-9, 7, -3, -2, -9], [4, 6, 4, 14, -4], [6, -2, -6, 6, 0], [6, -1, 0, -2, -3], [-1, 13, 1, 5, 8], [9, 2, 4, -5, 5], [7, 5, 11, 4, -1], [16, 8, -1, 0, 10], [0, 2, 8, 0, 6], [1, 5, -1, 8, -7], [-11, -7, 4, -4, -8], [12, 8, 2, 8, 2], [-2, 5, 1, -1, -5], [-22, 0, -3, -5, 6], [-8, -8, 4, 1, 5], [12, 4, 4, 4, -6], [6, -20, 2, -6, 8], [-11, 7, 3, -2, 1], [4, 6, 1, 1, -9], [-17, -7, 9, -4, -7], [8, 6, 11, 10, 0], [-18, 2, 5, 4, 7], [-16, 6, 5, -3, -8], [12, 4, -2, 8, -4], [-10, 8, 7, 6, -6], [-33, 11, 8, 1, -8], [0, 2, -1, -1, -5], [8, -10, -10, 0, 18], [34, -16, -8, -5, 5], [0, 0, 1, 7, 9], [22, 1, 8, 4, -1], [21, -11, 3, -13, 2], [22, 1, -3, 6, 1], [7, -3, 0, -7, 1], [-5, 9, -1, -1, 6], [-2, 6, -3, 1, -5], [-3, -2, 6, 1, 11], [9, 11, -5, -9, -10], [5, -4, -5, -8, 2], [-12, 2, 3, -3, -13], [-7, -8, -4, -14, -1], [27, -16, -6, -4, 2], [12, 8, 8, 0, -12], [-2, 10, 2, 16, 2], [-30, 18, 8, 5, -1], [-3, 11, 3, 8, 5], [9, -21, 0, -8, 1], [33, 1, 9, 0, -3], [3, 6, 3, -2, 6], [-2, 2, 5, -1, -1], [30, -10, -4, 3, 7], [-27, -7, 3, -4, 1], [-2, 15, 1, 1, 13], [-33, 11, 1, 0, -7], [-13, -8, -9, 0, 0], [-10, 10, 8, 0, -12], [-4, -3, -17, 6, 3], [-25, 3, -6, -1, 1], [-10, -8, 0, -4, -6], [12, 10, 6, 2, 10], [23, 13, 6, 11, -1], [-9, -18, 3, -4, 12], [7, -13, -7, -16, 1], [9, 17, 5, 15, -2], [-4, 8, 4, 11, -5], [12, 1, 2, 0, -7], [13, -1, -3, 4, 3], [19, 11, 0, -1, 1], [-14, -2, 0, -6, -20], [13, 10, 3, 6, 10], [30, -1, 2, 0, 5], [-7, 16, 5, 12, -8], [-14, 2, -6, 6, 0], [-15, 3, 7, -2, -17], [-6, -10, 1, 9, -2], [0, 10, 2, 1, 3], [-11, 8, 14, 4, 9], [11, -5, -3, 0, 7], [15, -11, 7, -8, 9], [3, -15, -4, -18, 10], [-14, 18, 6, -3, -7], [6, 10, 10, 9, 17], [-10, -10, 4, -15, -7], [7, 6, -3, -10, -8], [5, 3, 3, 0, 7], [-19, 0, 1, -14, -12], [-3, 7, -3, -4, -7], [-36, 15, 9, 5, 2], [9, 7, 9, 2, 15], [-20, 21, 7, 11, -8], [-6, 4, -8, 12, 0], [0, 6, 18, 4, -4], [3, 3, 5, -1, -16], [5, -12, -5, 10, 22], [-14, 2, 1, 5, -14], [0, 1, 9, -7, -11], [1, -11, 0, -3, 4], [-19, 15, -1, 2, -7], [-8, 16, 6, 2, -24], [-13, 5, 6, 10, -9], [4, 12, 5, 22, -7], [27, 7, -5, 18, 15], [-17, 9, 16, -10, -14], [-26, -1, 5, -6, -25], [-22, 2, 0, 5, 3], [-6, 7, -2, 4, 15], [28, 4, 2, 4, -16], [-17, -11, 3, -6, -11], [19, -9, 1, 6, -15], [-8, 15, 3, -10, -14], [-9, 6, 4, -5, -4], [12, -8, 1, -6, 5], [-8, -12, 4, 4, -2], [-3, -17, -5, -22, -7], [2, -8, -8, 2, 18], [9, 3, 5, -2, 11], [7, 3, 0, 6, -17], [-3, -6, 0, 14, 12], [1, -19, 11, 2, 3], [-40, 4, -1, -6, 7], [23, 13, -4, 5, 10], [27, -7, -9, 6, 19], [23, 17, 2, 19, 17], [8, -1, 5, 2, -9], [9, -18, -8, -8, 16], [-25, -3, 8, -5, -9], [6, 10, -2, 11, 3], [29, -1, 1, 0, -11], [29, -11, -12, 2, 6], [5, -14, -11, 8, 14], [32, -6, -14, 0, 0], [-11, 2, -3, 2, 18], [2, 15, 1, 6, -20], [3, 8, 3, 4, 2], [2, -12, -4, -4, 12], [10, 11, 7, 27, -3], [-4, -9, 3, 4, -14], [-7, -1, 6, -2, 3], [-19, 5, 0, 9, -19], [-1, -5, -3, -9, -12], [-29, 30, 4, 20, -12], [43, -7, -3, -2, 19], [-34, -3, -7, 2, -16], [16, -5, -7, 2, -6], [-14, -6, 2, -6, -16], [-11, -7, -1, 7, 14], [-27, 18, 7, -14, -8], [-17, -9, -11, -5, -6], [-31, -5, 3, 5, 6], [-12, 0, 6, 1, -1], [23, -9, 1, 2, 7], [-20, 1, -2, 10, -11], [-50, -8, -1, -5, -9], [2, -14, -6, -10, -4], [-12, 8, -5, 13, -5], [21, -8, -1, -16, 6], [-16, 8, -12, -13, 1], [-11, -3, 7, 10, -3], [12, -24, -2, -3, -5], [3, -16, 1, 0, -4], [5, -3, -5, -5, 14], [10, 20, -6, 6, -4], [-36, -15, -6, -20, -1], [0, 3, -9, -10, -4], [-9, 13, 1, 8, 1], [22, 6, -8, 7, 17], [8, 7, 6, 2, -9], [-35, 27, 9, 14, -5], [2, 11, 10, -4, -9], [-1, -7, -3, 12, 3], [24, -29, -9, -18, 1], [-22, 0, 3, -5, 1], [38, -10, -8, -11, 1], [1, -27, 1, -10, 9], [31, -10, 2, -10, 11], [-1, 5, -7, -12, -1], [-27, -7, 11, -13, -8], [4, -8, 4, -10, 18], [-28, 4, 2, -6, -6], [2, -26, -9, -16, 13], [15, -27, -15, -8, 29], [-5, 9, 1, -1, -10], [-5, -12, 7, -4, 16], [7, -15, 10, -17, 4], [13, 4, 4, -8, 17], [-12, 10, -2, 7, -11], [-23, 28, 3, 12, -20], [-24, 8, -1, -16, -13], [-23, 13, -5, 5, 8], [29, -21, -9, -2, -15], [-12, -4, 11, -5, -18], [30, -3, -7, -1, 9], [-17, -11, 5, -19, -10], [18, 12, -11, -4, -9], [-5, 19, 4, 8, 5], [11, -13, -1, -18, -7], [-2, -12, -14, 0, 14], [-13, 7, 7, -1, -14], [1, -6, -6, 5, 4], [24, 22, 3, 17, 1], [-3, 9, -7, 8, -11], [-42, 12, 16, -12, -26], [11, 17, -5, 7, -6], [12, 15, 14, 16, 9], [18, -2, 4, 7, -3], [-15, -7, 6, 1, 11], [40, -9, 1, -9, 14], [-15, 2, -3, -10, -12], [-19, -14, -10, -9, -25], [-3, 8, 7, 6, 10], [17, -9, 12, 6, 5], [-25, 14, -6, 12, 1], [-5, -11, 1, -23, 4], [-35, 32, 9, 14, 0], [-23, 13, 7, 5, 0], [2, 8, 6, 28, 12], [-50, -6, 6, -12, -16], [15, 7, -1, -4, 7], [30, -10, 5, -24, 13], [-29, 1, 17, -9, -16], [-13, 7, 7, 2, 19], [-3, -16, -14, -1, -5], [-21, 25, 15, 2, -11], [24, -4, -6, -1, -11], [32, 20, 3, 20, 10], [26, -3, -13, 9, 13], [31, -13, 5, -19, 0], [-1, -18, 9, -6, 2], [-6, -3, 10, -22, -9], [54, 6, 6, -2, 0], [-12, -10, 4, -4, 8], [15, 7, -5, 12, 11], [-8, 8, 3, -6, -29], [11, 16, -14, 4, -6], [25, -21, -5, -8, 23], [1, 17, 15, 13, -8], [10, -4, 7, -1, -2], [-11, 23, 1, -9, -6], [-18, 16, -2, 14, 10], [-13, 10, 13, 14, 0], [9, -12, -3, -28, 0], [-6, -5, 3, 4, 3], [-16, -16, -6, -1, 3], [6, -6, 2, 26, 6], [-5, 21, 9, -1, -30], [-25, -9, -7, -15, 2], [-15, 21, 2, -4, 2], [-1, -5, 3, -24, -7], [-34, -23, 0, -20, -5], [19, -5, 0, 2, 8], [10, 0, 8, -12, 16], [39, -12, -7, 4, 10], [-41, -2, -4, 1, -2], [33, 2, -13, 8, 10], [13, -9, -17, -2, 3], [4, 15, -14, 8, -3], [-10, -16, -4, -12, 18], [25, 5, 11, 7, -8], [23, -8, -5, 4, 6], [-46, 31, 11, 3, -12], [28, 13, -1, 1, -5], [30, 8, 4, 8, 10], [0, -18, -5, -21, -1], [2, -5, -6, 6, 5], [-9, -14, -7, 12, 4], [2, -10, 4, 1, 33], [-12, -10, 13, -3, -12], [0, 12, -7, -9, -12], [-13, -3, 13, 2, 1], [22, -16, -4, 12, 18], [-28, 4, 5, 10, -10], [-4, 7, 8, -2, -29], [2, 4, 0, 5, 27], [-9, -1, 3, -12, -7], [3, -14, -10, 0, 25], [-9, 13, -7, 8, -3], [-60, 6, -4, 2, -8], [33, 9, 0, -2, -7], [-51, 1, 9, -18, -25], [40, -20, -2, -4, 10], [-36, 0, 6, 15, 1], [42, 0, -2, -2, -14], [5, -2, -11, -8, -10], [-10, 2, 6, -7, -5], [-24, -2, 4, -18, -16], [3, -13, -13, -7, 6], [-17, 6, 16, 1, -13], [13, -9, 5, 20, -1], [16, 2, 7, -2, -14], [34, 14, -3, 2, 25], [29, -7, -5, -5, 10], [47, 8, -7, 0, 6], [26, -5, 18, 4, -9], [12, -18, 0, 26, 4], [34, 6, -4, -3, -3], [-2, 0, 9, 13, -10], [6, 15, 7, -3, 2], [4, 24, -10, -7, 5], [-41, 20, -2, 8, -6], [-23, -3, -11, 19, 2], [-44, 7, -3, -6, -4], [-19, -17, -13, -22, 21], [12, 7, 11, -19, 4]];
    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_x();".
// 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_x(: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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6027_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-9, -1, 20, -6, -3, 1]>,<5,R![139, -171, 42, 18, -9, 1]>,<13,R![-1681, 473, 192, -56, -3, 1]>],Snew);
    return Vf;
end function;