
// Make newform 6039.2.a.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6039_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_6039_2_a_a();".
// 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_6039_2_a_a();".
// 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, 3, -1, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [2, 0, -5, 0, 1], [-2, 1, 5, 0, -1], [-3, -3, 5, 1, -1], [3, 3, -9, -1, 2]];
    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_6039_a();"
function MakeCharacter_6039_a()
    N := 6039;
    order := 1;
    char_gens := [1343, 5491, 5248];
    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_6039_a_Hecke(Kf)
    return MakeCharacter_6039_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 := [[0, -1, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 1, 0, 0], [-1, 1, 2, 0, -1], [-1, 0, 0, 0, 0], [-2, 1, 0, 0, -1], [1, 0, -1, 0, 0], [-3, 2, 1, -1, -2], [0, 1, 2, 1, 1], [3, -1, -4, 3, 1], [-1, 1, -2, 2, 1], [0, -3, -3, 1, 3], [1, 1, 1, -2, -2], [2, -3, -4, 2, 3], [1, 3, 1, 0, -1], [-1, 3, 1, -5, -3], [2, -2, 0, 3, 0], [1, 0, 0, 0, 0], [-1, 4, 0, 0, -4], [-1, -1, -2, -2, -2], [-2, -2, 2, 4, 1], [-5, -2, -3, 1, 0], [1, -4, -4, -3, 1], [4, -1, -5, -2, 0], [-1, 6, 4, -7, -2], [2, 0, 1, 3, 4], [5, -8, -10, 0, 4], [1, 1, 4, -5, 1], [-4, 4, 5, 5, 1], [-1, 7, 6, 5, -4], [5, 5, 0, -6, 3], [0, -6, -2, 0, 7], [-1, -10, -4, 6, 3], [4, -9, -3, -4, 6], [3, 9, 4, -6, -9], [-4, 1, 2, -8, -3], [-5, -11, -1, 1, 1], [0, 6, 1, -7, -1], [3, -1, -8, 3, 0], [-8, 3, 7, 4, 0], [6, 2, -5, 1, 0], [-2, 4, 2, 3, 5], [5, -4, 1, -9, 4], [-6, 3, 4, 4, 4], [1, -2, -6, 2, 10], [5, -2, 8, 1, 3], [-11, -6, 2, 0, 4], [-7, -5, -1, 5, 0], [-1, 3, -2, -1, 0], [-6, -5, 7, -4, -5], [8, 11, 5, 0, 0], [-4, 1, 2, 7, -4], [-3, 3, -9, -2, -1], [-13, 8, 1, -5, -6], [-5, -7, 3, -8, 5], [2, -15, -11, -2, 7], [1, 0, -1, -2, 2], [6, -11, -13, 3, 11], [-13, 3, 9, -5, -2], [1, 15, 17, -3, -1], [-11, -6, -9, 4, -4], [-2, 9, 10, -3, -5], [-1, -2, 1, 2, -5], [2, 1, 4, 0, 6], [0, -14, -16, 4, -1], [-3, 2, -5, -6, 1], [-10, 13, 13, 1, -6], [-6, -5, 2, -5, -12], [0, 4, -1, -2, -10], [-13, 2, 11, -9, -5], [-12, -5, -4, 10, 2], [-6, 12, 8, -4, -10], [9, 5, 0, -4, -7], [10, 8, 2, -14, -3], [2, -1, -4, 18, -1], [-15, -9, -6, 8, -6], [-16, -24, -6, 4, 5], [0, 9, 0, -8, 0], [7, 6, 2, -12, -7], [-18, -11, -2, 7, 9], [8, -3, 1, 11, 2], [-5, 16, -1, -3, -9], [7, -14, -9, 17, 9], [7, 3, -2, -6, 2], [5, -2, 4, -9, 6], [-6, -2, -2, -6, 2], [2, -16, -5, 2, 5], [-9, 10, -1, -3, -5], [-8, -16, -1, 2, 1], [-6, -7, -3, 10, 4], [-9, 6, 2, 9, 9], [4, 22, 4, -9, -7], [-2, -5, 0, 7, -8], [11, -5, -6, 8, 11], [-1, 16, 2, -10, -5], [3, 5, 7, -18, 2], [-18, 15, 16, -13, -6], [-12, -8, 3, 2, 3], [11, 15, 13, -14, 4], [-10, -5, -2, -11, 4], [-13, -5, 4, -4, -4], [-13, -19, -2, 3, -2], [-2, 9, 1, -4, 7], [6, -21, -27, 7, 12], [7, 4, 5, 4, -2], [2, 8, 2, -11, -4], [-13, -8, 17, -3, 5], [5, -10, -13, 2, 6], [5, 8, 2, -5, -16], [-8, -5, -6, -2, -5], [12, -5, -9, 2, 8], [-1, -13, -6, 17, -4], [6, -6, -14, -6, -6], [4, 13, 19, -3, -15], [8, -19, -24, 9, 0], [-7, -4, 16, -8, -4], [5, -1, 1, 7, 11], [-6, -6, -8, -6, 3], [-5, 2, 6, 11, 4], [11, 10, 11, -15, 10], [13, 16, 7, -2, 7], [-18, 7, 4, 13, -12], [-5, -26, -13, 5, 1], [-8, -14, -4, 13, 4], [18, -11, -12, 11, 1], [11, -19, -18, 15, 13], [-17, 18, 14, -10, -29], [-15, -11, -2, -3, 0], [-1, 17, 2, 9, 14], [13, 15, 18, 7, -11], [18, 16, 13, -15, 11], [3, 6, 0, -1, -6], [4, -4, -7, -12, 2], [-10, 22, 24, -11, -13], [-17, 7, 8, 10, -19], [-7, 10, 0, 0, -14], [-12, 13, -5, 4, -9], [30, -2, -8, 8, 1], [20, -12, -25, 3, -3], [15, -16, 8, 18, -4], [-4, 2, 13, 2, 6], [10, 3, -3, -13, -5], [2, 25, 26, -13, -4], [-10, -17, -4, 17, -1], [-7, -1, 2, 8, -1], [-14, -3, -13, 8, -2], [-23, 4, -1, 7, -6], [12, 1, 3, 9, 14], [20, -21, -27, 11, 4], [4, 20, 4, 15, -7], [10, 10, 0, -11, 8], [17, 32, 10, 5, -10], [1, -7, 8, 9, 0], [-12, 1, 19, -11, -16], [-1, -18, -2, -5, 8], [22, 17, -1, 5, 7], [-23, 6, 19, 11, 8], [9, 20, 23, -11, 11], [4, -12, -1, 12, 9], [1, 8, -10, -10, -1], [-24, 22, 34, -11, -9], [21, 17, 7, -3, -10], [-14, -11, 11, -15, -2], [-6, -20, -9, -1, -16], [-15, 12, -3, -16, -7], [-26, -5, 2, 1, -22], [-11, 22, 4, -13, -5], [30, -5, -15, 7, 6], [-30, 5, 0, 7, -18], [23, -10, 2, -7, 13], [11, 29, 21, -10, -26], [-19, -32, -16, 16, 19], [8, -3, 1, -15, 5], [-11, -31, -24, 10, 17], [2, -11, 7, -4, -20], [-11, -33, -16, 22, 14], [12, -14, -3, -2, 5], [27, 20, 18, -13, -1], [-4, 24, 6, -33, -7], [22, -14, -9, -7, 21], [-27, -18, 3, -12, -1], [-15, 20, 13, -1, 7], [14, 13, 5, -18, -10], [15, -2, 11, -8, 14], [-17, 5, 1, -8, -1], [-1, -4, 5, -3, 11], [10, 12, 3, -13, 12], [4, 5, -19, 0, -12], [-17, 1, -7, -14, -8], [40, -7, 1, 4, 9], [9, 14, 5, 12, 11], [20, -14, -5, 7, 16], [-11, -12, -9, 9, 4], [-40, 1, -2, 8, -3], [17, 28, 10, -22, -19], [-14, 18, 19, -7, -32], [-7, -14, -26, 15, -10], [27, 7, -10, 4, 12], [27, -16, -18, -6, 19], [-1, -15, -8, -6, 16], [-3, 1, 25, 7, -3], [2, 25, 17, -3, 3], [-5, 12, -7, 6, -9], [-15, 15, -7, 7, -24], [-19, 15, 26, 3, -22], [9, 9, -3, 9, -7], [8, 7, 13, 16, -17], [41, -10, -30, 17, 2], [1, 11, 5, -5, 3], [-23, 24, 25, 6, -15], [-15, 18, -5, -2, 5], [-6, -25, -23, 14, 17], [21, 23, 7, -4, -1], [-20, -13, 5, 2, -5], [16, -19, 8, 12, -1], [-23, -3, -3, -5, 13], [7, -26, -19, 22, 27], [8, -14, -36, 2, 0], [-9, 26, 6, -19, -7], [0, 28, 5, -22, -16], [-2, 6, 9, -6, 4], [-27, 13, -1, 15, -24], [12, 16, 2, -7, 14], [16, -17, -21, 1, 20], [21, 0, 23, 6, 3], [30, 4, -5, -1, -7], [-6, 24, -5, -14, 11], [2, -25, -2, 10, 8], [35, -1, 4, -13, 7], [26, 23, 14, -24, 2], [-20, -41, -9, 13, 2], [1, 8, 29, -3, 4], [11, -25, -6, 20, -10], [32, -12, -9, -3, 26], [-2, -2, -5, 11, 2], [4, -31, -36, -3, 27], [-5, -25, 10, 15, 10], [-3, -12, 0, -27, 8], [7, 28, -5, 2, -1], [20, 20, -3, -8, 2], [32, -5, -10, 18, 10], [10, -30, -26, 16, 4], [34, 11, 19, -9, 9], [9, -20, -34, 11, 7], [-16, 3, -6, 8, -6], [-17, -6, 13, -5, 9], [-15, 21, -3, 1, -8], [24, -24, -18, -5, 34], [17, 8, 9, 3, 22], [45, 19, 5, 3, 0], [22, -1, 5, -9, 8], [23, -21, -20, 39, 13], [-16, -26, -12, 17, 24], [7, -8, -7, 15, 20], [17, 0, -16, 6, -5], [-6, -18, -7, 14, -12], [-15, 11, 5, -15, -28], [-37, 17, 11, 9, 4], [-40, 13, 9, 3, -6], [20, 27, 5, 16, -13], [3, 19, -15, -14, -27], [-13, -13, 4, 14, -7], [18, -24, -19, 11, 1], [19, 19, 14, 15, -9], [22, -21, -12, -21, 17], [-11, 24, 1, -3, -18], [-28, 20, 21, -30, -13], [-18, 22, 35, -6, -11], [-19, -15, -13, 11, -14], [16, 16, 1, 14, -12], [-10, -13, -21, 19, -16], [-24, 0, 5, -5, -24], [2, 4, -8, -12, 14], [7, 2, 7, 12, -1], [-3, 15, -5, 7, 10], [-13, -2, 18, 21, 4], [38, 22, 12, -20, 0], [18, -6, -19, -7, 16], [-26, -14, -13, 10, 11], [0, 32, 5, 3, -1], [18, -16, -46, 2, -8], [16, 18, 31, 8, -8], [-8, -36, -16, 12, 28], [3, -39, -41, 5, 35], [-17, -23, 2, 5, 3], [-5, 12, 1, 18, 22], [-7, 4, -9, 9, 4], [8, -16, -22, 9, -6], [-8, 25, 14, 7, -17], [15, 21, 26, 1, -10], [-31, 19, 6, 29, -22], [6, 0, 13, 13, -5], [-16, 13, -8, 19, -6], [26, 20, -4, -11, 8], [6, 13, -8, 15, 23], [-1, 13, 0, -34, -18], [-23, 31, 23, -36, -31], [-46, -7, 1, 11, 8], [-10, 44, 41, -7, -30], [6, -30, -10, 30, 21], [-10, 1, 20, -31, 7], [-3, 18, 1, 14, -15], [-21, -50, -19, 31, 37], [13, -28, -37, -4, 1], [2, -7, -11, 15, 15], [5, -3, 6, 13, 17], [-24, -1, 4, 12, -24], [-4, 16, 5, -6, -8], [-24, 26, 50, 0, -7], [34, -9, -8, -12, 20], [6, 4, 21, 12, 9], [-23, 4, 31, -4, -3], [-30, -10, -10, 5, -11], [-2, -10, 26, 18, -1], [-14, -19, 8, 8, -24], [-11, 23, 20, 18, -1], [24, -33, -45, 9, 34], [-11, 3, 13, 8, -13], [-53, 9, 17, -3, -12], [-6, -7, 18, -2, -2], [-17, -9, -20, -6, 2], [-26, -18, 13, -7, 6], [1, 15, 6, -7, 17], [-8, 18, -31, 1, -13], [-16, -11, 16, 11, -11], [-28, 22, 13, 16, -38], [7, -19, -7, 18, -8], [39, 17, -7, 6, -8], [-29, 9, 4, 6, -30], [-10, -37, 15, 6, 32], [4, 31, -12, 5, 4], [-2, -19, -17, 5, 27], [29, -12, -42, 0, -5], [36, 6, -15, -1, -15], [14, 37, 28, -8, -13], [0, 10, -11, -7, 26], [10, -1, -12, -3, 0], [-3, -6, 25, -3, 23], [-8, -35, 15, 11, 6], [24, -7, -12, -28, -4], [-17, 35, 45, 1, -31], [-30, -4, -17, -1, 10], [-8, 15, -14, -18, -11], [-23, 4, 18, 0, -2], [20, 13, 9, 9, -8], [-12, 42, 29, -20, -30], [-9, 17, 25, -30, -15], [-13, 5, 12, -1, -12], [-7, -10, -17, -12, 27], [25, -27, -19, -5, 37], [22, 10, 12, 1, 22], [33, -16, -12, 18, 31], [6, 1, 10, -3, 19], [34, 19, 10, 3, -13], [-8, 25, 27, 22, -18], [28, 48, 20, -25, -3], [13, -9, -30, 0, 12], [-12, 25, 8, 4, -22], [-28, 14, 18, -1, -23], [-48, 18, 22, 12, 8], [8, -11, -19, 21, 3], [-40, -20, -11, 18, -21], [-23, 19, -16, 2, -27], [2, -26, -17, 29, 6], [-27, -8, 22, -14, 4], [-23, 11, 40, 8, -15], [-22, 17, -7, 26, -21], [26, 47, 8, 1, -9], [-5, 19, 18, -10, -1], [-22, -2, 18, -18, -11], [-1, -18, 13, -7, 7], [34, -4, -10, -3, 24], [-18, 14, 21, -3, -7], [-17, 22, 17, 0, 15], [-26, -45, -31, 32, -19], [-27, -17, 18, -14, -6], [13, 30, 30, -20, 10], [24, -32, -36, -11, 9], [-4, -20, -11, 36, 0], [-4, -40, -17, -4, 21], [-8, -13, 4, -12, 5], [-38, 25, 45, -34, -2], [-19, 13, 17, 16, -24], [8, -44, -50, 7, 45], [-20, -24, 9, -8, 10], [-35, -7, -4, 18, -13], [18, -11, -33, 22, -1], [23, -44, -22, 28, 58], [-5, -14, 23, 10, 17], [12, 45, 43, -13, -34], [0, 23, -19, -15, -24], [-9, -6, 12, -19, -34], [-8, 47, 21, -37, -43], [-9, -21, -31, -14, 23], [-29, 29, 26, -9, 9], [-16, 37, 31, -2, -27], [-29, -19, 6, -25, 11], [-22, -20, -30, -1, -5], [-5, 9, 24, -16, -4], [11, 7, 18, 2, 29], [29, -7, -28, -24, -2], [14, -17, -13, 18, 18], [-13, -3, 15, -16, -31], [23, -25, -11, 21, 15], [3, 2, -14, -8, 3], [10, -22, -10, -26, 13], [-39, -28, -35, 10, 8], [-6, -20, 7, 3, -14], [3, -15, 4, 18, 28], [-28, 16, -14, -5, -37], [8, 21, 7, -22, -17], [-19, 8, 11, -11, 13], [-4, -23, -32, 13, 4], [-13, -12, 16, -21, 15], [-24, 10, -15, 4, -15], [-39, -2, -18, 23, -12], [4, -34, -11, -3, -2], [25, -31, -1, -27, 13], [-6, -5, -15, 34, 13], [-26, 30, 57, -22, -5], [35, -1, -6, 4, 15], [-48, -16, 9, 5, 19], [-32, -30, -31, 10, 9], [27, -24, -42, -2, 2], [9, -32, -41, 17, 27], [-13, -25, -38, -2, -1], [-16, -19, -29, 7, 22], [-27, 11, 16, 12, -23], [0, -3, -4, 2, -7], [-19, 7, 17, 8, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6039_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6039_2_a_a();".
// 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_6039_2_a_a(:prec:=5)
    chi := MakeCharacter_6039_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_6039_2_a_a();".
// 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_6039_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6039_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 2, 4, -3, -2, 1]>],Snew);
    return Vf;
end function;