
// Make newform 6020.2.a.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6020_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_6020_2_a_g();".
// 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_6020_2_a_g();".
// 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, 77, 32, -160, -9, 83, 0, -16, 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, 1, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0], [-173, 360, 358, -442, -86, 136, -7, -12, 2], [234, 446, -309, -616, 153, 204, -34, -18, 3], [-190, -330, -7, 507, 165, -156, -68, 11, 6], [-43, -327, 296, 678, -424, -252, 136, 25, -12], [184, -331, -1142, 732, 800, -265, -183, 22, 12], [9, -583, -838, 844, 879, -285, -258, 21, 20]];
    Rf_basisdens := [1, 1, 1, 47, 47, 47, 47, 47, 47];
    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_6020_a();"
function MakeCharacter_6020_a()
    N := 6020;
    order := 1;
    char_gens := [3011, 4817, 4301, 1121];
    v := [1, 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_6020_a_Hecke(Kf)
    return MakeCharacter_6020_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, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, -1, 0, 0, 0, 0], [-1, -1, 0, 1, 1, 0, 1, 1, 0], [-2, 0, -1, -1, 0, 0, 0, 0, 0], [0, -1, -1, 2, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, -1, -1, 0, -1], [-1, 0, -1, -1, 0, 2, -1, -1, 1], [0, -2, -1, 0, 1, 1, 1, -1, 0], [-3, 1, 0, 1, -1, -2, -1, -1, -1], [-1, -1, 0, 0, -1, -1, 0, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, 1, 1, -1, 2, -2], [-3, 0, 0, 2, 1, -1, 2, 0, 0], [2, -1, 1, 2, 1, 0, 1, 1, 1], [-2, 2, 1, -3, -2, 0, -2, -2, 1], [-1, 1, 2, -2, 1, -1, 0, 1, 0], [1, 0, -2, -1, 0, -1, 1, -1, 3], [-4, 0, 0, -2, -2, -1, -2, -2, 2], [1, -3, 0, -1, 1, 2, -1, 2, 1], [-2, 0, 0, -3, 1, 2, 0, -3, 2], [0, 1, 2, 3, 0, -2, 0, 0, -3], [-3, 1, 1, -2, 1, 2, 0, 3, -2], [1, -2, 2, 0, -3, 1, 0, 2, 0], [1, 0, 2, -2, 0, 1, 0, 2, -2], [-1, -4, 1, 4, 1, -2, 3, 2, 0], [-1, 1, 0, 2, -1, 2, 2, 2, 0], [-2, -1, -1, 2, 2, 2, -1, 0, 1], [-2, -2, -2, 0, 3, 3, -1, 0, 0], [2, -2, 0, 3, 0, 1, 1, -1, 0], [-5, 3, 1, 0, 0, -3, -1, 0, -2], [-1, 3, 3, -2, -2, -1, 0, 1, 0], [0, 2, 1, 1, -1, 0, -1, -1, 2], [1, 1, 2, -4, 0, 0, -1, 0, 4], [0, -1, -2, -2, 2, 3, -1, -1, 1], [-3, -2, 1, 1, 0, -1, 1, -2, 1], [-4, 5, 0, -4, -3, 0, -2, -4, 3], [-2, 3, 3, -6, -1, -3, 0, -2, 4], [0, -1, 2, 1, 7, 1, 1, 1, -2], [2, 3, 5, -2, 0, -1, -3, 0, 1], [0, -4, -1, 3, 1, 0, 2, 2, -3], [-9, -2, -4, 4, 2, 2, 2, 1, -2], [-3, -1, -3, 1, -2, 3, 3, 1, 3], [5, 3, 3, 1, -7, -7, 0, -2, -2], [-5, 4, -1, 0, -6, -4, -2, -2, -1], [-1, 4, 3, -7, -4, -1, -3, -3, 1], [5, -3, 0, 3, 7, 2, 3, 2, 1], [1, 1, 1, 1, -2, -4, -1, 2, 0], [-6, -3, 1, 3, 2, 4, 1, 3, -4], [-4, 6, -2, 0, -5, -2, -2, -4, 0], [-4, -2, 0, -3, -3, 2, -2, 0, -1], [2, -4, -4, 4, 1, 4, -1, 2, 2], [-4, 3, -1, 6, -3, -2, -1, 3, -1], [-2, 4, 0, 1, -1, -1, 0, 0, -1], [-1, 2, -2, 0, 3, -2, 1, 0, -5], [8, 1, 2, 2, -1, -6, 2, -1, -3], [-8, 4, -6, -2, -3, 3, 0, -5, 2], [-1, 2, 3, -4, 3, 1, -4, 2, 0], [-2, 3, 1, -2, 0, -4, -1, -4, 2], [-9, -3, -2, 7, 2, -1, 4, 1, -2], [4, 1, -1, 8, 1, 0, 0, 5, -3], [-5, -1, -3, 1, 1, -4, 1, -3, 0], [-1, -3, 0, 1, 4, 3, 2, 1, -4], [-4, 5, 4, -3, 2, -1, -2, 3, -2], [-1, -6, 0, 2, 0, 3, -1, 0, 0], [-6, -3, -2, 4, 5, 0, 5, 3, -2], [-1, 3, -3, 4, -7, -2, -1, 1, -4], [-3, 4, -4, 3, -4, 3, 0, 3, -2], [-7, 5, 2, -4, -1, -2, -3, 2, -1], [2, -1, 2, 1, 2, 1, 0, 1, -4], [3, -1, -3, 6, 0, 3, -1, 1, -6], [-13, 4, -6, 0, -5, -2, 0, -5, 3], [-6, 5, -5, -4, 6, 5, 3, -1, 6], [-4, 2, 1, -2, 1, 4, 0, 2, 2], [7, 4, 1, 5, 0, 4, 6, 3, 0], [5, 3, -1, -3, 5, 5, 1, 1, -1], [-3, -4, -2, 2, -2, 4, 0, 2, 2], [-2, -4, -2, -6, 3, 5, 3, -1, 3], [-9, 9, -5, -7, -7, 4, -8, -6, 0], [-1, -2, 0, 1, 1, 1, 5, -3, 1], [5, -3, -3, 2, 7, 0, 5, -1, -1], [-8, 1, 3, -3, 5, -3, 1, 2, 4], [0, 4, 1, -5, 0, -1, -3, 1, -3], [-7, -2, -9, 0, -2, 5, 2, -1, 5], [0, 3, 0, 0, 8, 4, -2, 2, -4], [10, 2, 5, 2, 1, 6, 1, 4, -1], [-1, -5, 1, 13, 6, 0, 3, -1, -1], [3, 1, 5, 1, -4, 2, -1, 3, -3], [3, 6, -5, 4, -2, 1, 1, -4, 6], [5, -3, -1, 5, -3, -1, -3, 3, -2], [4, 2, 0, -5, -1, -2, -4, -4, 3], [4, 0, 1, -1, 2, 1, -5, 7, -4], [-7, -3, -1, -3, 1, -1, 3, 5, -5], [-5, 0, 4, -8, -6, 0, -1, -5, 2], [-4, 2, -2, -3, 0, -1, -4, 0, -4], [9, -2, -2, 0, 7, 4, 2, 1, 3], [-8, -1, 4, 5, 0, -4, -4, 4, -9], [-3, 0, 4, -4, -4, -8, -4, -2, -4], [1, -8, -6, 3, 6, 9, 4, 8, 1], [4, 6, 8, -7, -2, -1, -6, -4, 2], [-8, 6, 4, -13, -1, 1, -6, -11, 4], [-8, 8, 4, -6, -10, -9, -7, -3, 0], [-3, 6, -6, -3, 3, -2, 0, -4, 4], [-3, 1, -3, -3, -7, -5, -6, -7, 4], [2, -1, -1, 7, -3, -1, 8, 2, -3], [-13, -7, 5, 2, -5, -9, 3, 4, -4], [6, 0, 2, 0, -5, -5, 1, -3, 0], [-9, -9, -4, 14, 3, -3, 3, 2, -4], [4, 14, 0, -5, -6, -1, -4, -2, 1], [-4, -2, -1, 2, -5, 1, -2, 0, 2], [1, 2, 1, 1, -5, -7, -6, -8, 1], [3, -5, -3, 2, -3, 6, -1, -1, 6], [8, 1, 1, -1, -3, 3, 3, 3, 0], [6, 3, 6, 6, -2, -7, 5, 3, -7], [-7, 1, 1, 4, -3, -4, 4, 0, -5], [5, -6, -1, -2, 9, 7, 1, 0, 4], [-11, 2, -4, 0, 8, 0, 2, 0, -1], [-7, 1, -5, 0, 1, 3, 5, 3, -3], [-3, -3, 7, 3, 6, -3, -7, 4, -4], [-5, -6, 2, -1, -4, 2, -10, 1, 2], [4, 1, 1, -3, 0, 5, 0, 4, -5], [-4, 2, -2, -3, 2, 3, -4, 6, 0], [6, 2, 2, -6, -1, -1, -1, -1, 7], [-2, -5, -6, -2, -3, -1, -2, -5, 0], [-8, -4, -4, 7, 6, 8, 0, 1, -2], [-1, 4, -2, 5, -2, -8, -1, 1, -1], [-1, -4, -4, -4, 4, 5, -2, 0, 5], [-1, -2, -4, -2, 0, 3, -2, -3, 1], [-7, -2, 4, 3, 3, -2, 6, 5, -6], [-5, 0, -3, 5, 7, -1, 11, -1, 1], [1, 1, 4, -4, -3, 3, -5, 3, 2], [2, -5, 6, -4, 3, -5, 6, 4, 2], [12, -12, -2, 9, 3, -4, 1, 4, -4], [-17, 5, -9, -12, 0, 7, -4, -8, 7], [1, -5, -7, 1, 2, 4, 9, 0, 5], [-1, 7, 2, -12, 0, -3, -7, -2, 4], [-6, 7, 3, 2, -6, -1, 3, 3, -3], [11, -4, -3, 8, 9, 1, 3, 5, 2], [12, 1, 1, 8, 8, 3, 0, 8, -8], [-17, 6, 2, -11, -4, 2, -3, -5, 2], [-12, -5, -9, 4, 6, 3, 7, -3, 2], [-6, -4, -1, 9, 3, -1, 2, 3, 1], [14, 7, 2, 12, -4, -3, -2, 3, 0], [3, -4, 7, -5, -3, -3, -5, 1, -1], [-5, 8, 3, -13, 5, 0, 1, -2, 6], [13, -8, 6, -4, -6, -4, -1, -2, 2], [-20, -7, -9, -1, 7, 2, 6, -2, 1], [-2, -8, -7, 7, 3, 0, 13, 3, 2], [-3, 5, -3, -6, -5, 0, -2, 0, 10], [-3, 0, 3, -2, -2, -1, -1, 4, -1], [-9, 7, 4, -2, -9, -8, 1, -3, -1], [18, 5, 4, -2, 3, 6, 1, 6, -5], [-6, 3, 0, 0, -3, -2, 2, -5, 1], [4, -3, -11, 6, 4, 0, 4, -6, 6], [-14, -1, 2, -2, -6, -4, 7, -3, 2], [3, -7, -10, 5, 2, 2, 6, -6, 6], [-18, -7, -3, 3, 8, 4, 3, 2, -2], [6, 1, -5, 6, -5, -4, 4, 6, -9], [-4, -1, 0, -7, 4, 5, -4, 5, -4], [8, -6, -3, 2, 10, 9, -1, 3, 3], [4, -5, 7, 3, -1, -2, -1, -1, 5], [7, 9, 5, -3, 0, -8, -5, 0, 2], [2, -11, 0, 5, 8, -1, 10, 10, -3], [2, -5, 1, 3, 3, 0, -1, 13, -2], [19, 0, -2, 15, 10, -1, 5, 7, 3], [9, 1, 1, 10, 2, -8, 6, 9, -9], [-7, 6, 1, 3, -8, -11, 5, -7, -1], [0, -13, 4, 3, -6, -6, 2, -2, 3], [5, -2, -4, -2, -2, -3, -1, -10, 10], [-3, 5, 2, -3, -2, -2, -8, 2, 2], [4, -3, 9, -8, -6, -11, -2, 1, 0], [2, -1, 7, -1, 4, -4, -2, 2, -2], [9, -4, 1, -3, -5, 8, -1, 1, 3], [8, 1, 6, -8, 5, -4, 3, -4, 3], [0, 15, 3, -9, -6, -2, -4, -2, -3], [16, 10, 5, -2, -3, -12, 0, -1, -2], [-3, 5, -1, 2, -11, -2, -14, -5, -3], [-11, 0, -3, -1, 3, -3, 5, -3, 5], [9, 8, 3, 3, -1, -9, 4, 1, -4], [11, -4, -6, 5, 5, 6, 7, 0, 3], [-11, 2, 2, -4, 4, 2, 0, 2, -7], [-10, -11, -7, -9, 2, 4, 0, -3, 3], [9, 1, -3, -6, -6, -1, -10, -2, 1], [4, -5, -7, -9, 4, 9, 3, 1, 10], [-8, -3, -4, 10, 3, -3, 2, 6, -2], [-4, -2, -6, 11, 0, 6, 6, 0, 1], [1, -12, -4, 6, 5, 0, 4, -2, 1], [5, 3, -1, 9, 2, 2, -1, -7, 3], [-6, -5, 4, -16, -2, -3, -5, -10, 4], [2, 13, 5, -6, -4, 7, -9, -2, -1], [-9, -8, -4, 8, 8, 9, 5, -1, -4], [18, -2, 2, -15, -6, 5, -4, -5, 7], [13, 6, -3, -4, -2, 9, -5, 3, -4], [20, -5, 3, 5, -6, -2, 0, -2, -3], [-6, -1, -4, -10, 7, 6, 2, 0, -1], [-4, -3, 0, -5, 2, 9, -5, 3, 7], [-22, 3, -2, -15, -6, 1, -10, -11, 3], [1, -7, 3, -1, 2, -1, -1, 12, -5], [17, 17, -2, -12, -6, 3, -1, -4, 3], [20, 3, 1, 6, -8, -7, 7, -6, 1], [5, 3, 1, 5, 0, -10, 5, -1, -5], [4, 5, -5, 1, -11, -6, 0, -6, -7], [5, 9, -7, 3, -4, 5, -2, -3, 2], [20, -12, -5, 0, 3, 6, 7, 3, 7], [7, 11, 0, -1, -6, -6, 1, -4, -5], [11, -3, 1, -5, -7, -9, -2, -7, -1], [1, -4, 2, -7, -7, -1, 4, 0, 4], [-6, 13, -9, -1, -13, -5, -3, -5, 1], [2, -5, 1, 16, 2, -6, -4, -2, 0], [14, 1, -1, -7, -4, 3, -12, -8, 10], [2, 4, 0, 0, 5, -1, 0, -2, 1], [18, -3, -1, 11, -3, 3, 2, 2, -10], [-3, 7, -2, -4, -1, 1, -7, -14, 0], [-1, -8, 8, 1, -1, -4, 3, -5, -2], [-1, -15, -12, 13, 11, 13, 9, 7, -5], [19, -6, 2, 10, 10, -3, 11, 8, -4], [-6, 2, -10, 0, -3, 8, -1, 5, 1], [-6, -5, -2, -6, -1, 2, -10, -3, -7], [-29, 8, -4, -1, -1, -2, -5, 4, -9], [-8, 2, 6, -10, 1, -9, -1, -9, 9], [2, -4, -2, -9, -4, 5, -4, -10, 11], [-10, 6, 8, 3, 4, -1, -1, 4, -9], [9, 9, 7, -16, 0, 2, 2, -1, 1], [-12, 2, -4, 14, 10, -2, 2, 11, -13], [-6, 0, 4, -7, 0, -7, -6, -7, 0], [29, -5, 2, 15, 5, 2, 5, 12, -11], [8, 10, -9, 10, -4, -10, 0, 0, -2], [7, 2, 9, 9, -1, -9, 11, 1, -7], [-18, -11, 1, 1, 10, 8, 2, 3, -5], [12, -5, 7, -3, 3, 1, -3, 11, -1], [8, 13, 3, -13, -1, 5, -4, -7, -2], [27, 9, 6, -5, -2, -5, -4, 5, 0], [1, -1, 3, -3, -9, 0, 2, 6, 5], [26, -2, 3, 3, 4, -3, 2, -1, 5], [14, 4, 7, -9, 1, 0, -5, 5, 10], [-25, 1, 5, -2, 7, 2, 1, 2, 5], [-3, 18, 2, -5, 3, 5, -8, -7, -2], [26, 0, 8, -14, -1, 5, -5, 0, 2], [16, 12, 2, 0, 0, 3, -4, 0, 6], [-9, 15, -4, -3, -5, 5, -11, -1, -3], [5, -14, -7, 10, 21, 11, 9, 3, 2], [-7, 6, 12, -3, -1, -3, -3, 4, -5], [0, -6, -2, 11, 1, -6, 9, -3, -6], [18, -1, -6, -3, 3, 11, 4, 3, -7], [19, 6, -4, -4, 0, 2, 1, 0, 0], [-7, 10, 10, 6, 0, -2, 1, 1, -1], [-6, 10, 6, 8, 3, -6, -2, 5, -5], [-18, 4, 9, -3, -7, -2, 1, 0, 3], [-9, -5, -5, 8, -4, -9, -7, -4, -6], [2, 3, -10, -8, 1, 13, -12, -3, 5], [31, -14, 3, 10, 6, 1, 5, 6, 5], [-6, -3, 3, 6, 0, -6, 7, -2, -1], [5, 7, 5, -4, 3, -11, -6, -1, -5], [-1, -3, -7, 3, -8, 5, 1, -6, 11], [15, -7, 4, 16, -4, -4, 5, 5, -10], [8, -25, -15, 8, 12, 10, 11, 1, 2], [-13, -1, 2, 4, 12, 1, 0, 4, -7], [-10, -2, 11, 2, 5, 1, 5, -2, 0], [-9, 15, 4, -5, -6, -5, -12, -2, -2], [-18, -5, 0, -9, 4, 1, 0, -4, 5], [6, -3, -2, 4, 7, 8, 5, 10, -1], [-12, 4, -8, -3, -3, 7, 5, -15, 10], [1, -11, -10, 20, 10, 7, 10, 15, -3], [3, -6, 1, 2, -6, -5, -3, -15, 4], [14, 19, 3, -23, -1, -1, 0, -3, 5], [13, -10, -2, 3, -6, 2, -8, 0, 1], [-14, -16, 1, 7, -3, 4, 5, 5, -2], [5, -2, -9, -3, 16, 18, 1, 3, 10], [-10, -7, -14, 12, 6, 8, 12, 8, 3], [2, 3, -3, -9, -10, 0, 1, -9, 4], [6, 7, -2, -10, -7, 0, 5, -9, 9], [-22, 5, -4, -3, -11, -12, -7, -5, -2], [-18, -3, 1, -5, -2, 1, 12, -1, -1], [2, 8, 0, -17, -5, 5, -2, -15, 15], [10, -11, 7, -15, 5, -1, 4, -3, 8], [-23, 8, 2, -7, 5, 2, -4, -12, 6], [-15, 13, -4, 1, 3, 0, -1, 1, -2], [8, 11, 2, 2, -6, -8, 0, 5, 0], [3, 26, 17, -4, -12, -13, -10, 3, 2], [10, -1, 1, 16, -3, -4, 3, 12, -13], [2, 3, 5, 1, -9, 0, -1, 7, 7], [-2, 0, -2, 11, 16, 0, 5, 3, -6], [2, 4, 0, 4, 7, -13, 1, -4, -3], [-10, -16, -4, 20, 8, -7, 6, 11, -6], [-9, -24, -14, 9, 16, 8, 7, 2, 0], [8, -22, 2, 16, 11, 0, 5, 7, -5], [19, 6, 9, -18, -6, -8, -1, -5, -2], [8, 1, 11, -8, -11, 1, 1, 2, 10], [12, -4, -5, 17, 1, -2, 4, 1, -6], [16, -3, 2, -9, -6, -13, -4, 0, -2], [-19, 7, 6, -11, -4, 5, -14, -10, -2], [8, 7, -7, 0, 16, 13, -1, 4, -1], [-18, -4, 6, 3, -4, -5, 1, 14, -3], [15, 13, 0, -2, 2, 6, -1, 4, 5], [16, 15, -10, -14, 4, 16, 0, -10, 11], [-1, -4, -2, 13, -6, -4, -3, -1, -4], [3, -4, -3, -15, -5, 0, 0, -3, 12], [2, 4, -3, 13, -11, 2, -9, 8, -8], [-2, -4, 4, -16, -17, 1, -10, -5, 3], [-17, 7, -5, -4, 5, 4, 5, 3, 2], [8, -21, -7, 24, 7, 0, 10, 9, -7], [0, 10, -19, 2, 2, 13, 4, -4, 7], [-15, -2, 7, -6, 8, 3, -4, -2, 5], [6, -6, 6, 7, 2, -4, -3, -7, 0], [14, -2, 10, 1, 9, -4, -4, -5, 2], [8, -1, 0, 5, -8, -6, 7, -6, 2], [25, -8, 0, 8, -2, -4, 1, -10, -4], [-25, 9, 1, 5, -5, -6, -2, -2, 6], [-10, -4, -1, 2, 12, 5, 1, 3, 1], [9, 17, 3, 12, -6, -5, 7, 3, -4], [-11, 17, 12, -14, -12, -12, -13, -6, 7], [-31, -1, 2, -4, 11, -1, 6, 8, -5], [-36, -6, -5, -17, 7, 0, -3, -4, -5], [16, 7, -10, -13, -5, 7, 0, 0, 2], [-12, 1, 6, 11, 3, -12, -3, 3, -2], [11, -7, 1, 4, 15, 9, 3, 10, 0], [2, 8, 3, -19, -1, 2, 2, 7, -2], [10, -3, 8, 16, 6, -3, 5, 10, -7], [1, 6, 11, -10, -9, -4, 4, 1, 8], [12, 10, 15, -16, -4, -13, -8, -1, -6], [20, 8, 3, -12, -10, -2, -2, -13, 12], [-9, 11, -4, -9, 1, -8, -7, -9, -5], [12, -7, 7, -2, -5, 6, -2, 9, 0], [-22, -1, -10, 8, -8, -15, 3, -8, 6], [-27, -1, -8, -11, -3, 1, -5, -8, 9], [3, 25, -7, -14, -8, 7, 1, -5, 3], [-10, 8, -4, -16, 3, 5, -4, -11, 8], [8, 7, 13, 0, 0, -12, 11, 2, -3], [-32, 4, -7, -8, 1, 3, -2, 2, -2], [-10, -13, -15, 13, -1, 12, -1, -11, 9], [11, 0, -9, 3, -4, 4, 3, -18, 13], [-24, -8, 2, 9, 9, 5, 1, 9, -14], [25, -8, 0, -1, 12, 12, 12, 3, 10], [-15, 16, 3, -14, -2, -4, -7, 1, -3], [3, -9, -15, 14, 2, 3, 11, 7, 4], [-23, -25, -2, 18, 10, 5, 5, 7, -10], [7, -5, 9, -4, -2, 2, 4, 10, -3], [11, 4, -8, -7, -2, 6, -9, 0, -7], [-18, -7, 4, -5, -10, -6, -10, -5, -14], [-11, -15, 13, 6, -7, -8, -5, 7, -2], [16, 4, 17, -5, -3, -6, -4, 19, -5], [11, -10, -8, 20, 11, 8, 12, 6, 3], [0, -19, -1, -1, 15, 0, 4, 3, -2], [8, -2, 9, 11, -4, -12, -3, -11, -5], [5, 4, -1, 18, 4, -6, 0, -11, 7], [-6, -6, 4, 14, 2, -1, -4, -6, 6], [-23, 1, 1, 11, 5, -11, 6, 6, -8], [1, 7, 14, 4, -8, -10, -9, -3, 1], [-2, 1, 11, 9, 5, -3, -4, 4, -5], [0, 4, -9, -21, -6, 10, -4, -8, 13], [-7, -3, 2, -2, -7, -7, -1, 1, -4], [0, -12, -15, 6, 1, 8, 11, 1, 5], [-13, -12, 4, 11, -14, -13, 5, -2, -7], [1, -10, 0, -7, 10, 5, 1, -7, 9], [21, 7, 6, -34, 0, 11, -7, -9, 10], [-9, 15, 7, 4, 7, 4, -7, 3, -10], [10, 6, -4, -1, -1, 5, 0, 5, -6], [-10, -7, -13, 0, 5, -2, 0, -4, -9], [3, 5, 6, 4, -4, 0, -2, -7, 17], [-7, -4, 6, -6, -6, 13, 1, 4, -8], [2, 2, 8, -11, -2, -7, -3, -9, -6], [2, 10, 9, 9, -6, -16, -1, 3, 0], [25, 6, 16, -13, -18, -2, -15, -4, -4], [-26, -5, -1, -10, -13, -4, 10, -9, 4], [-24, 5, -8, -10, -4, 3, -2, 2, -2], [-11, -3, -4, 0, -1, -1, -5, -4, 14], [-16, -13, -15, 16, 12, -3, 1, -1, -9], [1, 4, 9, -24, -12, 1, -5, -16, 8], [-11, -17, -4, 10, 14, 11, 4, 15, -7], [18, -8, 3, 13, -3, -1, -5, 5, -4], [-6, 10, -11, 1, 15, 16, 3, 0, 8], [3, -14, -7, 13, 8, -1, 7, -1, 6], [-12, 0, 2, 8, 5, 1, 11, 15, -7], [13, 7, 10, -8, -12, -4, -4, -9, 11], [9, 10, 9, 15, 9, -13, 9, 10, -9], [-1, 24, 8, 3, -6, 8, -1, 5, -6], [1, -3, 5, -1, 2, -12, 4, 9, 0], [-12, 17, -2, 7, -7, 8, 1, 6, -6], [-10, -4, -14, 9, 9, 2, -3, -5, 1], [4, 4, -4, -6, -11, 8, -5, -4, 4], [3, -8, -7, -14, -7, -6, 3, 3, 3], [-6, -15, -9, 23, 10, -2, 14, 3, 3], [-5, -13, -3, 23, 20, 3, 2, 10, -3], [-9, -8, 1, 16, -4, -16, 12, 1, -2], [-4, -3, 0, 19, 2, 1, 5, 17, -20], [14, -12, -4, 22, 6, 3, 1, 4, -2], [-5, 3, 4, 26, -1, -18, 3, -2, -4], [-12, -8, -8, -10, 16, 6, -2, -10, -5], [-2, 15, 10, 8, 1, -7, -2, 4, 3], [11, -25, 7, 25, 23, -3, 9, 5, 1], [-16, 11, 15, -14, -6, -9, -1, -1, -3], [-6, -5, 7, 7, 7, 6, 1, 9, -7], [-15, 3, 7, -10, -1, -5, 7, 8, -2], [-14, -4, -3, 0, -3, -8, -2, -10, 12], [-3, 18, -2, 5, 17, -2, 0, -3, -5], [-24, 6, 4, -16, -1, 6, 4, -1, 8], [-1, -9, 8, 26, -9, -15, 9, 7, -16], [19, 5, 15, -23, 1, 9, -10, 2, -2], [6, -8, 14, 21, -4, -12, 3, 0, -7], [12, 4, 5, -2, -16, -3, -1, 3, -7], [14, -13, -3, 5, 8, 6, 8, 20, -12], [5, -4, 12, 9, 21, -1, 9, 11, -7], [-4, -3, -11, 21, 7, -5, 6, 10, -5], [2, -15, -10, 16, 3, -5, 12, 8, -4], [-27, -20, -6, 2, 16, 0, 11, 4, -4], [15, 11, 9, -13, 11, -4, -3, -3, -3], [-17, 2, -14, 0, 8, 6, 8, -4, 9], [-4, -6, 4, -13, -5, 5, -5, 1, 7], [3, -19, 10, 7, -1, -2, 3, 16, 1], [-39, -9, 1, -5, -3, 1, -8, -6, -8], [-9, -1, -6, -15, 0, 4, -8, 6, 10], [-15, -8, 8, -3, -6, -4, -2, -4, 11], [-30, 3, 7, -17, -12, -7, -11, -19, 6], [-14, -6, -5, 12, 15, 3, 7, 13, 0], [-8, -18, -5, 31, 6, 4, 4, 6, -14], [23, -1, -9, 7, -25, -10, 0, -4, -2], [5, 5, 9, -7, -5, -16, 9, -8, 15], [31, 3, -6, -8, -10, 10, 1, -12, 8], [7, -16, 0, -3, 2, 4, -1, -2, -5], [14, -9, 3, 3, 3, -5, 7, 10, 8], [-2, -22, 9, 4, -6, -4, 2, 1, -4], [7, -14, -4, 21, 11, -2, 9, -1, 0], [-21, 7, 15, -8, 8, 0, -11, 7, -18], [12, -1, 3, 7, -1, -5, 4, 11, -3], [2, -6, 0, 1, 10, 9, 1, -1, 14], [2, 8, 2, -23, -4, 1, -2, 1, 11], [-37, -5, -16, -1, -6, -1, 3, -11, 10], [8, -11, -17, -1, 6, 16, 17, 7, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6020_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6020_2_a_g();".
// 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_6020_2_a_g(:prec:=9)
    chi := MakeCharacter_6020_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_6020_2_a_g();".
// 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_6020_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6020_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 77, 32, -160, -9, 83, 0, -16, 0, 1]>,<11,R![1611, 2898, -1789, -2516, 275, 547, 0, -42, -1, 1]>],Snew);
    return Vf;
end function;