
// Make newform 6003.2.a.l in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6003_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_6003_2_a_l();".
// 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_6003_2_a_l();".
// 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 := [43, -101, -28, 182, -29, -118, 32, 32, -10, -3, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-22, 50, 3, -66, 16, 26, -8, -3, 1, 0], [-53, 83, 45, -98, -1, 36, -6, -4, 1, 0], [-22, 72, -47, -69, 82, 10, -34, 5, 4, -1], [-83, 146, 54, -172, 14, 63, -14, -7, 2, 0], [-17, -84, 226, 32, -253, 39, 91, -23, -10, 3], [-109, 153, 144, -200, -69, 89, 14, -16, -1, 1], [78, -226, 94, 225, -186, -49, 77, -7, -9, 2]];
    Rf_basisdens := [1, 1, 1, 1, 1, 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_6003_a();"
function MakeCharacter_6003_a()
    N := 6003;
    order := 1;
    char_gens := [668, 3133, 4555];
    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_6003_a_Hecke(Kf)
    return MakeCharacter_6003_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, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, -1, 0, 0, 0, 1, 0, -1], [0, -1, 1, -1, 0, -1, -1, 0, 0, -1], [-2, 1, -1, 1, -1, -1, 0, 0, 0, -1], [2, 1, 0, 1, -1, 0, 1, 1, -2, -1], [0, 1, 0, 0, 1, 0, 0, 0, -1, 2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 1, 0, -1, -1, 0, -1, 1, 0], [-1, 1, -1, 1, 1, 0, 1, 2, -2, 0], [2, 1, -3, 2, 3, -1, 0, -1, -2, 0], [0, 0, -1, 1, -2, -1, 1, -1, 2, 0], [1, 4, -1, -2, -2, 1, 1, 1, 0, 2], [5, 0, 1, -2, 0, 1, 3, 0, 2, 2], [1, -3, 2, 0, -2, -2, -1, 0, 1, -2], [0, -2, 0, 0, -1, 0, 1, 2, -1, -1], [1, 0, 0, -3, -3, 1, 0, 1, 0, -1], [1, 2, 3, -3, -3, 2, -3, 1, 3, 0], [-4, 0, -1, 0, 1, 0, 2, -1, -1, 1], [1, 2, 2, -1, -4, 4, 3, 2, 0, 1], [3, 2, 2, 0, 1, 2, -2, 0, 2, 2], [1, 0, -3, 3, 0, -3, 1, -1, -3, 1], [0, 0, -3, -1, -2, -2, -2, -1, 1, 1], [2, 1, -3, -2, 1, -2, 2, 1, -2, 1], [3, -3, 1, 1, 4, -5, 0, -3, -1, 1], [2, -2, -1, -2, 3, -1, 0, -1, -1, 2], [3, -2, -1, 3, 2, -3, 0, -1, -4, -2], [5, -2, 3, 0, -4, 1, 1, 3, 3, -2], [1, -2, -2, 1, -3, -2, -2, 1, 0, -3], [8, 0, 2, -1, -3, 0, 0, 3, 0, 0], [7, 4, 2, -3, 0, 3, 2, 4, -5, 0], [1, 3, -2, -1, 3, 3, 3, -1, -3, 3], [-4, 2, -1, 2, -2, 1, 3, -1, -1, 0], [3, 2, 0, -3, 3, 0, -1, 1, -2, -1], [-5, 5, 1, 1, -6, 2, -4, 2, 0, 2], [1, -6, -3, 4, 5, -8, -3, -4, -1, 1], [1, -3, 0, 1, 2, -1, -2, -4, 0, 2], [8, 0, -2, 1, -1, 0, 1, -2, 5, 2], [0, 4, 1, 3, 2, -3, -3, 0, -2, 2], [3, -2, -3, 0, -1, -1, -1, -2, 4, -4], [-1, 1, -3, 2, 5, -2, 3, -2, -1, 4], [-3, -1, 0, -1, 2, -3, -7, 4, -3, -3], [8, -6, 1, 0, 4, -2, -2, 1, -1, 0], [4, 0, -3, -3, 7, -2, 4, 3, -3, 4], [-3, 3, -3, 2, 1, -4, -5, -6, 0, -2], [2, 1, 6, -7, -3, 2, -2, 0, 2, 2], [-8, 5, 1, -4, -4, 1, 3, 3, -4, -4], [1, -1, 1, -2, 0, 0, -5, -2, -2, -1], [-2, -6, -1, 2, -6, 0, -6, 1, 2, -6], [-3, -1, -10, 7, 4, -6, 3, -3, -8, -2], [5, 1, 8, -3, -4, 6, -1, 6, 4, -5], [-1, -1, 4, 0, -1, 4, 1, 0, 6, -1], [3, -6, 3, 0, 0, 2, -3, 0, 7, -1], [7, -1, -8, 0, 6, -7, -1, -1, -2, -1], [4, -4, -2, 6, -2, -3, -7, -1, 1, -2], [5, -1, -3, 3, 6, 0, -2, -2, -2, 0], [-9, 3, -1, -2, 2, 3, 1, 4, -2, 2], [-3, 0, 0, 1, -7, -3, -5, 1, 3, -3], [0, -4, -4, 1, 10, -6, -1, -3, -8, 0], [4, -6, -8, 10, 7, -10, -7, -7, -3, -4], [1, -3, 6, -1, -2, 3, 2, 0, 4, 3], [9, 1, -2, 2, 2, -1, 3, 0, -6, -2], [-4, 0, 8, 1, 0, 1, -1, 4, 0, 2], [-2, 5, -3, -1, -1, 2, 1, -2, 5, -3], [-7, -2, 1, -6, 3, 4, -4, 0, 5, 6], [4, -11, -2, 3, 7, -4, 1, -6, 5, 5], [0, -3, 1, 3, 10, -2, 1, 2, -6, 4], [-1, 11, 2, -2, -8, 6, 2, 7, -1, 0], [7, -1, -5, -3, 5, -6, -1, 1, -8, 0], [7, -1, -6, 0, 1, 1, 5, 4, -6, 1], [-4, -7, 2, 2, 6, -5, -3, -1, -1, -5], [2, -2, 0, 2, -4, 0, -1, -7, 1, 1], [0, 1, 1, 7, -4, -7, -3, -2, 3, 0], [3, 1, -2, 2, 9, 1, -2, -3, 4, 12], [2, 1, 3, -2, 1, 0, -10, 1, 1, 2], [-1, -1, 2, -7, 8, 4, 4, 5, 0, 1], [-4, -3, -5, -2, 6, -8, -2, -6, -1, 4], [-2, -6, -9, 8, 0, -4, -3, -8, 6, 1], [-5, 4, 4, -4, -3, 10, 1, -3, 6, 0], [-1, 1, 0, 2, 4, 1, 0, 3, -7, 0], [1, -3, 1, -5, 2, 7, 6, 4, 0, 6], [-6, 6, 4, -3, -2, 1, 3, 1, -7, -1], [6, -9, 1, -1, 8, 1, -4, -5, -2, -1], [12, -5, -1, 2, 7, 2, 2, -1, -5, 3], [-10, -2, 2, -2, -6, 1, 3, 8, -1, -5], [-3, -3, -7, 7, 0, -6, -1, -3, -3, -6], [13, 5, 5, 0, -4, 5, 0, 4, -2, -5], [6, 13, 1, 3, -1, 5, -1, 3, -2, 1], [6, -4, -5, 3, 2, 3, 1, -8, -1, 6], [4, -3, -5, 7, 1, -9, 0, -9, 1, -3], [3, -3, 8, -6, -6, 5, -4, -4, 6, 1], [-9, 0, 3, 1, -2, -3, -5, -3, 1, 1], [-8, 3, 0, 1, -12, 2, -5, 2, 4, -7], [-12, -6, -2, 6, 4, -8, 2, -2, 6, 1], [13, -1, -2, -1, -6, 6, 6, 6, 1, 2], [2, -5, 8, -1, -8, 2, -5, 7, -2, -4], [-1, 8, -9, 5, 10, 1, 6, 1, -1, 3], [4, -1, 1, 2, 4, -2, -4, 4, 2, 5], [6, -3, 0, -6, -2, 1, 10, 0, 3, 5], [7, -2, 4, -4, -8, -4, 3, 4, 3, -1], [-2, 5, 8, -4, -3, 9, 0, -1, 7, 6], [-8, 2, -12, 8, 4, -12, 12, 5, -11, -6], [4, -8, -11, 11, 9, -14, -5, -9, -3, -8], [4, -1, 13, -11, 1, 5, -11, 3, 7, 6], [0, 8, 9, -10, -1, 4, 9, 10, -7, -1], [-2, -10, -6, 4, 7, -12, 3, -5, -2, -1], [-9, 5, 1, -6, -6, 5, 8, 8, 0, 2], [-3, -2, -1, 5, -3, -2, 4, -9, 1, 3], [-5, 14, 6, -6, -12, 7, 1, 13, -5, -1], [11, -13, 9, 6, 5, -3, -4, -7, 7, -1], [14, -1, -2, -3, 2, 1, 7, -1, -3, 7], [12, 5, -4, 1, -7, -1, 1, 3, 1, 4], [9, -1, 2, -7, 1, 1, 14, -2, 2, 9], [2, 14, 0, 4, -4, 4, -1, 5, -4, -3], [16, -1, 1, 10, -2, 0, -7, -2, 7, 1], [14, -3, -2, 4, 1, -4, -6, 6, 5, -2], [11, -8, -5, 0, 6, -6, -3, 0, -9, -6], [-6, 6, -6, 7, -4, -5, -4, -7, 3, 3], [1, -3, 8, -5, -8, 5, -2, -1, 4, 3], [-1, -10, 2, 6, -2, -6, -10, -1, 5, -4], [2, -1, -2, -2, 9, -2, -4, 2, -16, -4], [3, -8, 0, -3, 4, 2, 0, -6, -7, -1], [-11, 3, -5, -4, -15, -6, -5, -2, 6, -5], [2, -3, 1, 4, 5, -2, -4, -5, 7, 10], [10, -1, 0, 1, 7, 0, 7, 0, 0, 3], [13, -8, 9, -2, 0, -1, -4, 0, 8, -5], [-2, 2, -1, 3, 2, 6, -3, -3, 8, 11], [-6, 6, -6, -6, -3, 0, 8, 5, -10, -5], [-7, -4, -8, 11, -3, -4, -1, -4, -1, -8], [8, -2, 8, -4, 3, -2, 2, 8, -6, -5], [-2, 1, -4, 8, -6, 1, 11, 4, 0, -5], [-7, -5, 0, -4, 1, 5, -3, -5, -3, 5], [-7, -3, 1, -9, 9, 9, 2, -3, 4, 10], [-5, -6, -1, 0, 10, -8, 0, -5, -3, 1], [11, -2, 7, -9, -12, 11, 3, -4, 3, -2], [4, -9, -6, 2, -6, 0, -6, -1, 1, 1], [5, 11, 1, 6, -6, 7, 0, 5, 7, 6], [-3, 0, -3, 8, 10, 1, 3, -8, -3, 9], [2, -2, 9, -7, 5, 2, -2, -6, 9, 5], [10, 2, 6, 0, 5, 7, 3, -2, 7, 4], [3, -7, 7, 3, -13, -1, -1, -3, 3, -2], [-1, -7, -7, -2, 1, -3, -2, -12, 4, 5], [-5, -3, 6, 0, 0, -4, 4, 4, -7, -7], [-22, 10, -6, 7, -2, -4, -1, 2, -4, -3], [0, -7, -2, 5, 2, -3, 0, 1, -12, -11], [1, 8, -6, 8, -9, -9, 0, -5, -1, -3], [-16, 0, -7, -7, 3, -6, 10, 5, -5, -2], [0, -5, -2, -1, -3, -9, -12, 2, -4, 0], [1, 9, -10, -1, 5, 0, 5, 0, -16, 10], [8, 6, -1, 2, -5, -5, 8, -3, -3, -4], [-2, -3, -4, 3, 12, -5, 3, 8, -8, 0], [8, -3, -3, 6, 4, -11, -2, 0, -1, -8], [4, 5, 10, -12, 6, 17, 11, 8, -1, 6], [-3, 10, -3, 5, -4, 6, -2, -8, -4, 1], [7, 10, 1, -2, 0, 6, 2, -7, -1, 13], [9, -3, 6, -6, 8, 1, -2, 3, -1, 4], [-3, 6, -8, -2, 4, 1, 2, -3, -5, 12], [-4, 1, -5, 4, 5, -3, -5, -1, -7, -2], [-1, 2, -1, 12, 3, -1, 2, 2, 8, 1], [-6, -10, 10, -3, -6, 3, -16, 1, 12, -3], [-1, 5, -5, 0, 3, -10, -9, -8, -3, 0], [9, -13, 3, 7, 5, 0, 1, -9, 8, 6], [-15, -15, -6, 4, -2, -7, 4, -3, -11, -18], [3, -7, 19, 0, 1, 8, -12, -9, 16, 1], [-1, 5, -2, 0, -1, 1, -2, 10, -14, -7], [-3, 5, 0, 5, 8, -8, 10, -5, -2, 4], [-21, -3, -10, 1, 4, -11, -5, -3, -1, -6], [13, -4, 2, -6, 10, 6, -3, 3, 3, 8], [1, 6, 8, -10, 2, 3, -8, 10, -8, 2], [-2, -2, 12, -7, -10, 13, 10, 10, -5, 1], [-2, 9, -17, 4, 10, -7, 1, -9, -7, 5], [12, 2, 3, -4, -2, 9, 22, -1, 1, 10], [0, -12, -7, 5, 15, -8, 0, -10, 2, 13], [-1, 2, 2, 7, -2, 3, -8, 4, 0, -2], [-6, -10, -7, -8, 0, -14, 3, 0, -8, -5], [-7, 2, -3, 11, -3, -13, -5, -11, 2, 2], [-7, -1, -2, -4, -11, -1, -3, -7, 3, -7], [7, 2, 4, -4, -1, 4, -1, 6, 1, -8], [-3, 0, 4, -3, -8, 1, 5, -3, -2, -5], [-2, -1, -1, -7, 2, 10, 0, -7, 8, 11], [-4, 6, 3, 6, -3, 3, 13, 2, 3, 1], [1, -1, 5, -4, -2, -5, -4, 0, -6, -4], [-6, 7, -4, 4, -2, 1, 6, 7, -9, 6], [9, 2, 5, -15, -6, 0, 1, 8, 0, -1], [-9, -9, 1, 13, 13, -3, -9, -3, -5, -5], [7, 12, 7, 9, -2, 8, 7, 8, -1, 1], [8, 7, 4, 0, -7, 6, -13, -6, 5, 1], [5, -7, -4, -5, 8, -10, 0, 0, 2, 6], [2, -2, 4, -2, 5, 9, 1, -1, 9, 13], [-9, 13, -6, 0, -3, 13, 7, 5, 5, 8], [-3, 8, 9, -4, -2, 5, -9, 1, 6, 10], [-11, 18, 1, 0, 1, 7, -6, -3, -4, 6], [-10, 6, -3, -12, 11, -4, 2, 5, 4, -1], [0, 7, 4, 3, 5, 3, -6, -3, 2, 0], [2, 7, -4, -8, 5, 0, 0, 5, 1, 2], [-1, 13, -7, -1, -10, 12, 11, -3, 12, 5], [-11, -3, 2, 0, 8, -5, -7, 0, -1, 6], [11, -7, 4, 0, -15, 4, 3, 8, -4, -8], [-3, -19, -1, -7, 3, -3, -1, -6, 6, 6], [-11, -10, 6, -6, -1, 1, -2, 0, 10, -5], [-20, 5, -8, -4, 9, -5, 2, 8, -15, 0], [-4, 0, 19, -12, -8, 18, -2, -1, 9, 5], [-7, -4, -11, 8, 5, -4, -4, -9, 8, -3], [13, 12, -3, -3, 7, 1, 15, -2, -3, -2], [-10, 13, 2, -2, 4, 2, -1, 4, -4, 5], [-3, 7, -6, 11, -15, -1, 1, -10, 8, 1], [-5, 5, -13, 13, -1, -12, -1, 3, 2, -3], [-21, 15, 1, -4, -14, 11, 12, 7, 10, 4], [-16, 5, -9, 1, -3, 2, 3, -7, 3, 4], [24, 5, 2, -7, 3, 7, 5, 3, 7, 10], [3, 6, 11, -8, 0, 10, -3, 3, 4, 0], [1, -1, 8, -5, -8, 11, -13, -3, 10, 5], [-3, 6, -4, -8, 0, 8, 5, -1, -6, 14], [-9, 4, 0, 0, 1, -5, -4, -2, 8, -6], [15, -9, 8, -12, 0, 9, 0, 5, 9, 8], [-17, 7, 13, -7, -15, 6, 7, 13, 0, -13], [9, -6, 0, -1, 5, -5, -1, -8, 2, 12], [-3, 12, 7, -5, 5, -7, -3, -4, 2, 2], [-9, 15, -3, -10, 0, 1, -7, 17, -9, -6], [14, 9, -10, 0, -6, -1, -7, -9, 1, 15], [-4, 9, -6, -10, 3, 8, 12, 13, -9, 5], [-14, 12, -7, 5, 5, -1, 2, 6, -10, -2], [-8, -1, -14, 0, 8, -5, -8, -6, -11, -5], [5, -6, 16, -7, -5, 3, -7, 5, -8, 6], [5, 7, -7, -7, 11, -11, 0, 0, -14, -3], [-6, 8, 8, -1, -6, 15, 5, 1, 8, 7], [-22, 7, -20, 9, 7, -12, 1, -6, 4, 7], [-4, 8, 13, -5, -7, -2, 6, 15, -14, -13], [7, -24, 4, -2, 15, -5, -2, -13, 5, 10], [6, 3, 17, -6, -1, 4, 6, 15, -5, -9], [-3, 9, -1, -5, 15, -4, -10, -7, -8, 11], [-4, 5, -6, 6, -13, -7, -9, 7, 3, -10], [14, 0, 20, -7, 1, 17, 0, 3, -3, 6], [-3, 2, -11, 0, 3, 1, 1, -17, 12, 8], [20, -5, -1, -7, 1, 9, 8, 11, -7, -10], [-15, -19, -11, 12, 4, -20, -13, -10, -3, -13], [13, -12, -10, 7, 10, -11, 0, -2, 3, 3], [-2, 6, 13, -16, -9, 19, 9, 9, 10, 12], [7, -13, -1, 2, -2, -5, 9, -5, 4, -1], [-14, -4, -1, 6, -2, -2, 7, 7, 9, -10], [-19, -6, -16, 4, 2, -13, -5, -14, -2, 0], [2, 10, -4, -2, -1, -6, 18, 4, -7, -3], [-7, 14, 0, -12, 3, 5, -5, 4, 1, 3], [2, 1, 5, -2, -2, -2, -14, -6, 6, 13], [-9, 0, 8, -14, -3, 4, 3, -10, 20, 12], [2, 20, -10, 2, -1, -4, 22, 0, -17, 4], [2, -2, -17, -2, 3, -4, -6, -10, 9, -2], [9, 4, 8, -10, 12, 4, 1, 1, -7, 2], [-3, -3, -2, 0, 15, -2, 7, -10, -4, 15], [-5, 9, 4, -3, 3, -6, 6, 0, -2, 5], [-16, 9, -12, 5, -7, -11, -7, -2, -5, 2], [6, 0, 4, -11, 14, -1, -3, 1, 0, 0], [13, 0, -2, -4, 6, 2, 11, -4, 0, 22], [-10, -27, 13, 4, 3, 3, -3, 1, 5, -10], [11, 7, 12, -11, -15, -1, 5, 16, -8, -15], [8, 7, 1, -15, 6, 5, 16, 13, -3, 10], [20, -12, 13, 1, 4, 4, -4, 6, 0, 6], [-14, 14, -16, 2, 13, -7, 14, -3, -11, 7], [18, -16, 10, 0, 8, -7, -3, -13, 3, -8], [-8, -14, 9, 4, -4, -3, 3, -2, -8, -14], [2, -4, -14, 0, -7, -5, -4, 3, 1, 0], [-7, 0, -19, 18, 7, -10, -2, -9, -4, -7], [16, 6, -5, -3, 26, 1, 13, 2, -10, 11], [-3, 10, 8, 2, -2, 15, 8, -5, 16, 0], [8, -7, -2, -9, -3, 5, 7, -6, -10, -5], [14, 5, 4, -4, -8, 1, 6, 1, 18, 4], [-20, 4, -5, 3, -9, -8, -14, 3, -3, -17], [7, 17, 14, -1, -10, 19, 1, 10, 7, 4], [-12, 4, 4, 8, -3, 2, -7, 3, -2, 8], [-18, 7, -11, 0, -2, -2, 8, -1, -19, -4], [13, -7, -13, 13, 2, -6, 6, -2, 1, 4], [-15, 4, -1, 5, -5, 5, 3, 9, 4, -6], [6, 4, -8, 2, 6, -2, -1, 3, -4, -15], [-11, 7, -7, 10, -6, -2, 2, 9, -21, 0], [-5, 16, -1, 0, -3, -1, -3, 0, -5, 9], [-1, 16, 0, -4, -15, -8, 4, 14, -13, -12], [-9, 10, -9, 3, -7, -8, 2, -1, -9, -2], [9, 12, -14, 7, 8, -1, -10, -8, -10, 14], [-4, 9, 11, -7, -5, 4, -13, 10, -8, -8], [12, -4, 12, -12, -12, 12, 3, 8, 10, 0], [5, 2, -4, 18, 13, 1, -4, -1, 7, 0], [-16, 6, 5, -12, -19, -6, 2, 8, -6, -16], [-17, -13, 8, 0, -16, 7, -7, 4, 18, -15], [-20, -5, -3, 2, -2, -10, -18, 3, 0, -25], [-7, 8, -4, -16, -2, 8, 10, 11, -6, -3], [7, -24, 0, 8, 6, -16, -19, -16, -6, -20], [4, -9, 11, 7, -2, -3, 7, 2, 9, -5], [-15, 5, -17, 2, 5, -11, -1, -16, 4, 11], [-6, 9, 17, -21, 3, 8, 15, 13, -6, 0], [13, 21, -12, -3, -10, 2, -1, -1, 6, 12], [-23, 8, 11, -13, -9, 16, 3, 18, 13, 1], [-4, 21, -3, -3, -5, -4, -7, 13, -6, -12], [-22, 3, 15, -15, -2, -2, 2, 13, 1, -13], [13, -5, 13, -15, -21, 15, 12, 13, -3, -8], [-2, -27, -5, 7, 0, -15, -7, -16, 6, -10], [30, 27, 3, -4, -7, 20, 2, 8, 1, 6], [3, 11, -3, -8, 5, 5, -9, -1, 2, 2], [0, -7, -7, 4, 18, -6, -2, -9, -6, 14], [8, -7, 8, 7, 4, -12, 2, 3, -11, -14], [14, 9, 0, 3, -2, 20, 19, -1, 7, 13], [-18, 5, -5, 6, 7, 2, -2, -13, 9, 17], [13, -18, -4, 0, 6, -16, -5, -17, -4, 10], [17, -15, 14, 4, -2, 6, -4, 8, 11, 0], [5, 22, 1, -9, -4, 29, 4, 1, 3, 13], [2, 3, 6, -17, -11, 3, 9, 4, 3, -1], [10, -13, 9, -6, -2, 4, 14, -2, 12, -7], [-4, -6, -6, -1, 21, -17, -13, -14, -5, -3], [27, -10, -2, 0, 15, -10, 20, -7, 4, 3], [-7, 9, 10, 5, -4, 3, 4, 8, 5, -18], [-2, 14, 5, 4, -1, 21, -4, -3, 7, 7], [-4, -6, -4, 2, 19, -7, 13, 4, -2, 0], [14, -22, 16, -6, 6, -5, -7, 7, -3, -14], [35, -21, 8, -3, -6, 10, 11, -7, 17, 9], [13, -8, -12, 14, 8, -1, -14, -17, 19, 8], [0, -3, -4, 10, -10, -16, -9, 0, 6, -5], [-15, -14, 11, -6, -12, 6, -17, 7, 14, -1], [19, -16, -3, 19, 11, -7, -14, -7, -10, 0], [12, 15, 6, 2, -11, 13, 0, 4, 8, -2], [-4, 1, -9, 19, 6, -6, -13, -9, -12, -10], [8, 19, 9, -5, -7, 21, 9, 10, 5, -3], [-19, -18, 0, 0, -12, -8, -5, -11, 15, -4], [-2, -15, 5, 14, 14, 1, -10, -3, 5, 17], [-1, -13, -6, -1, 9, -8, 7, 9, -12, -18], [12, -16, -5, 1, 3, -16, 12, 3, 3, -8], [-7, -9, 1, 5, 9, -2, -7, 0, -6, 10], [-1, -20, 12, -1, -8, 12, 4, -4, 17, 4], [-12, 25, 6, -10, -4, 22, 7, 3, -2, 24], [-3, -7, -7, 0, 6, 9, 23, 11, -8, 8], [2, 18, 3, -3, -8, -5, -11, 8, 8, 2], [-7, 7, 29, -9, -5, 16, -3, 19, 5, -2], [1, -17, 0, 19, 17, -13, -17, -14, 16, 7], [12, -17, -2, 4, 12, -12, -2, -21, 1, -5], [-6, 18, 2, -1, 16, 1, 4, -14, 8, 16], [28, -13, 9, 8, 3, -13, -18, -16, 0, -7], [-23, -8, 6, 2, 13, -1, -18, -21, 15, -1], [-20, 13, -6, 6, 10, -5, 2, -2, -1, 5], [3, 10, 11, -2, -11, 2, 3, 15, 9, 6], [-25, 7, -11, 0, -10, -1, -1, 19, -4, -5], [4, -2, -15, 7, 8, -15, 2, -5, -6, 5], [12, -5, -16, 17, 23, -16, -13, -13, -18, 3], [-2, 2, 14, -11, -14, 0, -15, 9, 4, 1], [11, 15, -1, -16, -17, 6, -7, -3, 13, 8], [25, 6, 18, -7, -8, 12, 1, 6, 16, 15], [1, 10, -16, 4, -4, -10, 15, -2, -9, 4], [4, -4, 22, -16, -13, 5, -8, 3, 24, -13], [5, 19, 5, -10, 9, 7, 13, 8, -13, 4], [-9, -13, -6, 2, 7, -1, -3, -26, 11, 5], [23, 5, 1, -7, -11, -14, 2, -3, -10, -2], [25, -15, 2, 6, 2, -11, 9, -6, 4, -13], [28, 2, 2, 6, -3, -6, 0, -4, 1, -18], [9, -8, -8, 16, 2, -9, 6, 0, -23, -12], [16, 11, -11, 1, 10, -1, -2, -10, -5, 26], [0, 5, -2, -1, -18, 1, -8, -4, 8, -10], [9, -9, -6, 3, -3, -28, -15, 5, 1, -18], [3, -1, -13, -15, -4, -11, 14, 3, -5, -2], [2, -16, 13, -11, 15, -3, -1, 6, 0, -2], [8, 4, -8, 0, -4, 10, 12, 7, 8, 13], [-13, 5, -5, 22, 26, 4, -1, -5, -6, 13], [18, -4, 7, 4, -20, 13, -6, -6, -1, 3], [-10, 6, 1, -1, 0, 1, -10, 0, -7, 11], [11, 8, -7, -5, 2, 12, 27, -8, 7, 16], [-13, 15, -1, -7, 17, 13, 9, 5, -1, 9], [2, -11, -11, -11, 0, -10, -2, -13, -8, 8], [-3, 7, 16, -21, -6, 8, 3, 18, 4, 3], [-5, 8, 1, -2, 8, 5, -8, -1, 3, -8], [20, -1, -7, -4, -4, 0, -3, -7, 3, -9], [12, -12, 9, -9, -6, 6, -3, 4, 10, -3], [7, 11, 0, 3, -5, -11, 9, 7, -8, -1], [-24, -11, 15, -2, -12, 5, 1, -8, 23, -1], [8, -9, 4, 1, -14, 0, -12, -2, 7, -16], [0, 10, -8, 0, 0, 14, 27, 14, -5, 0], [10, 33, -11, 9, 2, -1, 3, 2, -16, 5], [4, -2, -22, 17, 1, -8, 3, -1, -19, -1], [-25, 21, 0, -8, 7, 15, 1, -11, 5, 20], [-35, 19, 4, -12, -12, 8, 2, -1, -2, 10], [8, 10, -16, 15, -10, -5, 21, 6, -10, -10], [5, -21, 10, 1, -10, 9, 19, 5, 3, -3], [-12, -4, 5, 4, -4, 1, 0, 4, 13, 4], [1, -11, -19, 3, 10, -5, -5, -25, 15, 5], [-15, -6, 8, -17, -7, 10, -6, -14, 24, 18], [-2, -12, 16, 2, -21, 0, -10, -4, 7, -1], [-13, 7, 0, 4, -11, -3, -10, 5, 3, -7], [48, -16, -11, 0, 9, -1, -1, 2, 1, 3], [-14, -25, 3, -2, 4, -7, 12, -14, 2, 6], [5, 2, -6, 10, -13, 9, -13, -3, -2, 0], [8, 7, 0, -2, -4, -1, -12, -11, 12, -9], [-7, 13, -18, 11, 23, -14, -5, 5, -9, 10], [-14, -10, 4, 11, -8, -14, -23, -5, 5, -9], [-4, -9, -16, 11, 1, -7, 16, 8, -20, -8], [-20, -2, -1, -3, 0, -9, -18, 2, -1, -24], [-4, -25, -2, -5, 5, -6, 12, -9, 10, 12], [-17, -4, -12, -14, 4, 1, 9, 3, -3, 7], [-15, -27, 7, 0, -22, -2, -11, -10, 13, -12], [49, -3, -13, 4, 13, -13, 0, -5, -20, 7], [-35, -2, -1, 6, 1, 5, -3, 0, -13, 2], [10, -1, 17, -12, -10, -4, -4, 15, -3, -2], [-27, -3, -5, 5, 16, -9, 2, -8, -3, -2], [-14, 2, 11, 1, -2, 18, 4, 8, -6, 9], [0, 3, 5, 15, 5, 4, 4, -3, 7, 6], [12, 14, 4, 5, 5, 4, 6, -2, -2, 3], [-22, -4, 7, 2, 12, -11, -23, -14, 0, -9], [15, 0, 9, -14, 4, 7, -13, 3, 5, -3], [-5, -16, -9, -11, 1, -13, 5, 9, -6, -23], [4, 6, -19, 9, 20, -14, 15, 17, -33, 0], [-19, 9, -34, 13, -7, -14, 4, -12, 8, -3], [-6, 8, 5, -3, 1, -5, 3, -3, -26, -11], [-3, -13, 13, -4, -17, -5, -10, -18, 10, -3], [16, 1, -1, -6, 3, 26, 5, -1, 13, 18], [21, 11, -24, 8, 6, 8, 0, -6, -4, 9], [18, -3, -1, -12, -8, 9, 13, 2, 9, -6], [-29, -13, 8, -10, -4, 0, -8, 9, 2, 6], [18, 4, -1, 7, 18, 0, 5, -4, 2, 2], [24, 4, 13, 1, -30, 14, -18, 6, 4, -8], [-9, -11, 2, 11, 17, -15, -1, -2, -18, -25], [-14, 10, -12, -9, -12, -1, -1, -2, 8, -10], [24, 9, -15, 2, 1, 4, 10, -9, 2, 21], [4, -14, -21, 15, 21, -21, -3, -15, -4, -8], [8, -15, 8, 3, -19, 10, -11, -14, 3, -1], [-16, -7, -10, -12, 2, -14, -5, -6, -13, -14], [4, 1, -7, -6, 21, 13, 22, -2, -2, 23], [-3, 24, -8, 9, -12, 8, -1, -4, -3, 26], [13, -3, 3, -13, -7, 12, 0, -9, 16, -7], [-6, 25, 18, -18, -30, 29, 13, 22, 13, 7], [29, 7, 7, 21, -10, 7, 14, 9, -13, 2], [-21, 3, -17, 7, 28, 0, 3, -11, 3, 0], [-13, -7, 8, 4, -6, 10, 16, -3, -1, -8], [-29, 12, -23, -2, 5, -8, 3, -5, -11, 3], [15, -15, 1, 0, 18, 11, 3, -6, 7, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6003_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6003_2_a_l();".
// 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_6003_2_a_l(:prec:=10)
    chi := MakeCharacter_6003_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_6003_2_a_l();".
// 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_6003_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![43, -101, -28, 182, -29, -118, 32, 32, -10, -3, 1]>,<5,R![-349, 1352, -961, -884, 1109, -27, -310, 82, 20, -10, 1]>],Snew);
    return Vf;
end function;