
// Make newform 6003.2.a.i in Magma, downloaded from the LMFDB on 23 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_i();".
// 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_i();".
// 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, -5, -20, 19, 10, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-7, 10, 14, -11, -7, 2, 1], [-1, -26, -10, 15, 7, -2, -1], [-5, -2, 20, 1, -9, 0, 1], [5, -6, -50, 9, 25, -2, -3], [-5, -46, 46, 23, -25, -2, 3]];
    Rf_basisdens := [1, 1, 4, 4, 2, 4, 4];
    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, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 1], [-1, 0, -1, 0, -1, -1, 0], [2, 0, 0, 0, 0, 1, -1], [-1, 2, 2, -1, 0, 1, 0], [2, 0, 1, -1, -1, -1, -1], [-1, -3, 0, 2, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0], [-2, 2, 1, 0, 1, -1, 0], [-3, -1, -2, 0, -2, 0, 0], [-4, -1, -1, 2, 1, -4, -1], [2, 2, 3, -2, -2, 1, 0], [-5, 2, -1, 1, 1, 0, 0], [-1, 0, -1, 2, 3, 2, -1], [-2, -2, 0, 2, 2, 0, 0], [-1, -1, -2, 1, -1, -3, 2], [-4, 0, -1, -1, 3, -1, 1], [3, -1, 0, -1, -1, 1, 0], [-6, -2, -1, 3, 2, -2, -4], [-6, -2, 1, 0, 1, 1, -2], [-1, 2, 5, -1, 1, 3, -1], [1, -3, 0, 2, -3, -3, 1], [-7, 4, 2, -2, 0, 1, 1], [2, -4, 1, 1, 1, 1, 5], [-2, -3, -1, 1, -2, -1, 1], [4, 6, -2, -4, -5, -4, 2], [0, -3, 1, 1, 2, 6, 0], [4, -1, 1, -3, -3, 2, 4], [-2, 0, -3, -5, -2, 0, 4], [4, -1, 3, 4, 0, 1, -4], [-5, 1, -5, 0, -2, -1, -2], [-9, 0, -2, 1, 4, -3, 0], [-2, -2, -4, 3, 8, 3, 2], [-7, 4, -3, -3, -1, -1, -3], [-4, 0, -4, -1, 0, 1, -6], [-1, -1, -4, -1, 2, 1, 0], [-2, -2, 2, -4, 1, 4, 6], [1, -1, -3, -1, -8, -2, 0], [5, -8, -1, 2, 1, 2, -5], [-6, 3, 1, -4, -1, 1, 2], [2, 1, 1, -1, 4, -1, 5], [6, 5, 1, 1, 1, 1, -7], [4, 4, 1, -5, -8, -6, 4], [-6, 0, -5, -3, -4, -7, 3], [-5, -1, -2, 2, 1, 0, 4], [-2, 2, 8, -6, 1, 7, 1], [-1, 9, 3, -4, -3, 1, 0], [6, -2, 2, 5, 5, 7, 0], [-5, -2, -4, 3, 1, 2, -1], [11, -5, 0, 1, 0, 7, 0], [-3, -2, 3, -1, 4, 2, 0], [0, 3, 4, 7, 2, 1, -4], [-8, 9, 2, 4, 4, -1, -1], [1, -3, -1, -1, 8, 2, 2], [-8, 0, -7, 6, 4, -6, -1], [-2, -3, 1, 0, -10, -6, -3], [-2, -7, -5, 6, -9, -11, 0], [-1, 3, -1, -3, -1, 0, -8], [3, 6, -6, 0, 2, 1, 3], [4, -8, 7, 5, 4, 5, -1], [-2, -3, 1, 2, 10, 2, 1], [9, 0, 2, 1, -1, 3, 6], [-9, 0, 8, 9, 1, -3, -4], [1, -2, -4, -10, -5, -2, 4], [8, 7, 4, -13, -3, 2, 3], [3, -5, -3, 4, 0, -1, -2], [-1, -8, -5, -2, 1, 4, 1], [5, -5, 1, 5, 2, -2, 2], [-10, 3, -8, 1, -2, -8, -9], [-4, -2, -5, 5, -2, -6, 4], [-3, 9, 0, 7, 8, -1, -2], [-4, 5, 0, 2, -1, 3, -5], [-9, -3, -2, 7, 0, 1, 4], [1, 10, 6, -1, -4, -8, -1], [7, 5, 9, 3, 6, 7, 1], [-3, 7, 2, 5, 0, 3, -4], [0, 3, -6, -9, 8, 6, 1], [5, -15, -4, 12, 6, 6, 2], [12, 9, 12, -14, -1, 7, 3], [-8, 9, 1, 1, -3, 3, -3], [12, 2, 9, 3, -1, 4, -10], [10, -6, 1, 1, -1, -2, 2], [-15, -2, 7, 1, -3, -2, -4], [1, -2, 5, -11, -4, 8, 4], [1, 2, 7, -4, 0, 3, 4], [1, 6, 6, 0, 7, 10, 4], [-1, 0, 7, -8, 3, 8, 13], [6, 3, 4, -3, 2, 9, 6], [-16, 11, 1, 1, 3, 3, -3], [-2, 2, -5, 0, 6, -1, 2], [9, -3, 0, -3, -19, -9, 4], [-11, 2, 4, -6, 6, 3, 1], [-2, -11, -5, 9, 4, 0, 0], [-3, 5, -5, 0, -8, -2, -1], [2, 2, -7, -1, -9, -12, -2], [5, -3, -4, 1, -5, 7, -2], [7, 7, 11, -2, 6, 6, 3], [11, 10, 4, -5, 1, 10, 1], [1, 5, 4, 7, -4, 0, 3], [-3, 7, -1, -8, -7, -3, 4], [8, 5, -6, -4, -14, 0, 4], [-13, -3, -7, 10, 9, -4, -3], [1, 0, 4, 9, 5, 2, -9], [9, 8, 7, -7, -11, -3, 1], [-16, 8, 4, 0, -3, -4, -2], [-13, -3, -2, 7, -6, -7, -4], [-1, -1, -8, 2, -2, -11, -5], [1, 4, 1, -5, 4, 1, -3], [18, 1, -4, -12, -8, -7, 9], [-6, -10, -4, 0, -2, -3, 3], [-9, 2, -6, -1, -2, 6, -15], [-3, 2, -3, -3, 1, -1, -5], [-8, -2, 2, 16, 3, 1, -15], [-4, 0, -9, 10, -3, -4, -1], [-3, -4, 1, -2, 9, -1, 8], [-8, -2, 9, 11, -7, -8, 0], [-7, 2, 17, 5, -7, -2, -6], [5, 2, -8, -1, 6, -3, 4], [-12, -6, -2, 0, -9, -1, 1], [-3, -9, -12, 13, 3, -2, -9], [1, 2, -4, -4, 8, 8, 4], [11, -2, 10, 17, 10, 10, -1], [2, 2, -7, -5, -13, 1, 3], [-18, -8, -10, 12, 8, 4, -4], [14, -2, -4, -17, -6, -3, 6], [11, 5, 0, -6, 11, 10, -4], [-22, -5, -2, 10, 1, -5, -1], [5, 6, 7, -8, -21, -2, 9], [-2, -8, -15, 1, 2, -8, 2], [7, 2, -7, -3, -6, -12, 2], [-24, 11, 2, 1, 1, 0, -1], [-18, -3, -1, 7, 10, 13, -7], [-4, -4, -14, 5, -3, -5, -2], [-6, 6, -14, 6, 3, -10, 2], [-5, 1, 1, 1, -4, -14, -6], [9, -8, -2, -2, -2, 3, -5], [1, 7, 5, 5, 13, 12, -6], [0, -7, 4, -8, 7, 6, 6], [17, 1, -1, -2, -17, 2, 3], [-1, -6, 6, 14, -9, -9, 7], [4, -6, 11, 5, 5, 8, 0], [-5, 1, 9, -9, 5, 12, 12], [7, 6, 14, 1, -1, 0, 1], [-2, 4, -4, -6, -11, -19, 3], [-15, -7, -6, 4, 0, -6, -2], [15, 6, -3, -14, -6, 4, -1], [4, -1, -2, 17, -7, -6, -5], [3, -7, 7, 11, -6, -4, 6], [3, 9, 6, -11, 13, 13, 8], [8, 3, 15, 1, 1, -2, 6], [0, -9, -4, 5, 10, 8, -13], [-8, 0, -5, -15, 14, 5, 7], [5, -11, 9, 0, -4, -1, -6], [18, -7, -8, -10, 0, 0, 8], [-15, -5, -6, 17, 11, -10, -3], [-5, 11, 14, -4, 13, 18, -2], [-3, -1, -4, -14, -5, -7, 7], [-6, -12, -7, 4, -10, -9, -2], [10, 8, 3, -26, -2, 6, -5], [-17, -19, -13, 11, -5, -14, 8], [3, 1, 1, 1, 7, -4, 12], [0, -19, -10, 12, -1, -7, 1], [-4, -3, -7, 9, -4, -5, -5], [5, 3, 4, -5, -23, -9, 2], [0, -16, -7, -2, 6, 8, 5], [-15, 10, -14, 3, 8, 2, -5], [-14, 8, 10, 8, 10, 11, 3], [-18, 3, -6, 1, 8, -7, -4], [-7, -15, -1, 8, -2, 3, -8], [-4, -2, 2, -18, 8, -1, 7], [6, 8, 0, -3, -5, -7, 8], [7, 10, 22, -13, -9, 10, 3], [-28, -15, -17, 17, 6, 1, 7], [24, -7, 8, -10, -13, -7, 1], [-3, 8, 3, -12, 6, 9, 10], [1, 3, 17, -9, -4, 9, 17], [18, -9, -2, 4, 6, 5, 7], [-10, 16, -4, -14, -5, -7, -9], [-10, 1, 4, -2, -2, -7, -7], [2, -12, 4, 8, 8, 1, -5], [-6, 9, 7, 10, 3, 11, -2], [-15, -9, 0, 8, -8, -18, -4], [7, 9, 2, -2, -7, -4, -12], [-11, 20, 2, -18, -15, -15, 3], [19, 7, 7, -11, -5, 2, 0], [24, 10, 4, -6, -4, -6, 12], [11, 15, -11, -7, 0, 2, -4], [22, -3, 11, 0, 12, 16, 11], [-6, -5, -1, 7, -1, -12, -2], [12, 5, 15, -24, -11, 17, 6], [-3, 12, -4, 1, 6, 12, -11], [-4, -17, -10, 12, -1, -8, -2], [12, -16, -1, -2, -4, -10, 5], [3, -20, -5, 10, -6, -10, 1], [2, 9, 14, -11, 10, -2, 7], [-16, -11, -6, 5, -12, -19, -12], [-10, -9, -8, -1, -8, -10, -3], [-6, 3, 2, -21, 11, 2, 9], [14, 7, 10, 1, -10, -6, 7], [1, -5, -14, 1, 2, -10, -3], [0, 3, 16, -1, -12, 0, 3], [15, 3, -3, 8, -11, 5, 4], [7, 11, -6, 5, 2, -2, 5], [-14, 16, 14, 1, 10, 11, 2], [-7, 21, 3, -21, -5, -6, 16], [13, -11, -4, -7, -2, 15, -8], [-18, 10, -2, 1, -6, -5, 4], [23, -5, 13, -10, -10, 11, -6], [24, 2, -5, -21, -4, 14, 6], [-12, 12, 2, -2, 5, -6, 6], [-22, 18, 14, -9, -8, -3, 12], [17, -8, 9, -2, 3, 5, 0], [-3, -18, -13, 12, 13, 15, -4], [-21, 9, -16, -3, 5, -6, -11], [-40, -1, -7, -3, -7, -8, 4], [-3, -5, -16, -17, -13, -12, 1], [8, 8, 7, -3, -10, -10, -6], [-15, -18, -4, 12, -7, -8, -12], [-9, 4, -4, -5, -19, -26, -5], [1, 9, 10, 10, 13, -5, -3], [9, 15, -8, -23, 3, 8, -1], [-11, -18, -7, 11, -11, -9, 9], [9, -19, 5, 14, 7, 7, 8], [-25, 2, -5, 22, 19, -4, -15], [19, 8, 3, -12, 6, 18, -7], [-4, 6, -10, -17, -16, -9, -2], [-7, 2, 17, 8, -1, -8, 5], [-3, 1, -22, -13, -12, -19, 8], [6, -4, 0, -10, 5, 14, 12], [-9, -4, -10, -2, 27, 15, 1], [-14, -2, -16, -7, -21, -33, -10], [-20, 5, 0, 6, -4, -3, 3], [-8, 8, 12, -16, -9, -1, 13], [-6, 7, -8, -8, -7, 8, -12], [-20, 7, 4, -6, -1, 11, -9], [-12, 23, 11, 10, 14, 6, -1], [10, 0, 1, -12, -16, -17, -2], [13, 11, -8, -2, -17, -3, 1], [-7, 3, -9, -2, 12, 2, 9], [15, -4, 4, 1, 17, 16, -3], [-20, -1, -18, -3, -8, -1, 10], [6, -12, 11, 3, 12, 8, -2], [6, 13, 9, -12, 1, 4, 5], [-6, 28, 8, 10, 11, -2, -4], [29, 8, -3, 1, 1, -1, -11], [-7, 10, -6, 1, 1, 8, -15], [-4, 11, 0, 8, 1, 10, -4], [14, 7, -2, -31, -18, -1, 10], [3, 3, 13, -16, -6, 11, 18], [-13, -4, -1, 26, 15, 4, -9], [17, -9, 5, -12, 3, 0, 9], [43, 0, 2, 4, -1, 7, 7], [-13, -9, -10, 7, 10, -5, -10], [10, 11, 11, 6, 23, 24, 11], [-3, -15, 2, 14, 21, 4, -8], [20, 0, 15, -16, 0, 5, 12], [-15, -4, 4, -14, 12, 19, -5], [19, -6, -6, -4, -17, -12, 4], [-5, -1, -3, 9, -4, -2, -6], [-6, -14, 0, 11, 22, 16, -5], [3, 9, 4, -6, 7, 23, -5], [8, 5, 8, -12, 11, 7, 11], [-36, 9, -5, -5, 12, -7, -3], [-3, 4, 9, 11, 9, -7, -9], [8, 7, 17, -15, 5, 18, 0], [-7, -5, 4, 11, 3, -3, 0], [-14, -8, -10, 19, 12, -17, 2], [-1, 19, 10, 12, 7, 10, -8], [18, 9, 10, -24, -5, 4, 4], [21, 25, 8, -28, 1, 19, -3], [-10, -12, -27, 2, -4, -21, 2], [-21, 10, -2, -6, -3, -9, -5], [-13, -10, -3, -1, -5, -1, -23], [-23, -1, -17, 12, 3, 1, -2], [23, 7, 1, -19, -8, 14, 2], [-13, -1, -13, 10, 11, -6, -11], [47, 15, 10, -17, -4, 3, 4], [1, 9, 5, 7, 9, -6, -12], [30, 13, 4, -2, -8, -1, -7], [-4, -19, -7, -1, 8, 0, 2], [-28, -1, -1, 4, -23, -22, -19], [5, -2, 2, -6, -3, -2, 18], [6, -2, -6, -27, -4, 9, 8], [18, 15, -2, 3, -18, -19, -4], [-14, 5, 12, 2, 23, 35, -5], [-32, 0, 9, 8, 4, -12, -7], [-12, -9, -16, 8, -17, -20, 0], [-1, 12, 18, 3, 3, 16, -1], [-15, -5, 7, 6, -8, -17, 0], [10, -23, 3, 14, -2, -2, 9], [-28, 6, 3, -8, -6, -15, 2], [-16, -15, 1, 13, 20, 11, -13], [18, -21, -5, -7, 1, 15, 11], [4, 20, -11, -4, -3, 11, -12], [-6, 0, -10, -2, -11, 4, 2], [-37, -23, 6, 19, 9, -7, -6], [4, -3, 0, 20, 19, -5, -1], [21, -14, 9, 3, -3, -3, -5], [23, 21, 10, -18, 0, -2, 14], [-2, -8, -11, 14, 6, 9, 8], [0, -22, -1, 22, 4, 7, -14], [-8, 14, -9, -14, -9, -15, -12], [-17, 15, -3, -12, -1, -13, -6], [3, -1, -15, 12, -5, -7, -28], [-2, 21, 6, 1, 1, 11, -6], [13, -20, 8, -7, -4, 10, 1], [17, 0, 11, 5, -18, -15, 3], [-4, -7, -21, 3, -7, -24, 6], [4, -4, 18, 22, 6, 6, -8], [24, 5, 18, -15, 3, 3, 22], [-16, 3, 4, 24, 8, 1, -9], [-1, 11, 23, 11, 4, 3, -7], [26, -3, 8, 2, 9, 11, -7], [46, -4, 13, -2, 3, 16, -1], [29, -16, -2, 8, 12, 13, -1], [-13, -32, -15, 26, 2, -4, 5], [-17, -5, 0, -6, 2, 5, -15], [-2, 15, 17, -19, 13, 22, -2], [13, 17, 20, -26, -10, 12, 6], [-5, -5, 22, 2, 8, 1, 3], [2, -1, 6, 5, -1, -10, 21], [16, -21, -9, 1, 6, -5, 9], [0, -7, 11, -11, -3, 11, 9], [-17, 8, -14, -7, -3, -20, -13], [24, 18, 14, -39, -20, 1, 18], [-20, 11, 17, 22, 10, -2, -7], [-1, -15, -17, 27, 8, -15, -13], [-10, -14, -4, 29, -1, -11, 2], [-6, 8, 0, -1, 4, -7, 14], [-5, -5, 1, 20, 12, 5, -24], [-46, -8, -9, 3, 5, 2, 0], [-3, -10, 17, 6, 10, 1, -4], [11, -8, 0, -6, 10, 9, 23], [-12, -3, -6, -8, -9, 3, 3], [-17, -17, -12, 6, -1, -1, 5], [-2, -4, 14, -9, 0, 4, -1], [8, 16, 1, -16, 11, 5, 14], [2, -21, -16, 0, -4, -8, -4], [11, -3, 20, 17, -11, 1, -8], [-10, -13, 9, 4, 1, -20, 3], [-5, -2, -2, 5, -16, 1, -12], [-29, -20, -24, 1, -11, -19, 14], [3, -8, 2, -10, 25, 11, 15], [31, 10, -6, 2, -3, -11, -5], [11, -5, -10, -2, -19, -30, 4], [-10, 8, -10, 7, -18, -19, -16], [51, 15, 12, -24, -9, 11, 3], [6, -19, -12, 11, 10, 15, -8], [2, -8, 2, -2, -16, -2, -14], [-8, -14, 10, -1, 3, -2, 19], [-4, -11, -13, 9, 21, 25, -3], [-9, -16, -11, -10, -7, 14, -5], [-4, -6, -11, -16, -3, -1, -6], [-14, -10, -29, 17, 16, 6, 0], [18, -29, -12, 18, 6, 8, -28], [23, 13, 0, 4, -16, -6, 4], [-19, 8, -9, -7, -11, -4, -18], [-24, -9, -28, 10, 4, -17, 13], [5, 13, -5, 1, -12, -13, 11], [-11, -8, -18, -4, 25, 7, -9], [42, -13, -7, -14, -17, 1, 6], [16, 19, -4, -14, 20, 23, -9], [-31, -6, -11, 2, 4, -8, 9], [-16, -23, -1, 5, 12, 15, 17], [-29, -2, 1, 5, 12, -6, -4], [16, 26, 6, -7, -21, -10, -11], [-28, -8, -20, 10, -8, -16, 16], [4, -18, -7, -18, -31, -5, 4], [4, -10, 10, -6, 19, 25, 13], [-51, -2, -23, 18, 18, -24, -11], [21, 1, 17, -4, 17, 24, 11], [6, 9, 16, -16, 18, 24, 12], [-26, 2, -1, -2, 34, 27, 4], [-8, 14, -7, -11, 9, 1, 7], [-1, 22, 11, -18, -24, -25, -10], [-34, 1, -10, -1, 2, -18, -11], [-4, 7, 12, 0, -2, 19, 13], [-8, -31, 5, 17, 7, 7, 5], [28, 1, -20, -19, 8, 8, 15], [-29, -28, -3, 34, 19, 6, -11], [7, -3, 16, 0, -37, -14, 4], [-1, 7, 14, -9, 11, 24, -5], [-22, -4, -12, 10, 12, 12, 10], [-25, -4, 16, -9, 25, 32, -1], [35, -10, 3, -15, -13, -15, 9], [-31, 3, -31, -17, 10, -9, 13], [15, -4, 4, 6, 6, 21, -11], [-26, -38, -1, 31, 14, 3, -3], [11, -3, 8, 12, 12, -2, 14], [14, 14, 22, -21, -18, -3, 14], [11, 4, 16, -11, -17, -9, 20], [-10, -1, -18, -1, 8, 2, 5], [3, 9, 23, 22, 18, 10, -15], [1, -1, -5, 14, 28, 14, 9], [12, 25, 8, -16, -23, -21, -3], [22, 12, 23, -19, 7, 2, 22], [29, 25, 24, -32, -7, 27, 9], [-52, -33, 3, 12, 11, -1, -2], [-17, 25, 18, 15, 0, -11, -12], [25, 5, 6, -29, -33, 5, 22], [46, -10, -9, -21, -9, -10, 6], [36, -18, 4, 16, 8, 14, -12], [-13, 9, -9, -9, 4, 8, -8], [14, 12, 23, 0, 7, 11, 0], [11, -1, 13, -4, 18, 12, 7], [-17, 20, -20, -4, 3, 0, -20], [5, -9, -4, 11, 9, -17, 4], [23, -42, -7, -2, -19, -10, 5], [-6, 29, 15, -2, -16, -11, -10], [-1, -28, -5, -4, -5, -16, 5], [9, -1, 8, 24, 14, 6, -12], [-1, -3, 35, -3, 19, 16, 8], [24, -12, -11, -21, 8, -2, 6], [-4, 23, -15, 4, 11, 1, -2], [3, -24, -5, 9, 22, 4, -22], [-43, -16, -13, 8, -3, -25, 2], [15, -34, -19, 17, -12, -14, -2], [-15, -3, -20, 33, 12, -19, 8], [-24, -24, -20, 24, 6, -12, -4], [-8, -15, -12, -2, 11, -5, 27], [42, 9, 11, -21, -8, -7, 9], [8, 40, 17, -18, -10, 3, -14], [10, 6, -7, 11, 16, 11, 9], [-8, -7, -5, 12, 33, 6, -13], [7, -18, -12, 14, 22, -1, -15], [6, 16, -3, -37, 0, 9, 9], [10, 31, 20, -16, -9, 18, -6], [-10, -13, -28, -2, 7, 1, 5]];
    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_i();".
// 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_i(:prec:=7)
    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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![23, -24, -29, 29, 10, -10, -1, 1]>,<5,R![28, 36, -87, -15, 40, -3, -5, 1]>],Snew);
    return Vf;
end function;