
// Make newform 6003.2.a.k in Magma, downloaded from the LMFDB on 20 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_k();".
// 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_k();".
// 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 := [-11, -23, 60, 101, -106, -88, 69, 23, -17, -1, 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], [-357, 172, 1920, -1465, -1705, 1581, 466, -481, -14, 31], [-450, -87, 2027, -493, -1710, 587, 448, -182, -23, 11], [703, -566, -3466, 3297, 3051, -3065, -810, 871, 24, -55], [-505, 272, 2870, -1999, -2647, 1865, 720, -533, -30, 33], [54, -3, -279, 155, 256, -161, -70, 48, 3, -3], [-2227, 2, 11238, -2645, -10007, 2809, 2694, -871, -148, 51], [785, -256, -4252, 2367, 3737, -2299, -982, 663, 42, -41], [-2033, 490, 10830, -5435, -9717, 5263, 2594, -1521, -120, 93]];
    Rf_basisdens := [1, 1, 52, 26, 52, 26, 2, 52, 26, 52];
    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, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, -1, 0, 1], [0, 0, -1, 0, 0, 0, 0, 0, 0, -1], [-2, -2, 0, 2, 1, 0, 0, 0, 0, 1], [0, 1, 1, -1, 0, 1, 0, 1, 1, -1], [0, 0, 0, 1, 0, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -1, 1, -1, -1, 0, -2, -1, -1], [0, 2, -1, -1, -1, 0, 0, 0, -3, 0], [0, 0, 1, -1, 2, 0, -2, 2, 0, 0], [0, -1, -1, 0, 0, 2, 0, 0, -1, -1], [-2, 1, 2, -2, 0, 1, -1, 2, 1, -1], [-4, -1, -1, 1, -1, -2, 0, 0, 0, 2], [-5, -2, 1, 2, 0, 0, 0, 0, 0, -2], [-1, 3, 1, 0, -3, -2, 0, 1, 2, 1], [5, 0, 0, -1, 2, 0, 0, 1, 0, -1], [-4, -1, 2, -2, -1, -3, -3, 2, 0, 0], [-2, 0, 0, 2, 1, -2, -1, -1, 0, 1], [1, -2, 4, -1, 1, -3, -2, 0, 1, 1], [-2, 3, 3, 0, 0, 0, 0, 1, 3, 1], [-2, -1, -1, 1, -2, 1, 2, 1, 0, 1], [-4, 3, -2, 2, -2, 2, 0, -1, 2, -1], [1, -4, 2, 1, 0, 0, -1, 1, -1, -2], [1, -2, 2, -3, 2, 0, 0, -1, 0, 1], [-3, 0, 1, -3, 1, 0, -1, 2, -2, -1], [1, -3, 1, -3, 2, 1, -3, 3, -2, -3], [-2, 3, 0, 0, -1, -1, 0, -1, 2, 2], [2, 1, -2, -2, 1, 0, 0, -2, -1, 2], [0, -2, -1, 1, 1, 0, 0, 1, -2, -1], [8, -1, -3, 0, 1, 4, 1, 1, 0, -4], [-2, 6, -1, -3, 1, 2, 2, 3, 1, -1], [-3, -2, -4, 2, -1, 1, 1, 1, -6, -4], [3, 0, -4, 4, -2, 0, 1, 2, 4, 0], [-1, -1, -3, 1, -1, 2, 3, 2, 0, 0], [2, 5, 2, -3, 2, -2, -2, 3, -1, 3], [0, 1, 0, 1, -3, 1, 4, -2, -2, 0], [1, -3, -1, 6, -2, 0, 1, -3, -2, -3], [-2, 3, -4, 0, 1, 3, 2, 2, 2, 0], [-5, -1, 2, 0, 0, -3, -2, 1, 2, 5], [-10, 1, 1, -3, 0, 2, -1, 0, -1, 0], [-4, 3, 1, -2, 1, 2, 3, -2, 0, 4], [1, 4, -1, -2, 1, -1, -1, -2, 1, 1], [6, -2, -3, -4, -3, 2, 1, -1, 0, -2], [0, 3, -1, 0, -2, 0, 3, -1, 1, 3], [-10, 1, 3, -1, -1, -1, -1, 0, 5, 6], [-4, -1, 3, -1, 2, -1, 0, -4, 1, 3], [-7, -5, 4, 3, 2, -5, 1, -4, -1, 4], [-2, -2, 2, 2, 0, -2, 0, -1, 0, -2], [-4, -1, -1, 6, 0, -1, 2, -1, 1, 1], [3, 1, -3, -7, -3, 1, 2, 2, 1, -2], [4, 1, -4, 0, -4, 3, 5, -5, 0, 2], [2, -4, 1, 7, 1, 2, 0, 0, -1, -5], [-3, -5, 4, 1, 8, -3, -3, -3, -2, 1], [2, 1, -5, 1, -4, 1, 5, -2, 0, 1], [6, -1, 6, -5, -1, -3, 0, 1, 1, -6], [-3, -7, -1, 1, 2, -1, 2, -4, -1, 1], [-2, 2, 5, -2, 1, -2, -5, 5, 6, 1], [-8, -2, 2, 0, -2, 0, 2, 2, -2, -3], [3, 6, -2, 0, -3, 0, 4, -1, 1, 5], [-3, 1, -2, 2, -3, 3, 0, 2, -4, -4], [-7, 1, -2, 6, 0, 1, 0, 3, 1, -3], [-1, -6, -2, 6, 4, 0, 1, -3, 3, 0], [0, -5, -4, 6, 1, 4, 2, -2, 2, 4], [5, -8, -1, 4, 3, -3, 7, -3, 0, 1], [4, 0, -1, 0, -3, 0, 3, -4, 2, 2], [-10, 5, -4, -4, -3, -1, 0, 0, 0, -4], [1, 3, -3, 2, -2, 3, 5, -4, -1, -1], [0, 5, 4, -5, -2, 0, 5, 3, 6, 1], [-4, -11, -1, 1, 3, -2, 2, -3, -2, 2], [5, -5, 0, 4, -2, 3, 0, 3, -1, -1], [-14, 3, 3, 0, 2, 0, 2, -3, 2, 3], [-2, 1, 5, 0, 4, -2, 0, 0, 4, -5], [-4, 0, 4, -7, -1, 5, -3, 2, -1, -1], [9, -1, -4, 0, 2, -5, 0, -4, -10, -5], [-9, 3, -10, -2, -6, 3, 4, -1, -5, -1], [-9, 0, 4, -4, 4, -2, -1, -3, -3, -5], [-4, 2, -2, 2, -1, -2, 1, 1, 3, -3], [-2, 2, -9, 2, -4, 5, 3, -1, -2, -6], [3, -1, 8, 2, 3, -6, 4, -4, 5, 3], [-7, -2, -3, 9, 0, 6, 4, -3, 4, 6], [-4, 1, 3, -4, 4, 1, -5, -2, -3, 3], [3, 6, 5, -4, 2, -3, -4, 3, -2, 2], [1, -6, -3, 5, 1, 0, -1, -4, 1, 7], [-1, 9, -1, -4, 2, 2, 6, 2, 1, -5], [-2, 8, -2, -6, 3, 0, -3, 3, 1, 1], [-4, -10, -1, 7, 3, 0, 1, -1, -1, 7], [-1, 1, -2, -5, -4, 2, -4, 5, 1, -1], [-5, -8, 6, -2, 5, -5, -3, 0, 1, -2], [4, 8, 6, -9, 0, 1, -2, 3, 5, 5], [-2, -2, 3, 3, 3, 2, 0, 1, -1, 5], [-5, -1, -4, 1, -10, 0, 6, -2, -3, -1], [4, -2, -2, -1, 0, -1, -4, -2, -3, -6], [-7, -6, 6, 5, 7, 0, -1, 1, 1, 3], [-4, -5, 4, 1, 12, 0, -7, 2, 5, -1], [-5, 2, 3, -2, 5, -4, -4, 0, 6, 7], [7, 5, -5, -2, -1, 3, 1, 0, 3, 1], [-7, -5, -1, 2, -2, 2, -1, 2, -1, 5], [-8, -1, 2, -6, 2, 7, -5, 7, 2, -3], [-3, 5, 5, -12, 2, -2, -1, 0, -6, -8], [3, -3, 4, 0, -1, -3, -5, -3, -4, -7], [2, -1, -6, 3, -4, 0, 1, -5, -3, 7], [7, -4, -3, 0, -2, 2, 3, 3, 3, 1], [-2, 3, 3, 0, 6, -4, -4, 1, 3, 3], [0, -4, -2, 3, -9, -1, 5, -5, -3, -5], [2, -2, 7, -6, 6, 1, -7, 0, 0, 0], [-8, -12, -1, 8, 3, -5, 2, -6, -7, -1], [-3, 1, 8, -9, 3, -2, -5, 11, 1, -4], [1, 6, 2, -3, -7, -7, 0, -1, 2, 9], [-4, -4, -9, 10, -6, -7, 5, -13, -4, 4], [-4, 0, 0, -9, 1, 1, -5, 8, -1, 3], [-5, 2, 4, 4, -2, -6, 5, -4, 6, 6], [8, -4, 2, -2, -1, 2, -3, 1, -5, -2], [-7, -7, 9, -7, 5, -5, -6, 1, -5, 0], [-2, -1, 6, 5, -1, -7, 1, -7, 0, 4], [-12, 8, 2, -9, -2, -2, 2, -1, -2, -2], [-7, 1, -6, 3, 0, 4, -5, -4, -5, 3], [-2, 2, -2, 0, -5, -7, 4, -11, -5, 5], [-7, 20, -6, -3, -9, 7, 4, 3, 6, 1], [-7, -2, 3, -5, -3, 2, -9, 3, 2, 1], [-13, 4, -5, 7, -2, -2, -2, -9, -1, 8], [-6, 3, 0, -3, -3, -3, 4, -5, -2, 2], [7, -6, -6, 7, -10, -3, 1, -6, -4, -8], [0, -1, 2, -2, 4, 2, -4, 1, 2, -6], [1, -7, -8, 14, 5, -7, 5, -8, 0, 4], [-7, 7, -6, -2, -6, 5, 5, -4, 0, 1], [8, 0, 1, 0, 4, 3, -9, 5, 0, -2], [1, -2, -7, -6, 3, 4, -2, 2, -7, 2], [0, -10, -8, 2, 1, -3, 5, -6, -7, 1], [-10, -8, 0, -1, 1, -2, -6, 4, 2, 6], [4, 6, -8, 2, 1, 6, 7, 0, 5, -1], [-19, -5, -3, 2, -2, 0, -1, 1, 0, 0], [-1, 2, 10, -5, 0, 1, -5, 11, 9, 0], [-1, -6, 6, 8, 9, -2, 0, -7, -6, 0], [-4, -4, 12, -4, 14, -8, -2, 0, 3, 6], [17, -8, -4, -1, 0, 2, 2, -1, -9, -3], [-9, -7, -1, 9, 11, 1, -2, 0, 3, 4], [4, -3, 12, -12, -2, 0, -7, 1, 3, 3], [2, 1, 10, -9, 6, -12, -7, -1, -3, 1], [1, 10, -2, 2, -7, 2, 6, -3, -6, -3], [-4, 5, 3, 2, 2, -4, 3, 3, 4, 5], [11, 5, 0, -8, -7, 1, -5, 1, 3, -3], [-17, -5, 9, 0, 4, -2, -11, 2, -5, -7], [-3, 7, 4, -1, -7, -3, -4, 2, 10, 3], [6, -1, 8, 2, 1, -5, 0, -1, 3, -10], [6, 7, -16, -3, 0, 4, 5, -5, -7, -1], [-7, 1, 5, -2, 3, 0, -6, 7, 10, -6], [7, 0, 2, -3, 8, 7, 2, 2, 1, -2], [3, 10, 4, -3, -5, -1, 2, -2, 0, -9], [5, -4, 5, -8, 13, 5, -3, 6, -3, -1], [-15, 1, -5, 3, -12, -3, 3, -8, 3, 5], [-2, 2, 9, 0, 3, 3, -4, 0, 8, -3], [12, 6, -3, -2, -1, 9, 8, -5, 4, -1], [-7, -11, 0, 14, 3, -1, 3, -7, -6, -3], [13, -11, 2, -2, 5, -1, -3, 6, -8, -7], [10, 17, -6, -3, -4, 8, 5, 2, 4, -5], [-17, -4, 10, -7, 4, -7, -7, 10, 3, 8], [14, -10, 0, 5, 6, 5, 0, -3, 5, -1], [-3, 3, -9, 6, -6, -4, 3, -4, -4, 9], [-6, -1, -3, 2, 7, -3, 0, 1, -10, -2], [-7, 4, 3, -9, -3, -2, -7, 2, -7, 5], [-15, -5, 7, 5, -4, -3, -7, 1, -3, 2], [3, -3, -2, -9, 4, 9, -5, 3, -4, -10], [-5, -1, -12, 7, -1, 13, 10, 3, -2, 3], [7, 7, -7, -6, -8, 12, -5, 4, -2, -1], [-8, 4, -8, 3, -9, 3, 5, 2, 1, 3], [-8, 2, 9, -4, 4, 5, -7, 15, 4, -10], [-22, 4, 3, 9, 8, -6, -9, 2, 2, 14], [-5, 3, 2, 6, -3, -5, -5, 0, 1, -4], [-1, 0, 6, -16, 2, 0, -3, -2, -1, -2], [4, 4, -9, 11, -4, 4, -1, 4, 8, -4], [-15, 14, 0, 1, -4, -2, 0, -2, 3, 8], [3, -6, -2, 15, -6, 5, 7, 6, 8, -8], [-4, 6, -5, -6, 0, 4, 10, -6, -6, -9], [7, -3, -14, 1, -10, 4, 0, -1, 0, -9], [-14, -3, -7, 8, -6, 0, 0, -9, -3, 11], [25, -1, -10, -7, 1, 10, -1, 5, -2, -8], [-25, 7, -4, -1, -3, -2, 5, -3, -6, -2], [-1, 0, 6, -3, -1, -2, 4, -4, -1, 5], [17, 7, -6, -4, -6, 4, 1, 9, 3, -12], [-5, 8, -4, -8, -2, -1, 5, -8, -1, 4], [-13, -2, 11, -6, 1, -3, -7, 2, -2, 5], [1, 1, 13, -11, 8, -5, -5, -1, -5, 3], [4, -17, -1, 16, -3, -1, 4, -9, 1, -3], [1, -14, 7, 3, 11, 0, -3, -2, 1, 3], [-10, 4, 4, -6, 0, -4, -4, -4, -2, 2], [10, 2, -12, 8, 2, 8, 2, -6, -1, -6], [-4, 0, -2, -3, 1, 1, -5, -2, 0, 7], [-3, 0, -7, -6, -15, 4, 1, 5, 4, -9], [3, 0, 2, -6, -4, 4, 3, 7, 6, -8], [10, -8, 6, -1, 5, -7, -9, 10, 1, 3], [-4, -2, -8, 14, -6, 8, 8, -4, 12, 2], [-7, 7, 8, -7, -4, 10, 4, 11, 6, -12], [-21, 1, -6, 6, -8, -6, -1, 2, -2, -3], [15, 0, -12, -4, 0, 2, 9, -5, 1, 1], [-8, -13, 9, 7, -4, -9, -2, -4, -2, -1], [-5, 10, 2, -10, -10, -5, -6, 2, 1, 2], [20, 2, -4, -2, 7, -7, -1, -3, -16, 1], [11, -8, -7, -1, 10, 8, 8, -2, 1, 0], [-4, -7, 2, 8, 8, -1, 5, -7, -12, 1], [2, 2, 6, -1, -4, -3, -2, -3, -8, -16], [22, 4, 0, 0, 4, 5, -1, 8, 3, -12], [-7, 7, 8, 7, 4, -1, -5, 8, 9, -11], [0, -13, -5, 2, -1, -6, -3, -3, -7, 12], [12, 15, -2, -5, -9, 4, 4, 3, -4, -12], [-14, 9, 3, -2, -4, -3, -10, 7, -1, 1], [2, 3, -1, -8, 11, 6, 5, -1, 5, 4], [-4, 13, 4, -1, 7, -4, -1, -8, -1, 2], [-10, -2, 4, 4, 1, 4, 1, 9, 16, -5], [-2, -2, -8, 7, -1, -3, -1, -1, -3, -5], [2, -10, 5, 14, 13, -9, -4, -3, 5, 17], [3, -4, 9, 1, 3, 1, -6, 12, 1, 1], [7, 1, 12, -16, 3, -5, -1, 0, 3, 4], [18, 13, -8, -1, -11, 4, 4, 0, -5, 2], [1, 5, 13, -10, 1, -8, 2, 0, 11, 6], [15, 5, -8, 14, -1, 5, 2, -4, 7, -6], [2, -6, 8, 4, 13, 4, 3, 1, 8, -9], [-16, -19, 2, 12, 9, 4, -6, 8, -4, 2], [-3, 1, 3, -10, -14, 0, -3, 2, 1, 7], [2, 7, -8, 4, -14, 3, 7, 0, 4, 8], [11, 4, 2, -4, -9, -4, -3, 10, -1, -13], [7, -4, -6, -6, -12, -2, -7, 0, -4, 2], [-5, 12, -4, -7, -10, -3, -1, -11, 2, -2], [-21, 15, -5, -4, -7, 0, -4, -1, 1, 4], [15, -10, -5, 17, 10, 0, 4, -2, -8, -10], [3, -7, -2, 0, -1, 7, 1, 0, -8, -8], [-14, -6, -6, 18, -9, 0, 13, -12, 2, 13], [21, -4, -8, -12, -4, 3, -3, 6, -4, -10], [-31, 4, 5, 7, 11, 2, 5, -4, 6, 5], [-7, 4, -15, 15, -1, 9, 4, -1, 6, -7], [7, -1, -20, -1, -16, 10, 6, 3, -1, -3], [-7, 0, -6, 15, -3, 3, 2, -5, 2, -13], [-10, -9, 8, -2, 7, 7, -10, 3, -5, -1], [-3, -13, -6, 7, 8, 5, -5, 1, -13, -3], [8, 5, -14, 5, -12, 4, 17, -8, 4, 6], [11, 1, -13, 5, -17, -3, 16, -11, -7, 6], [14, -1, 7, 4, 6, -11, 6, -11, 2, 11], [-25, 12, -8, -5, -12, 5, 5, -4, -7, -4], [3, -8, 12, 11, 11, 2, -3, 9, 3, 1], [-14, 9, 0, 0, 4, 3, -5, 3, 4, 3], [-11, -5, 10, -1, 6, 6, -6, -5, -3, -5], [2, -8, 2, 2, 1, 2, -3, 3, 10, 1], [-9, 5, -3, 2, -12, 6, 13, 0, 4, 3], [-6, -20, -4, 15, 21, 3, -1, -1, 1, 11], [-2, 6, -3, 2, -4, 9, 13, 1, -3, -10], [3, 2, -13, 18, -6, -7, 6, -4, -11, -8], [8, -11, 0, 4, 12, 2, 6, -1, -8, -4], [17, 6, -4, 3, 6, -6, -2, -10, 5, 17], [-2, -1, -14, 13, -5, -3, -3, -3, -6, 6], [-18, 12, 3, -21, -17, 4, -2, 5, 6, 0], [30, 7, -7, 0, 6, 3, -1, -7, 3, 13], [11, -4, -3, 5, -1, 0, 9, -6, 6, 2], [-5, 10, -10, 14, 8, 2, -3, -1, 5, 6], [0, -11, 7, -1, 10, -7, -3, -2, 6, 9], [-14, 9, -1, 0, 2, -3, 3, 0, 6, 1], [-17, -5, -4, 16, 4, 9, 6, -3, 0, 11], [20, -5, 3, -1, 4, 15, 0, 11, 0, -3], [-19, 4, 7, -2, 3, -8, -3, 5, 3, 15], [12, -8, -4, 0, -4, -6, -6, 10, -3, -4], [15, 6, -10, 3, -11, 12, 9, 1, -6, -7], [0, -3, -5, 17, 6, 10, 2, -6, -7, 3], [11, -3, -5, 17, 18, 1, -1, -3, -1, -2], [-1, -7, 6, 7, 4, -10, -14, 8, -2, -3], [-7, 8, -4, -10, -2, -6, -3, 2, -4, -4], [3, -2, 7, -12, 1, 5, 5, -2, 3, 3], [7, 1, 12, -1, -3, -2, 7, 4, 6, 2], [-29, -12, 10, -8, -1, -4, -4, 4, 6, -15], [-1, -2, 3, -9, 10, -8, -12, 3, -3, 2], [2, -2, -1, 13, 8, 0, -2, 1, 3, 12], [8, 3, 8, 6, 4, -10, 6, 3, 12, -5], [15, 7, 6, -24, 0, 1, -14, 11, -10, -11], [-7, 13, -8, 2, -9, 1, 9, -3, -2, 8], [-14, 0, -5, -2, -12, -3, 1, -5, -1, 4], [0, 9, 5, -21, -6, -7, -6, 2, 1, 3], [8, -14, -2, 12, 0, 4, 2, -2, 1, -4], [-18, -7, -6, 17, 2, 4, 3, -7, -1, 23], [-7, -7, 12, 4, 12, -12, 3, -2, 7, 10], [5, 0, -9, 7, 7, 8, -11, 2, -1, -9], [16, 7, 14, -6, 9, -5, -11, 7, 0, -2], [-4, 2, 13, 4, 8, -4, -6, 0, 11, -8], [-9, -1, 9, 2, 10, -2, 1, -12, 0, -1], [8, 0, 4, -4, -9, 4, 9, -1, -1, -7], [-11, 10, 6, -14, -7, 0, 4, 10, -2, -13], [-4, -14, 3, 3, 14, -12, -11, -8, -10, 4], [12, 1, -6, -3, -6, -1, -4, 9, -16, -15], [-2, 2, 6, -6, -16, -6, -2, 1, 7, -4], [6, -8, -16, 0, 6, 8, 0, -2, -8, 6], [12, 6, 0, 4, -12, -6, 4, 3, 12, 4], [-4, 20, 4, -7, -7, 2, 2, -1, 12, 7], [10, -3, -8, 13, -3, -2, 10, -16, -2, 8], [6, -4, -3, 8, -4, -1, -1, 6, 1, -4], [11, -14, -1, -16, 6, 12, -5, 4, -10, -12], [-10, 13, -5, 0, -15, -2, 5, 0, 12, 16], [-1, -19, 9, 0, 9, -12, -4, -1, -3, -7], [-10, 5, 2, -11, -9, 4, -2, -5, 3, -12], [11, -3, 16, -17, -7, 2, -9, 10, 0, 4], [7, -4, -7, -9, -12, 6, -1, 3, -8, -18], [-26, -8, 9, 2, 6, -5, -9, -1, -9, 0], [-22, 16, 5, -3, -5, -6, 0, 0, 11, 18], [1, 20, 2, -4, 4, 1, 2, -8, 8, 5], [1, -3, -14, 10, -10, 7, 18, -5, -2, 5], [-12, 2, 3, -3, 3, 8, -6, -7, 2, 3], [-14, 11, -5, -11, 2, 3, -4, 10, -3, 13], [-1, -21, 4, 13, 18, -2, -12, 9, 4, 5], [23, 11, -2, -8, -5, 9, 7, 2, 10, -12], [-8, -3, 12, -3, 11, -4, -10, 0, -1, -10], [-17, -3, -14, 11, 7, 6, 7, 1, -5, 2], [-1, 7, 9, -24, 3, -2, -6, 12, -9, -22], [10, 7, 10, 0, 12, 0, -7, 1, -5, 1], [-15, -8, -6, 4, -9, 6, 10, -10, 2, -3], [-10, 8, 4, 4, -5, -4, 1, -5, -1, -16], [13, -18, -4, -4, 4, -4, -1, -2, -12, 0], [-28, -7, 5, -4, -6, -8, -10, 1, 8, 3], [12, -4, -2, 3, -1, -3, -3, -8, -16, 1], [-27, -7, -8, 17, -4, -4, 4, -6, -7, 7], [15, 13, -4, -26, -10, 3, -5, 19, 3, -17], [0, -7, -3, -4, 1, 6, 15, 4, 1, 0], [16, 7, -24, 0, -5, 10, 8, -6, -9, -18], [14, 17, 6, -10, 9, -3, 0, 3, 9, -10], [6, 25, -13, -10, -5, 2, 7, -6, 1, -8], [-8, -11, 0, 6, 1, 3, -8, 0, -5, 4], [-31, 7, 18, -5, 4, 2, -8, 14, 0, -3], [-31, -1, -1, 3, -5, 0, -9, 9, -9, -12], [10, -6, -5, -1, 10, -8, -3, -2, -12, 1], [12, 7, 18, -11, -1, -4, -14, 8, 7, -1], [7, 8, -15, -11, -7, 6, 11, -5, -1, 7], [-5, 1, 1, 3, -4, 15, -5, 3, 8, 4], [-12, 5, 3, 7, -10, -16, -1, 3, 0, 2], [23, -4, -6, 4, 11, 10, -6, 2, -4, -3], [20, -5, -3, 12, 5, -2, -1, -13, -8, -2], [30, 2, -8, -3, 14, -5, -4, -4, -13, -2], [8, 28, -7, -15, -12, 12, 1, 6, 13, -8], [26, -12, -1, -9, 12, 4, -5, 5, 4, 4], [30, 10, -5, 2, -4, 13, -3, 9, 2, -6], [23, -5, -10, 9, 6, 8, 18, 2, -5, 1], [-9, -2, -1, -8, -7, 1, 1, -5, -10, -3], [24, -13, -1, 13, 9, 9, -1, 3, 10, -6], [24, 19, 1, -20, -11, -4, 3, 2, 7, -11], [-12, -9, 21, -7, 13, -19, -11, -5, -3, 14], [-14, -8, -7, -7, 4, 2, 1, 2, -7, 17], [-18, 3, 3, 3, 9, 8, -7, 8, 2, -6], [-2, 2, 10, 9, 4, -11, -8, 3, 10, 17], [-13, -5, 13, 9, 1, 0, -3, -1, -8, 2], [-8, -1, 3, -8, 5, -8, 1, -2, -19, -2], [21, 2, -3, 3, 8, 10, 6, -3, -11, -8], [1, 10, -4, 4, -11, 19, 4, 7, 3, -13], [12, -1, 8, -4, 0, 3, -3, 4, -6, 6], [-13, 6, -2, 15, -4, -7, -3, -5, 1, 0], [-5, 4, -17, -6, -14, 15, 12, -5, -3, 4], [-16, -5, -9, 22, 2, 2, 9, -4, -5, 9], [-36, -16, 5, 6, 5, -4, -2, -3, 9, 17], [-44, -4, 5, 9, -11, -6, 2, -9, 6, 16], [-3, -17, 5, 0, 3, 3, -3, 8, 7, 0], [-5, 6, 2, -7, 5, -1, -5, 1, -6, -9], [29, -5, -6, 8, 3, 1, 5, -1, -4, 15], [44, -7, 6, 2, -1, 9, -2, 7, 1, 4], [-5, -1, -20, 19, -18, 0, 13, -14, -5, -1], [-33, 4, 3, -6, 1, -5, -8, 4, 1, 17], [12, -15, 17, 16, 4, -6, -4, 1, 7, 1], [-12, 3, -1, 2, -9, 2, 11, -2, -9, 6], [3, 22, -2, -12, -14, 0, 5, 1, -12, -10], [-15, 6, 14, -3, -9, -9, -10, 1, 14, 1], [11, 17, -10, 12, -14, -1, 15, 1, 1, -1], [-11, 0, -2, -9, 6, 21, -3, 12, 0, -12], [-5, 3, 25, 1, 1, -11, -10, 6, 15, 6], [-7, 7, -2, -2, 2, 3, 14, -7, 13, 10], [36, -3, -3, -3, 0, 10, 5, -4, -12, -13], [-26, 15, -1, 3, -5, -1, -1, -7, -2, 24], [8, -1, 5, -3, 1, 5, -7, -6, 5, 6], [-27, 16, -9, -1, -12, 14, 16, -8, 4, 6], [43, -8, -5, -13, -4, -2, -6, 0, -6, -4], [1, -7, 13, 7, 7, -10, -13, 1, -4, 3], [-9, 5, 5, -23, 4, 7, -6, -2, -3, 13], [17, -11, 0, 3, 6, -10, -10, 10, -10, -5], [-9, -2, 19, -2, 10, -27, -6, 4, 6, 15], [11, -1, -11, 7, -26, -4, 12, -7, -5, 2], [-18, -4, 4, 4, 13, -4, -1, -7, -7, 8], [19, -9, 4, -2, 21, 9, 1, 7, -5, -8], [-21, 2, 11, -11, 6, 2, 0, 10, 7, -14], [-21, -1, -11, 4, 1, 11, 5, -6, -7, -3], [14, -5, -5, 26, 6, -12, -1, -7, -5, 1], [4, 4, 6, -7, 7, -12, -9, 1, -4, 3], [-9, 12, 12, -17, -12, -14, 6, 5, 7, 0], [-29, 2, 10, -11, -2, -3, 2, -13, 8, 16], [-20, -5, -3, -2, -2, 8, 4, -13, -15, 19], [-18, -12, 7, 2, 8, -5, 5, -5, 1, 4], [-10, -20, 5, 23, 6, 0, -1, -13, -10, 0], [-5, 0, -8, 8, 1, -1, 3, -14, -1, 14], [-8, -18, 19, 9, 13, -6, -4, 4, 5, 14], [-9, -6, 4, -10, 5, -10, -10, -4, -4, -1], [-5, 11, 4, 0, 0, -3, 8, -7, 13, 21], [6, -15, -2, 15, 16, -1, 3, -2, -7, -15], [-19, 10, -12, -17, -13, 5, -6, 6, -3, -18], [9, -12, -3, 8, 1, 9, 11, -3, -7, 9], [5, -27, 5, 9, 4, -4, -8, 10, -3, 7], [8, -9, -14, 3, -8, -10, 15, -8, -17, -9], [30, 22, -18, -6, -6, 14, 8, 0, -4, -21], [9, -16, -2, 24, 2, 2, 15, -11, 8, 2], [12, 11, -12, -13, -8, -2, 2, -9, -5, -11], [3, 11, 5, -12, 7, -4, -1, -1, -7, -4], [-2, 12, 14, -13, 10, -1, -14, -6, -3, 4], [24, 25, 4, -10, 10, 4, -2, 5, 4, -2], [-24, -4, -2, 6, -14, 4, 10, -18, -1, -8], [-1, -8, -2, 11, 1, -3, 8, -12, -9, 19], [-5, 21, 3, -18, -9, 8, -8, 21, 1, -14], [-6, -6, 6, 4, 3, -5, 7, -14, -16, 5], [3, 0, 2, 5, 1, -15, 0, -16, -4, 21], [-19, -15, 2, -3, 1, -10, 6, -18, -6, 14], [-13, 5, 2, -1, -7, 1, -17, -7, -2, -8], [-1, 20, 3, -20, -7, -5, 9, -3, 1, 5], [6, 7, 6, -18, -1, -13, -10, 7, 7, 12], [10, -2, 15, -10, -7, -10, -21, 13, 0, 7], [17, 9, -11, -8, 13, 5, -9, 5, -7, -11], [-19, -19, 8, 8, 0, 5, -8, 1, 10, 15], [19, 5, -7, 13, -1, 1, 0, 4, 1, -2], [0, -6, 9, 12, 11, -5, -4, -1, -4, 3], [-2, -4, -8, -15, -7, -9, 7, -5, -2, -11], [11, -1, -12, 18, -2, 18, 13, 0, 1, -20], [-19, 21, 8, -11, -2, 4, 8, -2, 11, 24], [-22, 15, -4, 7, -2, -7, 6, 7, 2, -5], [-3, 4, -8, 5, -22, -5, 7, 0, 6, 12], [15, 15, 16, -16, 6, 1, 8, 7, 6, -17], [3, 4, 8, -10, -3, -13, -4, -2, -4, 23], [-11, -13, 3, 12, 6, -4, -1, -14, 6, 7], [9, -10, 6, 11, 16, -1, 7, -4, 9, 24], [0, 23, -7, -12, -17, 0, 7, -2, 16, 10], [6, 2, 0, 12, -6, 4, 10, 0, 2, 6], [4, 6, 3, 10, -18, -11, -5, -5, 16, 2]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-16, -160, -441, -388, 89, 254, 45, -49, -14, 3, 1]>,<5,R![-2, -38, -185, -180, 158, 223, -27, -69, -5, 6, 1]>],Snew);
    return Vf;
end function;