
// Make newform 1003.2.a.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1003_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_1003_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_1003_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 := [15, -56, -56, 178, -3, -129, 28, 34, -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], [3, -31, -11, 188, 20, -162, -13, 45, 2, -4], [5, -96, -30, 199, 31, -116, -10, 26, 1, -2], [-20, 34, 197, -96, -173, 58, 47, -13, -4, 1], [-25, 39, 178, -134, -162, 111, 50, -32, -5, 3], [-43, 99, 160, -352, -107, 271, 30, -71, -3, 6], [-51, 107, 243, -326, -200, 227, 60, -58, -6, 5], [-51, 135, 243, -333, -200, 227, 60, -58, -6, 5]];
    Rf_basisdens := [1, 1, 1, 7, 7, 7, 7, 7, 7, 7];
    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_1003_a();"
function MakeCharacter_1003_a()
    N := 1003;
    order := 1;
    char_gens := [768, 120];
    v := [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_1003_a_Hecke(Kf)
    return MakeCharacter_1003_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, -1, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1, -1, 0, 0, 1], [-1, -1, 0, 0, -1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 1, 0, 0, 1, 1, -2], [-1, 0, 0, -1, 0, 2, -1, 0, 0, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 0, 0, -1, 0, 0, 2, -2], [0, 1, 1, 0, 1, -1, 0, 0, -1, 2], [1, -1, -1, -1, 0, -2, 3, -1, -1, -2], [2, 0, -1, 0, -1, -2, 0, -1, 1, -2], [2, 1, -2, 0, -1, -1, -1, 1, 0, 2], [-1, -1, 2, 0, 2, -2, -1, -1, -1, -1], [-3, 1, -2, 2, 1, 1, -2, 1, 3, 1], [-1, -1, 1, 0, -1, 2, -2, 1, -1, 2], [0, -2, -1, 0, 0, 1, 1, -1, 1, 2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, 3, -2, 2, 0, 2, -2, 1, -3], [1, 0, 3, -2, -2, 0, 1, -2, -2, 1], [2, -3, 2, -2, -1, 4, 0, -2, -2, 3], [-6, 1, -3, 2, 2, 0, -1, 1, 2, -2], [2, -4, 2, -1, -3, 2, 1, -1, -3, 3], [0, 1, 1, 1, 2, -2, 2, -3, -2, 0], [-5, 1, 1, 0, 3, -2, 1, 0, -1, -3], [0, 2, -1, 0, -2, 1, -3, 0, 0, 0], [-6, 0, 2, 0, 0, -1, 0, -2, -4, 1], [1, -2, 2, -1, 0, 1, 0, -1, -2, 3], [0, -2, 1, -1, 3, 3, -4, 1, 1, 1], [2, -1, -2, 0, -5, 1, -2, 2, -2, 4], [-6, 3, 1, 3, -1, -3, 4, 0, -1, -1], [2, -2, -3, 0, 2, -1, 0, 4, 3, -4], [-1, 3, 2, -1, 2, 1, 1, -2, 0, 4], [-12, 1, -2, 1, -3, -1, 3, -1, -1, -1], [0, -2, 3, 1, -1, 0, 0, 2, 0, -1], [-2, 3, -2, 3, -1, -2, -1, -1, 2, 3], [1, -3, -3, 1, -4, 3, -1, -1, -2, 4], [-4, 1, -2, 2, -3, 3, 0, 1, 1, 3], [2, 1, -2, -2, 5, -3, -3, 0, 2, -1], [3, 5, -3, 2, -2, -5, 1, -1, -2, 3], [1, 2, 1, -1, -4, 0, 3, -2, -1, 8], [-8, 5, -3, 1, -2, 2, 3, 2, 2, -2], [-2, -4, 3, 1, 4, 4, 0, 0, 0, 1], [-4, -4, 7, 3, 0, 2, 2, -1, -2, -2], [3, -4, 5, -1, -2, 4, -1, 0, -2, 4], [-11, 3, -1, 0, 0, 1, 0, 3, 5, -1], [4, 4, -5, 0, -2, -1, 0, 2, -3, 0], [5, 2, -2, -3, 4, 3, -4, 1, 0, 0], [-5, -1, 1, 1, -2, -5, 6, -2, 1, -5], [2, -4, 4, -1, -2, 0, 2, 1, 0, 1], [-2, 1, 3, -1, 2, 3, -4, 5, -4, 3], [-1, -2, -4, 2, -5, 6, -5, 1, 0, 5], [10, -1, 5, 0, 0, 1, -3, 2, -1, 5], [-2, 3, 1, -1, 8, 0, -7, 4, 1, -4], [5, 4, 5, 1, 5, -2, -3, 1, 0, -2], [-4, -3, 1, 0, -3, 2, -3, 5, 2, 2], [-6, -1, -5, -2, -1, -4, 10, -1, 1, -5], [4, 2, 9, 2, 5, -1, -3, 2, -3, -1], [-2, -2, 0, -3, -2, -3, 3, 3, -3, 1], [-8, 6, -7, 1, -6, 4, 5, 2, 5, 0], [2, 2, -9, -1, 2, -9, 2, 1, 4, -10], [7, -7, 1, -6, -1, 4, 3, -1, -7, 4], [-5, 5, 1, 0, 2, -1, -5, -1, 3, -2], [-1, 2, 2, 0, 1, -2, 4, -6, -1, 1], [-5, -4, -2, 0, -2, 1, 6, 2, 3, -6], [7, 2, 1, -6, 2, -5, 5, -2, 0, -4], [-7, -2, 0, -3, -4, -2, 0, -6, -6, 0], [6, -6, -2, 4, -7, 8, -3, 2, 2, 2], [0, 2, 1, 3, 5, -4, 0, 0, 4, -5], [-8, 1, 4, 8, 0, 2, -3, 2, 2, 0], [-4, 7, 4, -1, 2, 2, 2, 2, 3, -4], [-7, 2, 3, -4, 4, 2, -3, -2, -1, 1], [6, 6, -10, 2, -4, -3, -6, 8, -1, 2], [1, 5, 2, 0, 2, -3, -5, -6, 0, 3], [-9, -1, 7, -2, 0, 4, 2, -3, -1, 9], [9, -4, 4, -1, 4, 1, -1, 0, 4, -6], [-10, -3, 1, 3, -3, 5, 3, 0, 1, 1], [-6, 0, -4, -3, -3, 0, 0, 3, -6, 0], [6, -5, 7, -4, 8, -2, -3, -3, 1, 0], [-5, -1, 0, -1, -2, -6, 0, -1, 6, 3], [4, -6, 4, 3, 1, 1, 0, -1, -7, 5], [10, -3, -4, -4, 0, 2, 1, 4, -3, -2], [-7, 3, 5, 3, 3, -3, 1, -3, -2, 0], [8, -2, -6, 1, -2, -7, 5, 0, 0, -12], [14, 1, 3, 0, 1, -2, -8, 2, -4, 1], [-5, 1, -3, -1, -3, 6, -2, -3, 2, 8], [9, -3, 5, 0, 3, 2, -2, 0, 0, -3], [-3, -1, -6, 5, 2, -5, 4, 1, 7, -7], [5, -1, -1, -2, -3, -1, -2, -4, -1, -5], [-13, 3, -2, -1, -4, -1, -1, 0, 4, 4], [-10, -3, -6, 1, -6, -1, 4, -1, -4, 1], [-7, 4, 2, 1, 5, -3, 2, -1, 3, 8], [-2, 0, 5, -4, -1, -1, -3, -2, -9, -1], [9, -6, 8, -6, -10, 4, 3, -7, -12, 6], [4, 0, -1, -6, 1, -1, -3, -1, 6, 1], [-4, 4, -3, -1, 6, -7, 3, -6, 2, -14], [-2, 3, -5, -6, -4, 1, -2, -2, -5, 2], [-14, 0, -6, -7, -3, 0, 2, 2, -3, -1], [-11, 1, 1, 5, -10, 1, 3, 0, -6, -1], [0, -2, 3, -7, -4, -5, 3, -4, -14, 4], [1, 5, -7, -1, 7, -3, 0, -1, 8, -3], [-10, 9, -7, 6, -2, -4, 8, 6, 5, -4], [-5, -7, 1, 8, 0, 11, -6, 6, 8, 3], [-2, -10, 8, -1, 4, 0, 4, 0, -1, -13], [9, 0, 17, -2, 7, -2, 0, -1, -5, 0], [7, 0, 2, 1, 6, 6, -2, -2, 8, 4], [-9, -9, 7, 2, -4, -1, 13, 0, 2, -5], [-16, 9, -3, -3, 1, 3, -4, 6, 2, 4], [6, 2, 4, 2, -3, -6, -3, 3, -10, -7], [-4, -2, -11, -2, 1, -1, -2, -1, 10, -7], [2, 7, -1, 5, 10, -5, -7, -2, 1, -5], [10, 5, -2, 3, 1, 0, -8, 6, 5, 10], [-7, 7, -9, 0, 3, 1, -1, 5, 0, 3], [-6, -7, -5, -4, -1, -1, 7, -3, 5, -7], [10, -7, 5, -8, -1, 8, -4, -2, -5, 11], [-12, -8, 5, 1, 1, 5, -5, 3, -1, -5], [-7, 5, -9, 5, -4, 1, 5, 8, 5, -7], [5, -4, 4, -1, 1, -2, 4, -11, -2, -1], [9, -1, 8, -4, -6, 4, 0, -1, -2, 10], [-2, 5, -3, -5, 10, 1, -9, 1, 7, 1], [5, 8, -5, -4, -2, 3, -7, -4, -1, 16], [6, -11, -5, 2, -7, 3, 10, -7, 2, 1], [10, -6, 5, 3, 0, 12, -6, 3, 7, 9], [0, 4, 10, -4, 1, 12, -7, -2, 2, 8], [0, -10, 10, -3, -1, 16, -5, -5, -12, 11], [-2, -7, -10, -3, -4, 12, -2, 4, -5, 4], [2, 6, 5, 2, 5, -1, -9, 6, -1, 8], [2, 5, 1, 5, 9, -12, 8, -4, 7, -18], [8, 9, 2, 1, -4, -5, 3, -8, 2, 4], [-6, -3, -2, 7, 1, 3, -1, 0, 0, 3], [-17, 4, -8, 4, -5, -1, 5, 0, 9, -3], [6, 6, 6, -4, -4, 8, -4, 0, -6, 5], [-4, 7, -1, 6, -1, 0, 0, 0, 5, -2], [0, 10, -6, 1, -2, -15, 5, 3, 5, -7], [-4, -6, 1, -4, -8, -1, 5, -3, -8, -5], [11, 1, -1, 0, 12, 1, -8, -4, 2, 2], [5, -14, 4, -4, -4, -2, -6, -7, -6, 4], [-10, 0, -7, -3, -7, -2, 3, -4, -7, 6], [9, -1, 6, 4, 12, -6, -3, -3, -1, -4], [-16, -8, -6, 2, 2, 4, 4, 0, 2, 4], [-3, -3, -1, 2, -2, -8, 8, 0, -5, -8], [1, -2, 1, 3, -4, -1, 13, -7, -5, -9], [5, 3, -5, -2, -7, -4, 3, 9, -2, 0], [3, -3, 0, -3, 0, -6, -9, -4, -11, -2], [6, -4, 8, -2, 3, 8, -6, 3, 5, -5], [-1, 4, -15, -3, -2, -1, 0, -2, 5, -8], [-12, 16, -16, 13, 4, -9, -1, 7, 15, -8], [9, 1, 9, 9, 6, 3, -8, 1, 1, 6], [-4, -13, 7, 2, 2, 4, 5, -4, -3, 7], [-13, -12, 1, -8, -3, 2, 9, -3, -1, -7], [7, 2, -2, -4, 1, -5, 7, -1, 1, 3], [10, -8, -5, 4, -3, -8, 3, 2, 2, -16], [0, -1, -3, 2, 2, 1, 1, -2, -4, 5], [-2, -3, -3, -1, -4, -9, 10, -1, -5, -6], [-9, -8, 11, -6, 11, 0, 1, 0, -6, -8], [10, 11, -8, 2, -2, 6, -13, 2, -5, 12], [12, 1, -3, 1, 8, -2, -7, 7, 9, -8], [-5, 9, -5, -6, -5, -3, 2, -1, -9, -2], [-2, 11, 3, 6, 0, 8, -7, 5, 8, 6], [8, 2, 6, -7, 9, 3, -3, -8, -5, 9], [-15, 1, 5, 1, 0, -8, 5, 5, -11, 0], [0, -15, -8, 0, -12, 11, 8, 0, 0, 8], [5, -2, -7, -3, -5, 5, -9, 3, 3, 4], [7, 4, 4, 1, 2, -2, 8, -3, -1, -13], [-3, 4, -7, 9, 4, -8, -6, 1, 2, -6], [1, 10, -4, 7, -1, -7, -6, 3, 9, 8], [0, 6, 3, 0, -2, -7, 4, 1, -11, -6], [11, 18, 0, 2, 17, 6, -17, 6, 9, 0], [-5, -2, 0, 3, -10, 7, 6, 2, -3, -5], [0, -2, -5, -4, -3, 4, 0, -1, -6, -7], [-32, -3, -1, 4, 0, -2, 10, 4, 6, -22], [1, -9, -1, -2, 3, -2, 4, -7, -4, -11], [10, -10, 3, -2, -5, 6, -6, -3, -5, 5], [14, 12, 3, 5, 2, -13, -3, -1, 3, -5], [2, 8, -3, 4, 6, 6, 0, -3, 4, -9], [-7, -1, 0, 3, 2, -3, 12, 4, -5, 1], [-14, -10, -4, -2, -2, 5, 11, 0, 14, -4], [-1, -1, 5, -1, -4, 4, 15, -1, 4, 3], [-6, 0, -2, 1, -6, 7, -1, 5, 4, -6], [-3, 14, 4, 8, 9, -4, -14, 1, 7, 2], [1, -2, -10, -9, 1, 6, -5, 6, 1, 3], [-3, -3, -3, 2, 1, 5, -6, 2, 16, 6], [9, 10, -6, 2, 4, -9, 0, -8, -4, 1], [-8, 14, 3, 5, -1, -13, 7, -5, -11, -9], [-5, 15, 4, -1, 6, -1, 3, 1, 4, 0], [7, 4, -6, 11, 6, -2, -1, -3, 10, -5], [-18, -16, 6, -4, -5, 7, 9, -1, -4, 6], [13, -10, 8, -3, 0, 10, 1, -3, -10, 6], [9, 8, -11, 1, -5, 1, -11, 10, -7, 19], [-23, 6, -8, 3, -1, -6, 7, 6, -3, -6], [10, -17, 11, -2, -10, -6, 10, -7, -17, -4], [-12, -6, 6, 5, 1, 10, -10, 9, 8, -7], [-15, 4, 4, -1, 0, 10, 9, -6, 10, 1], [1, 3, 5, 2, 13, -11, 1, -6, 11, -16], [0, -13, -8, 0, -15, 7, 10, 0, 3, 2], [-5, 6, -10, 2, 6, -15, 11, 1, 1, -25], [17, -3, -3, 2, 12, -3, -8, -2, -4, -3], [16, -8, 4, -6, -13, 1, 4, 0, -1, 2], [9, 3, 3, -6, -9, -9, 13, 1, -9, 5], [-9, 14, -5, 10, 8, -10, -9, 9, 5, 1], [-23, -4, -9, 10, -13, 2, 19, 2, 5, -8], [-7, 2, 3, 2, 1, -2, 1, -6, -3, 7], [13, 15, -6, 2, -5, 0, 0, 3, 7, 12], [-7, 0, -5, 3, 6, -8, 1, 11, 13, -22], [-6, -4, -1, 1, -13, -4, 8, -2, -12, -2], [-5, 11, -5, -10, 6, 6, -4, -3, 2, -5], [-3, 4, -11, -3, -1, -1, 3, -2, 6, -5], [-13, 8, -7, 5, -12, 1, 10, -2, 5, 4], [4, 7, 13, -4, 12, 2, -11, 5, -12, 13], [-21, -6, 9, -2, 2, 4, -7, 10, -3, 9], [-9, -5, 7, 1, 8, 2, -10, 1, 2, -4], [-2, 3, 6, 12, -1, 9, -2, 3, 8, 1], [5, -9, 8, -2, -5, 15, -16, 5, -4, 8], [-17, 12, -9, 5, -4, 4, 0, 7, 13, -11], [-25, 3, -5, 3, 5, 9, -8, 8, 14, -5], [0, -3, -19, -6, -9, 0, 6, 0, 16, -3], [-8, 12, -7, 4, -3, -11, -2, 6, -10, 0], [30, 5, 12, -4, -3, -1, -2, -4, -22, 24], [0, 1, 1, 1, -12, 0, 15, -10, 2, 10], [-12, -6, -8, -3, -6, 2, 8, 2, -8, -6], [-29, 8, 2, 1, 2, -8, 6, 3, 7, 4], [-6, -12, -14, 3, -9, -1, 8, 3, 8, -17], [1, 6, 13, 0, 5, 8, -10, 3, -11, 16], [-10, 7, -1, 2, 10, -14, 9, -2, -4, -19], [-7, -1, 6, 8, -3, -9, 9, -8, 3, 8], [-8, 6, -8, 4, -1, -5, -8, -1, 5, -2], [-2, -5, 6, 0, -4, -2, -1, -9, 0, -9], [-3, 5, -8, -5, -3, 1, 8, 9, 0, -20], [3, 4, -9, 0, 8, 2, -18, 13, 7, 2], [-7, 6, -8, 2, 0, -3, -10, -7, -2, 4], [6, -11, 12, 0, -2, 2, -2, -16, -2, 15], [1, -3, -4, 3, -4, -2, 0, 1, 5, -11], [-6, 2, -3, -1, 1, -3, 2, 3, 18, 1], [-5, 0, -12, 2, 4, -11, 7, -2, 2, 1], [6, 14, 1, -1, -13, -12, 18, -8, -6, 9], [-17, 5, -3, -6, 6, 4, -2, 4, 6, 9], [-3, 14, -4, -3, 4, -2, -2, -4, 7, -9], [5, -8, -7, 1, -22, 7, 4, 12, -3, 6], [-6, -9, -13, 3, -1, -5, -7, 12, -3, -10], [7, 20, -4, -1, -1, -3, -5, -2, -13, 14], [14, -4, 12, 3, 9, 2, 5, 2, -7, 3], [13, 12, -5, 7, 0, -2, 2, 6, 17, 9], [-4, -11, -11, 4, -16, 5, -4, 7, -2, -4], [-8, 1, 6, -11, 6, -7, 4, -4, -3, 11], [-25, -7, 4, 6, -6, 4, 0, 10, -9, 1], [-12, 10, 2, 6, -3, 12, -7, 2, 14, 16], [7, 0, 7, 3, 15, -6, -3, 6, 5, -26], [19, -6, -6, 8, 1, -8, -8, 1, -17, 8], [12, -1, 7, -10, -2, 2, -8, -1, -10, 3], [8, 1, -5, 2, 2, 5, -13, -4, 0, 2], [13, 1, 9, 4, 6, -9, 9, -5, -3, -9], [-8, 4, 0, -8, 7, 10, -15, 5, -4, 7], [1, -5, -9, 1, -1, 1, -9, 11, 0, -15], [-8, -4, 9, -5, 6, 10, -7, -7, 5, 15], [5, 9, 3, -2, 2, 14, -4, 7, 0, 5], [6, 21, 6, -1, 9, -21, -2, -9, -6, -2], [2, 6, -5, -2, 3, -12, 11, 9, 6, -13], [2, 8, -2, -3, 6, 0, 0, 5, -1, -8], [-10, -11, 5, -3, -4, 9, 0, 5, -10, 1], [-16, -10, 0, 1, 1, 10, -6, 3, 5, -6], [0, -1, -6, 9, -4, -8, 5, 2, 20, -1], [-15, 0, 13, 4, -7, 13, -2, 5, -4, 18], [-6, -2, 3, -5, -9, 18, 2, 2, -7, 0], [21, -4, 8, -1, 3, -18, 3, -10, -17, 1], [2, 2, 13, -10, -12, 5, 10, -11, -10, 2], [-7, 13, -4, -3, -3, 6, -8, 6, -4, 21], [6, -15, 2, 5, -5, 5, 0, 5, 6, -1], [-20, 2, -3, 2, -7, 6, 7, -10, -1, 13], [-4, -1, -19, -1, -5, -4, 1, 5, 6, -15], [-8, -13, 8, -10, 9, 1, 7, 3, -7, -7], [2, -19, 11, 0, -9, 14, 18, -5, 0, 4], [6, -13, 6, 0, -8, 22, -12, 8, 5, 23], [-14, -19, 0, -8, -13, 5, 8, -6, -17, -1], [5, -1, 17, 4, -7, -4, 11, -1, -4, 6], [-2, -7, 25, -9, 8, 11, -9, -1, -11, 4], [-1, 3, -6, 8, -4, -8, -3, 4, 4, -9], [-9, 2, -6, -10, 10, -5, -1, -9, 0, -8], [-7, -5, -2, 2, 13, -2, -8, -4, 7, -7], [-11, 6, -2, -9, 4, -2, -2, 2, -7, 7], [7, 9, 7, 7, 7, 5, -14, 3, -2, 14], [10, -7, 9, -1, 5, -11, -3, -12, -1, -7], [24, -9, -6, -3, -6, -5, 5, 8, 1, -2], [-28, 2, 12, -5, 1, 0, 2, 0, -5, -4], [5, 5, -12, -1, -3, 1, 3, 1, 19, -9], [-7, 0, 15, 4, 16, 0, -6, -9, 1, 9], [-5, 0, 7, -9, 6, 16, -5, -3, 4, 9], [18, 4, -3, -9, 0, 2, 10, -9, -8, 6], [10, 5, -4, -4, 13, -12, -9, -7, 4, -9], [-11, 3, -18, 7, 0, 2, -5, 15, 31, -18], [-17, 5, -10, 1, 9, -3, -10, 4, 6, -11], [19, 2, 18, 1, 9, 2, 6, 4, -11, -11], [19, -12, 3, 0, 2, 4, -3, -13, -9, 0], [-5, 23, -17, 9, -8, -11, -12, 8, 12, -3], [-19, 12, 3, 5, 8, -13, 15, 3, 3, -6], [-34, -2, -4, -5, -1, 17, 5, 4, 16, -7], [18, -10, -3, -3, -5, 1, 2, 0, -1, -9], [-19, -1, 8, 1, 0, -1, 21, -3, 9, 2], [-8, 10, 1, -1, 24, -8, -1, -1, 2, -13], [-31, 15, -2, 6, -1, 5, -11, -5, -6, 17], [-4, 0, 12, -7, 2, 8, -17, -11, -7, 21], [22, 2, 12, -5, 8, 3, -3, -1, 8, -9], [-11, -2, 7, -1, 16, 11, -15, 6, 6, -9], [-39, 0, 0, 0, 5, 1, 10, -1, 10, -21], [-26, 7, -22, 10, -13, -4, 6, 4, 18, -13], [25, -13, 2, -1, 0, -6, -7, -1, 1, 7], [-35, -6, -7, 6, 9, 3, -9, 6, 11, -5], [2, -5, -2, -5, -1, 2, 9, -18, -1, 9], [10, -10, 23, -9, -3, 2, 11, -8, -18, 6], [9, -18, 17, -9, -1, 26, -3, -6, -4, 19], [17, -10, 17, -10, -1, 15, 0, -3, 1, 15], [3, -3, -8, 6, 6, -17, 10, -10, 13, -9], [-13, -2, 0, -11, -10, -2, -4, 5, -13, 19], [3, -1, 6, 8, -18, 14, -4, 9, -5, 12], [12, -5, 6, -14, -4, -4, -1, -8, -26, 9], [-28, 3, 3, 3, 5, 5, 4, 0, 16, -11], [-9, 9, -6, 5, 2, -8, 1, 3, 2, -3], [15, 18, 15, 1, 22, 7, -24, 6, 2, 11], [11, -3, -15, 11, -11, -1, -6, 1, -4, 6], [4, -20, -1, 0, 1, -5, 2, -2, 9, -1], [-3, -3, -8, -7, -12, 4, 7, 1, -3, 15], [12, 1, -12, -4, -11, 9, -1, 4, 0, 0], [-13, -1, 2, 5, 7, -1, -8, 3, -14, 2], [3, 0, -2, 13, 19, -4, -7, 10, 16, -12], [9, 20, -6, 8, 2, -2, -18, 1, 3, 7], [4, 6, 11, -14, 23, 0, -7, -5, 7, -5], [25, -1, 8, -9, 10, 8, -9, -14, -2, 17], [0, -4, -2, -5, -5, -8, 17, -15, -6, -5], [-1, 7, -20, -7, 1, 4, -5, 8, 3, -5], [-33, 1, -1, 9, 9, -1, -4, 4, 6, 11], [9, 3, -24, 7, -15, 2, -2, 7, 8, 6], [11, 4, 5, 3, -3, -10, 11, -2, -13, -3], [-9, 10, -2, 9, 17, -15, 13, -2, 25, -23], [-2, 3, 3, -6, 19, -15, 9, -6, 2, -19], [-1, -2, -7, -3, 5, -14, 8, -5, 10, -8], [8, -5, 10, 4, 16, 4, -10, 4, 8, -6], [7, -11, 4, 8, 9, 6, 2, 9, 7, 4], [22, -17, 3, 4, -13, 9, -4, -4, -5, 8], [3, 13, -15, -3, -11, -6, -10, 1, 1, 9], [-11, 0, 12, 11, 0, 4, 5, 11, -1, 0], [-7, 2, -5, 2, -2, 0, 18, 2, -6, -18], [2, -28, 17, -8, -5, 14, 1, -9, -15, 19], [23, -4, -17, -2, -5, 5, -9, 12, 7, -8], [-3, -8, -27, 7, -11, -4, -2, 9, 4, -14], [9, 10, -13, -11, -4, -12, 5, 3, -2, -18], [-12, 2, -9, 8, -12, -14, 14, 4, -13, -1], [-6, 12, 13, -3, 4, 10, -12, 7, -4, 6], [25, -14, 1, -11, -9, 7, 2, -9, -12, 12], [21, 1, 2, -3, -7, -10, -3, 8, -10, 12], [-2, -13, -13, -5, -15, 6, 11, 12, -9, -10], [-1, 11, -1, 1, 17, -21, -13, 5, 0, -11], [-30, -12, 7, -13, 10, 10, 13, -8, 6, -6], [-36, 5, 9, 8, 4, 14, -15, 7, -7, 20], [-9, -6, -11, 1, 2, 5, 2, -13, 6, -5], [-40, 4, -5, 2, 12, 9, -6, 2, 11, -2], [-11, 4, 1, 4, 1, -14, 3, -5, 3, -7], [28, 5, 5, -18, -1, -8, -5, -14, -15, 4], [-1, -15, 13, -3, 17, -4, -8, -11, 0, -1], [20, -4, 8, -8, 4, 6, -3, -5, -8, 3], [13, 22, 11, 1, 5, 9, -12, 11, 0, -4], [7, -3, -10, 10, -2, 8, -18, 4, -7, 12], [-14, 4, -15, 5, 0, -5, 4, 3, 20, -12], [-12, 9, -4, 11, 6, 2, 1, -5, 29, -7], [3, -1, 12, 4, 1, 6, 5, -9, 14, 1], [-1, -1, -6, 7, 0, -3, 4, 16, -4, -8], [-11, 11, -13, 10, -3, -11, -9, 2, 11, -6], [-32, 7, 11, 9, 2, -3, 9, 9, -5, -17], [5, 6, 8, -5, 0, 6, 4, -6, 6, 16], [12, 0, 26, -9, 8, 15, -12, -7, -20, 29], [20, -7, -2, 0, -5, -3, -10, -1, -17, 5], [-34, -5, -12, -1, -8, 12, 5, 14, 6, -9], [0, -1, -15, 11, -12, 12, -15, 10, 10, 18], [-15, 8, -4, -1, 0, 15, -9, 17, 9, 6], [-12, -5, 13, 8, -7, 20, -21, 6, 0, 20], [34, 11, 2, -17, 14, -16, 6, -10, 5, -15], [-2, -6, 3, -6, 1, -3, 9, -13, 0, -20], [-11, -2, -7, -7, 6, 24, -7, 2, 8, 14], [10, -19, -9, 7, 2, -15, 0, 6, -5, -18], [-7, 22, -12, -2, 9, -11, -10, 8, 9, -25], [-3, 15, 5, -3, 10, 14, 2, 0, 2, 12], [1, -15, 18, 8, 0, 2, 21, 0, 9, -10], [31, 12, -10, 2, 0, -11, -14, -2, -1, 20], [10, 10, 1, -5, -7, -10, 12, -15, -6, 10], [-6, 28, -12, 14, 9, -11, -12, 13, 26, 0], [30, 22, -15, 0, -2, -11, -13, 2, 0, -3], [16, -29, -6, 4, -20, 3, 6, 2, 7, 11], [32, 4, 7, -4, -4, -15, 18, -11, -13, 3], [-20, -17, -4, 9, -6, -3, 17, 8, 6, -9], [21, 1, 4, -7, 20, 1, -17, 5, -5, -17], [-5, -11, 5, -1, 16, 4, -9, -3, 6, -21], [-7, -15, -7, 9, -7, -1, -10, 13, -4, 2], [12, 2, 5, -3, -3, 5, -14, 6, -7, 15], [15, 11, -3, 6, 10, -9, -6, 1, 19, -15], [-38, 5, -22, 7, -5, -13, 20, 9, -7, -27], [19, 2, 4, -8, 16, -18, 5, -14, -10, -2], [-2, -5, 18, 6, 10, 8, -16, 0, 9, -7], [13, -9, 23, 8, 3, -6, 12, -5, -2, -7], [1, 15, -12, -14, 2, 6, -11, 0, -7, 10], [-19, -10, -12, -1, -1, -5, 7, -3, 25, -9], [35, -2, 15, 5, 9, -5, 1, -1, -3, -14], [3, -27, 8, -8, 0, 2, -2, -7, -8, -6], [22, -20, 24, -13, -5, 10, 1, -13, -16, 5], [3, 5, -18, 6, -22, -17, 12, -6, 6, 14], [-16, -7, -8, 2, -1, 1, 5, -1, -11, 6], [13, -8, 34, -5, 15, 2, 11, -6, -18, 6], [-14, -1, -13, 0, -3, 4, 2, -1, 23, -19], [-29, -6, -9, 4, 1, 5, 2, -3, -2, 6], [1, 24, 8, 9, 20, -9, 4, -5, 16, 2], [-28, -10, 8, -4, -5, 18, 12, 2, 3, -6], [10, -9, 19, 2, 0, 1, 4, -20, -4, 6], [-11, -9, -15, -3, -15, 3, -4, 11, 8, -13], [-42, 7, -5, 4, 3, -1, -12, -1, 17, -8], [28, -29, 20, -15, -5, 25, 11, -11, -25, 15], [-2, -3, 6, -14, -1, 12, -21, 4, -17, 13], [-28, 15, -6, -8, 4, -11, 6, 5, -7, -4], [-23, -4, -1, 5, 10, 0, 9, 14, 13, -39], [20, -29, -6, -14, -20, 15, 13, -14, -10, 19], [28, 5, -22, 6, -19, 2, -14, 14, 4, 15], [6, -4, -10, 4, -9, -19, 12, -16, -2, 2], [21, 19, 11, 8, 0, -10, -7, -6, 4, 10], [-3, 24, -13, -14, 2, -4, -18, 6, -13, 2], [-14, 10, -15, 9, -5, -19, 24, 1, 20, -11], [4, -5, 2, 5, 5, 15, -14, 14, -6, -1], [16, -16, 8, 3, -8, 9, -3, -1, -9, 33], [10, -12, 1, 13, 6, 10, -10, 2, -6, 2], [-12, 9, -18, -6, -11, 17, -17, 10, 12, 14], [0, -6, -13, -21, -13, 5, 15, -10, 0, 1], [23, -9, 7, -13, -11, -3, 3, -9, -29, 8], [10, -16, 3, 0, 14, -10, 10, 2, 8, -27], [-19, 20, -17, 9, -16, -11, -1, -1, 13, 2], [-24, -11, 8, 6, 3, 8, -6, -2, 1, -25], [-34, 1, 14, 3, 6, 2, -9, 13, -10, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1003_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1003_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_1003_2_a_g(:prec:=10)
    chi := MakeCharacter_1003_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_1003_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_1003_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1003_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-81, -45, 357, -37, -308, 44, 108, -12, -17, 1, 1]>,<3,R![-1, 18, -39, -66, 94, 67, -56, -34, 8, 7, 1]>],Snew);
    return Vf;
end function;