
// Make newform 4005.2.a.q in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4005_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_4005_2_a_q();".
// 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_4005_2_a_q();".
// 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, 11, 52, -17, -75, 13, 31, -6, -4, 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], [-3, -1, 1, 0, 0, 0, 0, 0, 0], [2, 4, -4, -2, 1, 0, 0, 0, 0], [0, -12, 3, 13, -4, -3, 1, 0, 0], [3, 8, -5, -12, 4, 3, -1, 0, 0], [1, 10, 12, -28, -16, 23, 1, -5, 1], [-2, 22, 19, -44, -15, 27, 0, -5, 1], [4, -5, -35, 24, 35, -25, -5, 6, -1]];
    Rf_basisdens := [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_4005_a();"
function MakeCharacter_4005_a()
    N := 4005;
    order := 1;
    char_gens := [3116, 802, 181];
    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_4005_a_Hecke(Kf)
    return MakeCharacter_4005_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 := [[-1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 1, -1, 0, 0, 0, 0, 1], [-1, 1, -1, 1, 1, -1, -1, 0, 0], [1, 0, 0, 0, -1, 1, 1, -1, -1], [-3, -1, -1, 0, -1, 0, 1, 1, 1], [-1, 0, 0, -1, 2, 1, 0, 0, -1], [-3, 1, 0, 1, -1, 0, -2, 0, 0], [-1, 2, -1, 0, 0, -1, -1, 1, -2], [1, -3, 0, -1, -1, -2, 0, -2, 0], [0, 1, 0, 1, -1, 2, 0, 1, 0], [2, -3, 2, 1, 0, -1, 1, 0, 1], [-2, 1, -2, 1, 1, 2, 0, 3, 0], [-1, -1, -1, 0, 0, -1, 2, -2, -2], [-5, 1, 1, 0, 0, 1, 0, 0, 0], [-2, 0, -1, 3, 1, -3, -2, 3, 2], [-4, 2, 1, 2, 2, 1, 1, 0, 2], [-1, 1, 1, -1, 1, 1, 1, 0, 0], [2, -1, -1, 0, -1, 2, 1, -2, -3], [2, 1, -3, 0, -3, 0, 3, -2, -1], [-1, -3, 0, 2, -2, -2, 1, 2, 4], [-2, 1, 1, 2, 3, 0, -1, 2, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [3, 1, 4, -3, 1, 0, 2, -4, -2], [0, -4, -3, -1, -1, 1, 0, 3, 2], [-1, 0, -2, -2, -1, -3, -1, -3, -1], [-7, 3, 1, 0, 1, 4, -1, 1, 1], [-2, 5, 2, -1, 4, 3, -1, 1, -1], [-6, -3, 0, -2, -5, -3, 0, -2, 1], [2, 3, 2, -3, -1, 0, 2, -5, -4], [1, -2, 2, -1, 2, 1, 0, -2, 1], [-1, -1, -1, -3, -1, 1, 1, -2, -2], [-8, 5, 0, -1, 4, 3, -3, 1, -3], [-8, 8, -2, 2, 4, 0, -4, 4, 0], [2, -1, 4, -2, 3, 1, 0, 2, 5], [1, -4, 1, 1, -4, -2, 5, 2, 3], [-3, 0, 2, 0, -3, 3, 3, -3, 1], [-3, -5, 1, -2, -4, -1, 0, -1, 0], [-3, 1, 5, -2, 2, 5, -2, 1, 0], [2, -4, 1, 0, 0, 1, -3, 0, 0], [-1, 0, -2, 5, -2, -3, -2, 2, 1], [0, -2, 3, 1, 5, -3, -2, 3, 0], [2, 3, 1, 1, 4, 0, -4, 4, 1], [-5, 8, -4, 5, 6, -1, -4, 4, -1], [-2, 2, 4, 1, 4, 1, -2, 1, 1], [-6, 1, 2, 2, -1, 3, 4, 0, 1], [-1, -4, 0, -1, -2, 3, 0, -2, 3], [-1, -5, 1, -2, -2, -3, 4, -6, 0], [5, -5, 0, -3, -5, 0, 6, -8, -2], [-13, -3, -1, 0, -4, 1, 2, 1, 2], [8, -3, 6, 1, 0, -1, -1, -2, -1], [-4, -2, -1, -2, 2, 3, 3, 4, 2], [1, -2, -2, -3, 2, -1, 0, 0, -1], [-3, -1, 1, 2, -2, -3, -2, -4, -4], [-11, 3, -5, 4, 1, -2, 1, 5, 3], [8, 2, -1, -2, 4, -3, -3, -2, -2], [3, 1, -2, -2, -2, -4, -3, -2, -2], [5, 2, 3, -1, 2, 0, -1, 4, 3], [-3, 0, -5, 0, -2, 1, 3, -1, -2], [-3, -4, 3, -3, -4, -4, 3, -6, -1], [-3, 9, 1, -1, 3, 1, -3, 0, -4], [-6, 2, -2, 4, -2, -4, -6, -2, 2], [-2, 1, -2, -2, 1, -3, -4, 2, -3], [2, -1, 2, -3, 1, -4, -4, -5, 0], [-5, 7, 3, 4, 3, -2, -3, 3, -3], [4, -1, 4, -1, -2, -3, -5, 5, 3], [6, -1, 1, -5, 0, 4, 2, -2, 3], [-6, 2, 2, 4, -2, 6, 2, 2, -2], [-6, -2, 1, -2, 4, -1, -1, 0, 0], [-4, -2, -6, 0, -4, -2, 0, 2, -4], [-8, 5, 4, -1, 2, 5, 1, 6, 5], [-2, 5, -1, 2, 3, 2, -1, 2, -5], [-1, 0, -3, 5, 0, -2, -3, 10, 7], [-1, 13, 0, -1, 5, 0, -2, -2, -6], [6, -14, 4, -2, -8, -6, 10, -4, 6], [12, 1, 2, -1, 0, 1, 7, -4, -1], [2, 3, 1, 1, -6, 0, 4, -2, -1], [-14, -5, 2, 0, -3, -1, 2, 2, 3], [3, -11, 2, 0, -6, -2, 5, 0, 6], [1, 4, 3, 0, -4, 3, 5, 3, 2], [-1, 5, 0, -1, 1, 4, 2, 0, -2], [1, 10, -3, 0, 6, 9, -7, -1, -2], [7, 0, -2, 6, -3, -3, 1, -1, 3], [-4, -6, 1, 0, -4, 1, 7, 0, 2], [-1, -3, 3, -4, 4, 1, 0, 3, 0], [-10, 7, -7, 4, 1, -2, -3, -2, -1], [4, 2, 0, 4, 6, -4, -10, 2, -2], [-1, 6, 1, -5, 0, 2, -3, -2, -1], [-2, -1, 2, -4, 3, 9, 2, 4, 1], [-3, 3, 1, -6, 0, -1, 2, -1, -2], [0, -3, 2, 0, -5, 5, 6, -6, 1], [-4, 4, -8, -2, -2, -4, -2, 0, 0], [2, -6, 0, -8, -2, -4, 0, -6, 0], [4, -3, 0, -8, 1, -3, 4, -4, -3], [2, 2, 0, -6, 6, -4, 0, -2, 0], [0, -10, -6, 0, -10, -4, 6, -2, -2], [6, -5, 2, -9, -2, -3, -1, -6, -1], [10, -2, -1, 6, -4, -5, -5, 2, -2], [-2, 6, -3, 4, 0, -3, -5, 0, -2], [-9, -8, -6, -2, -7, -7, 1, 1, -5], [6, -1, 3, 0, 3, 8, 7, -4, -3], [9, -7, 7, 1, -5, 1, 7, -4, 2], [-10, -6, 5, -1, 1, -3, -2, 1, 8], [-10, 2, -10, 0, -6, -8, -2, -2, 2], [-12, -3, 3, 7, 0, -2, 4, 6, 9], [4, -4, -2, 4, -12, -6, 4, 0, 4], [-4, 6, -8, 0, -4, -4, -2, -10, -8], [-8, 1, -6, 5, 0, -1, -1, 6, 7], [0, 0, 8, -1, 4, 9, 0, 5, 1], [-6, 5, 5, 0, -1, 6, 1, 2, 5], [7, -5, 4, -5, -9, 0, 2, -4, -2], [-1, -9, 6, -3, -7, -4, -4, -4, -2], [-5, 5, 2, 4, -6, 0, 1, 0, 4], [-18, -5, -2, 5, -4, -3, -5, 1, 3], [-5, 13, -5, 9, 13, -1, -5, 6, -4], [-4, -4, -1, -6, -6, 9, 11, 0, 0], [-5, 8, 2, -5, 0, -3, 4, -10, -5], [1, 10, 0, -1, -4, 3, 4, 4, 3], [-14, -5, -1, -8, -3, -4, 7, -4, 3], [4, 4, -11, 4, 2, -5, -1, 0, -4], [19, -1, 6, 3, 3, -2, -4, 0, -2], [-4, 5, 3, -6, 3, 2, 1, -6, -1], [6, -2, 3, -2, 4, 3, -9, -4, 0], [0, -11, 4, -7, 0, -1, 5, -1, -3], [-8, -7, -2, 4, 1, -1, -8, 12, 7], [-8, -9, -5, 2, -3, -14, -7, -6, 1], [5, -6, 1, -6, 2, 1, -5, -3, -6], [-4, -1, 0, 7, 3, 2, 6, -1, 2], [6, 2, -8, 4, -2, -6, 0, 2, -6], [2, 9, -3, 0, 7, 4, 7, -2, 1], [-3, -1, -1, 1, 5, -5, -1, 8, 8], [-8, 14, -4, 3, 4, 5, -4, -3, -7], [13, 7, 1, -1, 9, 9, -1, 0, 0], [-4, -8, 4, -6, -2, 4, -2, 4, 4], [14, -2, -4, -1, 2, -5, -8, -1, -7], [-4, 3, -11, 6, 5, 6, -5, 8, 1], [0, -13, 3, 1, -8, -6, 6, -6, -1], [-3, 9, -1, -4, 0, -5, -4, -1, -2], [22, -15, 11, -12, 5, -4, 1, -4, 7], [0, -16, 0, -1, -10, -9, 4, 1, 9], [1, -7, 0, 3, -11, -10, -4, -2, 2], [-3, 0, 2, 3, 12, 7, 0, 8, -3], [-10, 0, -4, 6, 4, 10, -4, 14, 6], [-3, 13, 3, -3, 15, 3, -5, 4, 0], [4, 1, 2, 5, 0, 3, 9, -6, 1], [-16, -7, -4, 9, 3, 2, -8, 13, 10], [-10, -3, -1, 0, -13, 0, 5, 2, 9], [-7, 0, -9, 9, -4, 0, -3, 6, 5], [-4, 7, -11, 4, -5, 2, -3, -4, -7], [8, 0, 2, 0, 4, 0, -6, 6, 4], [9, -1, -7, 5, 11, 3, -3, 4, 2], [6, 1, 5, -1, 2, 0, 2, -6, -9], [-17, 18, -3, 5, 4, 12, -1, 2, -1], [-14, 2, 0, 4, 2, 2, -4, 4, 6], [3, 6, 5, 1, 8, 4, 1, 4, 5], [6, 2, 3, -2, -6, 3, 3, -6, 2], [4, 3, 6, -2, 5, 1, -6, 2, 1], [12, -12, 6, -16, 8, -2, 2, -6, 0], [3, 2, -9, 2, 8, 1, 1, -1, -4], [-9, 1, -7, -2, -4, -9, -6, -3, 2], [-11, 2, 7, -7, 8, 2, 5, 8, 3], [-3, -2, -6, 0, 1, -11, -5, 5, 1], [-3, -11, 2, 3, -11, -8, -2, 6, 10], [-5, -11, -1, -2, -4, -1, 2, -2, -6], [-9, 7, -11, 2, 5, 12, -1, -3, -5], [-13, -2, 5, -1, 2, 2, -5, 4, -3], [4, -3, -1, -4, 3, 0, -1, -6, -3], [17, -2, 5, 3, 0, 4, 1, 4, 3], [15, 5, -11, -2, 9, -4, 1, 1, -7], [-8, 1, -3, 2, 1, -4, -9, 2, -7], [-17, 17, -8, 10, 12, 0, -7, 4, -2], [4, -10, -3, 5, -3, -9, 0, 7, 6], [0, 8, 6, 6, 4, 4, 6, -4, 2], [11, -1, -3, -9, 7, 3, -1, -6, -10], [-30, 22, 2, 4, 2, 8, 0, 4, 0], [-13, 1, -2, 12, -2, 0, -11, 12, 10], [19, -10, 5, -2, 0, -5, 3, -1, 8], [3, -2, 5, -7, -2, 8, 5, -12, -7], [-15, 1, 0, 4, 2, -10, -3, 6, 6], [-13, 2, -2, 0, 3, 11, 1, 3, -5], [4, 8, -2, -6, 0, 18, 2, -2, -4], [7, 4, 19, -3, 10, 2, -3, -6, 5], [16, -1, 4, -14, 3, 9, 6, -6, -3], [1, 2, 4, -3, -6, 7, -2, 4, -5], [0, 4, 5, -13, 5, 13, 4, -5, -2], [-1, 2, 2, -1, 0, 11, 2, -6, -3], [21, -1, 0, -1, -5, 0, 2, -8, -6], [11, 6, -2, 5, 4, -1, 6, -2, -5], [23, 4, 5, -4, 8, 3, 3, -3, 4], [7, 1, 2, -7, 9, 6, 0, -4, -6], [-2, -13, -3, -2, -5, -4, 1, -8, 7], [-11, -16, -10, 7, -16, -7, 4, 8, 9], [-12, 17, -4, 1, -2, -1, 3, -4, -3], [20, -12, 6, -6, 6, -6, 4, -4, 0], [-10, 4, 0, 2, 4, 8, -8, 18, 12], [-5, -9, -14, 2, -16, -8, 9, -2, 2], [-16, -10, 2, 4, 2, 2, 8, -6, 8], [5, -5, 19, 1, -5, -3, 1, -6, -2], [-10, 11, -6, 14, 5, 1, -2, 14, -5], [13, 11, -6, 13, -5, 0, 2, -6, -10], [-43, -9, 3, -3, -9, -1, 5, -2, 4], [12, -10, 9, 2, -6, -13, -1, 6, 8], [-12, -7, -9, -2, -7, -2, 3, 0, 1], [-11, -6, -5, 2, -14, -11, 3, -3, -2], [18, -17, -3, -2, -5, 0, -3, 6, 3], [1, 0, 13, -19, -4, 10, 3, -4, 1], [7, 8, -2, -5, 0, 1, -4, -4, -1], [35, -8, 3, -8, -2, 7, 5, -3, 2], [6, 2, -2, 3, 12, 7, 6, -5, -9], [-2, 17, 5, -5, 16, -2, -12, 4, 1], [4, -1, 3, -10, -11, -2, 7, -14, -3], [-4, -5, 5, -4, -3, -4, 5, -16, -5], [17, -7, -5, 3, -3, 5, 5, 0, -2], [7, 1, -10, 9, -5, -12, 0, -2, -2], [7, 6, -10, -1, -2, 7, 0, 6, -13], [-15, -6, -6, 7, -2, 1, 4, 4, -5], [-12, -4, -13, 1, 5, -9, -6, -3, -8], [7, 3, -3, 6, 1, -4, 9, -11, -11], [-19, 7, -12, 3, 5, -16, -10, 0, 0], [9, -11, 4, -1, -5, -10, 8, 6, 4], [26, -24, 8, -6, 0, 2, 6, -8, 2], [12, 4, 1, 5, 5, 5, -6, 9, 12], [-7, 17, -8, 1, 13, -4, -6, 12, -2], [-4, 4, -7, 6, 14, -1, -11, 16, -10], [9, -1, 2, -4, 16, -2, -7, 6, 4], [11, -21, 1, -4, -10, -13, 8, -8, 2], [-26, 12, 0, -2, 4, 14, 0, -4, -2], [0, 3, 0, -5, -9, 2, 12, -5, -8], [6, 2, 5, 3, 1, 5, 8, 3, 0], [-22, -13, -17, 5, -16, -18, -2, 4, 11], [11, 0, -5, -1, 6, 8, 3, 10, -7], [-12, 4, 8, 3, 6, -1, -10, 9, 7], [1, -17, 5, -13, -1, -3, 5, -6, 0], [0, -7, 3, 7, 2, -4, -10, 0, 1], [-8, -11, -6, 0, -11, -1, 4, 2, 1], [-14, -2, -2, -2, -6, 6, 14, 4, -4], [1, -5, 11, 4, 8, -7, 0, -1, 12], [12, -6, 23, -1, -1, 1, 0, -3, -2], [-16, 2, 7, -2, 0, 13, 1, 6, 6], [-9, -11, -3, 10, -10, -13, -8, 9, 4], [0, -8, 2, -6, -4, 10, 16, -2, -2], [14, 9, -6, 7, 1, -2, 2, 5, -8], [-4, -12, -6, -3, 0, -1, 0, 11, 9], [-27, 1, -5, 20, 1, -10, -13, 13, 17], [-32, 6, -14, 2, 2, -8, -12, 10, 0], [14, -3, 0, 5, -15, -6, -6, -1, -6], [12, 8, -4, 6, 0, -4, 4, -14, -14], [26, 7, -2, 6, 3, -5, -2, 8, 1], [-17, 7, 4, 1, -5, 16, 8, -8, -12], [1, 9, 11, -9, 7, -3, -15, 12, 10], [3, -15, 1, 1, 9, 1, -3, 6, -4], [-3, -13, 1, 0, 8, 7, 6, 1, 4], [-11, 23, -12, 11, 3, 6, -2, -2, -8], [7, 7, -1, -12, 4, 13, 8, -10, -12], [-7, -9, 3, -11, -9, 5, 11, -18, -2], [-3, -4, -1, 13, 6, -6, -1, 2, 19], [-17, 4, 7, -5, 2, 16, 1, 2, 3], [12, -19, -5, -6, -7, -4, 3, -2, 5], [28, 2, 3, -4, -2, 7, 1, 2, -6], [-33, 4, 0, -2, 11, -1, -3, 5, -1], [-9, -18, -9, 7, -6, -10, 3, -10, 3], [-11, -3, -6, 8, -2, -16, -9, 2, 6], [13, -16, 7, -5, -14, -4, 17, -10, -1], [5, 4, -12, 5, -2, 3, -8, 4, 3], [-15, 3, -6, -6, 10, -4, -7, -2, -2], [-5, 1, 6, -1, -11, 4, 14, 6, 10], [11, 6, 5, -2, 0, 5, 1, -5, 2], [-21, 6, 6, -1, 8, -1, 6, -6, -3], [5, -3, 17, -6, 2, 9, -2, -3, 2], [14, 3, 10, -3, 14, 1, -13, 7, 5], [-18, 5, -15, 13, 6, -8, -8, 4, 3], [2, -1, 3, -9, -4, 14, 10, 6, -3], [30, -4, 0, -7, -6, 15, 12, -11, -9], [4, -7, -8, 3, -1, 6, -2, 3, 0], [14, -16, 4, 0, -8, -2, -2, -6, -6], [-37, 15, -13, 11, -9, 7, 1, -2, 0], [-34, 13, -8, -5, 6, 5, -7, 5, -3], [14, -7, 2, 3, 2, -9, -3, 13, 15], [4, -7, 8, 3, -4, -11, -7, 4, 3], [13, -7, -6, -7, 3, 2, -4, 2, 4], [20, 3, 16, -2, -5, 1, 8, -10, 5], [15, -7, 5, -12, 0, 11, 6, -16, -20], [-8, 1, 8, -1, 2, 23, 9, 7, -7], [5, 7, 3, -7, -9, 3, 23, -16, 4], [23, -16, 8, -13, -14, 3, 16, -12, 1], [-3, -2, -1, -11, -6, 10, -5, 6, 1], [-3, 17, 11, 0, 13, -2, -13, 5, 1], [5, 21, -2, -4, 12, 22, -1, -4, -16], [-7, 9, -1, -5, -7, 9, 7, -4, -14], [11, 12, -2, 11, 4, -1, -12, 16, 7], [0, 16, 1, 2, 14, 7, -11, -8, -10], [2, 6, 6, -10, 2, 4, 2, -18, -8], [-4, 13, -6, 3, 16, 3, -3, 16, 11], [0, -7, -1, -16, 5, 10, 5, -8, -9], [19, -26, 10, -9, -8, -9, 14, -16, 3], [-11, -1, -11, 1, -5, 9, 11, 2, -4], [-17, 5, -5, 1, -3, -9, -3, -2, 4], [-9, -5, 1, -11, -9, -5, 13, -14, -10], [0, -9, 8, -1, 3, -2, -2, -13, -8], [2, -1, 13, 5, 14, 8, -4, 6, 9], [5, 9, -3, 0, -4, -3, 14, -1, 6], [8, 1, -11, 6, 11, -14, -5, 4, 3], [5, -5, 21, -6, -3, 8, -1, -1, 7], [-14, 6, 0, 2, 6, 12, -2, 4, 4], [-6, -11, 8, -1, 3, -16, 0, 1, 4], [17, -9, 11, 6, -3, 2, 11, -9, 3], [-8, 6, 2, 11, 14, -5, -12, 17, -5], [-15, 12, -2, -7, 0, -3, 0, -6, -11], [-15, 8, -20, 14, -3, -7, -3, -3, 1], [3, -1, 1, 12, 8, 1, -12, 15, 6], [16, 12, -5, 0, 6, 1, -3, -6, -14], [-18, 22, 3, 0, 6, 5, -5, 10, 2], [15, 5, 9, -3, 3, 1, -13, 0, 4], [1, -23, 5, 4, 2, -7, -2, -8, 2], [-2, 10, 6, 5, 10, -1, -6, -7, 7], [9, -4, 13, -2, -4, -3, -3, -3, 2], [19, -22, -8, 5, 0, -21, -12, 10, 1], [-12, -9, -3, 7, 4, 2, -16, 16, -3], [-10, -1, -16, 11, 8, -1, -7, 8, 7], [-34, 6, 4, 5, 0, -11, 2, 5, 9], [28, -20, -1, -2, -8, -9, 9, -4, 6], [14, -1, -9, 6, -15, 2, 3, 2, -7], [-30, 18, -10, 2, 10, 6, -22, 4, -6], [-19, -13, 3, -10, 16, 11, -2, -2, 6], [12, 5, 2, -13, 6, 5, 1, 7, 5], [5, -14, -12, -7, -14, -1, 10, 0, -7], [-24, 14, 0, 8, 4, -4, -10, 14, 4], [-29, -1, -15, 2, 0, -11, -14, 16, 6], [-7, 9, 14, -5, -3, 10, -2, -2, -4], [-11, 1, 0, 12, 2, -2, -15, 14, 0], [29, -20, -12, 1, -24, -3, -2, 10, 5], [15, 0, 9, 3, 12, -12, -5, -2, 3], [-23, -5, 5, -4, 10, 11, -10, 9, 6], [-1, -7, 18, -5, 5, -10, -8, 8, 18], [-36, 8, -4, -2, -4, 8, -2, 2, 0], [10, 2, 12, -5, 14, -1, -2, 3, 3], [2, -12, 4, 6, 0, -2, 0, 0, 14], [-2, 7, -1, -14, 5, 6, -11, 2, 1], [0, -10, 0, 4, 14, 6, -10, 12, 20], [-18, 13, -9, -10, 7, -2, -17, -4, -9], [3, -5, 3, 12, 3, -8, 1, 19, 7], [23, -9, 3, -7, -5, 13, 7, -16, -10], [28, -8, 2, 2, 2, -6, 4, 10, 4], [24, 3, 4, -6, -1, -3, -14, 0, -3], [-21, 9, -1, 5, 21, 11, -1, 8, 0], [17, -27, 14, -7, 1, 4, -4, -10, -4], [21, -22, 10, -6, 1, 5, -1, 7, 1], [12, -41, -2, -9, -26, -3, 27, -8, 3], [12, 8, -2, 8, 0, -2, -10, 6, 0], [6, 6, 2, -19, 8, 11, 10, -3, 1], [22, 5, 5, -1, 8, -2, -10, 4, 9], [15, 1, -16, 15, 9, -6, -8, 12, -2], [-10, -7, -7, 6, -1, 4, 1, -2, 3], [-9, 12, 1, 11, -6, -2, -19, 12, 1], [-39, 15, -9, 8, -17, 2, -3, -3, -3], [24, 6, 8, -2, 10, 12, 12, -2, 0], [-15, 28, -4, 9, 12, 3, -10, 16, 9], [-2, 7, -3, 12, -9, 2, 13, 6, 1], [-8, -10, 6, -4, -2, -12, 4, 4, 8], [-4, 9, 8, 14, 5, 15, 0, 16, -3], [-10, 1, 8, -7, 0, 1, 11, -4, -3], [11, -15, 7, 4, -5, 2, 13, 3, 3], [-24, 15, 0, 1, -2, -1, -9, 2, 1], [10, -13, 0, -6, 1, -1, 10, 4, 9], [-12, 12, -14, 0, 0, -4, 0, -2, -2], [11, -2, -14, -5, 4, 5, -6, 2, 3], [-3, 23, -7, 15, -3, 1, 3, -6, -8], [-8, -26, 4, -10, -16, -10, 18, -14, 14], [11, -23, 20, -9, -3, 6, 0, -6, 2], [-27, -1, -4, 0, 6, -10, -9, 4, 8], [-15, -10, 2, 9, -10, 1, -8, 2, 7], [5, -6, 19, 1, 6, 16, 1, 14, 19], [29, -10, 0, 7, -12, -13, 8, -10, 1], [-15, 16, -4, 12, -1, -15, -17, -9, 1], [6, 1, -19, 6, -17, -2, 9, -10, -1], [-11, 9, 17, -3, 5, -3, -3, -16, 0], [6, -23, 2, -11, -5, -4, 16, -21, -10], [-20, 26, -13, 11, -11, 5, 2, -1, -10], [28, 0, -3, 10, 16, 5, -13, 12, -2], [4, 10, 10, 8, 2, 4, 2, -2, -6], [19, 11, -10, -3, 13, 8, 0, -4, -10], [4, -4, -10, 14, 12, -10, -8, -2, -12], [21, -5, 2, -1, -5, 12, 6, -16, -6], [34, 0, -14, -1, 2, 17, 0, -9, -17], [18, -21, -6, 12, -11, -5, -6, 0, -3], [3, 5, -16, -6, -2, 6, 7, -2, -10], [18, -23, -3, 11, -14, -2, 6, 6, 7], [-6, 21, -18, -9, -7, 0, -2, -11, -14], [-34, 28, -8, 4, 16, 2, -14, 14, -6], [-19, 24, 6, 3, 14, 7, -26, 14, 3], [7, -25, 0, -9, -9, -6, 4, -6, -10], [-38, 29, -4, 13, -6, -5, -5, 10, 9], [27, 10, -2, -4, 1, -1, 9, -13, -13], [13, 0, -3, -12, -16, 3, 7, -21, -8], [3, -6, -4, 16, -13, -23, 1, 9, 7], [-4, -13, 16, 7, -18, 9, 3, 1, 3], [4, 6, -5, 5, 27, 1, -10, 17, -4], [-18, -10, 7, 0, 4, -17, -15, 0, 14], [5, 0, -3, 7, 4, -22, -9, 2, -1], [-28, -28, 2, -11, -16, 3, 16, 3, 11], [-7, -8, -7, 9, -2, 6, 19, 6, 1], [20, -21, 16, 2, 9, 7, 2, 6, 1], [13, 7, 3, -14, 6, -3, 8, -11, -6], [20, 28, -3, 9, 17, 13, -6, 5, -6], [-34, 29, -10, 13, 8, 7, -11, 9, -7], [-4, -16, 8, 4, -8, 2, -4, 10, 10], [-16, -7, 3, -10, -9, 12, 7, 2, 7], [10, -1, -15, -8, -9, 6, 11, -14, -11], [2, -11, 18, -5, 0, -11, 1, 6, 23], [-10, 3, -15, 7, -10, -8, 2, -4, -5], [5, -15, 5, -10, -15, -10, 11, -21, 3], [17, -3, 17, -6, 26, -3, -22, -2, 6], [18, -1, -21, -4, 9, -16, 1, -8, -21], [-38, 10, -10, 6, -12, 4, -4, 2, 2], [2, 22, 11, 11, 9, 11, -10, 13, -4], [14, 12, -3, -5, 19, -9, -22, 3, -2], [-29, 22, -11, 15, 16, -4, -17, 6, 1], [-5, 3, 19, -2, 3, 8, -3, 1, -5], [-16, -20, 2, -4, -4, -18, 4, -12, 12], [-9, -14, -5, 2, 8, 7, -5, 3, 8], [18, -4, -1, 8, 4, 5, 1, 8, 4], [25, -9, 1, -8, 17, -6, -7, 5, 15], [-13, -22, -3, 7, -4, -2, 5, -6, -1], [-1, 11, -4, 7, -5, 8, 10, 0, -10], [33, 7, 3, 2, 11, 4, 9, -5, -9], [-1, 22, -3, 1, 4, 12, 7, 2, -1], [27, -12, -13, 3, -14, -14, -1, -2, 5], [-2, -14, 12, -5, 24, 5, -18, 3, 3], [52, -34, 11, -19, -11, 1, 14, -17, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4005_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4005_2_a_q();".
// 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_4005_2_a_q(:prec:=9)
    chi := MakeCharacter_4005_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_4005_2_a_q();".
// 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_4005_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![5, -50, -56, 83, 91, -25, -39, -2, 5, 1]>,<7,R![-251, -1448, -2675, -1301, 795, 468, -90, -39, 3, 1]>,<11,R![-8768, -37824, -25968, 3064, 6072, 820, -300, -60, 4, 1]>],Snew);
    return Vf;
end function;