
// Make newform 4005.2.a.o in Magma, downloaded from the LMFDB on 18 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_o();".
// 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_o();".
// 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 := [6, -12, -10, 19, 6, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, 0, 0, 0], [2, -4, -1, 1, 0, 0, 0], [3, 0, -5, 0, 1, 0, 0], [-2, -2, 4, 1, -1, 0, 0], [-5, 8, 5, -6, -1, 1, 0], [-4, -4, 13, 5, -7, -1, 1]];
    Rf_basisdens := [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], [-1, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, -1], [1, 0, 0, 0, -1, -1, 1], [-2, -1, 0, -1, -2, 1, 1], [2, 1, 2, 1, -1, -1, -1], [0, -3, -1, 0, 0, 0, 1], [1, -1, -1, -2, -1, 1, 0], [-1, -2, 0, -2, -2, 1, 3], [-2, 1, 1, -1, -2, -1, 3], [-1, -1, 2, -1, -1, 0, 0], [2, -3, -3, 2, 2, 0, 0], [-5, 1, 1, 0, 0, 1, 0], [2, 0, 3, 0, -1, 0, -1], [0, -1, 2, -3, -2, 1, 1], [2, -2, 0, -2, 2, 2, 1], [-1, -1, -3, -3, -1, 0, 1], [1, -1, 0, 2, -1, 1, 0], [2, -2, 0, 2, -3, 0, 0], [3, 1, 1, 1, 1, -4, -3], [-1, -1, 1, -3, -1, 2, 3], [8, -4, 0, -1, 1, -1, -1], [-1, 0, 0, 0, 0, 0, 0], [1, -1, 0, 1, 2, 0, -2], [6, -1, -1, 2, 2, 4, -2], [-2, -4, -2, -2, -2, 2, 1], [5, 4, 1, 3, 3, -2, -2], [4, -1, 0, 0, 2, -2, -1], [-2, 0, -1, -2, 2, 4, 1], [-8, -1, 3, -2, -1, 2, 1], [13, 1, 3, 3, 1, 2, -3], [-4, -1, -4, -6, -2, 0, 5], [-4, 3, 1, -2, -3, 0, 0], [-6, 5, 7, -2, -6, 0, 4], [6, 1, -1, 5, 2, -3, 1], [-1, 2, 1, 2, 2, 3, -2], [-5, 0, 4, 2, 1, 1, 0], [-5, 6, 3, -2, -4, -3, 0], [-1, -4, -4, -2, -4, -3, 3], [-2, 4, 0, -4, 1, -2, 0], [0, -2, -4, -1, 1, -1, 0], [-5, 4, 0, -3, -2, -2, 4], [-2, 0, 1, -1, 3, 3, 1], [7, -1, -4, 3, 7, 4, -6], [3, -4, 4, 1, 0, 8, -1], [-2, 6, -2, 4, 1, -4, -3], [-1, 6, -1, -6, 3, 1, -2], [5, 2, -3, -3, -4, 0, 6], [-3, 0, -2, 0, -1, -7, -1], [-2, 1, -1, -3, -1, -3, 0], [-3, 1, 3, -1, 3, 0, 0], [5, 0, 0, 6, 4, -5, -3], [6, -4, 6, -5, -3, 7, 6], [0, -3, -1, 2, 1, 2, 6], [0, -1, -6, -3, 4, 7, 1], [-7, 1, 2, -4, 4, 5, 2], [-8, 4, 8, -5, -7, 5, 6], [-9, -4, 4, -5, -4, 0, 3], [-2, -1, -3, -2, 3, -8, 4], [5, 6, 7, 4, 3, 5, -4], [-10, 5, -4, 1, 1, -1, 5], [-6, 3, -2, 5, 1, -3, -5], [-4, 4, -4, 1, 0, -5, 0], [7, 3, -2, 0, 0, -5, 0], [4, 4, 6, 5, 2, 1, -12], [-1, 7, -1, -1, 7, 0, -9], [3, 1, -4, 2, 0, 5, 0], [6, -7, 2, 1, -2, 9, 3], [4, 5, 7, 0, -1, 4, 0], [-6, 3, 4, 2, 4, 2, -9], [6, -5, -1, 7, 1, -1, -3], [0, 3, 4, -4, -6, 8, 3], [1, 9, -1, 1, 6, -2, -5], [13, -13, -5, 3, 5, -4, 2], [-12, 7, 5, -4, -3, -4, 1], [5, -1, 1, 4, -5, 3, -5], [16, 4, -1, 14, 9, -10, -5], [-6, 0, 7, 9, -1, -7, -3], [10, -5, 4, 4, -9, 0, 7], [22, -5, -3, 2, -2, -4, 1], [0, 6, 0, 1, 1, 1, -8], [-12, 4, 2, 1, 0, 1, -1], [3, 3, 0, -1, 2, 8, 2], [4, 4, -4, -8, 0, 2, -1], [10, 2, -3, 9, 5, -5, -3], [-8, -7, 5, -2, -5, 10, 12], [-11, -10, -1, -6, -2, 5, 8], [11, 7, 6, -2, -3, -7, -4], [2, 1, -8, 2, 6, -8, -3], [14, -1, -4, 5, 3, -3, -1], [7, -9, -5, -6, 1, 5, 1], [-4, -4, 1, 2, -6, 4, 9], [11, -1, -11, 4, 0, -3, 2], [9, 12, -2, 8, 4, -3, -12], [1, -6, 8, 1, 0, 12, 6], [4, 2, 1, 1, 0, -9, -7], [2, -2, -4, 3, -4, -3, -2], [-6, 3, 10, -3, -9, -3, 5], [2, 2, 8, 7, 2, -3, -4], [0, -3, -5, -9, -1, 5, 5], [-3, -1, 8, -6, -2, 7, 6], [15, 0, 2, -2, -4, 1, 10], [3, 1, 5, 3, -6, -6, 3], [-11, -4, 2, 2, -4, -9, 4], [-6, -9, 0, -1, -8, -7, 7], [-2, 11, 2, 1, -2, -11, -3], [-1, -5, 8, -6, -6, 5, 6], [1, 2, -2, -3, 5, -4, 2], [-20, -2, 4, -4, -5, 2, 6], [-5, 1, 4, 0, -2, -3, 2], [13, -2, -5, 7, 5, 0, -4], [4, -11, -2, 4, -6, 2, 1], [12, -3, -1, -2, 7, 0, 4], [6, -9, -3, 1, 1, 3, 2], [-9, 9, 11, 1, -6, 8, 1], [0, -18, 7, -5, -2, 11, 9], [9, -10, 0, 1, -6, -8, 0], [-7, 0, 7, -8, -7, 3, -2], [6, 10, 0, 6, 7, -4, -8], [0, -16, -6, 6, 1, 4, 4], [17, 9, 2, 2, 4, -1, -10], [-9, 7, 2, -4, -1, 7, -4], [12, -15, -9, 5, 4, -1, -3], [-5, 2, -6, -7, -7, -12, 5], [5, 2, -2, 7, 8, 6, -3], [-4, -1, 7, -5, -11, 5, -2], [5, 3, -7, 7, 10, -12, -2], [-11, -8, 4, -4, -9, 1, 12], [4, -9, -2, 1, -1, -3, 3], [10, 2, 4, 3, 4, 1, -5], [2, 0, 4, 5, -11, -1, 1], [6, 2, -8, 3, 5, -9, -2], [-3, 14, -3, -3, 6, -2, -4], [-5, 2, -9, -3, 13, 4, -6], [11, 1, 0, -5, -5, -4, -4], [-16, 9, -4, -7, 3, 5, -1], [-4, -4, 0, -3, -1, 21, 9], [11, 8, -6, 14, 12, -11, -7], [16, 6, 6, 1, -2, -11, -10], [3, -12, -14, -6, 1, 3, -1], [2, 14, 1, -6, -1, -4, 1], [3, -4, 5, 4, 4, 3, -12], [19, -2, 8, 3, 4, 8, -2], [-4, 6, 0, -1, 4, 7, 6], [-16, 9, 9, 2, 3, 10, -7], [-10, 4, 11, -4, -4, 14, 7], [-17, -8, 7, 0, 0, 9, 4], [5, 12, 12, -4, -5, -7, -4], [-5, -3, 5, 13, -5, 0, 4], [-25, 2, 1, 0, -1, 1, -8], [2, -4, -1, 0, -2, 6, -5], [-29, 4, 2, -5, -10, 8, 6], [20, -11, -3, -5, 0, -3, -5], [14, -9, 2, 2, 4, 0, -7], [-20, -1, 5, 1, -6, -3, 4], [10, 5, -13, 1, 5, 1, -5], [0, 3, 3, -4, -1, 8, 8], [19, 1, 11, 7, 3, 2, -5], [0, -18, -10, -2, -6, -4, 8], [-22, -5, 6, -8, -5, -2, 5], [-16, -3, -8, -8, -5, -2, 1], [6, -1, 3, 8, 14, 2, -17], [6, -4, -2, -1, -1, -1, 2], [-10, 9, 1, 0, 5, -6, -2], [13, -12, -5, 4, 8, 1, -4], [5, -9, 3, 1, 1, 2, -8], [-24, 9, 1, -2, 1, -6, -8], [-8, -3, -9, 1, 0, 3, 0], [-6, 7, 2, -6, -6, -10, -5], [6, -6, 8, 7, 8, 3, -14], [-9, 4, 5, 14, 1, -1, -10], [-17, 7, 11, -1, -6, 2, -4], [-25, 8, -1, -4, -5, -7, 6], [11, 15, -1, -3, 3, -8, 0], [10, 10, 0, 19, 10, -11, -10], [5, 15, 13, 4, -5, 9, -7], [-10, 8, -6, -6, -6, -14, -4], [-7, 0, -1, 1, -4, -8, 2], [-6, -2, -1, -1, 0, 11, -7], [9, 3, 1, -2, 9, 1, 2], [12, -5, -1, -1, 4, 1, -9], [-7, -9, 2, -10, -3, 15, 6], [28, -20, 1, 4, 9, 4, -5], [-10, 15, 1, 2, 5, 8, -1], [2, 0, 12, 20, -1, -8, 0], [8, -10, -3, -6, 8, 2, 5], [10, -10, -11, 9, -1, -5, 3], [-19, 8, 6, -7, 9, 8, -7], [17, -4, -2, 3, 8, -6, -6], [25, 11, 4, 9, 10, 0, -18], [21, -4, 2, -3, -3, 4, -8], [-7, 3, 13, -2, -2, -3, -6], [15, 2, -2, -1, 10, -12, -11], [11, 13, -1, 5, 11, -12, 0], [-9, 19, -4, 4, 3, -1, -6], [23, 3, 3, -7, 5, -4, 1], [-3, 7, 2, 6, 12, 11, -8], [16, -3, 6, 3, 3, 7, -13], [28, 13, 3, 9, 8, -5, -11], [-13, 12, -11, -9, -2, -14, 0], [5, 3, 5, -14, -4, 1, 10], [-4, 3, 3, 3, -13, 1, 12], [-12, -1, 11, 6, -7, -2, 12], [17, 2, 0, -6, 2, -5, -10], [-4, -12, -1, 6, 2, -2, -3], [10, -2, 8, 6, -9, 14, -7], [-6, 0, -6, -6, 10, 14, -9], [24, 3, -5, 4, -5, -8, 4], [-10, 14, -10, -2, 1, -20, -4], [-11, -20, 5, -7, -7, 2, 14], [-4, -4, -7, -5, -2, 13, 7], [16, 4, 5, 13, 4, -7, -5], [11, 5, 5, 6, 0, 3, -18], [-16, -16, 0, 4, -1, -12, 6], [-4, -6, -12, -11, -2, 9, 16], [7, 11, -6, 8, 7, -15, -4], [13, 1, 1, 5, -3, 10, -9], [10, -5, -10, -8, -4, 2, 7], [6, -17, -16, 1, 8, 7, -7], [1, 6, 6, 15, 12, -2, -9], [-6, -2, -2, -6, 2, 4, -6], [-3, 3, 3, -5, 2, 16, -7], [-10, 0, -9, -4, -2, -8, 15], [-1, 8, 10, 5, -4, -2, 4], [18, -21, -5, 5, 1, -7, -2], [5, 3, -5, -1, -6, -12, 5], [-24, -10, 6, -19, -11, 9, 17], [31, 17, 5, 4, 10, -7, -10], [-12, -9, 3, -10, -1, 18, 19], [-32, -11, 6, -9, -4, -1, 11], [-22, -2, 8, -19, -12, 19, -1], [18, 7, -5, 0, -6, 8, 0], [10, -6, -4, -9, -1, 1, 12], [-23, 7, -15, -12, 2, -1, 6], [10, -3, -9, -4, -5, -14, 7], [-26, -8, -2, -3, -12, 3, -2], [-33, -6, -8, -1, 3, 8, 7], [11, -3, 7, -6, -8, 7, 6], [-4, -12, -14, -12, -4, 0, 6], [25, -19, 2, 6, -7, 1, 6], [11, 13, 11, -4, 5, -5, -9], [-32, 15, 9, -8, -6, 8, -11], [-14, 15, -8, -10, -4, -14, -7], [17, 10, 11, 8, 6, -9, -10], [-9, -13, -21, -7, 4, 8, 10], [22, -23, 5, -5, -7, -1, 3], [11, 2, 2, 8, 18, 11, -10], [-4, 3, 9, 15, 2, -11, -9], [-16, 13, 5, 3, -7, 5, 7], [-25, 20, -2, -17, 1, 0, -9], [-18, 12, -7, -14, 8, -4, -11], [-25, 7, -8, 3, -4, -10, -6], [1, 6, 4, 4, 8, -15, -5], [-3, 2, 7, 1, 4, 10, 2], [16, 7, -1, 6, 1, -4, -24], [-33, 6, 2, 9, -1, 2, 7], [17, -8, -1, 0, -8, 5, 20], [2, 2, 9, -3, -9, 13, 13], [9, 1, -2, 0, 6, -7, 2], [4, 4, 16, -8, -17, -2, 11], [-11, 9, 1, -10, 8, -3, -8], [-1, 7, 12, 9, -13, 6, -2], [10, -11, -4, 17, 11, 3, -11], [-4, -7, -2, -14, 3, 8, 11], [19, -18, 0, -9, 8, 6, -5], [-7, 17, -1, 0, 6, -1, -5], [3, 4, -4, 7, -9, 0, 1], [13, -5, 10, 16, 6, 3, 8], [-12, 13, 2, -7, -1, 25, 5], [-8, -13, 0, -20, -9, 14, 17], [18, -14, 2, -17, -14, 5, 5], [9, 1, -8, -8, 0, 5, 2], [20, -7, -7, 4, -1, -22, 4], [9, 0, 1, 16, 6, -1, -12], [5, -2, -20, -10, 0, 3, 6], [7, 13, 5, 15, -7, -10, -8], [2, -14, 6, -1, -4, 5, -4], [-8, -7, -9, -2, 4, -6, 16], [21, -6, -2, 1, -10, 0, 8], [35, 0, 0, 3, 3, -4, -2], [15, -12, 2, -4, -1, -5, -6], [26, 17, 6, 3, 6, -9, 1], [4, 4, 6, 8, 9, -18, -6], [-1, 11, 0, 1, 3, -16, -6], [-2, 11, 9, 5, 10, 3, -4], [10, 22, -3, 9, 20, 1, -17], [-13, 28, 10, -1, 0, 0, -7], [33, -13, 3, -7, 0, 0, -3], [5, -1, -13, -6, 5, -3, 19], [25, -3, -8, 17, 23, 0, -18], [-16, 7, 19, -3, -12, 11, 19], [-7, -7, 2, 4, 9, 25, -8], [-14, -2, 8, -12, -8, 0, 13], [-19, 13, -5, 4, 9, -21, 5], [-2, -19, 2, -18, 11, 14, 7], [4, -18, -16, -4, -12, 6, 5], [4, 11, -4, 11, 7, -17, -15], [34, 18, -2, 9, 2, -31, -8], [6, 19, -3, -6, 8, 0, 1], [-5, -14, -23, -2, 11, -1, -2], [-3, 25, -14, 11, 1, -18, -8], [17, -3, -5, 7, 5, 12, -14], [6, 3, -4, -2, 11, 12, -13], [-1, 0, 2, 1, -11, 4, 7], [7, 3, 0, -20, 5, 7, 8], [-12, 0, -5, -4, 0, -6, -1], [-28, 3, -8, 10, 12, -4, -13], [15, 8, 10, 14, -3, -1, -15], [23, 15, 18, 12, -2, -17, -10], [3, 25, 2, 5, 1, -12, -18], [9, -11, -5, -13, -8, 16, -13], [38, -2, -2, 23, 16, -15, -8], [11, 1, -4, 0, 2, 1, -12], [-9, 16, 27, -7, -4, 10, 4], [-25, -11, -3, -2, 5, 7, 5], [1, -23, -5, -3, 6, 22, 2], [6, -2, -4, -2, 11, 0, -14], [1, -10, -5, -4, -1, 11, -6], [6, -9, 1, -6, 9, 4, 8], [11, 11, 9, 7, -2, -12, 1], [-2, -17, -10, 10, 5, -6, -15], [15, 10, 3, 3, 2, -10, -8], [-16, 20, -10, 7, -1, -9, -11], [1, 0, 2, 3, 8, -6, 19], [-14, 27, 3, 10, 9, -4, -14], [2, -6, 4, 21, 8, 19, -11], [-10, -4, -7, -3, 2, -9, 21], [-4, 25, -4, -18, 4, 14, -5], [-17, 13, 8, -2, 5, -7, -12], [33, -7, -16, 13, 16, 8, -4], [-2, 17, 6, -4, 5, -12, 7], [8, 2, -14, 13, 15, 5, -11], [29, 2, -11, 7, 11, -6, -16], [-2, -12, 12, 11, -3, -3, -9], [22, 24, -3, -2, 3, 2, -15], [13, -4, -2, 14, 1, -9, -3], [26, 10, 1, 10, -7, -8, -1], [-37, 1, -9, -2, 3, -13, -9], [2, -20, 4, -16, -21, -4, 11], [-3, -16, 5, -1, -17, 20, 16], [-13, 5, -9, -6, -9, 23, 15], [-2, -3, -6, -8, -9, -12, -1], [-1, 1, 15, 3, -4, 10, 10], [17, -5, 10, 12, -8, -3, 4], [-21, 8, -4, -2, 2, 11, 9], [29, 0, 6, 4, 16, 17, 9], [-4, -28, -14, -8, 1, 14, -1], [18, -18, 0, 17, -5, 7, 1], [30, 9, 12, 16, -2, -4, -5], [4, 8, 0, -1, 15, 5, 2], [3, -22, 1, -10, -17, -9, 18], [-27, -2, 6, -10, -22, 3, 9], [13, -5, 12, -7, -10, 18, 8], [-7, -6, -16, 2, -9, -17, -9], [-17, 16, -4, 3, 1, -8, 1], [14, -27, -15, -4, 7, -4, 8], [17, 12, -10, -2, 10, 5, -11], [2, 0, 7, -16, -1, 6, 3], [12, -8, 15, 4, -2, 6, -21], [11, 19, -8, 11, 0, -16, -4], [24, -5, -13, 3, 17, -3, -8], [14, -15, 0, -2, 6, 12, 3], [19, 9, -17, 26, 20, -21, -10], [11, 6, -19, -7, -6, -6, 10], [-8, -12, -3, -11, -8, 3, 13], [33, -13, 0, -8, 3, -11, -4], [-6, -16, -3, -1, 4, 5, 13], [6, -10, 12, -2, 1, 18, 16], [17, 9, 13, 13, -16, -12, -9], [-22, 16, 6, -3, -10, -7, 18], [-21, 11, 19, -5, -23, -6, 16], [3, 0, 0, 0, -8, -9, -1], [-4, -13, 17, -15, -10, 11, -1], [19, 2, 3, 19, 3, -10, 2], [-1, -22, 6, 4, -17, 1, 23], [-9, -12, 8, -1, -11, 6, -3], [-38, 11, -2, -1, -7, 1, -5], [32, 7, 5, 12, 22, 8, -20], [12, -2, 0, -10, 10, 16, 8], [15, -5, -13, 0, 5, -15, 1], [-4, 7, 0, 1, 15, 1, -13], [-13, 21, -15, 8, 18, -3, -18], [-19, 11, -8, 1, 14, -20, -16], [11, -7, 1, 12, 11, 7, -5], [-10, -8, 0, 0, -9, 24, 7], [8, 4, -2, 5, -16, -21, -2], [-37, 24, 7, -20, -4, 5, -14], [12, 7, -11, 3, 13, -11, -3], [28, -7, -4, -3, 15, -5, -15], [-13, -4, -10, 2, 4, -11, 1], [9, -8, -9, 8, 1, 3, 14], [5, 7, -13, 12, 15, -21, -14], [26, 0, -10, 29, 16, -13, -15], [-1, -9, 7, 5, -5, -20, -8], [-18, 8, 17, 23, 7, -5, 1], [6, 4, 24, -1, -16, 11, 0], [24, 9, 19, -1, -12, 1, -4], [38, -5, -27, 12, 12, 6, -3], [-31, 12, -1, 9, -5, -2, -10], [-5, -1, 1, 19, -6, -2, 1], [-33, 17, 10, 10, -3, 5, 4], [-9, 28, 11, 17, 10, 10, -26], [0, -5, 4, 0, 13, 34, 1], [-6, -5, 9, 17, 3, 5, 18], [-16, 15, -21, 8, 13, -6, -10], [8, 3, -8, 20, 17, 6, -23], [1, -17, -3, -6, -4, 5, -5], [4, -3, 10, -10, 1, 22, 1], [32, -2, -4, 19, 14, 11, -13], [-6, -31, -16, -4, 4, 0, 5], [-26, -6, 8, 4, 6, 0, -18], [33, -16, -3, 2, 3, 3, -6], [12, 2, 0, -3, 0, 13, 7], [1, 26, -9, -10, -3, 5, 8], [12, 3, -1, 5, 2, -25, -4], [26, 1, 1, 18, -7, 4, -13], [15, 21, 12, 20, 20, 1, -18], [2, 5, 2, 23, 13, -15, -17], [-7, 7, 0, -16, 10, 11, -12], [-37, -10, 8, 2, -11, -3, 9], [23, -13, -15, 4, -4, 5, -3], [19, 5, -10, -3, -2, -24, -6], [-13, -9, -9, -15, -9, 14, 19], [-21, 5, -1, -3, -10, -4, -15], [-17, 0, 12, 16, 0, -5, 23], [8, -15, 7, -15, 0, 19, 13], [-12, 14, 5, 23, 7, 5, -9], [-39, 27, 3, -11, -3, 8, -13], [-30, 20, 24, -11, -3, 15, -1]];
    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_o();".
// 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_o(:prec:=7)
    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_o();".
// 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_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 6, -29, -8, 24, -3, -4, 1]>,<7,R![4, -76, -96, 189, 236, 94, 16, 1]>,<11,R![128, -608, 968, -639, 149, 14, -10, 1]>],Snew);
    return Vf;
end function;