
// Make newform 4005.2.a.p in Magma, downloaded from the LMFDB on 19 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_p();".
// 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_p();".
// 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, -27, -19, 34, 9, -11, -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, 1, 0, 0, 0, 0, 0, 0], [0, -5, 0, 1, 0, 0, 0, 0], [-7, 65, 12, -76, -8, 23, 1, -2], [-7, 27, 18, -34, -9, 11, 1, -1], [-27, 121, 50, -144, -26, 45, 3, -4], [27, -136, -61, 170, 34, -55, -4, 5], [-29, 175, 72, -212, -42, 67, 5, -6]];
    Rf_basisdens := [1, 1, 1, 2, 1, 2, 2, 2];
    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 := [[0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 2, 0, 1, 1, -1, 0, -1], [-2, 0, 1, 0, 0, 0, 0, 0], [-2, 1, 0, 1, 0, -2, -2, -1], [-2, 1, -1, 1, 1, 1, 0, 0], [3, -2, 0, -1, 0, 1, 1, 1], [1, 1, 0, 0, 1, 1, 2, 0], [-2, 2, 0, 0, 1, -1, -2, 1], [0, 0, 1, 0, -1, 1, 1, 2], [0, -1, -1, -1, 0, 2, 2, 1], [-2, -1, -1, -1, 0, 0, -2, 1], [-1, -3, 1, -2, -4, 0, 0, 1], [-1, 2, 1, -1, 0, -1, 2, 0], [2, -3, -2, -3, -2, 2, 2, 3], [-3, -1, 1, -2, -2, 1, 1, 0], [1, -1, -1, -2, -3, 2, -2, 2], [-3, 1, 2, 2, -1, 2, 4, 0], [-4, 0, 1, 0, -1, -1, 0, -1], [-1, -1, -3, 0, 1, 2, 0, 0], [-1, 1, 1, 0, 1, -4, 0, -2], [-6, 2, 0, 2, 1, -4, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 3, -2, -1, 1, 1, -2, 0], [-2, -5, 1, -1, 0, 2, 0, -1], [1, -2, -2, -1, -1, 1, 4, 1], [-2, 0, 0, -4, -2, 2, 2, 4], [-5, 1, 1, 4, 1, -6, -6, 0], [-3, -2, 0, 3, 1, -2, -2, -3], [-7, -1, 0, 4, 0, 2, -2, -3], [-9, 1, -1, 0, 1, 0, -2, 0], [-3, 5, 3, -2, 1, -4, 2, 0], [4, -1, 0, 3, 3, 1, -2, 0], [2, -3, 1, 1, -2, 2, 0, -5], [0, 2, -3, 2, 5, 1, -1, -4], [1, 0, 2, -1, -2, 1, 4, 0], [1, -2, -1, 1, 3, -1, -2, 3], [-1, -4, 0, -1, 2, 1, -4, -2], [0, 6, 2, 2, -1, -1, 0, 1], [4, -2, -1, -2, 5, 1, 0, -1], [-1, 2, 0, 1, 0, 1, 4, -2], [-2, -1, -2, 3, 2, 4, 3, 2], [-2, 2, 0, 2, -1, -3, 2, 3], [-2, -1, -1, 1, -6, -2, -2, -5], [3, 0, 1, 3, -4, 1, 0, 0], [7, 1, 2, -2, -1, 2, 1, -1], [-1, -2, -1, 1, 3, 0, -2, -5], [-6, -4, 1, -4, -3, 3, 0, 1], [2, 0, -1, 2, 6, 4, 0, -2], [-3, 5, 3, 2, 0, -7, -4, -3], [-8, -6, -2, 4, -1, 1, -1, 2], [0, -2, 0, 2, -1, 5, -4, 1], [-3, -4, -4, -1, -2, 3, -4, 2], [0, 5, 2, 3, 3, -1, 4, -4], [-8, 5, 2, 1, 0, 2, -2, -1], [-3, -7, 1, -4, -2, -1, -2, 5], [-1, 0, 0, -7, -2, 5, 0, 2], [-9, -6, 3, -9, -6, 1, 0, 4], [-2, 1, 0, 1, -1, 3, 0, -4], [9, 2, -3, -1, 3, -2, -6, -1], [-6, -5, -3, -5, -1, 1, 2, 0], [-4, -5, 1, -1, -5, 3, 0, 6], [-5, 0, -1, 5, -1, -6, -2, -3], [-5, 1, 3, 2, -4, -1, 0, -3], [-1, 4, 3, 7, 1, -10, -12, -3], [-5, -3, 1, -4, -9, -2, 0, 0], [-9, -3, 3, -6, 0, 3, 2, 3], [-6, -1, 4, 1, 4, -4, 2, -3], [8, -7, 0, -1, 3, -1, -2, -2], [-3, 5, 1, -6, 1, 0, -2, 8], [8, 0, 0, 0, 0, 2, 9, 3], [-9, 5, -1, 2, 9, 2, 2, -4], [3, 5, 0, 6, 12, -3, 2, 3], [0, 0, 0, -2, 3, -1, -1, 0], [-3, 7, 2, 8, 8, -6, -2, -7], [0, -3, 0, -3, -2, 2, 6, -3], [-15, -4, 1, 1, -2, -7, -8, -6], [12, 2, 0, 4, 3, -1, 2, -1], [-7, 5, 0, 6, 0, -5, 4, 5], [-7, 0, 0, -1, 4, -5, -3, -1], [3, -10, 0, -7, 2, 9, 8, 4], [0, 17, 0, 1, 3, -5, -1, -3], [-6, 13, 0, 11, 5, -9, -4, -10], [5, -17, 3, -10, -10, 11, 7, 6], [10, -4, -4, -10, -1, 1, 2, 5], [10, -1, 0, 3, -5, -3, 0, 2], [-5, 6, -2, 7, 10, 3, 2, -6], [11, -5, -4, -8, -3, 8, -4, 8], [-9, -9, 1, -4, -7, -2, 2, -2], [-14, 7, 5, 1, -3, -5, -4, 0], [-10, 3, 2, 15, 6, -4, -4, -7], [-5, -16, -2, 5, -1, 8, 0, 3], [-15, 4, 4, 5, -1, -6, -1, -4], [5, -1, 1, 8, 1, -10, -7, -5], [-10, 8, -1, 10, 3, -6, -2, -11], [6, -4, 1, -4, -2, 0, 0, 2], [-5, 12, 3, 3, -3, -6, 4, 1], [0, 7, -3, 7, 7, -1, 0, 2], [13, -2, -5, -5, -5, 10, 6, 7], [4, -7, -1, -9, -1, 6, -2, 2], [11, -9, 3, -12, -8, -3, 2, 1], [-17, 8, 0, 13, 0, -12, -8, -4], [9, -5, -2, -6, -4, 1, 2, 5], [3, 5, 5, 4, -3, -10, -5, -5], [10, 15, -2, 7, 10, -10, -4, 1], [-10, 1, -4, -5, 2, 8, 6, 1], [5, -1, -1, -14, -4, 1, -6, -3], [0, -3, -1, 3, 5, 5, -3, 1], [-6, 6, 7, 6, -1, -5, 6, 3], [-13, -7, 3, -4, -2, 3, -2, 13], [0, -6, 0, -2, -6, 0, 2, 8], [7, -9, -5, -10, -5, 8, 6, 8], [2, 11, 4, 9, -1, 3, 4, 2], [7, 13, 1, 2, 4, -11, -2, -9], [-1, 11, -2, 4, 0, -3, 2, -11], [2, 10, 5, 8, 6, -14, -4, -4], [0, -8, 1, -4, -1, -2, -2, 1], [1, -8, 3, -9, -1, 8, 10, 1], [4, 4, -5, 4, 5, 5, 0, -3], [12, -6, -3, 0, 3, 7, 8, -3], [3, 1, 3, -2, 10, 5, 6, -1], [-3, 5, 4, 4, 5, -12, -4, -2], [6, -13, 0, -9, -6, 9, -4, -3], [7, 12, -2, 7, 3, -8, -4, -9], [7, 6, 3, 7, 4, -9, -4, -10], [13, -3, 5, 2, -2, -5, 2, 5], [-2, -10, -5, -2, -4, 6, 3, 3], [-5, 6, 1, 7, 3, 1, 6, -1], [20, 7, 5, -3, 1, -1, 0, -6], [2, 5, 0, -1, -5, -5, -3, -9], [18, 2, 0, -10, -3, -2, 4, 11], [1, 6, -2, 5, 2, -9, -12, -4], [-18, 4, 5, -2, -3, -3, -4, 3], [-8, -2, 4, 2, -6, -4, -6, -2], [8, 3, -5, -5, -4, 0, 0, 3], [-10, -1, 7, 3, -1, -7, 2, 4], [4, 0, 0, 2, -7, 0, -2, 3], [14, -14, -2, -8, 1, 3, 0, 7], [-9, -8, -1, 3, 5, -2, 0, 1], [0, -2, -1, -4, 0, 6, -6, 0], [13, 2, -3, 3, 8, 1, 2, 4], [-7, -10, 0, 3, -4, 1, -8, 6], [-6, -12, 1, -10, -3, 6, 2, -1], [-5, -10, -5, -3, -1, -2, -2, -1], [-1, -12, 3, -15, -7, 14, 13, 4], [11, -8, 4, -9, -3, 10, 0, 9], [3, 0, -8, -3, 6, -1, -2, 8], [1, 3, -2, 4, 2, 3, -1, 6], [10, 9, -3, 19, 8, -1, 0, -7], [3, -16, -1, -7, -3, 4, -8, -3], [-3, 20, -6, 5, 13, -6, -4, -3], [4, 6, 3, -6, -1, -8, 6, 3], [-12, 11, 0, 1, -2, 5, -4, -3], [1, 21, -1, 16, 11, -4, 2, -14], [-3, -15, 2, -4, -15, 4, 0, -2], [4, -12, -2, 0, -2, 10, 3, 11], [0, 1, 0, 15, 5, -7, -12, -14], [7, -5, -3, 0, 3, 0, 12, -6], [6, -4, -4, -8, 6, 14, 10, 8], [1, -13, -4, -6, -8, 1, 12, -1], [3, 5, 2, 0, 4, 5, 4, -7], [-19, -3, 1, -10, -5, 5, 0, 12], [3, 14, -6, -1, 2, -3, -10, 2], [8, 1, 0, 7, 1, 1, 8, 2], [25, -2, -6, 3, -4, 11, 12, 6], [-2, 18, 2, 10, 1, -7, 1, -6], [-2, 13, -2, 1, 9, -13, -4, -6], [1, 17, 0, 2, 5, -10, -2, -8], [1, 13, -1, -2, 3, -2, 10, 0], [-19, 16, 5, 3, 1, -10, -4, -7], [-3, 6, -3, 9, 3, -6, -2, 3], [-10, -16, -3, -8, -8, -2, -11, 7], [5, 6, -7, -7, -1, 6, 8, 1], [18, 8, 0, 4, 5, 7, 15, -6], [3, 8, 1, 7, 15, 0, -8, -5], [-4, 21, -2, 11, 12, -12, -6, -13], [-12, 2, -2, 12, 10, 0, -2, 6], [10, -2, -1, -18, 1, 7, 0, 3], [-10, 12, 3, 12, 4, -10, -12, -4], [-15, -5, -2, -6, -9, 3, -8, -2], [-24, 2, 3, 10, 5, 3, 1, -2], [3, -15, -8, -2, -1, 14, 4, 2], [19, -4, -1, -1, -4, 7, 8, -4], [9, 7, 2, 4, -4, 4, 2, -3], [2, 2, 7, 6, -5, -1, 4, -3], [1, 2, 4, 3, -11, 6, 8, -3], [-18, 16, -4, -2, 11, -3, -2, -3], [15, 16, 4, -3, 5, -2, 6, -1], [-12, 15, -10, 9, 4, 2, -9, 0], [-12, 11, 2, 3, 5, -23, -6, -8], [-8, 2, 2, 6, -4, -13, -14, 0], [19, -4, -2, 7, 7, -4, -5, -4], [15, -16, -3, -19, -16, 9, 6, 14], [-38, 3, -5, 9, 2, -3, -6, 3], [-1, -3, -4, 10, 3, -6, 8, -10], [15, -19, 1, -6, -3, 6, -8, 2], [9, -17, -3, -18, -8, 11, -2, 21], [4, 11, 9, 1, 1, -17, -2, -10], [0, 3, -6, -3, 10, -3, -16, -3], [4, -8, 3, -4, -3, -13, -4, -3], [-5, 28, 0, 27, 13, -14, -1, -10], [5, -1, -5, -16, 2, 11, 12, 9], [-14, 1, 4, 9, 5, -17, -16, -18], [1, 15, -3, 20, 13, -2, 3, -11], [-9, 6, 0, 9, 5, -14, -14, -3], [7, -23, 0, -2, -5, -2, -3, 11], [-3, -14, 4, 3, -3, -13, -4, -9], [-24, -7, 0, -13, -11, 5, 0, 6], [0, -2, 7, 2, 5, 1, 14, 1], [0, 0, -4, 20, 6, -4, -2, -12], [14, -14, -1, -6, -16, 18, 0, 6], [-24, 8, 1, -8, -8, 14, 0, -4], [-27, 3, 3, -6, -6, -2, -6, 7], [-2, 20, 5, -10, 3, -11, 2, -3], [-9, -8, -11, -1, 3, -2, 0, 11], [-9, 13, 2, 4, 5, -12, 4, 6], [-1, 3, -5, 8, 7, -4, -16, -6], [15, 17, 3, 2, 3, -6, 10, -10], [-6, -1, -4, 7, -2, 4, -8, -3], [11, 8, 7, -7, 0, -15, 2, 2], [-4, 8, 2, 16, 0, -4, -2, -14], [-13, 13, -2, 2, 10, -9, -10, -3], [-19, 4, 4, -5, 3, -8, 8, -1], [0, -4, 1, 4, -9, 12, 4, -5], [-15, 7, 5, -4, 6, -5, 8, 1], [-5, -9, 0, -4, -8, -3, 2, 9], [0, -7, 1, -7, -6, 8, 11, 8], [-7, 13, 7, 12, -4, -4, 4, -11], [-5, -24, 1, -23, -23, 12, -1, 10], [0, 1, 8, -13, -10, -2, -4, -1], [10, 7, 2, 1, 3, 15, 3, 7], [-2, -29, 3, -23, -16, 12, 4, 7], [9, -12, 8, -11, -16, -5, 6, 6], [-23, -9, -1, -14, -8, 2, 4, 7], [-7, -1, 8, 4, -6, 6, 12, 13], [-13, 2, 1, 17, 7, -6, -20, -3], [-1, 26, -6, 23, 17, -12, 2, -3], [33, -10, -2, -5, -3, 14, 11, 12], [-4, 2, 4, 18, 4, -8, -14, -16], [13, -25, 4, -8, -17, 18, 8, 2], [6, -7, 2, 5, 10, 0, -4, -1], [-3, -1, -1, -4, 3, 5, 10, -4], [-11, -15, -7, -6, -9, 10, -2, 6], [5, 0, -8, -1, 12, -1, -4, 2], [-12, 22, -7, 12, 12, -10, -13, -9], [8, 5, 1, -13, 7, 8, 6, -2], [31, -11, -5, -4, -3, 8, 17, 9], [-2, -2, -1, -4, 11, 15, 13, 8], [16, 7, 5, 5, 1, 0, 20, 8], [11, 0, -2, -7, -1, 22, 22, 5], [-3, 32, -6, 25, 19, -20, -8, -13], [8, -9, 2, -11, 5, 13, 2, -2], [-8, 0, 0, 0, -9, -3, -4, 3], [-12, 8, 3, 8, 13, -9, -4, 11], [-12, 5, -2, -3, 1, 11, 12, 2], [9, -12, 4, -9, -1, 4, -12, -5], [-3, -26, -2, -21, -20, 11, -8, 14], [32, 0, -4, -6, -3, 15, 6, -5], [-15, 17, -5, 16, 14, -5, 10, 7], [7, 16, -1, 3, 8, 12, 6, -2], [-5, -33, -3, -16, -4, 12, 2, 9], [-4, -3, -13, 1, 0, 0, -4, -7], [-4, 25, 3, 3, 7, -11, 12, -2], [-11, -5, -4, -4, 2, 1, -6, -5], [3, -6, -1, -15, -10, -5, -8, -4], [10, 9, -5, -5, 1, -7, 2, -8], [-3, -12, -2, -9, -23, 2, 10, 15], [15, -13, -3, 6, 8, -7, -10, 1], [-6, -15, -5, -3, -1, -1, -4, 10], [-3, 9, 4, -10, 10, -7, 0, -5], [14, -10, 5, -2, -8, 9, 14, -8], [-1, 15, -3, 4, 2, -3, 16, 9], [10, 19, 0, 17, 5, -3, 8, 0], [-7, 4, 2, 11, 4, -9, -2, -14], [-33, 10, 8, 15, 8, -24, -12, 6], [-4, -1, -1, -9, 2, -19, -2, 3], [-15, 12, -5, -3, 11, 6, 0, 3], [6, 7, 3, 7, -6, -20, -26, -1], [2, 9, 6, 1, -2, 0, 11, -4], [-2, 6, 10, 8, 2, -11, 2, 10], [9, 1, 0, -10, 2, 11, 9, 18], [-20, -13, 0, -5, -4, 6, 6, 1], [6, -28, -9, -20, -19, 17, 6, 23], [-34, 17, 1, 13, 2, -14, -12, -17], [11, -5, 4, 0, -1, -4, 10, 0], [-14, 6, -9, -6, 0, 10, 16, 16], [-15, -6, -1, 1, -4, 13, 8, 6], [-13, -31, 8, -16, -14, -7, -2, 1], [8, -5, 0, 3, -4, -14, -8, -13], [-6, 3, -3, 17, 8, 6, 4, -5], [-8, -3, -10, -1, 12, 1, -12, 1], [11, 11, 4, 0, 7, 6, 8, -12], [-19, 19, -5, 14, 18, -9, 17, -4], [26, -13, -12, -11, 6, 22, 0, -3], [-25, -1, -4, -2, -2, 3, 8, 5], [19, -11, -15, 10, 8, 15, -9, -2], [10, -11, 3, -19, -5, 5, 2, -2], [-3, -4, -11, -21, 3, 8, 4, 11], [5, 7, -5, -8, -1, -1, 2, 0], [-5, 0, -7, 9, 5, 2, -12, -7], [-8, -9, 3, -1, -4, 4, 0, 5], [-8, 6, -2, -16, -1, 1, -1, 2], [23, 19, 0, -16, -4, -7, 4, -1], [7, -4, 8, -9, -5, -8, 14, -3], [17, -15, 2, -4, -1, 8, -8, 0], [15, 16, -10, 13, 5, -2, 8, -3], [17, 17, 1, 18, 21, -8, 6, -20], [12, -18, -3, -12, -14, 8, 10, 2], [5, -23, 5, 8, -12, -1, 0, 1], [-28, 27, 3, 25, 16, -16, -8, -21], [21, -21, -2, -10, -18, 21, 6, 15], [-3, 16, -5, -11, 3, 10, 4, -5], [-9, 1, 9, 0, -8, 5, -6, -13], [26, -10, -1, -2, -9, 9, 4, 5], [4, 21, 2, 29, 14, -12, 4, -5], [6, -18, -13, -14, 0, 22, 1, -3], [4, 8, 5, 0, -5, -1, 0, -13], [-19, 14, -1, 21, 5, -16, -10, -7], [-2, 1, 2, -3, 21, 9, 16, 2], [21, 3, -4, 8, 4, 1, 0, -13], [3, -9, 12, 2, -8, 13, 0, 1], [20, -22, -7, -14, 1, 19, 0, -1], [-12, -14, 2, -12, -7, -6, -12, 19], [-7, 10, -1, -15, 2, 11, -4, -8], [-22, -3, 4, -3, -17, 13, 0, -4], [10, 3, -8, 13, 1, 5, 7, -7], [0, 8, -1, 12, 4, -6, -4, 0], [9, -9, 1, 6, -3, 10, -6, -2], [-9, -33, 6, -20, -23, 10, 4, 4], [8, -1, 4, -9, -3, 9, -14, 2], [-39, 19, -4, 16, 2, -9, -10, -25], [38, 7, -1, 9, 10, -6, -11, 0], [4, -2, -6, 4, 4, 2, -18, 4], [10, -3, -9, 15, 4, 14, 5, 4], [-15, 22, 13, 1, 0, -9, -6, -8], [2, -14, -3, -20, -18, 12, 18, 14], [3, 18, -1, -11, 18, 5, -8, -8], [8, -15, -6, -7, -10, -2, -16, -1], [17, 0, -1, 1, 12, 4, -2, -14], [-8, 16, -4, 4, -12, -10, 2, 0], [20, -5, -4, -9, -4, 6, 4, 7], [-3, 27, -2, 12, 4, -17, -16, -9], [-26, 18, 7, 2, 0, -16, -15, -15], [-13, -5, -1, -4, -6, -5, -14, -1], [-8, -6, -12, -12, 1, 19, 8, 27], [-10, -28, -4, -6, -5, 7, -10, 9], [-3, -10, 9, 11, -8, -4, 8, -10], [-2, 9, 11, -5, -8, 0, -9, -6], [21, -11, -1, -26, -1, 12, 4, 24], [5, -10, -2, -15, -6, -1, 2, 6], [-5, 20, 1, 5, -3, -10, -4, 1], [-26, -4, -2, -2, 13, -1, -8, 13], [6, 15, -2, -5, 7, -23, -14, -16], [-4, 26, 7, -6, 8, -20, -4, -2], [-15, 4, -6, -9, 9, -10, 2, -1], [-16, 5, -2, 1, 3, 5, 10, -10], [-16, 4, 8, -6, -13, -3, 10, -3], [29, 16, 5, -11, 10, -9, 8, -2], [-1, 4, 4, 5, -5, 8, 8, -1], [14, -13, -2, 1, -13, 5, -2, -10], [-1, -5, -1, -18, -2, -15, 0, -3], [1, -9, -3, -14, 3, -12, -6, 4], [-5, -16, -6, 19, -1, 6, 1, -10], [20, 4, 15, 6, -9, -15, -8, -5], [18, -12, 9, -4, -2, 8, 8, -6], [-21, 33, 4, 18, -7, -18, -16, 2], [-14, -4, -1, -10, -8, 16, 8, 24], [-14, -2, 9, 18, 1, -9, -2, -15], [-13, 11, 8, 10, 16, -1, -18, -3], [-27, 12, 7, -1, 7, -8, -10, 1], [6, -9, -3, -7, -8, -15, -8, -5], [20, -2, 0, -10, -1, 11, 28, -3], [14, -18, -1, -4, -15, 27, 5, -6], [-4, -6, 6, -2, -2, -14, -18, 6], [14, -30, 3, 2, -14, 10, -12, 4], [-17, 4, 6, 11, 15, -4, 6, -7], [-4, 7, -3, 1, 21, 7, 8, -8], [-14, 9, 7, 13, -2, -6, 10, 13], [-6, 6, -2, 10, 16, -12, -8, -18], [-2, 19, -6, 17, 8, -4, -12, -5], [10, 5, -7, 9, 11, 23, 6, -6], [15, -23, 1, 14, 4, 0, 8, 9], [38, 23, 2, 9, 15, 5, 6, -4], [-24, 5, -2, -1, -4, -8, -12, -19], [-15, -12, -1, -3, -12, 20, 6, 14], [3, 22, -1, 3, 11, -2, 4, -7], [13, 28, -4, 35, 20, -11, -6, -10], [-26, 5, -1, -15, 11, -4, 8, 4], [8, 13, -3, 5, 3, -5, -4, 4], [6, -10, -12, -12, 15, 7, -4, -7], [21, 4, -17, 11, 7, 0, -6, -3], [7, 20, 5, 11, 24, -9, 17, -1], [4, -12, -21, 2, -4, 9, -6, 6], [-2, -2, -2, -6, -13, 3, 1, 12], [34, -22, 0, 6, 3, -3, -2, -1], [5, -12, 7, -17, -15, 2, -2, -1], [-13, 9, 6, 16, -9, -6, -14, -2], [1, 8, 10, 19, 4, -3, 9, -1], [-2, 0, 4, -10, -16, -12, -22, -2], [1, -7, 2, -24, -10, 9, 18, 9], [-15, -11, 6, -4, -17, -4, -10, 2], [10, -22, -1, -22, -21, 13, 8, 13], [-37, 23, -10, 18, 14, 3, -8, 3], [-13, 9, 15, 16, 22, -9, -2, -7], [28, 23, -6, 23, 19, -23, -10, -24], [23, 19, -2, 2, 6, -7, -16, 1], [8, -21, 5, -17, -23, 3, -6, 10], [-43, -19, -2, 12, 0, 5, -4, 13], [18, -27, 8, -29, -23, 1, 5, 9], [-41, -11, -7, 0, -13, -4, -12, 4], [-4, 18, -8, 12, -8, -6, -12, -2], [17, -4, 1, 9, 19, 4, 2, 1], [28, -5, -6, 3, 11, 11, 11, 3], [-3, 18, -1, 25, 15, -18, 6, -7], [-23, 5, 6, 6, 1, -2, 0, 6], [-11, -14, -7, -3, 3, -14, -17, 6], [-11, 29, -1, 18, 12, -35, -22, -1], [16, 17, 3, -9, -3, -13, 12, -4], [-13, 11, 5, 10, -8, -9, -2, 9], [6, 38, -1, 32, 32, -36, -8, -6], [24, -1, -1, -21, 1, 3, 6, 18], [-12, -3, -2, 11, 9, -5, -18, 8], [-29, -9, 7, 6, -13, -10, 2, 12], [-9, -7, 4, 10, -6, 3, -2, -15], [-34, 14, 12, 12, -7, -13, 8, -5], [-4, 3, -2, -19, 14, -11, 4, 5], [-14, -32, 0, -4, -13, 21, -2, 1], [5, -23, -9, -6, 3, 24, 16, 16], [-10, 5, 3, -5, -6, -14, 5, -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_p();".
// 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_p(:prec:=8)
    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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -11, -27, 19, 34, -9, -11, 1, 1]>,<7,R![-676, -856, 244, 536, 11, -102, -12, 6, 1]>,<11,R![-2752, 704, 2192, -112, -599, -75, 50, 14, 1]>],Snew);
    return Vf;
end function;