
// Make newform 4005.2.a.s 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_s();".
// 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_s();".
// 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 := [4, -67, 78, 138, -163, -75, 85, 15, -16, -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], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-95, 1140, 59, -1823, -168, 733, 77, -106, -8, 5], [94, -743, -137, 1803, 56, -919, 0, 160, -1, -9], [94, -1128, -137, 1880, 56, -919, 0, 160, -1, -9], [292, 1270, -1171, -2399, 749, 1182, -154, -214, 10, 13], [-65, 340, 111, -684, -75, 351, 22, -65, -2, 4], [1, 340, 34, -684, -64, 351, 22, -65, -2, 4], [-531, 2830, 855, -5384, -413, 2720, 77, -509, -5, 32]];
    Rf_basisdens := [1, 1, 1, 77, 77, 77, 77, 11, 11, 77];
    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, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [-1, 0, 0, 0, -1, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, 1, 0, 1, 0, -1], [-1, 0, 0, -1, 0, 1, -1, 0, 1, 1], [0, 1, -1, 1, 0, 0, -1, 0, -1, 1], [1, 0, -1, 1, -1, -1, -1, 0, 0, 0], [-4, 1, 0, -1, 1, 0, 1, 0, 0, 0], [1, 0, 0, -1, 1, -1, 0, 1, 0, 1], [1, 1, -1, -1, 0, 0, 0, -1, 0, 0], [-2, -2, -1, -1, -2, 1, 0, 0, 0, 1], [-1, 1, 1, 1, 0, -2, 0, -1, 0, 0], [2, -1, 1, 1, 0, 0, 0, -2, -1, 1], [-4, 1, 1, 1, 2, 0, 0, -2, -1, 1], [-4, 2, 2, 0, 2, 0, 0, 1, 1, -1], [2, 0, 0, -1, 0, 1, 0, -1, 2, 0], [-3, 2, -1, 2, 3, 0, -1, -1, 2, 0], [-3, -1, 1, 1, 1, -2, -1, 0, 1, 0], [-1, -1, 0, -1, 1, 0, 0, 1, -1, 1], [4, -3, 0, -2, -2, -1, 1, 2, -1, 0], [1, 3, 0, 1, -1, 0, 0, 3, -3, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, -1, -1, -1, 1, -2, -2, 0], [-6, 2, -1, 2, 0, 0, 1, 0, -1, -2], [0, -3, 0, 2, -2, -1, 0, -1, 0, -1], [-1, 2, 0, -1, 0, -3, 1, 2, -1, 1], [-1, 1, -1, -1, 1, 2, -1, 1, 2, -1], [-1, 3, 0, -2, -3, 1, -2, 2, 1, -1], [3, -1, -1, -1, -2, 0, -2, 1, 2, 0], [-4, -1, 1, -1, 2, 0, 3, -2, 1, 1], [-3, 0, 2, -2, 1, 0, 2, -2, 0, 1], [-3, -3, -1, 1, 3, 0, 1, -1, 0, -1], [-4, 2, 1, 2, 2, -2, -1, -1, 0, 3], [-3, -1, -1, 0, -1, 1, -1, -3, 1, 0], [-6, 1, 0, -2, 4, 1, -2, 0, 1, 0], [0, -1, 2, 4, 2, -3, 0, -1, -2, -1], [5, -2, -1, -1, -5, 1, -1, 3, -3, 1], [1, 2, 3, -1, -3, -1, 1, 3, -1, -3], [-4, 0, 2, -1, 4, 1, 4, -1, 0, -2], [0, 1, 1, 1, 2, 0, -1, 4, -3, -1], [-2, -2, -2, 2, 2, 4, 0, -3, -1, 1], [2, -4, 2, -4, 0, 0, 1, 2, 2, 0], [-2, 1, -1, 1, -2, 0, 1, 2, 1, -1], [-2, 0, -1, 1, -1, 1, -2, 1, -1, 1], [-3, 3, -1, 2, -3, 3, -1, 1, -1, 0], [-8, -5, 6, -1, 2, 0, 2, -5, 1, 2], [-2, 1, -1, -1, 2, 2, 0, -4, 5, 1], [-1, 2, -2, 1, -1, 3, -2, -1, 0, 3], [-5, 0, 1, 1, 3, -1, -1, -1, -1, -1], [-4, -2, -1, 1, -4, 1, -2, 2, -2, 1], [2, -4, 4, 1, 0, 3, 4, -3, 2, 0], [4, -5, 1, 1, 0, -2, 1, 2, -1, 1], [0, -3, 1, -3, -2, 2, -2, 6, 1, -3], [3, 2, -2, 1, 2, -1, 1, 0, -3, 1], [-8, 6, -2, 3, -2, -1, -4, 1, 0, -2], [-6, 1, 1, -2, -4, 1, 0, 3, 1, -5], [-8, -1, 2, 2, 4, -1, -2, -4, -3, 4], [-6, 1, -1, 3, -3, 4, 2, 1, -2, -3], [-6, -3, -1, -4, -6, -1, -1, 3, -1, -3], [1, -2, 1, 2, 1, 2, -1, -3, -2, 2], [-6, 0, 1, -2, 2, 2, 1, -1, 0, 5], [7, -5, -3, -4, -1, -1, -3, 1, 1, 2], [-1, 1, -3, 1, 2, 4, -4, -3, 2, 4], [5, 2, 0, 1, 0, 1, -3, -2, 1, 1], [-7, 3, -2, 7, 1, -2, -4, -2, -2, 0], [6, 0, 0, 2, 6, -4, -1, 2, -2, 0], [0, 0, -6, 4, -2, 0, 0, -2, -2, 0], [4, 0, 0, -1, -2, 1, -2, -1, -2, 2], [-12, 0, 1, -2, 4, 2, 3, 3, 4, -5], [-8, -4, 6, -5, 0, -1, 5, -5, 2, 0], [-7, 1, -1, 3, 1, -2, 3, -2, 3, 0], [4, 1, 0, 2, -4, 1, -2, 0, 1, -4], [-1, 4, 4, -1, 3, -5, -2, 1, 0, -1], [-6, -8, 2, 2, 0, -2, 0, 0, 0, 2], [-12, -4, -2, -1, -2, -3, -1, 1, 2, 0], [0, 2, -4, 0, -4, 4, -3, 0, 4, -2], [-5, -5, -5, 2, 3, 1, 1, -3, 1, 0], [6, -5, 2, 0, -2, -3, 3, -4, -3, 4], [0, 3, -4, 3, 1, -2, 1, 2, -4, 2], [5, 2, 0, 3, 1, -3, 0, -1, 0, -1], [-6, -3, -7, 1, -5, 0, -4, 3, 0, 3], [10, 3, 2, 2, 4, -3, 0, -1, -2, 3], [0, 2, 4, 1, -6, 1, 2, 5, -4, -2], [-3, -6, 3, 1, -1, -1, 3, 1, -1, -1], [-11, -1, 1, -1, 7, 0, 5, -2, 1, -2], [8, -4, -4, 0, 0, -2, 0, 4, 0, 0], [-4, -9, -3, 0, 4, 3, 3, 1, 1, -1], [1, -5, -3, 0, -5, 1, -3, -1, -3, 0], [3, 2, -3, 1, 3, -7, 1, 1, -3, 3], [-9, 1, 1, 2, 5, -1, -1, -1, 7, 0], [6, -2, 3, -6, -4, 8, 7, 3, 0, -5], [-4, -12, 5, -2, 2, -2, 3, -1, 2, 3], [-11, 5, -3, 6, 3, 1, -1, -1, -3, -4], [-4, -2, 0, -2, 4, 0, 6, 0, 0, -2], [-4, 0, 2, 2, 0, -6, 0, -6, -2, 4], [0, 0, 4, 1, 2, -5, 1, -1, -2, 2], [-6, -6, -4, 1, -2, -1, 0, -1, 2, 2], [-2, 2, 2, 1, 2, -3, 4, 3, -4, 0], [2, 1, 4, -4, 0, 1, 4, 1, 2, -1], [7, -3, -2, -7, -3, 2, 2, 7, 5, -7], [3, 0, -1, 2, 1, 0, 1, -5, -4, 4], [-12, 6, 0, 2, -2, -2, 0, -1, -3, -3], [-6, 8, 4, -2, -4, 2, -2, 4, -2, -2], [8, -2, 4, 2, 2, -4, 9, 4, -4, -4], [2, 0, 8, -4, -6, -2, 6, 4, -2, -6], [0, 0, -3, 4, 0, 4, -1, 1, 0, 3], [-4, -2, 0, -3, 2, 1, 9, 1, 0, 0], [-8, 2, -1, -3, -7, 7, -2, 1, -1, 1], [3, -5, -2, 7, 3, -4, 0, -5, -1, 1], [5, 4, -5, 5, 1, 5, -1, 2, -2, -4], [9, 2, -3, 3, 1, -1, -3, -4, 2, 4], [-12, -3, -4, 2, 2, -7, -1, -4, 1, 4], [3, 5, 1, -3, 7, 2, -1, -5, 6, 3], [-7, 2, 0, 0, 1, -2, 2, 2, 0, -5], [0, 2, -5, 5, -8, 3, -9, -2, -2, 1], [10, 1, -2, -3, 0, 0, -2, -7, 5, 2], [-4, -3, -4, -1, -2, 6, -4, -1, 3, 4], [7, 5, -7, 5, -3, 0, 3, 2, -7, -6], [2, -8, 2, -3, 0, -1, 0, -5, 2, 2], [1, -8, -3, -1, 1, 7, 1, 2, 2, -2], [-1, 3, 3, 3, 1, -6, 1, 2, -5, -4], [-4, 16, -1, 3, 8, -1, 1, 2, 2, -7], [7, -11, 7, -7, -3, 0, 1, 5, 0, 1], [-11, -1, 5, 2, 5, 1, -1, -7, 3, -2], [-3, 7, -3, -1, 3, -2, -1, 2, -1, -2], [-2, 5, 5, 3, 3, 4, 2, 3, -2, -1], [-9, 5, -3, 3, -2, 0, 2, -3, -2, -4], [2, -4, -5, -4, 2, 4, 3, -1, 0, -1], [7, 7, -7, 3, -3, -2, -5, 6, -5, 4], [-3, 2, -1, -2, -1, -2, -7, 3, 2, 2], [6, 2, -4, -1, -1, 7, -7, -2, 3, 8], [1, 0, -3, 2, -9, 6, -5, 1, 0, 2], [-2, -6, -4, -2, -2, 4, -2, 2, -2, 2], [2, 6, -1, 1, -5, 3, -2, 1, -5, 5], [11, 3, -2, 1, -5, 4, 0, -1, -1, -5], [6, -4, -4, 2, -4, -2, -1, 6, -6, 0], [3, 2, -1, 9, 9, -5, -3, -3, 1, 3], [1, 3, 3, -7, -11, 6, -5, 4, -1, 0], [-4, 0, -5, -1, -3, -3, -4, 1, 5, 1], [7, 0, 2, 1, 1, -5, -2, 3, -2, 1], [0, -11, 4, -7, 2, 0, 0, 1, 9, 2], [-2, 4, 3, 2, -4, -2, -1, 5, -4, 1], [-17, 12, -2, 4, 5, -2, 0, 2, -4, -3], [6, 6, 1, -1, -4, 1, 0, -4, -2, -1], [1, -3, -1, -5, -2, -8, -2, 5, -2, 2], [-1, 9, -3, 5, 5, -6, 3, 2, 1, -2], [-24, 1, 5, -6, -2, 7, 1, 1, 3, -1], [-3, 1, 1, -1, 3, 4, -1, -2, 3, 6], [-4, -6, -6, 0, -4, -2, -4, 2, 0, 2], [3, -6, 4, -10, -11, 4, 4, -2, 2, -3], [-6, 0, 4, -4, 0, 2, 1, 0, 8, 0], [4, -3, 6, 0, -2, -1, 2, 6, -7, 2], [4, -6, 3, -4, 4, 2, 5, -3, 8, -1], [-8, -1, -5, -2, 2, 1, -5, 3, 3, 1], [6, 2, 0, 1, 2, -7, 2, -5, -10, 8], [7, 5, -3, 4, 7, 1, -1, -7, 3, 6], [-2, 6, -1, -2, -8, 4, -5, -3, 6, 1], [-2, 5, -6, -1, 1, -2, -5, 0, -4, 4], [15, 6, -9, 3, -5, 1, -7, 7, -7, -1], [-8, 3, -2, 2, 12, -5, 4, -2, 1, -2], [6, 13, 0, 4, 2, -7, -2, 3, -2, -1], [-13, 4, 0, -1, -7, 9, -8, -1, 6, -1], [-6, -3, -3, -1, -6, 2, 2, -4, 1, -7], [1, -12, -2, -9, -6, 11, 7, -2, 1, -1], [16, -13, -1, 2, 10, -3, 5, -7, -1, 7], [-23, 3, -4, -3, 3, 6, -10, 3, 3, -1], [10, -1, -1, -2, -12, 5, 3, 1, 3, -3], [5, -4, 4, -5, 6, -1, 1, -2, 3, 1], [15, 3, -9, 1, -5, -2, -3, -2, 1, 0], [0, 5, 10, 0, 4, -1, 1, -4, 3, -2], [-6, -6, 3, -4, -10, 2, -3, -8, 3, 2], [2, 2, -8, 0, 4, 2, -8, -4, 2, 8], [-13, -2, -2, 4, -1, 0, -8, 0, 4, 1], [14, 10, -7, 0, 0, 0, -1, -3, 4, 1], [-16, 5, -1, 2, 2, 1, 2, -1, -3, -5], [-10, 5, 3, 5, -1, -4, 6, 9, -6, -5], [4, -3, 0, -6, 0, -7, 0, -6, -1, 2], [-10, 9, 11, -2, -2, -5, 10, 1, -3, -7], [-2, -1, -4, 0, 6, -1, 2, 1, -2, 1], [2, -8, -1, 2, 0, -2, -3, -7, 2, 7], [-4, 1, 5, 4, 4, 3, 7, -7, -5, 1], [-5, 5, 2, -10, 1, 3, 2, 0, 7, 5], [-6, -1, 8, -3, 4, -2, 10, 5, 1, -6], [10, -2, -4, 2, 2, -6, -2, -5, -7, 7], [-6, 5, -7, 1, 2, -4, -5, -6, 1, 1], [-29, 4, 1, 5, 1, 1, 5, -4, 2, -2], [6, 3, 3, -3, 4, -10, 1, -6, 3, 3], [2, 9, 0, 3, 3, -8, 1, 4, -6, -4], [13, 0, 1, -7, 3, 1, 5, 8, 6, -4], [9, -9, 4, 1, -3, 0, 8, 1, 3, -5], [-2, -7, -3, -1, -2, 6, -1, 4, 3, -3], [10, 2, 9, -5, 4, -1, 8, 4, 4, -3], [-14, 6, -1, -8, -4, 2, -1, 5, 4, -1], [16, 4, -8, 0, -2, 0, 0, 6, -8, 2], [2, -3, 6, 4, 0, -5, 5, 0, 1, 2], [-12, 0, 0, 2, -10, 8, -2, 4, 2, -6], [15, -4, 3, -6, 7, -4, 5, -3, -2, 4], [13, -7, 1, 0, 1, -7, 5, -1, 1, -4], [3, 4, -2, -3, 3, 3, -2, 3, 2, -1], [19, 0, 6, -4, -1, -6, 12, 0, -2, 1], [-4, 2, 2, -1, -4, 1, 0, -1, -6, -2], [-11, -13, 3, 7, 9, -2, -3, -10, -3, 4], [6, 13, -6, 1, 1, 0, -5, 6, 6, -8], [11, 1, -12, 1, -5, 2, -2, -1, -9, 7], [-6, -1, 1, -8, -8, 7, 3, 5, 7, -9], [0, -21, 2, 1, -2, 2, 0, 3, -3, -2], [-10, -1, -1, 3, 15, -4, 2, -7, 4, -5], [-14, -8, 8, -3, -1, -1, 5, -4, 5, -6], [-8, 8, 0, -12, -8, 10, 9, 6, 2, -10], [5, -7, 7, -5, -5, 6, 7, 0, -1, 0], [-3, -1, 2, -1, -17, 4, 2, 9, -3, -13], [-7, -6, 7, 4, -1, 4, 3, -7, 2, -2], [3, -2, -4, 1, 3, 1, 2, 1, 8, -7], [-4, 1, 1, 3, -2, -2, -3, -2, 5, 3], [18, -9, 0, 5, 0, 2, 0, -3, 3, 0], [4, 10, -6, 0, -10, 8, -2, 11, -9, -7], [-25, 4, -4, 1, 2, 3, -3, -8, 5, 1], [-15, 8, 5, -1, 5, 5, -7, 0, 8, 4], [13, -4, -2, -5, 5, -5, -2, -5, 4, 3], [-26, -10, 3, -4, 4, 4, -5, 1, 6, -1], [-6, 2, 6, 4, 4, 2, -4, 7, 1, -1], [3, 12, 10, 3, -3, -9, 8, -3, -4, -5], [-8, 10, -10, 5, 4, -5, -2, 1, -2, 0], [-2, -5, 3, -2, -2, -1, -4, 1, 5, -3], [-14, 3, 7, -3, 4, 6, 2, 0, 5, 1], [4, -2, 2, -14, 0, 0, -2, 4, 10, 6], [-7, 9, 1, -3, -10, 4, -6, 9, 2, 0], [-26, 0, -7, 6, 6, -6, -5, -4, 1, 4], [26, -8, -8, 2, -10, -2, -3, 0, -4, -4], [-24, -13, -1, 0, 0, -1, 1, -3, 5, -1], [14, 0, -3, -3, -7, 5, -2, 3, 5, -1], [-9, 6, -2, 8, 3, -6, 4, -4, 4, -5], [10, -2, -6, 6, -4, 0, 5, 2, 0, -2], [7, 13, 0, 2, 7, -5, -4, 6, 3, -11], [22, -8, 8, -12, -10, 4, 4, 8, 4, -6], [5, 8, 1, -7, -1, 7, 1, 11, 7, -3], [-6, 2, 1, -4, -6, 2, 11, 0, 1, -6], [-8, -14, 6, -1, -2, 7, -8, -3, 4, 2], [21, 2, 7, -7, -1, -5, -5, -1, 5, 5], [8, -12, -9, 8, -8, -6, -7, 1, -6, 3], [23, 1, -1, -5, -12, 2, -4, 5, -10, 2], [4, -10, -5, -7, -3, 9, 0, -9, 5, 5], [23, -2, -16, 5, -2, -5, -5, 0, -5, 3], [-22, 6, -3, -4, 6, 4, -1, 7, 8, -5], [-15, 13, -3, -3, -4, 2, -6, -5, 2, 2], [-10, 16, 0, 0, 0, 0, -6, 0, 6, 4], [-1, 1, 3, 10, 1, -3, 1, -7, 1, -8], [9, 10, 5, 5, -1, -7, 7, -6, -10, 2], [11, 10, 4, 0, -1, -6, -4, -4, 0, -3], [-11, -10, 2, -2, -3, 0, 4, -6, 4, 1], [-3, -2, 5, 5, 11, -3, -1, -3, -5, 7], [-5, 2, 8, 3, 5, 9, 10, 1, -2, -3], [18, -3, 5, -3, 6, 0, 4, -2, 5, -3], [-3, 8, -2, 4, 7, 12, -8, -4, -2, 3], [4, -11, -1, -6, -8, -1, 1, -1, 7, 7], [-10, 3, 10, 2, 6, -5, 2, 2, 7, -4], [-17, 7, -4, 3, 9, -10, 2, 1, 1, -3], [-18, -8, 1, -1, -2, -1, 5, -8, -2, 7], [20, -7, -8, -4, 2, 5, 4, 5, -4, -1], [-6, -9, 5, -4, 2, 5, 1, -1, -3, 9], [18, -7, -8, -4, -8, 1, -7, 10, -1, 4], [0, 7, 7, -6, 4, -5, 5, 1, -3, 7], [-12, -3, -6, 1, -8, 4, -6, 3, -1, -4], [-12, -5, -8, -8, -4, 7, -3, 2, 3, -2], [1, 12, -8, 11, 1, 5, 0, -1, -6, -5], [-2, 5, 4, -11, -1, 6, -3, 0, 0, 6], [-2, 5, -2, -7, 0, 4, 6, 9, 3, -4], [-7, -10, 3, -1, -13, 7, -1, -7, -3, 3], [7, 7, 3, 7, 1, -4, 7, -1, 0, -1], [6, 8, 6, 6, 0, -6, 5, 2, -6, 2], [12, -4, 6, -10, -6, 2, 1, 2, 2, 6], [-6, 2, 5, -1, 7, -5, 2, -3, -5, 3], [1, 9, 1, 1, 8, 4, 4, 5, -6, -6], [24, -14, 4, -4, 0, 4, -6, -6, 0, 8], [-1, -10, 0, -8, -11, 4, 4, 2, -6, -1], [19, 5, -1, -3, -1, -6, -9, 9, 4, -5], [-7, -7, 3, -3, -1, 8, 7, 3, 4, -9], [-2, -22, 1, 3, -2, -5, 12, 0, -2, -7], [-25, -2, -1, 11, 1, -1, 5, 0, -8, -4], [-9, 3, 3, -4, 7, -3, 3, -3, 1, 0], [-12, 11, 3, -3, -2, 0, -4, 6, 7, -3], [5, 7, 3, -1, -1, -6, 1, 3, -8, -1], [3, 6, -6, 6, 7, -8, -8, 4, -4, 3], [-14, -9, 5, -3, 4, -2, 3, 6, 3, 1], [20, -5, -6, -4, 2, 5, 8, 0, 3, 4], [-13, -2, 4, 9, 10, 3, -1, 0, -1, 3], [-14, 1, 14, 2, -2, -11, 5, 4, -1, -6], [-17, 6, 8, -10, 3, 12, 0, 2, 2, 3], [-14, 15, 5, -1, 14, 0, -3, -4, 3, 11], [-8, -10, 7, 1, 10, -1, 5, -8, 8, 11], [-14, -6, -2, -14, -2, 12, 8, 4, 4, 0], [-8, -14, 12, -1, -2, 1, 3, 5, -2, -4], [7, 1, -8, 9, -1, -8, 2, -3, -5, 1], [24, -3, -11, -5, 6, 10, -1, 4, 1, 3], [3, 4, -4, 2, 3, -4, 2, 4, -2, 3], [15, 10, 5, 6, 5, -6, -5, 3, -4, 4], [7, -8, 4, -6, -1, 6, 6, -4, 6, 5], [3, -7, 1, -13, -8, 4, 0, -9, 12, 2], [-4, -2, -6, 4, 14, 0, 5, -4, 2, 2], [3, -2, 5, -5, -7, 5, -1, 9, 1, -11], [9, -13, -7, -7, -1, -8, 1, -2, 5, 0], [-11, 6, 4, 7, 8, 1, -1, -14, -3, 5], [12, -4, 3, -8, -10, 2, -5, 13, 0, -7], [1, -5, -11, 9, -6, -4, -4, 1, -8, 8], [-25, 10, -2, -1, -2, 3, -1, 8, 3, -7], [2, 4, 2, -1, 5, 9, -3, 2, 9, -2], [-20, -1, 9, 1, 0, 2, 3, 4, 1, -5], [18, 9, 6, -4, -8, -5, 2, -1, -2, -9], [-3, 12, 11, -3, 3, -7, 3, 1, 3, 1], [8, 2, 2, 1, -4, 1, -8, -5, 0, 8], [-12, -8, -12, 1, -8, 3, 0, 5, -2, 6], [-1, 4, 1, -4, 1, -8, 15, 3, -4, -4], [12, 7, 5, 7, 10, 0, 6, -2, 1, -1], [30, -2, -3, -7, 5, 3, 2, 1, -3, 15], [4, 7, 7, 7, -5, -4, 8, -3, -4, -9], [-32, 1, 3, 3, 16, -2, -1, -8, 7, 9], [10, 2, 10, -6, 6, 0, 11, 10, 6, -10], [-10, 2, -1, 3, -6, -7, 2, -4, -4, -11], [-6, 0, -11, -3, -5, 9, 0, -1, 3, -5], [8, 6, 1, -11, 6, 7, 7, 0, 6, 5], [-1, 1, 5, -1, 3, -10, 1, -6, -5, 6], [10, 0, -6, -4, 0, 0, 0, 0, 6, -2], [0, -1, 3, -11, 12, 4, 12, -4, 5, 5], [-1, 3, -5, 7, -11, 0, -5, -1, -2, 5], [-6, 5, 3, -13, 0, 2, 9, -2, 3, -3], [16, -12, 4, 0, -10, 0, 4, -6, 6, 8], [-2, -5, 7, -7, -8, 2, 4, -2, 5, -1], [9, 4, -9, 13, -5, -1, 3, 6, -2, -2], [-14, -7, 6, -2, -6, -1, 5, 8, -7, -8], [22, -13, 11, -11, -18, 6, 5, 0, 1, -3], [-20, 19, 7, -2, 10, -3, 7, 7, 1, -15], [13, -2, 5, 1, -7, -1, -17, 3, 9, -1], [-3, 2, 10, 1, -3, 3, -6, -1, 4, -7], [-4, -12, -6, -10, 4, 4, 8, 6, 6, -10], [0, -2, 13, -9, -1, 3, 4, 3, -1, -5], [16, 2, -4, 2, 4, 0, -8, 6, -6, 0], [-5, 7, 3, 3, 7, 2, 9, -4, -1, -4], [-16, 6, -8, 2, -2, 10, -16, 4, 14, -6], [7, -1, -7, 7, -9, 0, -13, -2, -5, 2], [-5, -2, 4, -9, -4, 7, -1, 0, 7, 3], [1, -18, -6, 0, -7, -6, -2, 2, -2, 3], [4, 4, 2, 8, -4, -8, -6, 12, -2, -4], [5, -5, 7, 10, 9, -11, 7, 1, 9, -8], [15, -8, -2, 8, -9, -12, -6, -2, -4, 9], [-13, -2, -2, -5, -9, 9, -4, 3, 14, -7], [8, -3, -12, 0, 4, -7, -6, -7, 6, 5], [-20, 0, -3, 13, 2, 3, 10, 0, -6, -7], [-18, -6, 7, -2, -12, -4, -1, 1, 2, 1], [-20, 12, 9, 1, 13, -1, -2, 3, 1, -11], [14, -8, -2, -2, -6, 2, -3, 14, 8, -10], [31, -2, -4, 5, -1, -7, 6, -1, -10, 3], [-23, 9, 7, 9, 9, 4, 7, 0, -5, -4], [14, -11, -1, -2, -6, 15, 7, 5, 1, -9], [5, -16, 0, -3, 6, 3, 11, -4, -3, 5], [14, 14, 6, -6, 0, -8, 4, -2, 8, 0], [-22, -3, 7, -13, 6, 6, 11, -10, 9, 5], [-15, -17, -7, -9, -7, 6, -3, 0, 7, -6], [-8, 2, 4, 8, -2, 12, 0, -2, 2, -10], [7, -21, -1, -4, -3, 5, 3, 9, -7, 0], [10, 2, -6, 5, 0, -5, 1, 11, -10, 6], [15, 14, -2, -5, -2, 9, 5, 4, 9, -3], [-28, 4, -1, -3, 0, 5, 0, -10, -2, 13], [-25, -11, -1, 4, 1, -1, -7, -11, -1, 4], [10, -6, 8, 2, -8, 6, -2, 0, 0, 0], [16, -9, -4, 13, 0, -6, -2, -13, -5, 4], [3, -6, 0, 6, -5, 4, -10, -4, -10, 3], [-2, 6, 0, 6, 14, 4, 0, -4, -4, 16], [-27, 16, 4, -11, 9, 3, 4, 1, -2, -1], [2, 7, 10, 2, 0, -7, -7, 6, -7, -8], [-4, -23, -9, -1, -8, 10, 5, -6, -1, -5], [8, 5, 9, -12, -6, -3, 1, 5, -1, -3], [-18, -11, 5, -11, 12, 4, 7, -4, 9, -7], [-22, 1, -6, 6, 8, -7, 10, -1, -4, -3], [9, -5, 1, 9, -7, -2, -11, -4, -3, -4], [13, 20, 0, -6, 3, -10, 0, 2, -4, 11], [7, 19, -5, -1, -6, 8, -10, 7, -4, -4], [-20, 0, 6, -4, 14, 0, 0, -11, 1, 11], [-20, 2, 3, 7, -14, 5, -7, -4, 2, -5], [32, -4, -2, 10, 0, -10, 8, -14, -6, 12], [5, -18, 9, -17, 1, 3, 5, 12, -2, 0], [-8, -4, 2, 8, 12, -18, -4, -4, -6, 6], [-11, 8, 8, -3, 9, -1, 8, 7, -2, 3], [-10, -6, 18, -5, -11, -3, 3, 2, -5, 6], [-3, 5, 8, -2, 9, 11, 8, 8, -3, -3], [10, 1, -2, 8, 14, -17, -1, -4, -1, 6], [-4, 10, -4, 14, -2, -10, 13, 4, -2, -8], [29, -15, -1, -7, -4, 0, -2, -1, 4, -4], [0, -6, -6, 12, -16, 8, -12, 4, -8, -2], [40, 11, 5, 3, -2, -16, 3, 2, -5, 3], [41, -18, -7, 7, 1, 7, -7, -10, 0, 6], [-22, 6, -3, 7, 6, -5, -14, -14, 2, 9], [-10, -17, 4, 4, 6, -3, 0, 1, 2, 7], [14, 11, -4, 3, -3, -14, 3, 2, 2, -12], [60, 9, -2, 10, -4, -11, -8, 11, -12, 3], [-19, 15, -3, 15, -1, -10, -7, -5, 0, -7], [-2, 4, 0, 4, -2, 10, 6, -5, 3, 3], [18, 14, 0, -1, 6, 3, -10, 15, 0, -2], [4, -11, 7, 4, 4, -7, -1, -9, 3, 1], [36, 10, -4, 5, -11, 1, -7, 0, -3, -6], [-4, 3, -4, 4, -12, 1, -4, 4, -13, -6], [3, -1, -3, 4, -1, 5, -3, -7, -5, 2], [-5, 20, -11, -3, -5, 5, -11, 3, 5, -5], [-18, 18, 13, 8, 4, -6, -5, -6, 1, -2], [-21, -13, -3, -5, 3, 6, -13, -11, 6, 1], [-16, 2, 2, -6, -14, 4, -10, -2, 2, 8], [15, 11, -3, -1, -15, 8, 1, 14, -7, -12], [-23, 17, -1, -5, 13, 2, -7, -10, -1, 12], [-6, 4, -13, 1, 4, 5, -14, 4, 0, -3], [20, 6, -8, -2, -8, 4, -1, -12, -6, 6], [13, -2, -4, -5, 10, 3, -1, -12, 7, 1], [-6, -21, -3, -5, 0, 8, -6, -4, 9, -3], [-17, -5, 3, 13, -3, 0, -5, -8, -7, 0], [16, -2, 1, 4, -8, 6, 3, -15, 2, -1], [10, 14, 6, -2, 0, 4, -12, 17, -1, -15], [14, -14, 5, 12, 2, 0, -3, -2, -7, 0], [-4, -11, -3, -4, 10, 7, 7, 9, -1, -1], [-7, -22, -4, -1, -2, 3, -5, -6, 11, -7], [-6, 0, -6, 16, -6, -2, 12, 2, 2, -8], [-10, 1, -2, -9, 5, 8, 9, -10, 14, 6], [-32, -8, 2, 3, 16, -1, -2, 1, -2, 8], [3, 22, 18, 3, 10, -9, -5, 4, -5, -5], [-12, -9, 11, 0, -4, 3, 1, -1, -7, 5], [-23, -4, 3, -7, 1, 1, 11, 6, 6, -12], [53, 0, -10, -1, -4, -7, -1, -6, -5, 7], [16, -3, -14, 0, -14, 7, -2, 6, 1, -2], [-14, 5, 5, -2, -6, 7, 11, 7, 3, -15], [8, 0, 2, 5, -11, 9, -11, 10, -1, -6], [-14, -10, 8, -10, 12, 0, 16, -3, 5, 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_s();".
// 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_s(:prec:=10)
    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_s();".
// 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_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 67, 78, -138, -163, 75, 85, -15, -16, 1, 1]>,<7,R![-588, -2359, 9864, -1345, -4589, 459, 732, -40, -47, 1, 1]>,<11,R![36608, 40256, -16384, -22128, 3160, 4080, -164, -328, -12, 10, 1]>],Snew);
    return Vf;
end function;