
// Make newform 8016.2.a.ba in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8016_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_8016_2_a_ba();".
// 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_8016_2_a_ba();".
// 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 := [-10, 51, 152, -83, -166, 67, 45, -16, -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, 1, 0, 0, 0, 0, 0, 0, 0], [68945, -635601, -563660, 716568, 247955, -175482, -17171, 10903, -253], [-301330, -988001, -38080, 878658, 67645, -178372, -2266, 9923, -428], [-130065, 200063, 208150, 207921, -199070, -99334, 48718, 7666, -3031], [255125, 1605297, 520415, -1646101, -79645, 387944, -35953, -23831, 3811], [445920, 296533, -1469865, -184819, 751965, 65121, -118627, -5129, 5684], [59190, -769633, 148730, 1137454, -367380, -311796, 101842, 20594, -6589], [286160, -396149, -2000020, -446023, 1500975, 290982, -293609, -24668, 16128]];
    Rf_basisdens := [1, 1, 102665, 102665, 102665, 102665, 102665, 102665, 102665];
    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_8016_a();"
function MakeCharacter_8016_a()
    N := 8016;
    order := 1;
    char_gens := [3007, 2005, 5345, 673];
    v := [1, 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_8016_a_Hecke(Kf)
    return MakeCharacter_8016_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, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, -1, 0, -1, 0], [-2, 1, 0, 0, 1, 1, -1, 0, 1], [0, -1, -1, 1, 1, 0, 0, -1, 1], [-1, 0, -1, 0, 0, 0, -1, -1, 0], [1, -1, -3, 1, 1, 1, 1, 0, 1], [0, 1, 0, 0, 0, 0, -1, 1, -1], [-5, 0, 1, -1, 3, 1, -2, 1, 2], [3, 0, 0, 1, 1, 1, 0, -1, 0], [1, 0, -1, 0, 0, -1, 1, -3, 1], [-3, 0, 2, -1, 1, 0, -2, 1, 1], [4, 0, 0, 2, 0, 0, 0, -1, 0], [1, 1, 0, -1, 0, 0, -1, -1, 0], [0, 0, -3, 3, -3, 1, 1, 0, -1], [-2, 2, 1, -2, 0, 0, 0, 0, 2], [-1, 1, -1, 1, -2, 0, 1, 1, -1], [-1, 2, 4, -3, 1, -1, -2, 0, 2], [0, 1, -2, -1, 1, -1, 0, 0, 2], [2, 1, -1, 3, -3, -2, 0, -1, 0], [4, 0, -2, 0, -2, 0, 0, -1, 2], [2, 0, 3, 1, -1, -1, -1, 1, -3], [-6, 4, 4, -4, 4, 1, -4, 1, 1], [4, -1, -2, -1, -1, -2, 4, -3, -1], [-6, 1, 1, -2, 4, 2, -1, 2, 0], [10, -3, -2, 3, -1, -1, 2, 1, -2], [-1, 2, 2, -2, 1, -2, -1, -2, -2], [-3, 0, -2, 0, 4, 1, -1, -1, 1], [5, 1, 4, 1, -3, 0, 3, 2, -7], [1, -1, -2, -3, 3, 3, -3, 0, 0], [-4, 2, 2, -1, 3, -1, -3, -1, 0], [-1, 0, -4, 0, 2, 0, 1, 0, 4], [3, -2, 0, -1, 1, -3, 0, -2, 3], [-8, 1, 2, -4, 4, 3, -3, 1, 0], [7, 1, -3, 0, -4, -2, 2, 0, 1], [-5, 3, 4, -3, -3, 2, -1, 5, -4], [6, -2, 1, 2, -2, -4, 0, -1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-9, 3, 2, -3, 1, 2, -1, -1, 2], [0, -4, -3, 2, 0, 3, 2, 3, -2], [-7, 1, 4, -4, 7, 0, -4, 3, 2], [-1, -4, -2, 3, 2, 4, -2, -2, 3], [-5, 2, -2, 1, -1, 2, -2, 0, 2], [-4, -3, -3, 0, 4, 1, -3, -3, 4], [8, -3, -3, 1, -3, 2, 2, 3, -2], [5, 0, 4, -2, 0, -1, -1, 1, -3], [11, -1, 4, 0, -2, 0, 0, 3, -5], [-7, 2, -4, -2, 2, 0, -1, -2, 3], [-8, 4, 7, -4, 5, 4, -6, 2, 3], [3, -4, -1, -2, 6, 1, 1, -1, -3], [1, -1, -5, 3, 2, 4, 3, -1, 4], [-3, 1, -9, 0, 2, 0, -2, -3, 2], [0, -4, 1, 0, 0, 3, 2, 3, -1], [2, 2, -1, -1, 3, -2, -3, -3, 5], [-6, 6, 5, -6, 0, 2, -2, 2, -1], [8, 3, -2, 4, -6, -3, 1, -1, -1], [3, -3, -1, 1, -1, 3, -1, 1, 4], [8, -1, 1, 3, -9, -3, 2, -4, -2], [5, 3, 2, 1, -3, 3, 3, 4, 0], [5, -3, -6, 3, 5, 4, -1, -5, 6], [3, -4, 2, 3, -1, -2, -4, 0, 3], [-1, 2, 1, -5, 3, 0, -2, -1, 2], [-7, 4, 1, 2, 6, 7, -3, 1, 3], [0, -2, -9, 5, 5, 4, -3, -2, 4], [2, -3, -2, -1, 7, -1, 2, -4, 3], [-3, 7, -2, -3, -1, 2, -1, -2, -2], [-3, 4, -3, -3, 5, 2, -2, 0, 3], [-10, 3, 2, 1, 3, 0, -4, -1, 0], [12, -4, -3, 9, -3, 1, 5, 1, -2], [1, 4, 6, -4, 2, 2, -1, 2, -2], [4, -1, 0, 1, -1, -3, 0, -2, 0], [12, -5, 6, -2, 4, 0, 3, 3, -1], [-2, -1, 0, -3, -3, 0, 2, 1, 2], [10, -2, 1, 9, 1, 1, -1, -2, -3], [-6, -1, 7, -3, 12, 1, -4, 0, 0], [20, -2, -2, 4, -10, -4, 8, -5, -7], [-1, 3, 7, -4, 1, -1, -4, 0, 2], [8, -1, 6, 2, 4, 2, -3, 0, -2], [4, -3, 3, 1, -4, 5, -2, 8, -4], [-4, 2, -8, 3, 0, 2, -1, -2, 4], [4, -4, 1, -5, -1, 2, -1, 2, 1], [-9, -2, 2, -2, 4, 2, -7, 6, -5], [2, 1, -7, 2, -1, 3, -1, -4, 0], [15, -1, 3, -1, -3, -1, 3, -1, -1], [-9, -1, -3, -6, 10, 5, -2, 2, 9], [12, 2, -1, 1, -5, -3, 5, 5, -5], [6, -7, -2, 6, 2, 0, -1, -3, 0], [8, -3, -7, 1, 1, 0, 6, -1, 1], [1, 2, 8, -5, 5, -2, -6, 3, -1], [5, 0, 4, -6, 0, -7, 1, -3, -4], [14, -2, 2, 1, 3, -1, -1, 1, -5], [3, 0, -6, 2, 4, 7, 1, -2, 1], [-14, 1, 7, -8, 1, 0, 1, 5, 2], [-5, 3, 6, -8, 8, -1, -8, -3, 6], [24, 0, -5, 5, -10, -7, 7, -9, -2], [2, -4, -3, -1, -1, 2, -5, 4, -4], [9, 5, 0, -1, -1, 0, 1, -4, 2], [5, 2, 6, 1, -7, -4, -2, -6, 2], [-8, 2, 10, -5, 7, 0, -9, 10, 3], [10, 9, -4, 0, -6, 1, 1, -1, -6], [-4, 1, 3, -5, 9, 2, -4, 5, -1], [0, -3, 2, -5, 7, -7, -4, -6, 2], [9, -2, -13, 9, 1, -1, 0, -6, 7], [-6, -1, -2, -4, 6, -3, 1, -3, 10], [0, -1, 1, -3, 4, 4, 0, 9, 3], [1, -7, 5, 1, -3, 2, 1, 5, -7], [-11, 5, -5, -4, -2, 0, -2, -6, 5], [9, 5, -3, 7, -5, 0, -1, -5, -5], [-10, 6, 0, -9, 7, 0, -1, -4, 6], [7, 0, -2, 5, -1, 2, 0, 1, -2], [13, -10, -13, 9, 8, 2, 0, -10, 5], [12, -6, -17, 9, -3, 0, 7, -4, -1], [10, 2, -2, 4, -14, -2, 4, 3, -10], [-11, -4, 5, -8, 10, 1, -1, -1, 7], [15, 6, -4, -1, -9, -2, 4, 3, -7], [13, 6, 8, -4, -2, -1, 3, 6, -5], [5, 2, 8, -10, -4, -5, -1, 3, -6], [-4, -2, 8, -3, -1, -3, 1, -1, 2], [-3, -4, -8, 5, 1, 9, 0, 1, 1], [5, 3, -5, -3, -3, -7, 5, -4, 0], [1, 0, 5, 5, -5, 3, 4, 1, -1], [6, -2, 2, 1, -5, 1, -1, -1, -2], [9, 2, 3, 5, -9, -7, 6, -3, -8], [-6, 0, 12, -3, 3, -1, -5, 4, -2], [-5, 0, -3, -3, 7, 1, -4, 3, 4], [17, -10, -5, 2, -4, -5, 13, -3, -3], [2, -3, -5, -6, 0, 3, -3, -4, 6], [0, 1, 7, -10, 8, -5, -7, 1, 0], [-16, 7, 0, -2, 9, 11, -1, 6, 3], [-11, -2, -12, 1, 3, 9, -4, 2, 6], [-2, -4, -11, 11, 5, 3, -3, -1, 1], [-18, -3, -5, 5, 7, 10, -10, 6, 2], [9, 3, -4, -4, -2, -3, 4, -2, 10], [16, -5, 8, 1, -7, -2, -4, 1, -10], [-11, 1, 15, 2, -2, 6, -4, 13, -4], [2, 2, 0, -1, -3, -6, 3, 5, -3], [19, -12, -14, 10, 4, 1, 11, -2, 5], [-4, -7, 9, 2, -4, -1, 3, 5, 1], [-14, 0, 8, -7, 11, 2, -11, 4, -3], [-32, 8, 10, -6, 10, 9, -6, 8, 6], [7, 5, 6, 1, -3, -4, -3, -4, -10], [14, 4, -3, 0, -2, 1, 8, -8, -5], [-1, -8, -4, 3, 7, -2, 4, -10, 4], [17, 1, -6, 1, -3, -3, 7, -12, 3], [12, -7, 3, 1, 4, 2, 6, 5, -4], [-20, 4, 8, -4, 9, 5, -12, 5, 6], [-21, 6, -5, -2, 4, 9, -7, 5, 0], [-7, -6, -10, 1, 1, 2, 0, -4, -1], [-10, 8, 9, -6, 6, -1, -8, 5, -2], [-1, 7, 2, 0, 2, -3, -2, 0, -4], [-2, 6, -1, -1, 1, -4, -7, -11, 5], [-1, -6, -1, 3, -1, 5, 4, 1, 5], [-8, -7, 8, -6, 4, -8, -3, -1, 7], [3, 5, 3, 4, -12, -6, 2, -5, -8], [-23, -2, -6, -6, 6, 0, -1, -2, 1], [-7, 5, 5, -8, 6, -3, -8, -2, 9], [11, 3, -4, 8, -8, 8, 4, 3, -2], [15, -6, -1, -3, -3, -9, 0, -9, -1], [-3, -10, 4, -5, 5, -6, -2, -3, 11], [-18, 1, -2, -2, -3, 5, -3, 4, 1], [-14, 7, 19, -3, -3, -2, -2, 14, -6], [4, 0, -9, 4, -2, -9, 4, -7, 0], [-16, -10, -12, 2, 2, 5, 0, 5, 1], [5, -7, -1, -3, 1, -3, 3, -3, -4], [4, -10, -1, 1, -1, 0, 1, 2, -1], [-20, 13, 22, -11, -3, -3, -12, 5, -5], [12, 4, -1, 9, -4, 1, -3, -3, -10], [6, 7, -5, 6, 1, 7, 5, -2, 0], [-1, -7, 1, -1, 9, -1, 5, -2, 3], [-2, -3, 3, 8, 4, 4, -3, 9, -2], [7, -1, -19, 7, 5, -1, -1, -8, 9], [-17, 7, -4, 5, 7, 3, -1, -2, 10], [9, -2, -1, 3, -22, -7, 6, -3, -6], [-8, 13, -3, 2, 10, 7, -3, -1, 8], [-9, 5, -1, -5, -7, -3, 5, 2, -1], [-14, -5, -1, -3, -3, -2, 6, 5, 8], [-4, 2, 4, -4, 4, 11, 0, 14, 1], [4, -14, -11, 12, 0, 5, 0, 7, -1], [-8, 5, 1, -5, 5, -7, -4, -8, 12], [8, 1, -14, 9, 1, 6, 6, -3, 9], [8, -12, -13, 9, 3, 2, 11, 2, 3], [15, 4, 8, 11, -9, -1, 2, 3, -11], [8, -9, -14, 11, -5, 5, 6, -9, -4], [2, 0, -18, 4, 0, -4, 2, -6, -2], [-22, -1, 10, -11, 19, 11, -14, 8, 9], [-15, -2, 12, -8, 1, 0, -11, 10, -6], [5, -7, -16, 5, 7, 0, -1, -7, 5], [-20, 4, -11, -1, 0, 6, 3, -6, 13], [-13, 0, -9, 1, 5, 15, -2, 9, -2], [-6, 3, 2, -2, 6, 12, -1, 1, 3], [16, -16, -7, 6, -8, -6, 22, 0, -3], [16, -8, -6, 6, -2, -4, 8, -10, -7], [-7, 2, 16, -7, 11, 2, -12, 6, 0], [11, -3, -3, 8, -5, -3, 6, -8, 3], [-16, 6, 3, -4, 4, 2, 4, 6, 1], [-2, -16, -10, 7, -2, 5, 1, -7, 9], [-18, -7, 9, -15, 18, 7, -4, 6, 2], [-14, 9, 5, -1, 17, 9, -6, 2, 6], [-21, 16, 17, -12, 5, 2, -11, 10, -2], [18, -17, -17, 19, 3, -4, 0, -12, 7], [25, 3, 0, 3, -9, -3, 5, -6, -11], [28, -3, -5, 2, -8, -2, 11, -1, -2], [-21, -1, 4, -1, 9, 6, -5, 11, 3], [-19, 14, 12, -3, -3, 6, -18, 6, 0], [-15, 7, 14, -5, 3, -4, -7, -1, 7], [27, -10, -9, 2, -8, -2, 11, 3, -7], [3, -4, 0, 7, 10, -3, 0, -11, 5], [18, -14, -15, 11, -1, 0, 3, -9, 2], [4, -5, -2, 2, -8, -2, 13, -1, -6], [1, -7, -4, -3, 0, -4, 7, -3, -2], [-21, 2, 3, -12, 12, 4, -19, 0, 2], [-1, 3, -1, 3, -7, 7, 7, 7, -11], [-17, -4, -5, 5, -3, 9, 2, 1, 10], [-8, 15, 10, -12, -5, -1, 3, 12, -9], [-11, -4, 9, -4, 3, -2, -9, 2, -1], [3, 4, 3, 0, 4, -6, -3, -5, 2], [4, -11, -13, 8, 12, -1, -1, -12, 10], [0, 2, -4, 10, 4, 1, -4, -13, 3], [-7, -9, 4, -2, 6, -2, -10, -6, 5], [6, 4, -2, -3, -17, -3, 7, 3, -7], [24, -3, 4, -1, -9, -2, -4, 7, -3], [-4, -6, -23, 5, -1, 2, 1, -5, 7], [-12, -4, 0, 2, 2, 0, -8, 4, 5], [1, -14, -6, 5, 3, 10, -6, 8, 0], [5, -2, 9, -5, -8, -8, -6, 4, -7], [2, -14, 5, -4, 4, -2, -2, 2, -7], [10, -2, -1, -8, -6, -5, 10, 9, 1], [2, -1, 3, -8, 0, -7, -3, 3, 5], [-30, 6, -3, 0, 8, 5, -6, 3, 7], [-4, -11, -18, 6, 6, -1, 5, -10, 11], [16, -6, 7, -1, -1, -10, -9, -3, 2], [3, 6, 0, 8, -2, 7, -5, 2, 6], [-1, -5, 1, 1, -5, -1, 5, 5, -7], [-18, -5, 0, 2, 18, 6, -5, 1, 8], [14, -9, 13, 4, 0, 3, 1, 14, -8], [20, -9, -6, 9, -13, -11, 8, -6, -9], [3, -11, -12, 7, -17, 2, 9, -1, -1], [9, 3, -9, -4, -12, -1, 4, -6, -2], [32, -8, -12, 10, -11, -7, 8, -13, 0], [-25, 4, 19, -22, 18, -6, -19, 2, 7], [-6, -7, -7, 1, 3, 10, 0, 1, -2], [-22, 9, -5, -11, 3, -4, -2, -8, 8], [6, -3, 2, 0, -6, -11, -5, -4, 6], [-34, 1, 7, -6, 8, 3, -3, 4, 7], [27, -12, 0, 3, -10, -3, 14, 3, -8], [22, 6, 5, 3, -15, -3, 3, -3, -11], [-26, 1, 13, -10, -6, -10, -3, 5, -7], [-4, 6, -15, 13, -9, -4, 5, -6, -7], [4, -6, 13, -3, 5, -2, -3, 0, 3], [-4, -5, 7, 0, 11, 8, 1, 3, 7], [35, -10, 0, 7, -5, -18, 6, -10, -9], [2, -1, 3, -8, -12, -1, 7, 6, -9], [13, -19, -5, 7, -7, -9, 9, 0, -12], [0, -10, -21, 9, 1, -4, 9, -9, -5], [17, -14, -6, 8, 2, 7, 3, 7, -5], [9, 11, -1, 3, -19, -8, -3, -1, -14], [-18, 8, 3, -5, 23, 8, -19, 4, 10], [6, -11, 9, -3, -1, -7, -2, -1, 4], [-2, -10, 1, 0, 0, 4, 0, 8, -2], [-8, 5, 6, -13, 9, -1, -4, 7, 10], [0, -8, 12, -2, 4, 3, -4, 11, -10], [23, -10, 9, 5, 13, -2, 0, -2, -8], [0, -12, -14, 9, -3, 8, 3, -2, 3], [22, 7, 7, -4, -16, -13, 17, -4, -15], [9, 7, 4, -2, -2, -5, -12, 6, -8], [-1, 4, 0, -3, 5, -10, 2, -7, -2], [9, -1, -1, 1, -9, -1, 13, 1, -11], [-13, 11, -7, 2, -6, 1, -6, -3, 10], [-30, 0, 10, -9, 7, 11, 3, 7, -8], [-7, 9, 1, -7, -1, 3, 5, -6, -1], [13, -6, -13, 15, -23, -1, 10, -1, -13], [-16, 13, 2, -10, 6, -4, -7, -3, 10], [22, 11, -3, -2, -4, -6, 5, -7, 6], [-6, 0, -20, 12, -3, 15, 4, 1, -1], [-2, -8, -1, 6, 8, -1, 4, -10, 1], [2, -6, -3, 4, 6, -2, 8, -10, 1], [-23, -5, 4, -4, 0, -2, -8, 9, 3], [-4, -6, 1, 11, -9, -2, -1, -12, -12], [20, -5, 0, -5, -10, -10, 6, -5, -9], [0, 3, -3, 7, 5, 12, -8, -7, 11], [-25, 2, -5, -9, -3, -1, -2, 0, 5], [0, 15, 19, -12, 2, -3, -7, -4, 9], [21, -14, -3, 9, 2, -10, 0, -6, -1], [-13, 9, 11, -5, -7, 4, 7, 8, 2], [41, -11, -1, 9, -9, 1, 11, -2, -19], [-9, 14, 14, -2, -9, -7, -7, 7, 3], [-37, 11, -2, 1, -7, 8, -3, 6, -4], [-7, 4, -9, 0, 6, 9, -9, 3, 17], [-11, 10, 15, -10, -10, -8, -3, 17, -10], [-6, 4, -2, -1, -5, -7, 1, -3, -5], [12, -12, -16, 18, 8, -1, 0, -17, 12], [-16, 7, 3, 1, 15, 5, -10, -10, 3], [8, -1, 16, -11, -10, -16, 0, -3, -3], [24, -14, -25, 17, 6, 1, 13, -7, 13], [-22, 19, 2, -3, 5, -3, -2, -6, 9], [4, 9, -10, -4, 12, -7, 3, -16, 14], [-13, 1, 4, 3, -23, -3, 5, 5, -11], [-23, 21, -4, -6, 6, 20, -10, 13, 13], [20, 2, 14, -2, -6, 12, -4, 6, -2], [-31, 3, -9, -8, 18, 0, -16, 1, 4], [-26, 0, -12, 11, -3, -4, 9, -11, -9], [5, 9, -2, -1, 11, -1, 5, 4, 6], [10, 12, -9, 6, -10, 5, 4, 1, -2], [-50, 9, 7, -13, 15, 0, -14, 7, 2], [17, 1, 16, -17, -3, -2, -1, 13, -9], [-7, -4, 4, -4, 8, -2, -15, 4, -4], [-25, 0, 23, -12, 8, -2, -15, 12, -5], [4, -4, -1, -3, 19, 6, -1, -2, -1], [4, -6, -7, 11, -9, 12, 5, 4, 13], [40, -9, -1, 10, -18, -7, 5, -10, -8], [-13, -9, -1, -5, 9, -9, 5, -8, 12], [22, 3, 11, -2, -2, 3, 5, 2, -4], [-19, -2, 19, -12, 0, 6, 1, 15, -5], [-40, 4, -9, -5, 5, 12, -15, 4, 5], [-27, 6, -4, -12, 6, 5, 1, 7, 9], [28, 3, -15, 10, -8, -6, 1, 1, -9], [0, 3, 6, 4, -12, -5, -1, -6, -5], [-15, 2, -12, 7, -11, 2, 0, -4, -1], [-18, -11, 6, 3, -1, -5, -2, 1, -1], [-19, 14, 6, -8, 0, 13, -7, 13, -8], [-28, -5, -9, 7, 13, 7, -6, -4, 16], [-11, -4, -2, -6, 4, 13, 1, 5, -1], [-22, -8, -19, 1, 11, 6, 5, 1, 14], [-28, 7, -12, 1, 9, 7, -10, -5, 22], [-3, 6, 0, -8, 0, 10, 1, 8, -4], [49, -15, -23, 9, -21, -13, 19, -8, -5], [8, 8, 5, 3, -11, -8, -1, -6, 4], [-32, 2, 14, -15, 24, 15, -15, 3, 15], [16, -11, -19, 2, -6, 1, 11, 0, -7], [-29, -3, 3, -8, -1, 7, 4, 16, -3], [-21, -13, -14, 5, 3, 4, 5, -3, 1], [38, 0, -5, 12, -20, -11, 20, -6, -10], [-15, 9, 3, -17, 12, 4, -1, 3, -5], [-15, 12, 1, -7, 13, -9, 0, -10, 3], [23, 4, -4, -4, -8, -3, 15, 5, -9], [38, -5, -7, -4, -14, -11, 9, -10, -7], [0, 3, -3, -14, 8, -9, 3, -5, 2], [-37, 18, 17, -16, 18, 8, -21, 2, -1], [39, -22, -34, 20, 0, -2, 9, -18, 12], [9, -14, -4, 3, 15, 12, -4, 4, -1], [-4, 5, -2, -8, 4, 0, -5, -5, -2], [1, -2, -1, 13, -9, -8, 0, -3, 1], [8, 20, 8, -4, 2, -6, -8, -9, -5], [-23, -3, -2, -8, -10, -9, 6, 6, -6], [1, 18, 4, 5, 3, -2, -10, -2, -4], [27, -11, 0, -4, -2, 11, -2, 10, 3], [-21, 3, 25, -2, -4, 0, 2, 7, -4], [11, -12, 9, -7, 3, -3, -2, -1, -4], [1, 13, -11, 7, -7, 3, -7, -4, -2], [-27, 0, 11, -21, 9, 5, -14, 7, -4], [-13, 16, 25, -8, -10, -10, -5, 8, -20], [-4, 7, 4, -3, 11, 8, -2, 0, -4], [-5, 11, 9, -4, 8, 6, -4, -2, 12], [-3, -14, -6, 0, 12, 8, -11, 2, -4], [-12, -11, 10, -21, 11, -5, -4, -1, -5], [11, -12, 9, 4, -12, 3, 15, 11, -6], [-2, 2, 8, -3, -5, -2, -13, -4, -6], [-6, 6, 1, 14, 4, 5, -4, 2, -10], [4, 15, -8, 0, -5, 8, -3, 7, -5], [-11, 6, 7, -6, 18, 2, -19, -4, 16], [-4, 9, -2, -6, -16, -8, 9, -10, 1], [36, 0, -11, 5, 3, -4, 15, -16, 9], [-20, 17, -5, -4, -3, 18, -3, 9, 9], [-3, -3, -11, 3, 5, -4, -5, 0, 5], [-7, 2, 5, 9, 9, 3, -4, -7, -14], [-9, 2, 1, -5, -1, 0, 2, -3, 2], [5, -17, -2, 7, -13, -4, 5, 8, -7], [37, -12, 7, -5, -5, -10, 14, 2, -22], [-4, 7, 11, -15, -21, -15, 0, 4, -7], [-9, -11, 3, -10, -4, 0, 14, 14, -3], [31, -8, -2, 4, -30, -3, 17, -3, -8], [22, -3, -1, -1, -7, -24, 10, -5, 1], [-13, 23, 27, -9, -7, -5, -17, 5, -2], [23, -21, -14, 10, -22, 0, 18, -1, -12], [3, 3, 4, 9, -17, 4, -1, 9, 1], [7, -8, 11, -1, -1, -5, 0, 13, 0], [-12, 10, 7, -5, 18, 10, -13, 6, 10], [-16, 2, 10, -9, 1, -7, -1, 4, 1], [6, -14, -21, 10, 0, -1, 10, -17, 7], [-5, 3, 20, 1, 7, 4, -9, 7, -10], [-13, 5, -6, 3, 7, 5, -3, -4, 14], [4, 10, 15, -8, 2, -5, 0, 0, -12], [-7, -13, 5, 6, 14, 19, -12, 12, 10], [17, -14, 17, -2, 2, 2, 5, 8, -9], [-31, 9, 22, -11, 17, 16, -7, 13, 6], [-25, 7, 10, 16, 0, 9, -6, 15, -3], [-10, 7, 21, -13, -1, 9, -4, 8, 0], [-16, 6, -5, -1, -7, -4, -7, -4, 10], [3, 16, -6, -2, 10, 15, -1, 3, 4], [-24, 12, 10, -12, 12, 8, -24, 0, 11], [17, -10, 3, 6, -14, 0, 1, 10, -6], [-36, 5, 12, -4, 24, 12, -13, 5, -3], [-22, 0, -1, -16, 0, -4, 2, 15, -3], [6, -16, -2, -1, 1, -9, 7, -1, -13], [-8, -12, -14, 8, -18, 0, 8, -4, -1], [42, -1, 10, -5, -15, 0, 0, 5, -13], [18, 0, 6, 6, 4, 7, 18, 5, -7], [-3, -1, -1, 19, -25, 8, 9, 16, -21], [11, -3, 1, 1, -5, 6, 3, 8, 0], [11, 0, -16, -4, 10, 5, -7, -11, 7], [40, -8, 9, -5, -9, -8, -1, 6, -15], [1, 16, 5, -1, -13, -11, -2, -2, -7], [43, -9, -9, 6, -24, -12, 20, -6, -9], [19, -19, -19, 10, 0, 11, 6, -11, 2], [1, -26, -24, 9, -5, 4, 6, -5, 9], [-20, -20, -12, 0, -6, 1, 22, 5, -5], [6, -12, 7, -6, 8, 9, -8, 10, -11], [25, -18, -15, 15, -13, -6, 30, -5, -14], [1, -4, -33, 22, 6, 8, 1, -12, 12], [-4, -9, -6, 13, 13, -3, -2, 2, 12], [-6, 8, 19, -5, -3, 2, 1, 19, -7], [38, -2, -1, 2, -5, -16, 14, -2, 5], [-21, -12, 20, -13, 9, 6, 0, 18, -2], [-2, 1, -1, 5, -3, 2, 16, 1, 1], [-16, 5, 9, 0, 12, 3, -13, 0, 2], [-11, 9, 26, -17, -1, -14, -11, 11, -1], [9, -4, -21, 17, -1, 18, 14, 2, 0], [-23, 3, -9, 10, 20, 23, -8, 10, 14], [14, -3, 11, 3, 24, 2, -6, 1, 0], [26, 2, 7, 7, -1, -8, -1, 3, -11], [23, 15, 1, 2, -14, -5, 12, -2, -5], [-18, 16, 4, -11, -2, 7, 1, 13, 1], [5, -9, -12, 1, 11, 0, -1, -5, 18], [2, -16, -3, -6, 23, 13, -4, 1, 9], [-52, 23, 1, -14, 10, 1, -3, 3, 3], [32, 12, 16, 2, -11, -7, 12, -1, -6], [42, -12, -1, 5, -5, -19, 11, -13, 6], [-5, 1, 18, 3, -17, -16, 1, -9, -12], [15, 19, -3, -2, -24, 8, 0, 15, -19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8016_2_a_ba();".
// 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_8016_2_a_ba(:prec:=9)
    chi := MakeCharacter_8016_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_8016_2_a_ba();".
// 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_8016_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![38, -80, -132, 179, 200, -41, -67, -4, 6, 1]>,<7,R![-80, 1064, -2148, 1265, 242, -446, 88, 27, -11, 1]>,<11,R![400, 780, -600, -1612, -71, 595, -18, -47, 1, 1]>,<13,R![-5360, 11568, -4048, -4140, 1539, 612, -152, -43, 4, 1]>],Snew);
    return Vf;
end function;