
// Make newform 8016.2.a.bd 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_bd();".
// 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_bd();".
// 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 := [-2, -48, 628, -67, -675, 42, 220, -3, -26, 0, 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], [17959625, -26831938, -62769314, -4209233, 31352276, 5651462, -4445172, -799085, 188403, 31274], [3445799, 45128857, 4461694, -31921253, -3054363, 9006576, 535650, -1096194, -26608, 45076], [-3074005, -128029606, 66704449, 125932814, -36535738, -37660725, 4959639, 4018521, -197290, -140074], [-20051371, -122806757, 61255546, 150476284, -25138542, -53041455, 2278935, 6474592, -49534, -252127], [15210925, -195374082, -5489909, 233226318, 2450805, -81378952, -2045962, 10079448, 156741, -400908], [3313559, -233347950, 36533942, 282874842, -17429855, -95483893, 1022842, 11479221, 13553, -446372], [24062029, -291434843, 37019495, 344299563, -24247147, -120466276, 1678767, 14786968, 4817, -581351], [-18894784, 458021977, -87928296, -472650162, 47856867, 151157770, -4911181, -17690130, 130521, 676598]];
    Rf_basisdens := [1, 1, 6386707, 6386707, 6386707, 6386707, 6386707, 6386707, 6386707, 6386707];
    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, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, -1, 0, 0, -1, 0, 0], [0, 0, 0, 0, 1, 0, 0, 1, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0, 0, 0, -1, 0], [-1, 1, 1, 0, 1, -1, -1, 0, 1, 1], [0, -1, 1, 0, -1, 1, 1, 1, -1, -1], [-1, 0, 0, 0, -1, 0, -1, 0, -1, -1], [-2, 0, 1, 0, 0, 1, 0, -1, 0, -1], [-1, 2, 0, -1, 0, 0, 0, -1, 0, 0], [0, 2, -1, 1, 0, 0, 2, -2, 0, -1], [-1, 0, 1, -1, 1, -2, 0, 2, 0, 1], [-1, 0, 1, 0, 1, 0, 0, 0, 0, 0], [-2, 0, 1, 0, -1, 1, 1, 0, -1, 1], [2, 2, -1, 1, -1, 0, 1, -1, 0, -2], [1, 0, 0, 2, -2, 2, 1, -1, -1, -1], [2, 0, -2, 0, -1, 2, 2, -3, -1, -2], [-3, -1, 0, 0, 2, 0, -1, 2, 0, 1], [-2, 1, 2, 0, 2, -1, -2, 2, 1, 1], [-3, 0, 1, 1, 0, 0, -1, 1, 1, 2], [-1, -2, 2, -3, 3, -1, -2, 3, 0, 1], [-5, -2, 1, -2, 1, -1, -2, 3, 0, 1], [-2, 1, -1, 2, 0, 3, 0, -2, 1, -1], [-6, -1, -1, -2, 0, -2, 1, -1, 0, 0], [0, -1, -1, 1, -3, 2, -1, 0, 0, 0], [2, -1, 0, 0, 1, -2, 0, 2, -1, 2], [-1, -2, 1, 0, 0, 0, -2, 3, -1, 1], [-7, -3, 0, -5, 4, -2, -2, 4, 2, 2], [1, 1, 0, 2, -2, -1, 4, -1, 0, 0], [4, 0, 1, 2, -2, 2, 1, -2, -2, 0], [-10, -2, 2, -4, 6, -3, -1, 5, 2, 3], [1, 0, -2, -2, -1, 1, 0, -1, 1, 1], [-9, -1, 1, -2, 2, 1, 2, 1, 1, 2], [-4, 1, -1, -1, 3, -3, 1, 1, 2, -1], [-5, -7, 1, -2, 0, 1, -1, 5, -1, 1], [4, 2, -2, 4, -2, -1, -3, -3, 0, -2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 5, -3, 4, -2, -1, 1, -3, 1, -1], [1, -6, 2, -6, 3, -1, -2, 7, 0, 5], [-5, -2, -3, -6, 4, -2, -1, 1, 2, 4], [-3, -1, 0, -4, 4, -1, -1, 1, 1, 3], [-4, 6, -2, -2, 3, -5, 0, -2, 4, -1], [-7, -1, -2, 3, -3, 3, -1, -2, -1, -2], [-2, 1, 2, 0, 1, -1, -1, 1, 0, 0], [1, 4, 1, 0, 3, 0, 3, -4, 0, -2], [-2, 3, -1, 2, 2, 0, -3, -1, 4, 3], [6, 0, 2, 3, -2, 1, -3, -3, 1, 0], [2, -3, -3, 6, -8, 7, 2, -6, -2, -4], [-6, 0, 2, 0, 1, 1, 1, 0, 0, -1], [2, -2, 0, 0, -4, 5, 3, 1, -3, 0], [0, -3, -2, 0, -4, 4, 3, -3, -2, -1], [12, 2, -2, 4, -5, 0, 4, -5, -4, -4], [-5, 8, 0, 3, 0, -4, 3, -3, 2, -1], [2, -4, -5, -2, -3, 2, 3, -3, 1, -3], [-8, 1, 1, -4, 1, -4, 1, 0, 2, 4], [3, -5, 0, 1, -3, 4, 2, -1, -4, 0], [1, 1, -7, -1, -3, -2, 2, -5, 2, 1], [1, -1, -4, 6, -8, 7, 0, -7, -2, -4], [9, 1, -3, 4, -2, -1, 1, -3, -1, -3], [-11, -6, 2, -2, 2, 2, -1, 3, -1, 1], [7, -2, -2, 0, 0, 3, -2, 0, -3, -1], [-8, -6, 2, -4, 5, -3, -3, 6, 0, 1], [-5, -4, 3, -4, 2, 1, -4, 4, 1, 4], [-12, 1, 2, -6, 6, -5, -3, 4, 4, 4], [2, 1, -2, 5, -3, 1, 1, -1, -3, 0], [2, -2, 1, 6, -6, 7, 3, -1, -3, -6], [4, -3, -3, -1, 0, 3, 4, -2, 0, -4], [0, 2, -2, -1, 0, 2, 5, -4, 1, -3], [-4, 4, 0, 2, -1, -5, 0, -6, -1, -4], [-9, -9, -4, -4, 3, -2, -2, 5, 1, 4], [0, 5, 3, 2, 2, -3, 4, -4, 2, 0], [2, -7, -1, 1, -3, 3, -1, -3, -1, -1], [2, 2, -5, -4, -1, 3, 7, -2, 1, -1], [0, 2, -1, 6, -9, 2, 2, -9, -3, -5], [-2, 2, 5, 1, 3, -1, 2, 4, -4, 4], [0, 2, 1, 2, 0, 3, -2, -1, 1, -2], [-2, -7, 2, 3, -4, 0, -3, 5, -1, 3], [-2, -4, -5, -2, -1, -2, 0, -3, 1, -1], [0, 5, -2, 4, -1, -3, 1, -3, 4, 0], [8, 0, -3, 0, 0, -2, 0, -4, -1, 1], [-1, -2, 0, 2, -3, 2, -4, 2, -3, 1], [-4, -2, 5, -4, 10, 0, -1, 6, 3, 3], [-12, 1, 2, -1, 2, 1, -1, -2, 1, -4], [-6, -5, -1, -1, -4, 4, -6, -3, 2, 1], [8, 2, -2, 4, -4, 4, 7, -2, -4, 0], [-3, -3, 1, -4, 8, -4, -2, 0, 2, 1], [3, 5, 0, 6, -4, 2, -2, -4, -3, -4], [-4, -2, 3, -3, 0, -2, -4, 6, 0, 1], [6, 2, -5, 2, 1, -1, 5, -4, 2, -3], [-13, 2, 1, -4, 5, -7, -6, 1, 5, 6], [4, 6, -4, 3, -2, 2, -1, -8, 0, -3], [0, -2, -2, -2, -2, 4, 0, 2, -4, 0], [6, 3, 3, 7, -4, -1, 3, 0, -2, -3], [-1, -1, 2, 0, -3, 0, 0, -1, -5, 0], [-5, 4, 3, -2, 5, -3, -1, 9, 2, -1], [-2, -9, -4, -1, -7, 3, -1, -5, -5, -4], [0, 0, -2, 4, -1, -1, -4, -2, 2, 1], [0, -4, 2, 3, 3, 4, -1, 1, -1, -5], [7, -3, 1, -2, -3, 5, -1, -6, 0, -1], [3, 7, 0, 8, -4, 6, 4, -6, -5, -6], [1, -1, 0, 4, -6, 0, -3, 2, 0, 3], [1, -2, 1, -5, 5, -3, -4, 3, 4, 2], [-7, 5, -1, -4, 0, -1, 0, -1, 5, 6], [-1, 4, 4, 0, 8, -3, -6, 10, 5, 2], [2, 1, -2, 3, 4, -4, -3, 3, 4, 1], [2, -3, -3, -3, 2, -5, -4, 0, -1, 2], [-3, 3, 4, 8, -3, 1, 2, 4, -2, 0], [12, 0, 2, 2, 0, -4, 0, 0, -1, -1], [1, -2, 9, -1, 4, 1, 2, 6, 0, 2], [0, -1, -5, -4, 5, 0, 2, 0, 2, 1], [-6, -8, -3, -4, 1, -2, -3, 1, 0, 2], [-5, 4, 3, 3, 1, -4, -4, 0, 2, 3], [-6, -2, 6, 0, 5, -3, 1, 12, 0, 1], [-2, 4, 3, 1, -4, 0, 0, 2, 0, 1], [2, -8, 2, -1, 4, -4, -3, 6, 0, 5], [2, 6, -1, -2, 7, -3, -3, 4, 4, -1], [-4, 0, 5, -2, 3, -2, -2, 9, -3, 7], [2, 6, -3, 1, -2, 7, 6, -3, -3, -4], [-6, 1, 3, 0, 5, -8, -5, 4, 9, 6], [6, -8, 2, -4, 3, 3, -1, 6, -1, 0], [0, 0, 1, 0, 0, 2, -1, 8, -2, 0], [-1, 4, -2, -2, 7, -2, 3, -6, -1, 1], [-7, 6, 0, -1, 4, -2, 0, 3, 7, 0], [11, 2, 0, 6, -4, 2, 8, -1, -2, 0], [13, -10, 10, -3, 3, 4, -1, 10, -3, 1], [6, -7, -5, 0, -14, 9, 5, -6, -7, -5], [5, -3, -2, 1, -5, 9, -1, 0, -7, -2], [-6, -6, 12, -4, 9, -4, -2, 15, 2, 6], [0, 6, -2, 0, 2, 0, -5, 0, 6, -1], [-6, 10, -6, -2, 4, -4, 1, -2, 4, -2], [-3, 9, -3, 5, -2, 1, 3, -7, -2, -9], [-1, -1, 6, 0, 4, -3, 0, 7, -5, -5], [-1, -1, -7, 0, -9, 4, 5, -9, -2, -7], [-4, -5, -2, 2, -4, 4, -5, -5, 0, -3], [7, -4, -5, 2, -3, 0, -2, -2, -8, -5], [12, 0, 8, 1, 4, -4, -3, 4, 4, 3], [-10, 1, -2, 1, 5, 2, -3, 0, 3, 3], [5, 8, -2, 6, -6, 5, 6, -12, 1, -6], [-10, 4, 2, -10, 11, -13, -5, 8, 10, 8], [8, -5, -2, -1, -11, 13, 5, -7, -7, -6], [-7, -2, 7, -3, 9, 2, 1, 12, -1, 4], [0, 12, 6, -6, 11, -7, -6, 6, 4, 1], [19, 7, -2, 9, -6, 1, 8, -13, -5, -7], [-10, 0, -6, 0, -3, 2, -7, -1, 4, 0], [-6, -7, 3, -6, 4, -3, -5, 4, 5, 5], [1, -6, 4, -4, 9, -6, -5, 14, 0, 6], [1, 4, -6, -4, -3, 2, 4, -8, 0, -2], [-9, 4, 2, 4, -4, 1, 5, -8, -3, 0], [15, 3, 6, 6, -12, 7, 4, -3, -7, -5], [-5, 8, -6, 1, 1, -8, 0, -4, 7, 4], [8, 16, -4, 4, -5, -1, 7, -6, 3, 2], [8, 7, 5, -2, 4, -3, -4, 8, 8, 8], [-4, -1, 2, 2, -4, 2, -3, -1, -6, -7], [-9, -4, 4, -2, 7, -2, -4, 10, -3, 7], [-1, 1, -1, -10, 4, 3, 4, -1, -1, -3], [1, 15, -1, -4, 9, -9, -2, -4, 7, 5], [4, -4, -2, 10, -15, 7, 7, -10, -5, -4], [-2, 4, -1, -3, -1, 0, -5, -7, -1, 2], [2, -10, 8, 2, 8, 0, 1, 10, -7, -7], [-1, 5, 3, -5, 4, -7, -7, 3, 0, 1], [-7, -2, 4, -8, 0, -3, -5, 0, 7, 4], [-2, -2, -5, 3, -4, -2, -1, -4, -4, -4], [-3, 3, -6, 7, -7, 0, -2, -1, -4, 0], [6, -6, -3, 8, -13, 12, 5, 1, -6, -8], [-9, 1, -5, -1, -8, -3, 0, -3, -1, 0], [9, -5, -2, 8, -16, 8, 7, -12, -8, -9], [-4, 4, -1, -2, 5, -8, -6, 1, 8, 0], [-7, 11, -1, 0, -2, 3, 10, -7, 4, 2], [9, 1, 10, 3, -2, -2, 0, 4, 1, 2], [5, -17, -5, 6, -10, 15, 2, -7, -6, -8], [-3, 5, -2, 4, -7, -1, 1, -16, -1, -3], [5, 7, 4, 2, 2, -5, -2, -7, 6, -2], [3, -7, 0, -1, -8, 11, -2, 3, -4, -1], [-13, -1, 9, 0, 8, -5, -4, 5, 8, 4], [-8, -9, -6, 6, -15, 11, 5, -11, -6, -6], [-9, 4, -4, 6, 5, -2, -7, -4, 6, 1], [8, 18, -2, 6, 0, 2, 7, -8, 1, -5], [5, 5, -3, -1, 3, -4, -6, -1, 12, 3], [-12, -13, -7, -12, 3, -3, -7, 3, 2, 4], [4, 0, 3, -10, 1, -6, 3, 5, 0, 8], [4, -6, -3, -5, 8, -3, -2, 1, 3, 0], [-10, -3, 5, -5, 6, 2, -12, 9, -3, 3], [-8, 2, -6, -10, 5, -8, -3, 1, 3, 9], [-2, -7, 10, -2, 2, 1, -1, 12, -2, 4], [10, 1, 4, 0, 4, 6, 11, -1, -2, -5], [10, -7, 5, 8, -9, 2, -2, -4, -10, -3], [2, 5, 3, -10, 7, -7, 3, 3, 4, 8], [7, 4, -6, -2, -9, 2, -1, -12, 1, -3], [2, 7, -5, 6, -6, 9, 10, -18, 0, -8], [-5, -2, -3, -5, 2, -3, 4, -2, -7, -6], [1, -4, -6, -2, 4, -3, -2, -6, -2, 1], [2, 4, 0, 0, -1, -3, 2, 8, 2, 1], [11, 2, -5, -2, -6, 0, -2, -5, 3, 5], [2, 0, -7, -2, -13, 8, 7, -9, -1, -5], [-13, 1, 1, -4, 12, -8, -1, 10, 10, 5], [14, -2, -1, 4, -9, 0, 0, -3, 1, -3], [-2, -15, -3, -6, -2, 3, 3, 8, -12, -2], [-17, -5, 1, 2, -5, -1, -6, 0, 7, 3], [-12, 5, 5, -3, 6, -2, -1, 5, 6, 0], [-10, 7, -10, -4, -4, -4, 4, -7, 2, 5], [-5, 9, 0, 3, 2, -4, -6, -2, 7, 6], [-14, 9, 5, 4, 2, -4, -1, -7, 1, 2], [-10, 2, -6, -2, 3, -11, -2, -2, 10, 7], [-18, -13, 3, 0, -7, 6, -2, -2, 2, -1], [7, 12, -4, 7, -2, -1, 4, -14, 5, -9], [3, -5, -7, 4, -15, 9, 5, -2, -5, -5], [6, 2, 8, 11, -2, -1, 1, 7, -2, 0], [-9, 1, 1, -4, 0, 2, -6, -2, 6, 1], [7, 2, 5, -4, 0, -4, 9, 1, 1, 3], [-7, 2, -1, 2, -2, 0, 3, -3, -2, -6], [6, -1, -4, 5, 1, -5, 2, 5, -2, -1], [-1, -8, 10, 6, -1, -2, -1, 10, -1, 3], [-14, 8, 0, -8, 2, -10, 3, 4, 1, 3], [-14, -3, -9, -6, 3, 2, 2, -2, 1, -2], [2, 4, 10, -7, 16, -4, -4, 16, 9, 10], [-2, -9, -1, 6, -4, -1, -1, -6, -1, -3], [-7, 10, 1, 4, 2, 3, 1, -2, 11, -2], [0, 1, 2, -4, 2, -11, -2, 6, -4, 7], [15, -8, -9, 4, -12, -1, -3, -10, 0, -1], [-16, 3, 9, -2, 13, -13, -3, 11, 4, 2], [7, 4, -6, 9, -5, -2, -2, -8, 5, -4], [7, 0, 6, 6, 2, 1, 6, -2, 0, -3], [16, 4, -2, 2, 7, -1, -4, 0, -2, 5], [4, 5, 5, 0, -4, 2, -3, 1, -1, 4], [-10, -12, 7, -8, 7, 2, -3, 13, 5, 7], [14, 2, -6, 6, -16, 6, 1, -10, -1, -7], [-18, 5, 6, -3, 9, -14, -9, 4, 5, 9], [-10, 0, 4, 6, -4, 6, -1, -8, 3, -7], [19, -5, -6, 9, -8, 4, 5, -10, -4, -3], [40, 0, 2, -5, 2, 1, 1, 5, 2, 4], [13, 4, -4, 5, -12, 3, 3, -6, 0, -6], [-12, -11, 8, -7, 13, -7, -8, 17, 0, 3], [-9, 3, 1, -4, 4, -4, -6, 2, 6, -5], [6, -5, 1, -4, 3, 1, 6, 7, -4, 2], [0, 1, -2, -2, -1, -13, 0, -5, -3, -1], [-4, -11, 5, -2, -1, 8, 12, -2, 0, -7], [-5, 9, -1, -9, 7, -3, -5, -6, 3, 1], [-16, -3, 5, 6, -8, 3, -3, -10, 3, -3], [10, -18, 1, 0, 4, 5, 1, 13, -8, 3], [-8, -9, 4, -4, -5, -3, -7, 5, 2, 2], [25, 11, -7, 9, -7, -7, 0, -8, -5, -8], [4, -3, 2, -3, -2, 5, 3, 4, -8, 0], [1, -9, -1, -12, 0, 6, 7, -8, -1, -2], [6, 9, -11, 6, -4, -2, 6, -15, -2, -1], [18, 0, 2, -2, 6, -8, -3, 10, -2, 8], [4, -5, 2, -1, -11, 11, 11, -5, -6, -14], [-14, -8, -5, -2, 7, -10, -3, 5, 10, 8], [6, 0, 1, 6, -5, 6, 11, 5, -6, 0], [9, -6, 10, -2, 8, -11, -6, 12, -4, 5], [-5, 4, 6, 6, -4, -4, 5, -1, -3, 1], [5, 3, 5, -8, 8, -4, -3, 4, 5, 0], [2, -3, -5, 10, -21, 6, 3, -18, -7, -10], [1, 0, -6, -9, 12, -6, 3, 3, 6, -1], [7, -2, 5, 4, 2, 6, -2, -1, -5, -3], [5, -1, 12, 1, 3, 1, -6, 12, -1, -5], [7, -8, -1, 0, 4, 7, 6, 2, -1, 0], [-16, 3, -6, -8, 13, -14, -3, 4, 5, 13], [-21, -6, -1, -14, 14, -7, -8, 16, 4, 13], [-8, -5, -9, -9, 12, 0, -6, -5, 5, -1], [9, 12, 1, -1, 12, -9, -3, -2, 2, 3], [-8, 12, -5, 9, -11, -3, -4, -14, 1, -6], [-2, 14, 4, 1, 6, -10, -6, -8, 8, -4], [-21, 7, 3, 4, 0, 0, -3, -8, -3, -2], [3, 5, -2, -4, 0, 1, -10, -5, 3, 3], [23, -14, -5, 5, -13, 5, 1, -3, -7, -7], [8, 5, 8, -9, 12, -7, -7, 10, 7, 6], [20, 11, -10, 8, -4, -7, 5, -8, -5, -6], [0, -4, 3, 2, -11, 6, 8, 3, -9, -1], [21, -15, 1, -2, 1, 9, -1, 4, -9, -1], [34, 2, 0, 10, -2, 13, 8, -3, -8, -11], [-9, -5, -5, -2, -5, -4, -9, -7, 7, 2], [16, 5, 2, 2, -3, -1, -11, 1, -1, 7], [-9, -3, -3, -12, 10, -6, -1, 8, -3, 6], [9, 8, 11, 3, -5, 0, 1, 2, -7, 0], [-3, -16, -1, -9, -2, -1, -6, 12, -1, 8], [-3, -11, 1, 7, -4, 7, 1, -1, 1, -7], [-21, -8, 3, -14, 12, -3, -12, 10, 5, 12], [12, 12, 3, -4, 7, -10, 3, 9, 2, 0], [-3, 9, 16, -2, 17, -10, -7, 11, 0, 7], [-19, 8, -15, 7, -10, -5, -1, -12, 1, 1], [-7, -17, 5, -8, 17, 0, -6, 15, -8, 5], [8, -11, -5, -6, -14, -2, 1, -1, -5, 4], [16, -11, 3, -6, -4, 1, -2, -6, 1, -2], [5, 3, -11, 6, -7, -1, 5, -16, 1, 1], [-3, -11, 5, -2, -11, 5, 7, 2, -8, 2], [-20, 3, 8, 1, 1, 5, 1, 5, 1, 0], [22, 14, -4, 24, -11, 1, 1, -16, -4, -11], [-16, 10, -2, 5, 0, 0, -2, 2, -3, 8], [1, 6, -7, -8, 5, -5, 4, -5, 3, -8], [9, 4, -1, -1, -7, -1, 12, -3, 5, 0], [27, 9, -9, -4, 2, -4, 9, -10, 8, 1], [8, 2, 7, 4, -8, 4, 4, 0, -11, 5], [8, 19, -6, 0, 1, 3, 16, -7, 5, -5], [8, -5, 6, -6, -3, -3, 10, 9, -3, 3], [-29, -19, -2, -2, 1, 4, -2, 1, -4, -3], [-12, 7, -3, -10, 6, 3, -3, 2, 6, 5], [9, -11, 5, -11, -1, -1, -4, 6, -2, 4], [36, -10, -12, 16, -28, 12, 0, -14, -14, -10], [-12, -7, 4, -2, 11, -8, -8, 2, 7, -4], [21, 10, 5, 14, -8, 8, 5, -5, -1, -6], [1, -1, 14, 6, 9, -9, -7, 14, 5, 1], [1, 2, -8, 12, -7, -1, -6, -9, -6, -3], [-4, 9, 4, 2, 1, -1, -7, -1, -2, 6], [18, 7, -5, 18, -14, 4, 5, -13, -9, -6], [11, -8, 2, -1, 1, 14, 4, -4, -6, -2], [8, -2, 5, 4, -2, 3, -3, 3, 2, 7], [-22, -12, 1, -6, -1, -2, -1, 5, 2, 4], [10, -14, 3, -2, 4, 10, 8, 12, -9, 3], [-2, 13, 9, 8, -4, 9, 9, 0, -3, -11], [-4, 3, 5, 6, 1, 10, 1, 4, -5, -6], [10, 13, -3, -12, 16, -11, 1, 2, 3, 3], [13, 8, 0, -7, 14, 1, 6, 0, 7, -1], [14, -18, -3, 6, -10, 18, 0, 0, -11, -5], [-13, -14, -5, -10, -5, 2, -7, 8, 2, 6], [16, -13, -10, -5, 1, 7, -1, -9, 2, 0], [-12, -19, -16, -6, -13, 7, 4, -9, -4, -4], [-23, -12, 4, -8, 8, -6, 3, 15, -5, 5], [-11, -3, 5, -7, 7, -3, -3, 2, 4, -1], [-14, -24, 5, -10, 5, -1, 3, 18, -2, 3], [6, -1, -15, 4, -10, -1, -5, -14, 4, -4], [17, -4, -7, 12, -11, 8, 5, -12, -4, -6], [28, -6, -6, 13, -15, 11, 16, -16, -11, -15], [-12, 1, 1, -3, -3, 4, -1, 2, 4, 10], [-8, 14, 14, 4, -6, 1, 7, -1, 2, -3], [11, 5, -5, -2, -4, -1, 2, 1, 6, 8], [-9, -2, -1, 0, -6, 9, 12, 0, -7, -8], [5, 7, 5, -4, 10, 0, 3, 10, 8, 7], [-13, 13, -5, 6, -5, -8, -2, -17, 8, -11], [35, 2, 3, 12, -4, 2, -8, -3, 1, -1], [-6, -3, 13, 2, 7, 8, 4, 16, 0, 3], [14, -6, 1, -3, 5, -3, -1, 0, -5, 3], [-5, 16, -10, 2, -1, -4, 4, -12, 4, -2], [29, 14, -5, 15, -9, -1, -2, -9, 4, -10], [19, -10, -3, -2, -6, 8, 8, -3, -1, -1], [-23, -11, -7, 0, -4, 8, -5, -4, 5, -2], [-19, -5, -2, 7, -6, -7, -16, -7, 2, 3], [6, 6, 1, 16, -8, 1, -1, 1, 4, -3], [24, -6, -4, 6, 3, -3, 0, 4, 3, 2], [1, 16, 0, 2, 3, 2, 4, 0, 4, -10], [45, 7, -7, 16, -22, 8, 10, -20, -8, -13], [20, 4, -8, 12, -8, 14, 2, -16, -9, -4], [-24, 12, 3, -9, 9, -1, 0, -2, 10, 0], [15, 1, -12, 10, -14, 9, 10, -21, -9, -5], [14, 8, -2, -4, 8, -11, -1, -11, 1, 2], [-20, 9, 10, -6, 1, -2, 7, 6, 8, -1], [-8, 7, -2, 0, -3, -2, 6, 2, -1, 0], [-19, -18, 6, -16, 17, -14, -9, 26, -1, 17], [-34, -7, -7, -18, -2, -8, 0, -3, 2, 11], [-11, -12, -4, -10, -9, 10, -11, -2, 7, 7], [-15, -2, -5, -8, 3, 0, 2, -8, 3, -5], [3, 13, -10, 6, -8, -1, -11, -17, 13, 4], [11, 1, -9, 1, -11, 0, -9, -3, 6, 4], [11, -16, -2, -5, -11, 12, 8, 4, -12, -4], [11, 13, -5, 3, 9, -5, 7, -12, 4, -9], [-31, 0, 8, -12, 21, -4, -5, 8, 10, 6], [-33, -2, 2, -3, -2, 5, -4, -6, 2, -1], [-8, -2, 1, -1, 9, -5, -7, 6, -1, 11], [-8, 8, -9, 2, -12, 2, 0, -6, 4, 2], [27, 4, -1, 4, -4, -2, -1, 5, 0, 0], [25, 1, 0, 17, -10, 15, 9, -3, -7, -6], [8, 0, -3, 0, -1, -4, 9, -7, 6, -6], [-17, -20, -6, -16, 2, -3, -4, 0, 4, 1], [45, 17, -5, 24, -14, 4, 6, -16, -9, -13], [1, -2, 14, 6, -4, 4, 6, -1, -4, 0], [29, 0, 4, 16, -21, 11, 7, -5, -10, -10], [-3, -9, 0, -5, -2, 18, -8, -2, 4, 0], [5, -2, 11, 4, 4, 2, -5, 11, -3, -9], [-12, 3, 3, -3, -7, -5, -10, -5, 8, 2], [22, -5, -3, 15, -18, 17, 2, -4, -9, -20], [-16, 10, 5, -2, 13, -17, -1, 6, 12, 7], [-1, 7, 0, 2, 9, -18, -13, 21, 6, 9], [3, 7, 14, -1, 8, -4, -1, 0, 7, 11], [0, 5, 5, -5, 5, -8, 0, 10, 2, 13], [-10, -7, -1, 6, -3, 10, -6, -4, 10, 1], [-17, 8, 6, -4, 15, -18, -13, 4, 5, 9], [-39, 1, 3, 0, 5, 0, 3, 11, -10, 3], [-10, -8, 6, 3, 3, 1, -3, 26, 2, 0], [14, -4, -4, 16, -18, 18, 9, -8, -7, -13], [1, -5, -17, 10, -18, 7, 9, -27, -7, -11], [23, 5, 8, 16, -10, 1, -10, -3, -6, -4], [34, -12, 1, 3, 4, 5, 4, 3, -2, -4], [-14, 19, 13, 2, 13, -4, 3, -2, -2, -3], [16, -8, -13, -6, -4, 1, 1, 7, -4, 7], [-29, 19, 5, 6, 7, -7, 10, 2, 4, -4], [-21, 5, 5, -7, 17, -6, 2, 11, -3, 9], [-7, 17, -5, -1, 2, -4, 7, -8, 9, 8], [3, -12, 5, -16, 9, -1, -3, 17, -2, 3], [22, -8, -5, 18, -16, 13, 6, -7, -15, -18], [3, 6, -6, 0, 11, -5, -7, 7, 1, 8], [-7, -6, -1, 6, -5, 6, -6, -2, 3, 3], [-14, 5, 11, -14, 19, -11, -9, 9, 5, 11], [-14, -14, -4, -20, 18, -9, -9, 19, 0, 6], [7, 20, -1, 14, 3, 7, 8, -13, -1, -12], [5, 4, 10, 6, -1, 3, 1, 7, -1, -8], [6, 5, 11, -2, 28, -5, 6, 20, 1, 7], [15, 4, -9, 8, -5, 11, 12, -13, -4, -3], [-11, 9, -23, -2, -4, -18, -2, -8, 3, 7], [-6, -1, -7, -6, -3, 11, 1, 1, 2, 5], [9, -4, 5, 4, -5, 4, 7, -2, 0, -8], [-4, -18, -7, -12, -3, 2, -15, 1, 2, 6], [11, -10, -9, 3, -9, 5, 2, 1, 0, -8], [4, 7, -7, 3, -2, -1, -8, -12, 3, 6], [-10, -7, 6, -16, 21, -23, -10, 29, 10, 11], [-4, -14, -3, -1, -15, 6, 1, 1, -7, 4], [20, -4, -17, -5, 1, -8, -6, -1, -7, 5], [15, 12, 9, 9, 0, 2, -4, -5, 4, 1], [17, -8, -6, -16, -2, 7, -2, 2, 3, 11], [-15, 8, 11, -10, 10, -12, -7, 21, 4, 10], [1, -9, -8, -4, 5, 6, -6, 15, -9, 2], [-13, -2, 2, -5, 16, -4, 11, 5, -4, -9], [-6, -5, -7, 2, -3, 5, 11, -5, -3, -1], [8, 4, -4, 2, 7, 3, 0, -6, 4, -3], [-15, -15, -4, 0, 1, -8, 1, 11, -6, 5], [20, -9, -3, 2, 0, 1, -1, -6, -9, -5], [-18, -13, -7, -18, 2, 0, -11, 1, 7, 8], [-7, 18, -1, 0, -12, 4, -1, -13, 1, 5], [-6, 13, 5, -8, 7, -16, -1, 6, 10, 1], [0, -20, 4, -8, 8, 0, 2, 16, -11, -3], [-21, -22, -6, -7, 14, 0, -1, 3, -2, -3], [-14, 5, 1, 3, -10, 10, 8, -15, -4, 1], [15, 7, -3, 0, 18, -16, -11, 8, 8, 15], [-3, -1, 13, 4, -1, 11, 12, 2, -6, -10], [-14, 5, 0, 14, -6, -1, -5, 2, -9, 7], [4, 33, 0, 7, 9, -14, 1, -6, 11, 1], [-20, 13, -3, 10, -6, -11, -1, -6, -3, -1], [25, -10, -12, 10, -18, 15, 21, -4, -12, -5], [-9, -5, 13, 0, 6, 1, -5, -1, 7, -3], [7, -23, 6, -16, 6, -7, -3, 21, -4, 2]];
    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_bd();".
// 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_bd(:prec:=10)
    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_bd();".
// 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_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![222, -262, -961, 116, 910, 137, -277, -85, 19, 10, 1]>,<7,R![-72, -1364, 3051, -134, -2125, 186, 467, -25, -38, 1, 1]>,<11,R![4656, -10432, 2548, 7060, -3290, -1395, 781, 76, -53, -1, 1]>,<13,R![-96, -2320, 20900, -5540, -8328, 2443, 920, -224, -47, 6, 1]>],Snew);
    return Vf;
end function;