
// Make newform 4016.2.a.i in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4016_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_4016_2_a_i();".
// 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_4016_2_a_i();".
// 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 := [104, 136, -462, -601, 537, 644, -317, -277, 106, 49, -17, -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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-728, 133, 2253, 457, -669, -551, -431, 331, 159, -72, -13, 5], [2132, 4424, -8631, -16503, 5894, 12521, -1801, -3424, 251, 385, -13, -15], [1612, 209, -8999, -710, 14389, -1033, -8390, 1601, 1743, -427, -117, 32], [4680, 6974, -18023, -27933, 11688, 21401, -3007, -5994, 331, 701, -13, -29], [10556, 18504, -37559, -74725, 14758, 59185, 1249, -18008, -1009, 2363, 91, -113], [5304, 10470, -17565, -41796, 3058, 33549, 3628, -10624, -1165, 1466, 91, -74], [-3757, -6382, 12802, 25875, -3603, -20676, -1669, 6515, 638, -895, -52, 45], [-6396, -14093, 19022, 56324, 4955, -46274, -10694, 15585, 2820, -2303, -208, 124], [20696, 29712, -77449, -118031, 48762, 90399, -11813, -25492, 1177, 3017, -39, -127]];
    Rf_basisdens := [1, 1, 1, 26, 52, 26, 52, 52, 26, 13, 26, 52];
    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_4016_a();"
function MakeCharacter_4016_a()
    N := 4016;
    order := 1;
    char_gens := [2511, 3013, 257];
    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_4016_a_Hecke(Kf)
    return MakeCharacter_4016_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, 0, 0], [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, -1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1], [0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [-1, -1, -1, 1, -1, 1, 1, 0, -1, -1, -1, -1], [0, 1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0], [-1, 0, -1, -1, 0, 1, 0, 1, -1, 0, -1, 0], [-1, 0, 0, 0, -1, 0, 0, 0, 1, 0, 1, -1], [0, 1, -1, 0, 0, 0, -1, 1, 0, 0, 1, 1], [-2, 0, 0, 2, -1, 1, 0, 0, 0, -1, 0, -1], [-3, 1, 1, -2, 1, 0, 0, 0, 0, 1, 0, 1], [3, 0, 1, 1, 0, 0, -1, -1, -1, 1, -2, -1], [-2, 1, -1, 1, 1, 0, -1, 1, 0, -3, -1, -3], [0, 1, 1, 0, 1, 0, -1, -1, 0, -1, 1, 0], [-2, 3, -1, -1, 1, 0, -1, 0, -1, -2, 0, 0], [-3, -1, 0, -1, 0, 0, -1, -1, 3, 2, 0, 0], [-2, 2, 0, -2, 2, 0, -2, 0, 1, 0, 1, 0], [-5, 0, 3, 1, -2, 1, 0, 0, 1, 1, 1, 1], [-3, 1, 2, -1, 1, -1, 1, -2, 1, 2, 1, 2], [2, 0, 1, 1, 2, -1, 1, 0, -2, 1, 0, 1], [0, -2, 2, 3, -2, -1, 1, 0, 1, 0, 1, 1], [0, 1, 3, -1, 0, 0, 0, -2, -1, 1, 1, 2], [-4, 0, -1, -3, -1, 3, -1, -1, 2, -1, -1, -2], [-5, 3, 0, -1, 1, -4, 0, 2, 3, 1, 2, 0], [0, -1, -1, -2, 3, -1, 0, 1, 1, 1, 0, -2], [-1, 1, 2, 0, 1, -3, -1, 3, 1, 1, 3, 1], [-2, 1, 0, 2, -1, 0, 1, 0, -1, -1, 1, 2], [0, -1, -1, 0, 0, 0, 0, -1, 0, 2, -4, -1], [2, -3, -1, -2, 1, 1, -2, 0, 3, 2, -3, -1], [-4, 2, 1, 1, -1, -2, -1, 2, 1, 0, 1, -2], [4, -1, -2, -1, 1, 2, 1, -1, -3, 0, -2, 1], [0, 0, -3, 1, 2, 1, 1, 0, -1, -3, -1, -3], [-1, -5, 0, 2, -4, 1, 0, 0, 3, 1, -1, -3], [-2, -1, 4, -2, 0, 0, -2, -3, 1, 6, -1, 2], [-3, 1, 3, 4, -2, -1, 0, 1, 0, 1, 3, 1], [1, -2, 2, -2, 0, 4, -1, -4, -1, 1, -1, 3], [0, 1, 2, -1, 2, -1, 0, -2, 0, 2, 0, 4], [1, -5, -2, 1, -3, 1, 0, -1, 3, 3, -4, 0], [-1, 1, 0, -1, 3, -2, -4, 2, -1, 0, 1, -1], [-5, 0, 2, 2, 0, -1, 4, 0, -1, -1, 1, 2], [4, -1, -2, -2, 1, 0, 3, 0, -4, -1, 1, 2], [-2, -2, -1, 3, 2, 0, 0, 0, 1, -3, 3, -2], [-1, 0, -1, -3, 1, -1, -1, 1, 1, 2, 3, 1], [-2, -1, 0, 1, 2, 2, -2, 1, 0, 1, 1, -1], [2, 1, -5, 3, -2, 2, 1, 4, -6, -5, -2, -3], [-3, 0, -2, 1, 1, -2, 2, 2, 0, -2, 2, 1], [-3, 0, 1, -1, 0, -4, 1, 1, 2, 1, 4, -1], [0, -2, 2, 3, 2, 1, -1, -4, -3, 0, -1, 1], [-8, 0, -1, -1, 1, 0, -4, 1, 4, -1, 1, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5, -2, -3, 1, 1, 0, 0, 1, -4, -4, -5, -2], [0, 2, 3, 0, 3, -2, -2, 0, -2, 3, 5, 6], [3, -2, 0, 4, 2, -1, 0, 0, -1, -1, 1, -2], [2, -4, 4, 3, -3, 0, 4, -5, -2, 2, -1, 1], [-3, 1, -1, -3, 1, -2, 0, 4, 1, 0, 2, 2], [-3, 1, 3, 2, 0, 1, 0, -1, -2, 5, 1, 3], [3, -1, 3, -1, -1, -2, 4, 0, -1, 1, -1, 3], [1, 1, 2, 2, -2, -6, -1, 0, 2, 3, 2, -1], [-1, 2, -3, 0, 1, -2, -2, 4, 2, -1, 2, -2], [2, -3, -4, 0, 2, -5, -1, 1, 2, -5, 3, -2], [-2, 2, 0, 1, 0, 0, 0, 0, -3, -2, -3, -2], [6, -1, -7, 1, 1, 2, -2, 1, -2, -7, -1, -4], [-3, -4, -4, 1, -1, 4, -2, 0, 5, 0, -1, -1], [-7, -1, 0, -3, 4, 1, 1, 2, -3, 0, 0, 0], [-1, -1, -1, 2, -4, 5, 4, -2, -4, 0, -6, -2], [-3, -2, 6, 0, -3, 2, 0, -3, 3, 1, 4, 4], [-2, 2, 1, 2, -7, -2, 3, 1, 0, -5, 1, -3], [-3, -3, 2, 0, -4, -1, 5, 1, -1, -2, 2, 3], [-2, -3, 0, 0, 2, 2, 2, -1, -1, -1, 0, -3], [-3, 4, -2, -6, 1, -3, 1, 4, -2, 2, 0, 4], [4, -3, -1, 5, -1, 0, -1, -2, 0, -2, -2, 2], [-5, -1, -3, -1, -3, 1, 1, 4, -1, -1, -1, -2], [-2, -2, -2, -5, -4, 3, 3, 0, 3, 1, -4, 0], [-3, -5, 1, 1, -3, 2, 2, 2, -1, 2, 2, 0], [-3, -1, 1, 1, -1, -4, 4, 0, -1, -3, -1, -3], [-9, 4, 2, 5, -2, -1, 4, 0, -6, -3, 2, -2], [-4, 1, 0, -5, 5, 0, 1, -1, -3, 2, 6, 4], [1, 1, 3, 3, -1, 2, -2, -4, -1, 0, -4, 0], [0, 1, 1, 3, -3, 1, 0, -3, 1, -4, -1, -1], [-6, 0, 2, -3, 0, -2, 0, 0, -1, 8, 1, 6], [2, -1, -1, 0, 3, -4, -1, 4, 3, -2, -1, 0], [-2, 3, 3, 3, -1, 1, 0, 2, -4, 2, 2, 3], [-1, -1, -2, -9, 5, -1, 0, 1, 1, 4, 1, 3], [0, 1, 2, -2, 1, -3, 2, -2, -5, 5, -3, 5], [8, -3, -7, -1, 5, -1, -2, -1, 0, -3, -1, 0], [-2, -3, -1, 0, -3, 2, -3, 4, 5, 2, -1, -6], [1, -2, 2, 3, -6, 3, 2, -2, 2, -2, -3, -7], [3, -3, 3, 0, -1, -2, 4, -6, -2, 1, 0, 1], [3, 3, -3, -3, 1, 1, 1, -4, -1, 3, -5, 2], [-1, -4, 0, -1, -2, 1, 0, 0, 3, 1, 0, -2], [3, 0, 3, -2, -1, -4, 6, 0, -4, 3, 2, 10], [2, -1, -11, -3, -3, 7, 0, 0, -5, 0, -10, -5], [2, -5, -1, -1, 0, 1, -3, 1, 1, 4, -3, -2], [2, -1, -7, -1, -1, -2, 3, 2, -2, -4, -6, -4], [8, 8, 3, -2, 7, -8, -1, 2, -1, -2, 3, 4], [0, -6, 4, 3, 1, -2, -1, -6, 6, -1, 4, 4], [-5, 1, 0, -5, 1, 3, 1, -5, 2, 3, -1, -1], [5, -6, 5, -3, -2, -2, -3, -1, 7, 5, 2, 1], [0, -3, -2, -4, -2, 5, -3, 0, 1, 2, -3, 1], [-1, -7, -9, 1, -7, 7, 5, 3, 1, -4, -2, -3], [4, 7, -2, -3, 3, -2, -1, 2, -2, 1, 0, 8], [-10, -3, 5, -4, 0, 5, -7, -2, 8, 9, 0, 0], [-3, 3, -1, -3, 6, -2, 1, 5, -2, 1, 2, 2], [5, -4, 0, 2, -4, -1, 0, -4, 1, 3, -1, 3], [-4, 1, -7, 1, -1, -1, -4, 6, 2, -2, -2, -7], [-7, -2, -8, 0, -2, 3, 4, 0, -1, -7, -5, -6], [5, -1, 3, 5, -5, 2, -4, -3, 0, 0, -1, -3], [1, 3, -3, 3, 3, 4, 6, -1, -8, -6, -3, -3], [10, -4, 2, 5, -6, -2, 0, 0, 1, -6, 1, -3], [4, -8, 2, 2, 2, 4, -2, -4, 5, 4, 0, 2], [2, 2, 1, -2, 3, -1, 2, -1, -4, 3, 9, 8], [4, -5, -1, 3, -6, 0, 2, 1, 0, -2, -7, -3], [-2, 2, -4, 0, -2, 2, 0, 0, 0, -4, 2, -6], [4, -1, -2, 5, -3, -3, 2, 4, -2, -3, 2, -5], [2, 0, -3, -3, -6, -4, 2, 1, 11, 0, -2, -5], [1, 0, 2, 3, 8, 0, -3, 2, 0, -2, 7, 4], [-2, 2, -2, -1, -3, 2, -7, 3, 3, 5, -1, -2], [4, -3, -5, 2, 6, -1, 1, -1, -7, 2, -4, 2], [10, 6, 7, -2, -3, -6, -1, 0, -1, 8, 7, 8], [-3, -1, 1, 6, 2, 0, -1, -1, 3, -5, 1, -2], [7, -5, 0, 2, -6, 0, 7, -5, 1, 0, -5, 0], [20, -1, -2, -1, 2, -1, 1, 1, -10, -5, -5, 0], [8, 1, -4, 2, -10, 1, 4, 2, -3, -6, -7, -6], [-12, -2, 4, -3, -3, -3, 3, -1, 5, -2, 4, -1], [-1, 5, 4, 3, 3, -7, 2, 1, 5, 3, 2, 4], [-2, -2, -3, 4, -11, 3, 0, 5, 0, -2, -6, -7], [6, -4, -6, -1, 1, 1, 1, 4, -3, 3, -9, -4], [5, 6, -1, -2, 0, -1, 0, 1, -4, 0, -2, 8], [3, -4, -12, 1, 2, 0, 0, 0, 4, -1, -6, -6], [12, -10, 2, -1, -6, 4, 6, -5, 4, -1, -7, -3], [-3, 10, -2, -10, 9, -2, -4, 1, -1, 1, 0, 0], [-7, -3, -4, 4, -6, 6, 5, -1, -3, -7, 0, -1], [6, 12, 0, 0, -4, 5, 5, 3, -7, -1, -4, 0], [7, -5, 1, 7, -4, -6, 3, 2, 2, 2, -2, -2], [8, -6, -12, 2, -3, -1, 2, 5, 4, -9, 0, -9], [3, 7, 9, 3, 2, -1, 0, -7, -1, -5, 6, 3], [-4, 3, -4, -1, -3, -2, -4, 5, 6, -6, 3, -8], [10, 4, 4, 3, 6, -2, 2, 0, -4, -2, 7, 2], [1, 1, 6, -1, -4, 3, -4, -3, 9, -2, 2, 1], [-4, 8, 4, -4, 8, 1, -1, -2, -6, 0, 6, 7], [7, -2, -4, -3, -5, -5, 1, 1, 4, 3, -5, -3], [-1, 7, 9, 1, 4, -5, -2, 4, -1, 2, 4, 3], [3, 4, 8, -4, 3, 2, -4, -5, 8, 7, 6, 4], [5, 5, -3, 3, 5, 2, 0, 6, 1, -1, 5, -1], [-1, -6, 8, 4, -1, 5, 2, -4, -7, 0, -5, 3], [-12, -8, 1, 1, -5, 6, 2, 0, 11, 0, -3, -7], [22, -6, -1, 3, 4, -3, -2, 1, -9, 4, 0, 3], [0, -5, 2, 4, -3, 4, -3, -4, 3, -1, 3, -4], [-3, -6, 4, 0, -7, -2, -4, -2, 8, 2, -2, -2], [-9, 1, -9, -4, 5, -4, -6, 8, 4, -1, 8, -7], [-16, -6, -3, 1, -7, 6, 1, -1, 15, 1, -2, -3], [1, -1, -4, 0, 3, 0, 4, 0, -2, 2, -8, 1], [-3, -6, -2, -2, -3, -2, -2, -1, 3, 1, 2, 4], [4, -4, 8, -4, -8, 4, 8, -4, -4, 4, 0, 8], [7, -3, -10, -3, 0, -9, 0, 3, 3, -2, -2, -5], [10, 8, 0, 2, -1, 6, 7, -1, -10, -1, -6, -2], [-4, -1, -4, -6, 9, 12, -2, -3, -5, 0, -2, 2], [7, -1, -8, -5, -1, 9, 3, -6, -5, -4, -3, 2], [4, -1, -1, -3, 6, 7, -5, -5, 1, 0, -4, 4], [-2, -7, 2, 4, -6, -4, 2, 1, 0, -3, 6, 3], [13, -3, -4, 3, -3, 5, 3, 0, 1, -8, -5, -6], [3, 1, 2, -6, 12, -2, -5, 1, -2, 4, 7, 10], [-9, 10, -4, 1, 1, -3, 1, 3, -4, -7, 5, -3], [5, 11, 5, 8, -2, 0, 3, 4, -6, -2, 10, 3], [-6, -1, -1, -2, 1, -3, -3, 2, -1, -4, 3, -10], [2, 3, -9, -6, 4, -5, -7, 7, 9, -4, 4, -4], [-5, -8, 6, 2, 6, 0, -3, -6, 8, 5, 5, 3], [-1, 9, 4, -8, 1, -8, -6, 2, 3, 0, 2, 3], [-12, 13, 9, 2, 0, -2, -2, 1, 0, -2, 10, 3], [0, -2, -6, 0, -12, 4, 0, 4, -2, -4, -12, -8], [-11, -4, -2, -4, 4, -1, -6, 5, 4, 1, 10, 0], [8, -2, -2, -1, 6, 2, -2, -3, 12, 2, -4, -4], [6, -2, 3, 4, 4, 0, 2, -4, -8, -3, -4, 2], [3, 11, 0, -2, 5, -2, -2, 5, 0, -1, 1, 2], [-4, 5, 9, 0, -4, 3, 1, -1, -1, 2, -2, 4], [-7, 3, 9, -6, -1, -10, 2, 3, 3, 8, 4, 2], [-5, -10, -6, 3, -7, 14, 4, -7, -3, -6, -10, -5], [9, -3, -4, -5, 3, 1, -3, 3, -8, 1, -3, -4], [8, 9, 0, -4, 7, -5, -2, 2, 0, 1, -7, 1], [22, 5, 0, -7, 0, -5, -1, 2, -1, -1, -6, 4], [-3, 7, 0, 2, 8, 3, 4, 1, -16, -6, 1, -3], [4, 1, -6, 7, -5, 9, -4, 2, 5, -7, -6, -7], [5, 2, -8, 3, 10, 1, 0, 4, -8, -13, 2, -2], [-4, -3, -6, -9, 4, 2, -2, 0, -5, 2, -1, 0], [-3, 3, -3, -2, 1, -6, -6, -4, -2, 3, 2, -2], [10, -4, 3, 6, -4, 9, -7, -5, 0, 2, -3, -4], [-8, 11, 1, 3, 2, 3, 1, -2, -13, -3, 6, 9], [1, -9, 1, 2, -4, -5, 6, -4, -5, 2, -7, 3], [-8, -3, 5, -3, 0, 3, 5, 0, -2, -3, 0, 3], [2, -2, 1, 4, 2, 0, -4, 5, -7, -8, 4, -6], [-3, 1, 7, 0, -1, -4, 2, 1, 6, 2, 5, 8], [3, 3, 3, -2, 7, -5, -6, -3, 6, 1, -2, -1], [-15, -3, 2, -2, 0, -5, -2, -1, 4, 4, 1, 6], [2, 5, -3, -10, 10, 0, -4, 0, 4, 3, -2, 2], [6, 1, 10, -5, 5, 1, -2, -6, 6, 9, 4, 7], [6, 8, -5, 6, 3, -9, 6, 6, -7, -8, 0, 1], [2, -2, 8, 7, -4, -5, 5, 0, -5, -2, 5, -1], [-12, -1, 1, -8, -4, 3, 1, 1, 2, 8, 2, 0], [6, -12, -6, -3, -9, 9, 5, -6, -7, 1, -11, -2], [-10, -3, -4, -6, -15, 3, 7, 0, 5, 3, -7, -2], [-5, -6, 3, -6, 1, 9, -1, -14, 8, 3, -4, 6], [14, -3, -1, 2, -2, -8, 12, 6, -4, -3, -2, 4], [-15, -4, 7, -2, 4, -9, -4, 2, 9, 6, 9, -4], [-6, 1, -2, -2, 8, -3, -5, -1, -6, -9, 10, 6], [-4, -5, 0, 9, -4, 7, 3, 1, -4, -3, 3, -6], [6, -8, -7, -3, -2, 0, -1, 1, 5, 6, -6, -2], [-5, -1, -8, -2, 3, -2, -4, 4, 2, -14, 6, -7], [6, 0, -6, 4, 1, -5, 14, 5, -10, -11, 0, 1], [-2, 8, 0, -4, 14, 7, -8, -6, -2, 2, 0, 4], [-8, 6, 3, 1, 10, 1, -7, 5, -8, -2, 11, 2], [-9, -12, -2, -3, -14, 5, 1, 0, 6, -1, -10, -11], [-21, 1, 7, 2, 3, -11, 1, 0, 10, 6, 5, 1], [-12, 17, 11, -6, 13, -16, -9, 0, 7, 4, 11, 6], [-3, -2, -2, 6, -13, 1, -3, 0, 10, -2, -9, -12], [14, 1, 6, 3, 2, -6, 3, 1, -3, 3, -2, 9], [-10, -13, 1, 0, -14, 6, 2, -2, 10, 7, 4, 4], [-10, 1, -2, -4, 2, 4, -4, -7, 1, 6, 1, 2], [12, 1, 2, 5, -4, 0, 2, -5, -10, -1, 1, 8], [8, -3, -3, -4, 6, 4, 4, 0, -3, -7, -2, -3], [-10, -15, 2, 9, -12, 4, 0, 1, -2, 5, -1, -1], [9, 7, 1, 3, -5, -7, 5, -1, -2, -2, -1, 0], [0, -15, -4, 8, -8, 7, 10, -2, -3, -6, -9, -6], [-11, 2, 14, 2, -8, 3, -4, 2, 13, 3, 1, -4], [-12, 8, 1, -4, 8, 8, -2, -2, -6, 1, -4, 4], [5, 6, -5, -1, 2, 2, 13, 4, -15, -3, -5, 0], [-13, -3, -2, -2, 1, 4, -8, 1, 17, 1, -2, -10], [-18, -5, 3, 9, -4, 5, 1, -1, 1, 2, 5, -4], [-2, 1, -10, -4, 10, -1, -7, 4, 2, -2, 6, -3], [5, 9, 5, 8, -12, 1, 6, -1, -10, -5, -3, -1], [-4, -1, 8, 2, -3, -2, 5, -8, 1, 9, 1, 8], [-13, 9, 5, 0, -6, -5, -2, -4, 1, -2, -1, -1], [4, -10, -12, 5, 4, 4, 2, -1, 5, -7, 1, 0], [-14, -1, -6, -2, 2, 6, -2, 3, -7, -3, -6, -7], [5, 2, 8, 2, 6, -2, 1, 7, 1, -4, 12, 6], [-7, -9, -6, 5, 4, -3, 4, -1, -1, -7, -1, -4], [-1, 6, -1, 5, -5, 1, 4, 4, 2, -4, 3, -8], [8, 1, 8, 4, 4, -4, -9, 0, 13, 4, 3, 5], [3, 8, -5, 0, 8, -6, -4, 1, -5, -1, 8, 3], [11, 3, 2, 0, 5, -9, 1, 4, 6, 0, 6, 10], [0, 6, 14, 5, -3, -10, -3, 0, 2, 7, 2, 7], [16, 2, 5, -3, 7, -5, 6, -3, -7, 5, 0, 16], [19, -5, -5, 9, -2, 2, 3, -5, -13, -3, -10, 4], [-11, -10, -14, -1, -5, 2, 4, 5, 8, -7, -7, -10], [9, -4, 1, -4, 1, -6, 2, 9, 3, -6, 6, -4], [-14, -5, 2, -2, 1, -7, -4, 2, 16, -1, 3, -7], [-18, 5, -1, -4, 7, -3, 6, 6, 1, 2, 3, -1], [-7, -12, -6, 0, -4, 3, -2, -4, 0, -7, -1, -6], [-6, 9, 9, 6, -2, 1, -1, -9, 3, -2, -2, -2], [5, 10, 17, -5, 4, -2, -5, -10, 1, 9, -1, 4], [-15, 10, 10, -5, 4, -7, 2, -4, 7, 4, 13, 13], [-2, 13, 2, -12, 10, 3, -13, -2, -4, 4, 6, 10], [-2, -3, -8, -11, 5, 7, 0, -3, 9, -2, -2, 2], [-14, -3, 6, -6, -5, 4, 5, 0, 1, 11, 1, 8], [-8, 4, -6, -9, -4, 3, -5, 5, 10, -3, -3, 0], [14, 1, 4, -1, 4, -1, -4, -4, 4, 6, -4, -4], [-14, 6, 2, 8, -15, -2, 5, 3, 4, -4, -5, -9], [-4, 5, 8, 4, 4, 6, -6, -5, 1, 5, 6, 5], [10, 5, -4, 2, 3, 18, 0, -5, -11, -4, -8, 0], [-14, 11, 4, 3, -2, 11, 1, -10, -2, 6, -2, 3], [0, 8, -3, 4, 5, -4, -3, -1, -12, -4, -2, -2], [2, 16, -1, 3, 12, -5, 3, 4, -9, -1, 1, 7], [-5, -10, 0, -6, 6, -4, -1, -5, 5, 2, 2, 4], [-6, -2, -7, -2, 4, -4, -6, 2, 10, -1, 6, 0], [-2, 5, 3, -5, 3, 5, -4, 0, 6, 6, -2, -1], [-7, -2, -7, -4, 0, -3, 7, 13, -7, 3, 6, 2], [-24, 8, -3, 1, 1, -1, 2, 5, 3, 1, 0, -2], [-22, 1, 6, 9, -9, 5, 0, -12, 6, -2, -7, -4], [7, 5, 2, 5, 5, -3, 3, 1, -2, 1, 3, 5], [10, 9, -3, 6, 12, -3, -5, -1, -3, -8, 0, 0], [-5, -6, 10, -1, -2, -1, 4, -4, 4, 1, -10, -2], [-8, 20, 5, 2, -2, 7, 3, 2, -3, -1, -5, 1], [3, -9, -9, 10, -1, 5, 1, 0, -8, -7, 6, -2], [-10, -4, -6, 1, 4, 0, 0, 8, -9, -1, 4, -1], [-16, 5, -14, -12, 3, -2, -9, 10, 13, -6, 0, -5], [-26, -1, 2, 0, -1, -6, 3, 6, 3, 7, 3, -4], [9, 1, 5, -5, 12, -3, 4, 0, 10, 4, 12, 5], [4, 10, 4, -2, 9, -10, 1, -4, -6, -3, 11, 7], [6, 15, 3, 3, -1, 12, 1, 1, -7, 4, -7, 0], [3, 9, 4, -1, -4, 3, 4, -3, -7, 0, -6, -1], [-15, 13, 3, -3, 5, 19, -9, -1, 2, 0, -3, -4], [14, 11, 5, 4, 11, -3, 2, 1, -20, -3, -1, 2], [-12, 9, -4, -5, 1, -1, 2, 3, 7, -4, 4, 4], [-16, 7, -8, -4, -6, 11, -1, 1, 3, -1, -8, 4], [-7, 15, 6, 3, 2, -14, -3, 4, 4, -5, 14, -4], [1, -6, 8, -9, 1, -3, 3, 2, 4, 6, 1, 7], [-18, -1, 10, 0, -9, 6, 5, 0, -3, 5, 1, 6], [-12, 2, 3, 6, -1, -3, -4, 5, 15, 3, 1, 2], [23, -9, 7, 7, -7, -2, 4, 0, 5, -10, -2, -6], [5, 12, -1, -3, 7, -6, 2, 0, 5, -1, 2, 11], [5, -10, -11, 0, 1, -2, 2, 1, 10, 4, 4, 8], [8, -1, -2, 4, 5, -9, -2, 8, -1, -9, -3, -9], [-23, -7, -2, -1, -1, 4, 0, -4, 5, 6, 5, 11], [15, 3, 9, 2, 16, 3, -6, -4, -13, 0, -4, 12], [-15, -6, -5, 11, -15, 7, 3, 4, 6, -9, 4, -2], [-2, 12, 1, -5, -4, -10, 0, 5, -12, 4, 3, 1], [-11, -12, 10, -2, -3, 6, -4, -13, 13, 6, -1, 1], [-13, -1, 3, 13, 3, 0, -2, -6, -11, -5, 5, 7], [13, 0, -9, 4, -4, 1, -2, 3, 2, -1, -15, -11], [-19, 2, -3, -5, -4, -7, 4, -6, 2, -4, -4, -2], [-7, 6, -8, -2, -8, 7, 0, 0, -7, -6, -8, -1], [9, -8, 9, 8, 7, -2, 5, -8, -15, -7, 3, 2], [11, -2, -4, -2, -2, 0, 2, -3, 3, 12, 6, 7], [-9, -6, -11, 8, -8, 12, 9, -1, 2, -7, -3, -9], [4, -2, -10, -6, 3, -5, 1, 4, -4, -7, -2, 2], [3, -1, 4, -3, 0, -16, -1, 0, 8, -5, 5, -1], [7, -4, 11, 3, -10, 10, 14, -5, 0, 7, -7, 9], [-4, 4, 6, -10, 11, -16, -1, -1, 0, 6, 9, 1], [-12, 6, 4, -8, 3, 6, -7, 3, 4, -1, -4, -2], [-10, 1, -19, -9, 4, 13, -5, 1, -1, -5, -8, -9], [16, -10, -13, -5, 0, -13, -2, 5, -9, -2, -10, -1], [11, 7, 3, 1, 4, -5, -8, 3, 9, 3, -2, 2], [-14, -8, -7, -2, -7, -12, -7, 2, 19, 0, 1, -6], [-5, 1, -4, 6, -12, 4, 2, 2, 10, -7, 6, 2], [6, 9, 1, 3, -8, -8, -6, 1, 7, 6, -2, -3], [-3, -2, 3, 3, 0, 10, 3, -7, 15, 2, 1, -9], [5, -9, -2, 3, 4, -3, 11, -6, -15, 3, -3, 11], [8, 8, -5, 1, -5, 9, 0, 1, 0, -7, -16, -12], [-13, 13, 3, -6, -7, 6, 2, -4, 2, 3, -2, -3], [-15, 5, 7, -1, -2, 6, 7, -6, -11, 4, 3, 9], [-9, 4, 6, -3, -6, -4, 7, -3, -10, -4, 6, 3], [-5, 6, -7, -3, 4, -12, -1, 2, 8, -4, -2, -7], [-8, 0, -9, -2, -8, 11, 5, 7, -13, -4, -11, -2], [-19, 7, 8, 9, 2, 5, -8, -2, -3, -5, 3, -4], [-17, 5, -3, -13, 3, -8, 6, 4, 8, -5, 7, 1], [-19, 0, 7, -9, 0, -4, -1, -6, 10, 14, 16, 13], [-17, 8, -3, -14, 7, -6, -5, 12, 7, -3, 9, 2], [7, 9, 14, 7, 3, 4, 2, 3, -10, 1, 7, 5], [-11, -7, 7, -2, 8, -6, -7, -2, 15, 14, 1, 0], [-3, 14, -9, -3, 10, -2, -1, -2, -4, 0, -8, -7], [4, -1, 1, 9, -4, 5, -5, -6, 3, 7, -3, 0], [2, 4, -10, -7, 11, -2, 3, -1, -6, -2, -12, -3], [3, -4, 13, 0, -8, 5, -10, -4, 5, 16, -1, -2], [-1, -5, 12, 3, -3, -3, -3, 1, 0, 5, 1, -2], [1, 6, -4, -7, 0, -12, 9, 10, -18, 3, 2, -3], [-12, -10, -2, 5, -15, 8, -1, 2, 8, 3, -11, -6], [-3, -11, -8, 8, -2, -6, -3, -2, -1, -1, -3, -6], [-4, 5, 0, -9, 5, 3, 0, -3, -11, 0, 6, 1], [-5, -4, -2, -2, 1, 0, -4, 3, 10, -7, 6, -8], [-13, 8, 12, 7, -11, -11, 1, 5, 4, 1, 3, -1], [0, 8, -2, 8, 8, 1, -1, -2, -8, 6, 6, 8], [5, -9, -7, -9, 16, -3, -5, 0, -1, 8, -1, -1], [-3, 4, -3, 3, 8, 3, 0, 4, 5, -2, -2, -10], [2, 17, -1, 6, -5, -10, 1, 2, 7, -7, 2, 5], [-19, -3, -19, -2, -2, 6, -5, 12, 9, -6, -3, -15], [-12, 31, 10, -9, 17, -7, -7, -5, 4, 8, 5, 9], [0, -14, 2, 9, -4, 8, 4, -2, -3, 4, -1, -6], [15, 1, -7, -4, 11, 16, -2, -10, -10, 1, -8, 5], [-2, -12, -10, -6, 6, 6, 3, 3, -18, -8, 0, -5], [3, 1, 5, 0, -8, -7, -8, -5, 2, -1, -11, -5], [-11, 3, -6, 10, -5, -8, -4, 5, 9, -7, -1, -4], [8, -10, 5, -3, 3, -2, 3, -4, -2, 8, -10, 15], [25, -1, -12, -2, 1, -13, -5, 6, 10, -2, 0, -12], [-19, -12, 6, -1, -4, 6, 6, -3, -2, -1, 4, 8], [7, -6, -1, 1, 0, 7, -10, -6, 9, 2, -4, -4], [14, 7, -5, 4, 4, 3, -7, 3, -7, -4, -8, -10], [7, 7, 0, 5, -3, 3, -1, 1, -14, -2, 10, 6], [15, 7, 2, 10, -13, 0, 4, 7, 3, -9, 6, 3], [-19, 1, 8, -9, 8, -4, -1, -5, -3, 4, 10, 12], [-13, -3, 7, 5, 0, 0, 11, -3, 3, 11, 0, 16], [-12, 13, 24, 6, -3, 4, 5, -2, -15, 3, 1, 2], [13, 10, -13, -5, 8, -10, 1, 4, -3, -4, -8, -7], [8, 19, 16, -4, -6, -1, 4, -1, -13, 5, 4, 11], [19, -9, 13, 9, -6, -6, 11, -8, -8, 2, 0, 1], [-12, -11, -8, 9, -2, -4, -2, 9, 6, 1, 1, 1], [-2, -9, 4, 3, -2, 13, 7, -1, -24, 1, -1, 6], [15, -1, -4, 0, -2, 3, -6, 2, 11, 5, -5, -4], [-10, -1, 10, 6, -14, 13, -9, 0, 2, 4, 2, 2], [-24, -2, 11, -13, -3, 8, -3, -9, 19, 9, 2, 3], [4, -28, -8, 2, 3, -3, 2, 10, 7, -7, -7, -9], [-3, 30, 4, -11, 6, -9, -1, -1, -8, 3, 6, 11], [35, -9, 6, 8, 1, 4, 4, -12, -8, 5, -3, 8], [-1, 16, 1, 0, 6, 3, 2, 3, 2, -5, 5, -3], [-1, 8, 4, -10, -2, 5, -1, 2, 9, 9, 5, 5], [8, -13, -3, -7, 4, 19, 2, -15, -9, 6, -8, 6], [-12, -15, 3, -4, -13, -7, 8, 8, 7, 2, 3, 1], [23, -9, -3, 16, 0, 3, 4, 5, -10, -13, -3, -11], [-19, -4, 0, 8, -9, 14, 4, 3, -5, 4, 1, 5], [6, -10, -8, 5, 5, -4, 7, 2, -8, -15, 6, -4], [-12, -12, 4, 2, -6, 16, -6, -10, 0, 0, -10, -14], [10, 25, -4, -13, 23, -5, -7, 8, 5, -7, 17, 3], [8, -2, -8, -8, 11, 9, -2, -5, 4, 0, -9, -2], [16, 3, -6, 10, 4, -7, 6, 7, -1, -11, -6, -17], [-38, 5, 16, 3, -6, 0, 6, -5, 13, 9, 12, 1], [7, 1, 3, 0, -1, -2, 6, -2, 0, -14, 3, -6], [-13, 11, 16, -6, 19, 3, -5, -13, 1, 9, 10, 13], [-1, -7, 11, 5, -8, 7, 10, -11, 3, -3, -2, -7], [4, -21, -2, 1, -7, 17, 0, 1, 2, -2, -4, 1], [6, 5, 16, 10, -14, 5, 11, -11, -11, 4, -9, 5], [-5, 19, 6, -11, 0, -7, -8, 1, 10, 6, 8, 7], [-12, 9, 12, 7, -14, -12, -1, -4, 16, 10, 6, -1], [18, -2, -4, -6, 4, 4, 5, -5, -14, 3, -11, 0], [-10, -6, -12, 12, 6, -4, 6, 2, -4, -8, 4, -2], [-1, -13, 3, 0, -13, -1, 1, -3, 5, 6, -7, -5], [19, -23, 13, 9, -7, 9, 5, -18, 1, 7, -9, 7], [16, -2, -4, 16, -9, 9, 4, 0, -1, -11, -5, -3], [-11, -6, 2, -6, 3, -9, 7, 0, -1, -2, -5, -4], [7, -17, -8, -10, 0, 8, 5, 0, -5, -1, -11, 0], [2, -12, 8, -2, -6, 17, 4, -8, -10, 4, -8, 6], [-14, 0, 5, 0, 7, 8, -1, 7, -2, -3, 1, 5], [5, -1, -15, -1, 11, -2, -4, 4, 6, -7, 2, -9], [-12, 7, 9, -1, 2, 11, -4, -14, -2, 5, 2, 2], [4, -6, 13, 1, 1, -9, -8, -5, 13, 11, 12, 6], [13, 15, 9, -2, 9, 2, -16, -3, 19, 8, 6, 10], [0, 11, 1, -9, 6, -8, -12, 3, 4, 2, -3, 3], [-6, -8, -5, 0, -4, 10, 4, 5, -1, 10, 6, 2], [-6, 11, 21, 1, -6, 7, 1, -5, 0, 12, 9, 14], [15, -16, 12, 4, 1, -11, 11, 3, 9, 7, 8, 15], [29, -14, -6, 8, -13, -12, 16, 7, 7, -7, 2, 4], [2, -11, -1, -6, 8, -13, 5, -3, 3, -4, -6, -2], [-21, 24, 2, -8, 13, 0, -8, 7, -13, 2, 1, -1], [22, -7, -18, 0, 4, -5, 9, 7, -15, -5, 4, -4], [5, 0, 0, 16, -10, 7, -3, 9, 1, 0, -2, -2], [27, 5, -8, -1, 9, 3, -3, 0, -16, 3, -12, 5], [-3, 1, 1, -5, 4, -17, 6, 5, 13, 5, 18, 15], [27, -2, -3, -5, -4, -3, 2, 4, -2, 14, -4, 12], [-9, -4, -4, -3, -2, -6, 1, 5, 3, 6, 8, 8], [-16, 4, 14, 4, 7, -8, -13, -6, 14, 11, 14, 10], [-13, 14, -9, 1, 7, 9, -11, 1, 12, -2, -5, -3], [-12, -6, -4, -5, 3, -1, -8, 5, 1, -4, 0, -2], [-33, 10, 2, 0, -5, 11, -7, -2, -3, 2, 13, 8], [31, -4, -10, 0, 2, 4, 7, 4, -20, -10, -12, 0], [5, 8, 2, -9, 0, -7, 4, -14, -5, -1, 2, 8], [-17, -7, 0, 2, -9, -5, 1, -4, -6, 7, -5, 1], [14, -2, 0, -11, 0, -6, -12, 4, 8, -2, -1, 2], [-21, 7, 18, -2, -4, -10, -7, 0, 27, 15, 7, 2], [4, -7, 7, -10, 4, -9, -1, 1, -1, 20, -2, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4016_2_a_i();".
// 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_4016_2_a_i(:prec:=12)
    chi := MakeCharacter_4016_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_4016_2_a_i();".
// 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_4016_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![104, -136, -462, 601, 537, -644, -317, 277, 106, -49, -17, 3, 1]>],Snew);
    return Vf;
end function;