
// Make newform 9016.2.a.bo in Magma, downloaded from the LMFDB on 22 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9016_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_9016_2_a_bo();".
// 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_9016_2_a_bo();".
// 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 := [99, -85, -362, 256, 443, -278, -215, 125, 37, -20, -2, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-44740, 52347, 72584, -95955, -29490, 50081, 599, -8424, 784, 433, -63], [82658, -70941, -205766, 140533, 138036, -88017, -24598, 15987, 983, -851, 33], [70172, -62775, -166821, 135122, 92589, -84329, -7547, 15404, -1341, -831, 135], [68311, -46632, -192074, 93629, 155835, -55236, -43000, 9402, 4652, -465, -168], [12364, -81478, -2969, 139195, -27399, -68603, 20568, 11529, -3869, -606, 209], [-32062, 105, 99978, -7243, -97536, 3077, 37135, 50, -5309, -43, 244], [10449, -16143, 23106, 41493, -63246, -29093, 35453, 6002, -5993, -366, 303], [8358, 85414, -74938, -199549, 127406, 130730, -75388, -25055, 13394, 1422, -703], [9219, 90179, -97940, -210950, 171361, 143522, -97973, -28140, 17090, 1623, -887]];
    Rf_basisdens := [1, 1, 2147, 2147, 2147, 2147, 2147, 2147, 2147, 2147, 2147];
    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_9016_a();"
function MakeCharacter_9016_a()
    N := 9016;
    order := 1;
    char_gens := [2255, 4509, 1473, 1569];
    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_9016_a_Hecke(Kf)
    return MakeCharacter_9016_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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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], [-2, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0], [-1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, -1, 0, 1, 0, -1, 1, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, -1, 0, 0, -1, -1, 1, -1, 0, 1], [0, -1, 1, 0, -1, 0, 0, 1, 0, 0, 0], [1, -2, 0, -1, 0, 0, 0, 1, -1, 1, -1], [-2, 2, -1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 1, -2, -1, 1], [2, -2, 1, -1, -1, 2, 1, -1, 1, 0, -2], [0, -1, 2, -1, -1, 0, -1, -1, 3, 0, 0], [-3, -2, 2, 0, -2, 1, 0, -2, 3, 0, -1], [0, -1, 1, 0, -1, 1, 1, -2, 3, -1, -2], [-3, 1, -2, 2, 1, 0, 0, 0, -1, -2, 0], [1, -2, 1, 0, 0, 0, 1, 1, -2, 2, 2], [1, -2, -1, 1, 1, -1, 0, 1, -2, -1, 1], [-3, 0, 1, -1, 0, 0, 0, 1, 1, 1, 1], [-2, -3, 0, 1, -1, 0, 1, 0, -2, 2, 1], [-1, 2, -2, -1, 0, -1, 0, 0, -2, -1, -1], [-2, 0, -2, 0, -1, 2, 1, 0, -1, 2, -1], [0, -2, 1, 1, 1, -1, 0, 0, 0, 1, 1], [0, -1, -1, 1, 0, 0, 0, 0, -1, 2, -1], [1, 2, 1, -1, 1, 0, 0, -3, 3, -1, -1], [3, 0, 0, 1, 1, -2, -1, 0, 1, 0, 1], [-2, -2, 0, 0, 1, -2, 1, 1, -3, 0, 3], [-2, -1, -1, 2, 0, 1, 2, -1, 0, 0, 0], [-1, 1, 2, -2, -1, 1, 0, -3, 6, -1, 0], [1, -2, 2, -1, 1, -1, 1, -1, -1, 0, -1], [-3, 1, -2, -1, 1, -1, -1, 0, -2, -2, 2], [2, -2, 0, -1, -1, 1, -1, 3, 1, -1, -1], [-3, -1, 0, 1, 1, 1, -1, 1, 0, 1, 1], [-2, 2, -3, -1, 1, -1, -1, 0, -1, 0, -1], [-6, 1, 3, 1, -1, 2, 1, -1, 3, 2, 2], [-7, -1, -2, -2, 1, -1, -1, 2, 0, -1, -1], [2, -2, -2, 0, 0, 1, 1, 1, 1, 3, 1], [-3, 1, -1, 1, 0, -3, -1, 1, -2, 4, 0], [0, -3, 0, 2, -1, 1, -1, 1, -3, 0, 3], [6, 0, -2, -1, 0, 0, 0, 1, -4, 1, 0], [-2, -1, 0, -1, 0, 2, 2, 1, 0, 0, 0], [2, -4, -2, 1, 2, -2, 0, 4, -6, -1, 2], [-2, 2, 2, -3, 0, -1, 0, -1, -1, 0, -2], [1, -4, 4, -1, -4, 2, 1, -3, 1, 1, 0], [1, 1, -2, 0, 1, 0, 1, -3, 1, -1, 0], [-5, -1, 2, 1, 0, -3, 1, -1, 2, 0, -1], [3, -2, 2, 1, -2, -3, -1, 0, -3, -1, 1], [-4, 1, -2, -3, 2, 0, 0, 0, -3, 0, -1], [-4, 5, 3, -1, 1, -1, 0, -1, -1, -2, -2], [-8, 0, -2, 2, 0, 2, 0, 1, 2, -2, -3], [-3, -5, -3, -1, 2, -5, -2, 2, -4, 1, 2], [6, 0, -2, 2, 0, 2, 0, -1, -1, 3, -1], [-6, -1, 3, 0, -2, 2, 1, -2, -1, -1, 1], [5, -4, 2, -1, 1, -1, 3, 1, -2, 2, 0], [1, 1, 3, 2, -3, 1, -1, 1, 1, -1, 4], [6, -5, -1, 2, -1, 0, 2, 2, -2, 4, -1], [4, -2, -1, 2, -1, -3, 1, 2, -4, 2, 2], [-10, 2, -1, 1, -2, 3, -1, -1, 2, 1, 0], [-1, 1, -2, -2, 2, -1, -1, -3, -2, -3, 1], [4, -1, 2, -2, -3, -2, -1, -3, -1, 0, 3], [-2, -2, -3, 2, 3, -1, -1, 6, -6, 0, -1], [-6, 2, 4, -1, -2, 4, 1, -5, 4, 1, -3], [3, 2, 0, -1, 2, -3, 0, 1, -2, 0, 0], [10, -2, 1, -4, -1, -3, -2, 0, -1, 2, -2], [8, -6, 3, 0, -2, 2, 1, 0, -3, 3, 2], [-11, -2, 10, -2, -1, 4, 3, -4, 8, 0, -3], [-6, 2, 3, 0, 1, 1, 2, -2, 3, -2, -1], [-1, -1, 2, 1, -3, 6, -1, -3, 2, 2, -2], [-1, -1, -2, 0, 4, 2, 2, -1, 0, 0, 1], [-2, 0, 0, 0, 3, -4, 0, -1, -1, -3, -2], [-3, 4, 3, 2, 2, 2, 2, -1, 1, 1, 2], [2, 1, 2, 0, 2, -5, -1, 0, -2, 1, 0], [-2, 4, 2, 3, -1, 4, 1, -2, 2, 1, 1], [-8, 2, -2, -2, 1, -7, -2, 4, -7, -1, 3], [-1, -4, -2, 2, -3, 1, 2, 1, -1, 6, 2], [-6, 0, 4, -1, -2, 3, 0, -5, 2, -3, 0], [-4, -6, -4, -2, 0, -3, 0, 2, -4, 1, 1], [-5, 2, -2, 3, 3, -1, -3, 1, -2, -4, 1], [4, -1, -5, 3, 5, 1, 0, 4, -3, 0, 0], [1, -2, -2, 2, 6, -7, -1, 5, -7, -1, 5], [2, 3, 1, 5, -2, 2, 1, 0, 1, 4, 3], [-8, 5, 5, -2, 1, 5, -1, -1, 9, 1, -1], [2, 5, -6, -2, 2, 0, 0, -1, -5, -2, -1], [-2, 6, 3, -2, 4, 1, 0, -1, 3, -2, 1], [-7, 0, 0, 1, 4, -1, -2, 2, -4, 1, -2], [-2, 2, -6, 2, -2, -3, 0, 1, -1, 0, -1], [0, 0, 4, -2, -2, 4, 3, -3, 5, 4, -1], [-8, -6, 0, 2, -2, 4, 2, 2, -2, 3, -3], [10, 3, 5, 2, -3, 0, 0, 0, -1, 3, 4], [-4, 3, 3, 2, -2, 2, -2, -2, 4, 2, 0], [-3, 7, -5, 1, 0, -4, 2, 1, 0, -4, 1], [-3, -1, 1, -1, -2, 7, 1, 0, 2, 0, -3], [-10, -3, -5, 3, 4, -8, -2, 6, -8, -1, 2], [-2, -2, -1, 2, -1, -3, 0, 2, -4, 7, 4], [2, 2, 6, 0, -4, 4, 2, -5, 10, -2, -2], [-3, 2, 4, 0, -3, 3, -1, 0, 4, 5, -1], [-2, 1, 1, -3, -2, -3, 2, -1, 1, 0, -2], [3, 4, -1, 0, 1, -4, -4, -3, 0, -4, 1], [-4, 3, 6, 0, -3, 0, 0, -1, 4, 3, 2], [-7, 10, 2, -3, -1, 4, -1, -4, 4, -3, -2], [-2, 1, -5, 7, -2, -2, -3, 1, 3, -2, 0], [1, -1, 3, -2, -3, 1, 0, 1, 2, 6, 0], [0, -5, 1, 0, -4, -2, 2, 0, 3, 4, 1], [-6, -7, -4, -4, 4, 1, 3, 1, -5, 1, -3], [-1, 2, 1, -1, -1, 5, 1, -4, 9, -7, -6], [-7, 0, 4, -2, -4, 0, 2, 0, 8, 1, 1], [16, 3, -6, -1, 1, -4, -4, 4, -7, -1, 1], [9, -4, -1, 1, -1, 0, -1, 0, -6, 3, 1], [11, 0, 1, -3, 0, -1, -1, 2, 0, -4, 1], [-3, 1, -5, 3, 1, 1, 0, 4, 3, 4, -2], [-8, 8, 1, 2, 0, -4, 0, 4, 1, 5, 4], [-7, 4, -4, 4, 3, -7, 4, 4, -7, -1, 4], [3, -8, -6, 2, 1, -8, 0, 3, -8, 4, 1], [0, 0, -4, 2, -1, 6, 0, -1, -1, -2, 0], [-1, 2, 2, 0, 5, -2, -3, 2, -5, -2, -1], [0, 10, 3, 0, -2, -2, -3, 0, 7, -2, 2], [-7, 5, -2, 2, 0, 4, 0, 3, -4, 1, 2], [-5, -1, -2, 5, -4, 5, 3, 3, 0, -2, 1], [-3, -8, -2, 5, -4, 4, 2, -2, -1, 3, 2], [0, 1, 6, -3, -1, 1, 4, -3, 1, -2, 2], [14, 4, 4, -2, -2, -2, -2, -4, 3, -1, -4], [8, 3, 4, 0, 4, 4, -2, 1, -4, -2, 3], [-2, -1, 4, -3, -2, 1, 2, -3, 4, 0, -4], [-12, 3, -2, -1, 3, -3, -4, 3, -2, -5, 0], [-1, 7, -1, 3, -3, 2, -3, 0, 6, -4, -3], [-9, -2, -2, -3, 4, -3, 2, -5, -4, -6, 1], [6, 14, -3, -2, 3, -3, 1, 1, 0, -4, 0], [-2, -2, -1, -1, 2, 0, 1, 2, -11, 4, 1], [4, 0, -2, -2, 4, -1, 1, 0, 3, -1, -3], [6, -3, -3, 4, 1, 5, 2, -3, 2, 5, -1], [-16, 2, 1, 2, -3, 4, 2, -2, 1, 1, 2], [-4, 7, -8, 2, 5, -5, -2, 2, 4, -4, -2], [0, 2, 2, -3, -2, -4, 2, -7, 3, -2, 0], [4, 0, -2, 1, 2, -3, -1, 4, -2, 2, 7], [-3, -3, 0, -5, -1, -4, -2, 3, -3, -1, 2], [-9, 3, 7, -7, -7, 3, 3, -6, 11, 4, -7], [7, -9, 4, 2, 1, 2, 2, 4, -3, 6, -2], [-6, 1, 1, -1, -5, 3, -1, 3, 3, 5, 2], [6, -2, -1, 1, -5, 0, 3, -1, 8, 5, -3], [8, -4, -6, 1, 2, -3, 1, 4, -8, 2, 1], [-9, 0, 1, -2, 4, -3, -1, 9, -6, -1, 3], [-4, 8, 10, -1, 2, -4, 3, -5, 5, -1, 0], [-14, -1, -1, 2, -2, 1, 0, -1, -1, 0, -7], [-9, 7, -6, -1, 2, -2, 1, -2, 1, -5, -8], [10, -1, 2, 2, 2, -2, -3, -2, -1, -3, 4], [3, 3, 4, -1, -4, 6, -2, -3, 6, -1, 2], [0, -4, 10, 1, -3, 0, 3, -2, 3, 4, 7], [11, -8, -1, 0, 2, -2, -2, 8, -2, 3, 1], [-1, 5, 3, -5, 1, -5, 1, -7, 5, -2, -3], [8, -2, -3, -1, 2, -5, 1, 0, -4, -3, -4], [13, 10, 8, 1, -6, 0, -1, -6, 3, 1, 4], [-4, -3, 6, -7, -3, 8, 2, -12, 5, -2, -3], [1, 1, 4, 0, 5, -4, -1, -1, 5, -6, -1], [-4, 2, 7, -3, 2, 4, -1, 0, 6, -2, -3], [7, -4, -2, 2, -6, 1, 0, -1, 5, -1, 0], [-1, 8, -8, -5, 4, -8, -5, 3, -5, 1, -1], [-5, 13, -3, -4, 0, 5, 0, -6, 2, 1, -3], [0, 3, 0, 4, -3, -2, -3, 0, 9, 1, -1], [0, -3, 2, 0, 1, -10, 1, -1, -1, 3, 0], [-15, 7, -1, 4, -4, 7, -1, -2, 7, -6, 1], [-9, 5, 8, 0, 2, 4, 4, -3, 5, -1, -2], [-11, 7, 3, 0, -2, 3, -2, -1, 6, -10, -2], [1, -6, 7, -3, -3, 3, -1, -4, 6, -2, -4], [-9, -2, -3, 0, 2, 0, 3, 6, 4, 6, -2], [4, 3, 2, -4, -4, 1, -2, -3, 3, 2, -1], [-6, 13, 3, -3, 3, -3, -3, 0, 7, 0, 1], [-6, 0, -4, 1, -5, 7, 0, 0, 7, 7, -3], [4, -10, 2, -2, 4, -8, 0, 4, -3, 4, 2], [2, -6, -2, 4, 1, 1, 0, -1, -1, 1, 1], [-1, -5, 1, 3, -1, -1, -2, -3, -2, 0, 1], [-7, 1, -6, 6, 9, -6, -5, 1, -4, -2, 1], [1, -3, 2, 5, -5, 10, -1, 0, 4, 1, 6], [8, -1, -6, -5, 4, -11, -3, 7, -3, -2, 0], [0, 4, 5, -2, 2, 1, 2, -7, 6, 1, 2], [-5, 7, -3, 1, 1, 4, 0, -2, 11, -8, -3], [-8, -9, 8, -2, 2, 3, 3, 1, -5, 5, -1], [-4, -6, 4, -2, -7, 4, 0, -4, 4, -3, -4], [3, -1, 11, -4, -7, -1, 3, -4, 3, -1, 3], [-3, -12, 1, -1, 3, 2, 2, 0, -4, 1, 2], [-12, 10, -3, 3, 2, 0, -1, 3, 0, -8, 2], [-9, -1, 2, 3, 1, -7, -2, 8, -12, 3, 7], [-14, 0, -6, -1, 12, -9, 1, 4, -3, -6, 0], [7, -4, 6, 3, -4, 9, 4, -8, 7, 1, -6], [-8, 1, 6, -8, 4, 3, 2, -5, 4, -5, 0], [10, -4, -5, 3, -1, 10, 1, 8, -9, 2, -1], [7, 3, -1, 7, 0, -4, -6, 8, -12, 1, 12], [10, -9, -6, -1, 6, -7, -2, 4, -8, -4, -4], [1, 2, -6, -1, -1, 3, 0, -4, -3, -3, -3], [2, -5, 7, -2, -3, -2, 0, -2, 2, 5, -1], [-13, 0, 4, 0, -6, 2, -7, -2, 6, -2, 1], [3, -2, 0, 1, 6, -2, -1, 11, -10, 4, 4], [-15, -4, -4, 7, 0, -4, 1, 3, -1, 2, 2], [-8, -7, -4, 0, 4, -6, -3, 1, -6, -6, -4], [2, 2, -6, 5, 4, -6, -3, 1, 0, -7, -5], [8, -4, -5, 0, -7, 5, 0, -5, 6, -2, -5], [-6, -11, -1, -2, 4, -1, -2, 5, -2, -1, -1], [5, 2, 6, -1, -1, -1, 0, 6, -6, 3, 5], [7, 3, 2, -1, -3, 1, 2, 2, 7, 6, 5], [-5, 4, -4, 0, 0, -9, -2, -5, 6, -2, -1], [0, 7, 0, -4, -5, 7, 1, -7, 7, -2, -3], [-2, 0, 7, -3, -6, 0, 0, -1, 7, 3, 2], [8, 7, -2, 4, 5, -6, -4, 1, 9, -11, -1], [7, 0, -5, -3, 0, 3, -1, 0, 5, -1, -2], [16, -9, 11, -2, -5, 7, 5, -3, 0, 6, -2], [1, 6, -1, 3, 3, -8, 1, 8, -2, 9, 2], [8, 2, 4, -3, -12, 7, 0, -8, 6, 5, 3], [0, 4, -7, 2, 8, 1, -3, 6, -8, -5, -1], [-4, 4, -3, -3, -2, 0, -3, 0, -1, 3, 2], [2, 2, 0, 0, -7, 3, 3, 4, 9, 8, -5], [-19, 6, 5, -1, -5, -2, 1, 4, 5, 6, 7], [-10, 5, 4, -6, -4, 9, 4, -4, 9, 4, -6], [21, -7, 6, -8, -6, 9, -1, -4, 11, 4, -6], [-5, 0, 5, 7, 0, 1, -3, -3, 9, -3, -1], [-2, 3, 12, -4, -6, 8, 4, -3, 4, 6, 3], [-7, 8, -3, -2, 0, -4, 1, -6, 3, -6, -3], [4, 1, 12, -5, -8, 6, 3, -14, 19, 0, -8], [14, -3, -8, 1, 3, -5, 1, 0, -3, -3, 3], [8, 1, -1, 1, -1, -8, -2, 6, -13, -1, 10], [1, -1, 4, -4, 3, -3, 2, -7, -1, -8, 7], [4, -5, -11, 6, 5, 0, -1, 7, -19, 0, 0], [13, 1, -12, 3, 8, -2, -1, 5, -9, -4, -1], [-6, 5, -7, -1, 2, -4, 2, 1, -2, -4, 1], [12, -7, 2, 0, 0, 0, 3, 4, -8, 0, 3], [-12, 7, -17, 4, 8, -9, -4, 6, -7, -5, 0], [18, -7, 3, 0, 1, 0, -5, 2, -3, 4, 3], [-17, 1, -2, 2, 4, 0, -4, 1, -5, -8, 1], [-3, -1, -12, 6, 5, -6, 1, 13, -11, 2, 9], [14, -16, -7, 3, 5, -2, 0, 7, -6, -1, -1], [-1, -8, 4, 3, -6, 7, 3, -9, 6, -5, 0], [-13, 1, -6, 4, 3, 4, 1, -6, 0, -3, 5], [5, 4, 3, 2, -3, 11, 1, -2, 7, 2, -3], [-9, 1, -5, -3, 5, -6, -6, 4, -10, 0, 1], [14, 2, -1, 4, -1, -8, 3, 1, -4, -1, 6], [-7, -15, -6, 7, 5, -5, -1, 6, -5, -5, -1], [11, 19, -4, -2, 0, -3, 1, 2, -7, 1, 3], [-11, 11, -4, 7, -4, 0, -3, -2, 13, -4, 0], [-27, 2, -3, 3, 2, 3, -1, -1, 0, -6, 1], [15, -14, 0, -5, -2, 3, 2, 4, 0, -4, -4], [17, -6, -2, 0, 0, -3, 5, 0, 4, 8, -3], [6, 8, 2, -2, 8, -3, 2, 2, -7, -3, 6], [8, 14, -3, 0, -1, 9, -3, -1, 4, -1, -3], [-7, 6, 0, -10, 4, 3, 0, -5, 1, -5, -10], [10, 15, -5, 5, -2, 3, 4, 0, 7, 0, 2], [-15, 4, -5, 0, -3, -1, -2, -2, 11, -2, -4], [6, -4, 5, -3, 3, 0, 3, -1, -7, 2, 1], [6, -18, 5, 3, -2, 6, -1, 5, -2, 4, 5], [-4, 4, 12, -3, -8, 5, 0, -3, 1, -5, 1], [1, 13, -4, 1, 0, -9, -4, -5, 8, -6, 0], [-8, 3, -3, 2, -6, 3, -4, 0, 7, 2, -2], [21, -5, 5, -5, -1, -4, -5, 5, -1, 0, 5], [-15, 7, 7, -1, 0, 6, 4, -8, 1, 2, 1], [14, 10, -1, 4, 1, -10, -1, 1, -1, -11, 7], [0, 3, 6, -1, -5, 4, 0, -10, 3, 5, -4], [27, -1, -5, 1, 3, -9, -3, -1, -11, -6, 5], [-4, 2, -6, 3, 7, -2, -3, 2, -6, 2, 7], [37, -3, 7, 3, -8, 8, 2, -2, 10, 7, -2], [17, -9, 0, 3, 1, -1, 1, 1, -15, 4, 2], [4, 14, 2, -3, -2, 13, 2, -1, 8, 2, -1], [-13, -6, -7, 2, -1, 5, 5, -2, -1, 7, -3], [-14, 5, -5, 5, 2, 2, -3, 6, -6, 5, 2], [-11, 9, -5, 6, -3, 12, -2, -2, 1, -3, 1], [3, 2, -4, 1, -2, -5, -3, 5, 7, -2, 2], [-9, -4, 3, 2, -3, 10, 0, -2, 1, -2, 0], [15, -4, 1, -3, 0, -3, 0, 3, 0, -1, -6], [5, 6, -4, 1, 2, 3, -6, -4, 0, -5, 4], [-7, 13, 4, 4, -5, 4, -2, -13, 10, -11, 0], [0, -2, -2, -2, 1, -8, 2, -4, -2, -7, -5], [5, -15, 1, -3, 2, 1, -3, 5, -3, 14, -3], [-11, 1, -10, 8, 7, -4, 2, 2, -6, 2, -5], [-5, -1, -2, -2, 3, 5, -1, 5, 0, 7, -4], [21, 12, -15, -6, 4, -9, -2, 6, -12, -3, 1], [13, -3, -3, 2, -1, 5, 0, -5, 7, -1, -1], [-2, -9, -3, -10, -3, -2, -5, 1, -3, -3, -4], [-15, 0, -1, 6, -2, -8, 0, 5, -4, -2, 4], [14, -9, 14, -4, -2, 14, 1, -3, 5, 3, 2], [-11, 4, -1, 3, -5, 2, 0, 1, -5, -2, 2], [-26, 11, 1, 11, 2, -7, -5, 6, -4, -1, 7], [-5, 1, -12, -2, 10, -6, 1, 5, -16, -7, 6], [7, 0, 5, 1, 4, 3, 2, -1, -4, -2, 3], [-3, 2, -6, -6, 4, 7, 4, -4, -1, 3, -4], [10, -6, -7, 7, 10, -13, -5, 16, -10, 1, 10], [-3, -9, -2, 2, 6, 6, 1, 3, 9, 5, 0], [-5, 8, -5, -6, 7, -1, 2, -6, -6, -6, -7], [1, -2, -9, 1, 10, -11, -8, 9, -18, -4, 5], [13, 8, -14, 2, 9, -5, 0, 10, -13, -2, 3], [19, 6, 2, -3, 1, 4, 0, -4, -1, -1, -2], [-2, -3, -11, 1, 5, 0, 1, 8, -10, 5, 3], [-14, 4, -3, -10, 7, -5, 0, 6, 0, -1, -5], [7, -1, -8, 9, 1, -17, -3, 8, -6, -4, 5], [1, 3, -14, -3, 1, -8, -5, 2, -1, -8, 1], [-9, 1, 1, -1, 11, -10, 0, 4, -9, 4, -4], [7, -10, -2, -4, 7, -5, -2, 2, -10, 3, -8], [-1, 9, 9, -1, -3, 13, -2, -3, 11, 0, -4], [-3, 4, 0, 2, 2, -14, -1, -4, 2, -10, -1], [-2, 3, 0, 2, -3, 11, 2, -4, 8, 2, 0], [8, 0, -10, 5, -1, 8, -3, 7, -8, 5, 2], [4, 5, 2, -7, 1, 1, 1, -10, 14, -1, -7], [-1, -3, 4, 6, 0, -12, 0, -2, 3, 3, 4], [-6, 5, -8, -2, -2, 4, 4, -5, -7, 1, -2], [12, -13, 4, 7, -8, 12, 2, -1, 13, 11, -4], [-2, 6, 8, -2, -6, 10, 3, -7, 16, -8, -7], [8, -10, -5, 5, -6, 3, 1, -3, 0, 6, -1], [-12, 5, 8, 0, -7, 5, 4, -6, 9, 7, 3], [-20, -4, 1, -5, 2, 7, 2, 0, 1, 2, -5], [-18, 4, -2, -1, -4, -3, 5, 3, 10, 1, -2], [-14, -4, 0, 6, -3, 4, -2, -6, -3, 1, 11], [-2, 0, 5, 0, -5, -1, -2, -4, -2, -6, 7], [1, 2, -1, 2, -3, -2, -2, -5, 9, -4, -7], [-10, 9, 1, 13, 3, 2, -3, 6, -3, -3, 8], [13, -6, 0, -12, 8, -11, 1, 3, -18, 8, 0], [15, -2, -2, 2, 3, -1, 1, 0, -7, 3, 1], [13, -12, 5, 1, 3, -12, -7, 3, -5, 0, 0], [-18, 3, -2, 1, 9, -2, -5, 5, -8, -2, -3], [-9, 2, 4, -1, 8, -5, 1, 0, -2, -9, 3], [-3, -1, -3, -6, -1, -8, -2, 5, 3, 0, 3], [18, -8, -1, -2, 7, -8, 2, 7, -7, -2, 0], [-15, -3, 0, 1, 11, -3, -1, 0, -5, -2, -4], [-18, 7, -9, 3, -4, 7, 2, 2, -2, 10, 3], [-11, -10, -2, 0, 8, -2, 1, 12, -12, 5, 1], [3, 4, 1, 6, -4, 1, 0, 8, -1, 13, 1], [-9, -16, 1, 3, -5, 4, -2, 1, 4, 1, -3], [27, -5, 8, -7, 2, 3, 5, -5, -2, -6, -2], [-5, 0, 6, 3, -7, 13, 5, -10, 24, -1, -11], [1, 10, -12, 7, 3, -7, 2, 9, -18, 2, 3], [0, -5, -3, 2, 0, 15, 6, -11, 3, 3, -7], [-9, -7, -8, 6, 2, 2, -2, 5, 15, 0, -1], [5, -10, 15, -4, -9, 11, 3, 0, 6, 4, -4], [6, 4, 1, -9, 1, 6, 1, -9, 2, 3, -7], [-9, 1, -6, 9, 4, -2, 6, -4, -7, -9, 1], [-9, -7, 4, 4, -5, -3, 2, -3, 2, 10, -6], [26, -3, 1, 2, 6, -3, 0, 8, -14, 9, 2], [4, -4, 7, -1, -2, 3, 2, -1, 9, -4, 3], [-5, 5, 1, 1, -1, 2, 5, -7, 15, -7, -5], [-11, -6, 0, -7, 3, 0, -1, -7, 10, -7, -4], [2, -5, 1, 5, -2, 14, 2, 0, 2, 2, 0], [11, 8, 8, 3, -2, 6, -1, 1, 15, 5, -7], [3, 5, 3, -8, -3, -1, -3, -3, 15, -2, -2], [10, 13, 8, 3, -7, 15, 6, -5, 12, -1, -6], [2, -13, 11, 2, -11, 0, 2, -11, 12, 0, -5], [16, 7, 7, -11, -1, 18, 4, -9, 15, 2, -4], [-6, -10, 7, -2, 0, -1, 0, 4, 2, 0, 2], [4, 1, -13, -1, 2, -16, -3, 11, -4, -1, 1], [-24, 10, 11, -9, -1, 9, 1, -10, 17, -4, 2], [-28, -2, -2, -7, -1, 11, 1, -7, -4, -4, 1], [5, -2, -1, -3, 3, -12, 7, -2, 5, 1, -9], [-28, -13, 8, 5, -4, -5, -1, -3, -5, 0, 4], [-2, 3, 4, -9, 1, 12, 3, 3, 0, 8, 1], [-23, 10, 8, 6, -6, 14, 4, -6, 19, -1, -1], [-2, 7, -3, -2, 1, -9, -1, 6, -14, -4, 3], [16, -9, 5, -12, -2, 1, 4, 1, 2, 0, -1], [26, 17, 13, -4, -8, 14, -1, -6, 11, 1, 2], [-7, 0, -4, -1, -1, 3, 5, -13, 6, -6, 1], [-13, 15, 12, -6, -3, 15, -1, -7, 15, 2, -7], [-23, 1, -17, 8, 10, -1, -3, 9, -19, -4, 3], [-2, 5, -5, 5, 8, -8, -4, 3, -7, 3, 1], [33, -3, -3, -1, 0, -5, -2, 1, 2, -5, 2], [-4, 8, -1, 0, 2, -19, -5, 6, -5, -4, 9], [-14, 3, 3, -1, -1, 1, 2, -7, 7, 4, 4], [5, -6, 14, -4, -8, -7, 1, -6, 12, 7, -4], [-5, 4, -5, 2, 8, -1, 7, -1, -9, 1, 3], [3, 11, -5, -6, -4, -3, 3, -4, -3, -1, -5], [10, -1, 16, -9, -6, 12, 2, -18, 18, -6, 2], [-3, 0, 4, 2, -1, 2, 5, 2, 2, 7, 0], [3, 8, -5, -5, 0, 5, -9, -2, 0, -8, -6], [-35, -4, -8, 0, 5, 7, 3, 0, 0, -2, 3], [9, -19, -5, 4, 3, -3, 5, 3, -14, -4, 4], [15, 18, -3, 5, 3, 0, 4, -6, 6, -9, 0], [12, -9, 6, 1, -5, 0, 7, -3, 5, 0, -1], [17, -11, -11, -3, 0, -1, 1, 2, -4, 8, 0], [12, -3, 17, -1, -8, -10, 0, 1, -2, 0, 2], [-16, 11, 4, -6, -3, 2, 3, 1, 4, -6, -5], [-22, -5, -1, 6, 4, -1, 5, -1, -5, -5, -3], [9, 19, -8, 6, -8, -13, -6, 6, 10, -7, 4], [6, 20, -17, 0, 6, -4, -3, -11, -1, -14, 1], [31, -4, 0, 2, 3, -11, -2, 2, 2, -13, -3], [6, 3, 5, -4, -4, 10, -4, 5, 7, 9, 2], [17, -10, 0, 11, -3, -5, 1, 4, -6, 7, 6], [5, -1, -4, 0, -11, 9, 3, 4, 1, 4, 5], [28, -17, -5, 1, 0, 12, 4, 10, -7, 8, -1], [-8, 18, 8, -1, -6, 10, 7, -21, 26, -2, -4], [-6, 5, 5, -7, 6, 2, -1, 0, 10, 5, -4], [3, 5, 0, -2, 9, -2, -3, 6, -3, 2, -2], [-24, 8, 11, -2, 4, 1, 0, -2, 15, -8, -3], [14, -1, 9, 0, -8, 11, 4, -12, 15, -1, -11], [-9, 14, 4, -12, -5, 7, 1, -8, 0, 4, -3], [-7, -4, 1, -8, -4, -4, -2, -10, 11, -7, -3], [7, -4, 7, -9, -3, 7, -5, 2, 1, 5, -3], [-22, 6, 2, -1, -1, 12, 0, -1, -3, -3, -4], [7, 3, -5, 4, -6, 3, -3, 0, -4, 11, 1], [15, -3, 4, 2, 5, -10, -1, 6, -3, 10, 2], [-16, -3, -3, -6, 5, -8, -1, -3, -6, -1, -1], [-7, 14, -15, 5, 9, 0, 3, 8, -16, 2, 0], [-9, -1, 0, 7, 0, 13, -3, 1, -6, 1, 9], [25, 0, -5, -1, -3, -10, -11, 1, -2, -3, 0], [-3, -2, -3, -7, -2, -7, -2, 4, -7, -3, 8], [22, 12, 5, -7, 0, -12, 1, 4, 6, 5, -5], [23, 6, -6, 1, 7, -14, -7, 7, -2, -10, 2], [2, 6, 7, -2, 1, -5, -4, 7, 11, 6, 3], [9, 2, -1, 1, 5, 6, 4, -1, 5, -5, 1], [4, 7, -4, 4, 2, 9, 9, -1, 6, 5, 0], [-11, -1, 3, -5, -2, -11, 2, -2, -4, -1, 1], [4, -8, -8, -1, 2, -12, -5, 6, -24, 0, 12], [1, 8, 3, 1, -10, -1, 1, 2, 16, -4, 0], [10, 3, -8, -5, 6, -10, -4, 7, -20, -5, 5], [-14, 7, 8, -1, -7, 15, -1, -6, 16, -5, -11], [-12, 9, -11, 10, 2, -10, -3, -2, -5, -2, 11], [1, -16, 0, 2, -5, 12, -3, 0, 15, 3, -9], [-25, 2, 4, 8, 2, 6, 3, 5, -8, 2, -7], [22, 8, 0, -1, 7, -1, -4, -6, 7, -15, -10], [19, -9, -14, 6, -1, 0, -10, 13, -22, 6, 8], [-14, -12, 8, -2, -4, 7, 0, 0, 5, -5, -7], [-7, 4, 8, -3, 1, 9, 2, -8, 11, -4, 4], [17, 11, -2, -1, 4, -3, -3, -5, -2, -2, -4], [27, -2, 12, -5, -2, 8, 2, -2, 13, 4, 1], [-5, -3, 1, 1, -3, 12, 5, -1, 13, -8, -6], [-16, -14, 11, 1, -6, 7, 7, -6, 7, 9, 7], [-22, 11, 5, -2, -6, 9, 6, -14, 10, -4, 4], [-16, 19, -2, -2, 6, -2, -1, -5, 5, -1, 2], [26, -3, 6, 3, -8, -3, -1, -1, 1, -1, -4], [7, 14, -8, 2, 2, -18, -3, 8, -6, -6, 2], [3, 8, 2, -1, -8, 9, 4, -7, 13, -6, -5], [22, 3, -9, -2, -2, 6, 1, -7, 6, -3, -3], [15, -4, -8, 2, 5, -2, 1, 17, -21, 2, -1], [6, 6, 2, 9, 7, -3, -2, -2, -2, -5, 8], [10, -7, -11, 3, 4, -8, 1, 2, -15, -5, 5], [7, -20, -3, 3, 3, 3, 1, 0, -13, 4, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9016_2_a_bo();".
// 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_9016_2_a_bo(:prec:=11)
    chi := MakeCharacter_9016_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_9016_2_a_bo();".
// 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_9016_2_a_bo( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![99, -85, -362, 256, 443, -278, -215, 125, 37, -20, -2, 1]>,<5,R![-361, 1368, 573, -4059, -1807, 2109, 944, -308, -172, 2, 9, 1]>,<11,R![1349, -2948, -1171, 4963, -320, -2626, 269, 570, -23, -45, 0, 1]>,<13,R![-188779, -374796, -193348, 82893, 119926, 37468, -2484, -3654, -589, 30, 15, 1]>],Snew);
    return Vf;
end function;