
// Make newform 8008.2.a.q in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8008_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_8008_2_a_q();".
// 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_8008_2_a_q();".
// 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 := [-26, -69, 252, -53, -201, 64, 45, -15, -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], [773, 2939, -3917, -3127, 2232, 771, -419, -54, 25], [-841, -2956, 3934, 3127, -2232, -771, 419, 54, -25], [-1563, -6014, 7834, 6271, -4464, -1542, 838, 108, -50], [-3374, -13564, 17347, 14126, -9865, -3457, 1847, 241, -110], [-3965, -14789, 19596, 15461, -11095, -3794, 2071, 265, -123], [9008, 34825, -45333, -36393, 25758, 8932, -4821, -624, 287], [-11960, -46310, 60309, 48383, -34335, -11877, 6432, 830, -383]];
    Rf_basisdens := [1, 1, 17, 17, 17, 17, 17, 17, 17];
    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_8008_a();"
function MakeCharacter_8008_a()
    N := 8008;
    order := 1;
    char_gens := [6007, 4005, 3433, 4369, 4929];
    v := [1, 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_8008_a_Hecke(Kf)
    return MakeCharacter_8008_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], [1, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 0, 0, 0, 0], [2, 0, -1, 0, -1, -1, -1, -1, 0], [0, 0, 0, 0, 1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 1, 1, 1, 1, 0, 0, 0], [-1, 1, 0, 0, 0, 1, 1, -1, 0], [2, 0, -1, 0, -1, 0, 0, 0, 0], [1, 0, -1, 1, 1, 0, 1, -1, 0], [0, 0, 0, 0, -1, 0, 0, 0, 1], [2, -1, -1, 0, -1, 0, -1, 0, 0], [2, 0, -1, -1, 1, -1, 1, 0, 0], [0, 1, 1, -1, 1, 1, 1, 0, 0], [3, 0, 1, -1, 1, 0, 0, 1, 1], [2, -1, 1, 0, 1, 1, 0, 2, 1], [3, -1, -1, 1, -1, 0, -1, 1, 0], [-1, 1, 1, 0, 1, 0, 0, -2, -1], [1, 1, 0, 1, 1, 1, 2, -2, -1], [3, 0, -2, -1, 0, 0, 1, -1, 0], [2, 3, 0, -1, 1, -1, -1, -1, 1], [-1, 0, -2, -2, 1, 0, 2, 1, 0], [-1, -4, -2, -1, -1, 0, -1, 2, -1], [3, -1, 1, 1, -1, 0, -2, 0, 0], [-2, 0, 0, 0, 2, 1, 0, 1, 0], [-3, 1, -2, 2, 1, 0, 1, -4, -1], [5, 0, 1, -1, -2, 1, 1, -1, 2], [2, -2, -3, 0, -1, 0, 0, -2, 0], [4, -3, 1, 1, -1, -1, -1, 3, 0], [9, 0, 2, 0, -1, 1, 0, 2, 0], [-4, 5, 3, -1, 3, 2, 1, 2, 1], [-1, 2, -2, -3, 0, 0, 2, -2, -2], [1, -1, -1, -1, -1, 2, 0, 3, 0], [5, -1, 0, -3, 1, -1, -1, 2, 1], [4, -2, -1, -1, 0, 1, -1, -1, 1], [1, -2, 0, 0, -2, 0, 1, 3, 0], [-3, 1, -3, 1, 0, 2, 2, -3, 0], [0, -3, 1, 3, -1, -1, -1, 1, -3], [-1, 1, -1, -2, -2, -1, -2, -1, 3], [-1, -2, 2, 0, 2, 1, -1, 3, 0], [4, 2, 0, -2, 2, -2, 0, 3, 1], [0, 1, 1, -3, 4, 0, 2, 2, 0], [3, -1, -2, -1, 1, -2, 0, -2, 0], [1, -2, -2, -1, -5, 0, 0, 0, 1], [6, 2, 2, 1, -1, -1, 2, 3, 0], [0, -1, 0, 1, -2, -2, -1, -2, -1], [-5, 1, -1, 1, -4, 0, 0, -1, -2], [8, 0, 2, 1, -2, 1, -1, 4, 0], [-4, 4, 3, 5, 1, -1, -2, 0, -2], [10, -4, 3, 0, -2, 1, -1, 5, 0], [3, -3, 2, -1, 0, 0, -2, 3, -1], [8, -3, -3, -5, -2, -2, -1, 2, 2], [1, 1, 4, 2, 0, -1, -1, 5, 2], [-5, -2, 2, -1, 2, 2, 2, 1, -1], [-5, 3, -1, -4, 3, 1, 4, -1, 2], [-6, -4, -2, 1, 0, 1, 1, 1, -1], [4, -5, -4, 2, -4, 0, -2, -1, 1], [4, -1, 1, 1, 2, -1, 1, 1, 0], [-2, 4, 3, 2, 4, 0, 0, -3, -3], [-4, 0, 2, 0, 0, 1, -2, 3, 3], [5, -2, 1, 3, 1, 2, -2, 2, 2], [3, -5, 1, 1, -4, 0, -1, 3, -1], [6, -4, -1, 1, -2, 2, 0, 3, 1], [3, 2, 2, -2, -2, -1, 2, -2, 2], [5, -3, 5, 5, -3, 1, -4, -2, 1], [5, 1, 2, -2, 6, -1, 0, 1, -2], [-4, 2, -2, 2, 2, 0, 0, 4, -2], [-1, 4, -1, -3, 3, 0, 4, 2, -1], [-5, 3, 0, 1, -1, 0, 1, 0, 0], [3, 2, 5, 6, -3, 1, -1, 4, -1], [-1, 0, 3, 3, 6, 2, 3, 1, -1], [7, -5, -3, -3, 1, 0, -2, -1, -1], [-4, -1, 2, -2, 4, 1, -2, 6, 0], [9, -5, 3, -1, 4, 2, 0, 2, 1], [-4, 1, -4, -5, 0, -3, 1, -1, -1], [2, 6, 3, 2, -3, 2, -1, 4, 4], [2, 3, 2, -6, 6, 2, 2, 2, 2], [-2, -2, 4, 2, -3, -2, -3, -2, 1], [2, 1, 0, -1, 0, -1, 1, -1, -4], [2, -2, 2, -1, 5, -1, 0, 6, 0], [3, -3, 8, -1, 5, 2, 1, 3, 2], [6, 2, 1, -4, -1, -4, -3, 4, 1], [7, -6, -1, -5, 0, 0, -1, 2, 3], [4, 6, -5, -1, 0, -4, 2, -3, -2], [-3, 4, 4, 6, -3, 3, 0, 2, 1], [-8, 1, 3, 4, -3, 2, -2, -1, 2], [-1, -4, 3, 3, 2, 2, 3, 5, -3], [1, -1, 1, -5, -2, -1, -2, -2, 2], [0, -4, 0, -1, 7, 1, 0, 2, -5], [9, 0, 2, 4, 2, -1, 1, 2, 0], [1, 2, -1, 0, 0, -2, 2, 1, -4], [10, 2, 0, 2, 3, 0, 3, -8, 0], [3, -6, 1, 0, 0, -1, 3, 6, -2], [-3, 2, -3, -1, 0, -4, -1, 0, -2], [9, -7, -5, -6, 3, -1, 1, 1, 2], [1, -1, -3, -7, 2, 2, 1, 1, 3], [12, 3, 4, 1, -1, 3, -1, 5, 3], [5, -6, -6, -2, 0, -2, -3, -3, -2], [8, -4, 4, 0, 1, -2, -1, 6, -1], [2, -1, -6, -1, 1, 1, -1, -5, 0], [0, -3, 0, 1, -5, 2, -3, 6, 1], [-1, -7, -6, 1, -7, -2, -1, 0, -2], [3, 3, 5, 4, -1, 1, 0, -3, 0], [7, 4, 7, 3, 0, 4, 2, 6, 0], [5, 6, -3, 4, -3, -1, 2, -8, -2], [-6, 2, -2, -1, 1, 1, 4, -7, -1], [-1, -4, 0, -2, 6, 1, 3, -4, 1], [-6, 5, 9, 3, -1, -1, -3, 1, 3], [3, 2, 2, 1, 0, -2, -2, -8, -2], [-5, 1, -5, 2, -1, -5, -3, -3, -4], [-7, 4, 7, 4, -3, 1, -1, 1, 1], [4, -2, 6, -2, 3, 2, 0, 7, 3], [11, 0, -6, -6, 1, -1, 2, 0, -1], [2, -7, -9, 1, -2, -1, 2, -9, -5], [1, 4, 3, -2, 1, 5, 1, 1, 0], [10, -5, -4, -3, -3, 1, 1, 0, 0], [1, -2, 2, -2, 0, 6, 2, -1, 2], [1, -3, -2, -5, 5, 0, 0, -7, 2], [11, -5, -5, -6, 6, -1, 0, -4, 1], [-16, 3, 6, 0, 8, 4, 4, 4, -2], [-3, -4, 4, 2, -3, 1, -3, 4, 2], [-4, -6, -3, -3, 0, 1, 1, 0, 4], [0, -8, 3, -2, 3, 0, 2, 2, 0], [-6, 1, 2, 6, -3, 2, -5, -6, -1], [8, -8, 0, -6, 1, 1, 3, 5, 3], [5, -8, -4, -2, -4, -1, 0, -2, 0], [-2, -1, -3, 3, -2, -5, -2, -3, 0], [2, 1, -4, -5, -2, -1, 1, -9, 0], [-1, 4, -2, -5, 1, -4, 2, 0, 1], [0, -10, 2, 2, -7, 1, -4, -1, 2], [10, -5, 7, 4, -1, 0, -3, 3, 2], [-10, 0, -4, -1, -6, 1, -4, 2, -1], [-11, 8, 3, 2, 0, 1, 6, -3, 0], [2, -6, 1, 3, -7, 1, -6, -5, 2], [10, -13, -1, -4, -1, 0, 0, 11, 1], [5, -9, -6, 4, -4, -1, 0, -5, 0], [14, 3, 5, 7, -1, 2, -3, 4, 1], [2, -1, 3, 5, -5, 3, 1, 6, 0], [4, 3, 1, -2, -2, -4, -3, 3, 2], [-3, -4, -5, -1, -3, -2, 0, 2, 2], [-3, 1, 3, -6, -1, 1, -1, -1, 3], [8, -5, -4, -1, -2, -1, -1, -3, 4], [1, -1, 1, 1, -1, -6, -3, 1, -2], [-7, 15, 4, -1, 2, 1, 2, -4, 2], [-4, 7, -3, -2, -3, -4, 0, -5, -4], [4, -7, -7, 3, -8, 1, -6, -3, 1], [7, -3, -3, 4, -2, -3, -2, -4, -5], [-2, -5, -3, 1, -5, 3, -3, -3, 4], [-10, 2, -8, -7, 3, -1, 6, 2, 1], [14, -4, 9, 0, 7, -2, 0, 6, -1], [15, 1, 3, 0, -2, -3, -3, 1, 4], [-10, 14, 1, -1, 2, -1, 5, -4, -2], [18, -10, 4, 3, -1, 3, -2, 1, -3], [1, 0, -2, -2, 3, 1, 7, 9, 1], [-8, -9, -6, 0, -3, -1, 3, -4, -3], [7, -4, -6, -4, 8, -3, 2, 2, 0], [6, -3, -2, 0, -14, -4, -2, 0, 0], [-8, 3, 7, 2, 5, 0, 5, -3, -4], [-11, 7, 5, 4, 2, 1, 0, -9, -3], [5, -6, -1, -1, 3, -1, 4, 3, -3], [0, 9, 5, 1, 1, 5, 7, -3, 1], [-19, 0, -2, 6, -4, 1, 2, 4, -3], [-6, 1, 2, 12, -7, 0, -1, -7, -4], [-6, -5, -13, -6, -2, -5, 1, -6, -2], [-14, 0, -3, -5, 3, -4, 1, -9, 0], [5, -2, -3, 3, 5, 4, 0, -2, 0], [3, 7, -7, -2, -6, -3, -4, -3, 2], [-12, -10, 0, 3, 5, 1, 2, -4, -5], [9, 6, 10, 6, -2, 4, 2, 5, 1], [7, 10, 0, 3, -5, 0, 1, 5, 1], [-13, 4, 3, 4, -2, -3, 2, -5, -1], [1, -6, -4, -9, 0, -2, 2, 4, -2], [5, 3, 1, 0, -4, -1, 2, -5, 0], [-2, -15, -3, 3, -5, -1, -1, 9, -2], [-1, 5, 4, 0, 5, 3, 2, 3, 2], [18, -8, -4, -6, 2, 1, 2, -1, 0], [1, 13, 3, 1, -2, 3, 1, 0, 2], [1, -3, -4, -4, -3, -3, -4, -3, 2], [-6, -1, 8, 4, -1, 1, 1, 6, 2], [14, -5, 6, -5, -6, -3, -2, 11, 2], [-2, -1, -2, 3, 3, 1, -4, 0, 2], [-13, -1, -3, 5, -2, 0, 1, -5, 1], [-9, -5, -5, -2, 3, -1, 0, 3, 3], [10, -12, -2, 1, -2, 1, -2, 8, -2], [3, 7, 2, 0, -3, 0, -6, -10, 5], [-3, 12, 9, 2, 13, 3, 4, 6, -4], [11, 1, -8, -4, -9, -9, -6, -5, 4], [4, 14, -10, -1, 3, -5, 2, -8, -4], [-2, -6, -8, 0, -6, 2, -3, -6, 0], [18, -1, -5, -4, -3, -6, -3, -2, 4], [-4, -3, -2, 2, 9, 2, 2, -3, -6], [-8, 4, 5, 2, -6, 0, -5, 0, 3], [8, -7, -5, 3, -14, -7, -8, -2, -1], [-18, -1, -3, 0, -2, 2, 4, -6, 0], [-14, -1, 7, 1, 12, 5, 1, 7, 2], [11, 6, 13, 4, -3, 5, -3, 4, 5], [-1, 3, -3, -1, 1, 0, 8, -4, 2], [-1, 3, 6, 4, 7, 1, 0, 1, -1], [11, -7, 3, 3, 2, -1, -2, 2, -7], [-14, -4, -9, -5, 3, -2, -1, 5, -2], [12, -1, 4, 3, -2, -3, -10, 3, -1], [19, -1, -4, 1, -2, -4, -4, -3, -5], [-12, -2, 5, 0, 6, 6, 4, 12, 4], [-5, -1, 4, 2, -7, 1, -3, 1, 2], [4, -12, -7, 7, -7, -2, -5, -1, 0], [-7, 5, 6, 0, 11, -2, 2, 6, -4], [-17, -1, 0, -1, 5, 4, 6, 7, 0], [-14, 0, 6, 12, -6, 2, 0, -4, -2], [-1, -11, -5, 0, -2, -1, 4, 3, 0], [0, 8, -1, 0, -2, -6, -3, -4, 0], [4, 3, -1, 3, -10, 0, -3, 2, 0], [1, 7, 2, 1, -2, -1, 3, -6, -8], [-10, 16, 7, 7, -2, 1, -5, 3, -3], [2, -11, -3, -2, -5, 2, -6, 9, 2], [13, -4, -3, -2, -6, -7, -4, -1, 3], [-12, -2, 3, 1, 4, -3, -4, 6, 0], [10, 7, 5, 0, 7, 4, 9, -3, 3], [-5, -2, -4, -1, -1, 0, -7, 12, 1], [4, 13, -3, -2, -4, -4, 7, 1, 0], [-13, 17, 5, 5, 0, -1, -3, -6, 0], [16, 4, -4, -6, 3, 2, 1, -2, -3], [-27, 5, -2, 5, 2, 2, 3, 1, 0], [6, 11, 12, 2, 5, 6, 3, 5, 5], [7, 15, 3, 4, -3, 3, 4, -5, 1], [12, -6, 3, 0, -2, 0, 5, -2, 0], [13, 4, -2, 5, -4, -6, -6, -6, -2], [-28, 4, 7, -1, 4, 1, 4, -1, -1], [-10, -6, 0, -5, 1, -1, -4, 12, -1], [-14, 1, 0, 6, 2, 2, -2, 1, 2], [-18, 10, 1, -2, -2, -4, 5, -2, 0], [-17, 22, -1, 6, 2, -2, 0, -7, -2], [-4, 10, 1, 4, 2, -4, 0, -13, 2], [-14, 5, -11, 5, -9, -1, 1, -11, -1], [11, 14, -4, -3, -2, 1, 4, -1, -1], [-9, 1, 14, -2, 3, 3, -5, 7, 5], [4, -3, -2, 1, 0, 1, 5, 0, 4], [5, -5, 11, 1, 2, 4, 1, 5, -1], [4, -3, 10, 7, 9, 9, 8, 7, -3], [-13, -15, -4, -2, 0, -3, -1, 2, 1], [-18, -3, 4, 5, 2, 5, -3, 1, -2], [12, 0, 7, 1, -1, 1, 0, 10, -3], [16, -5, -9, -5, 3, -3, -5, 3, 4], [-23, 2, 6, 0, 2, 3, 0, -2, 1], [-11, 6, -7, 2, 3, -1, 2, -16, -3], [11, 9, 12, -6, 7, 1, 0, 5, 1], [7, 4, 5, -1, -3, -2, -5, 0, 5], [14, -9, 1, -3, -2, -6, 0, 6, -1], [-6, 12, 7, 1, -1, 5, -1, 2, 7], [-17, -1, 0, 0, 3, -1, 4, 2, -1], [1, -2, 2, 4, -14, -5, -5, 6, 0], [-3, 3, -3, -7, 3, 1, 0, -2, 1], [5, -16, -7, -4, -2, -5, 0, -4, -4], [-27, -6, 0, 8, 4, 3, 2, 1, -6], [-5, -1, -9, -11, 2, -5, 7, -4, -1], [-12, 6, 4, -1, 3, 3, 6, 4, -6], [-6, 15, 4, -4, 6, 4, 9, -3, 3], [-21, 2, -11, 4, -14, -3, -3, -5, -6], [-10, -10, 3, 9, -1, 3, -6, -1, 0], [-13, 0, -2, -4, 2, 7, 4, -6, 1], [3, 7, 3, 1, -2, 3, 7, 6, -1], [2, -7, -12, -5, -12, 1, -1, -4, 2], [14, 1, 0, 0, 0, -2, -2, -6, 8], [3, 8, -6, -8, -6, -5, -3, -6, 3], [-10, -13, 5, -7, 7, 1, 2, 10, 0], [-19, 0, 5, -4, 18, 3, 4, 4, 0], [-9, -6, -2, -2, -10, -3, -1, 6, 4], [-12, 0, -3, 1, 2, -7, -6, -4, -4], [3, 3, -8, -2, -7, -6, -2, -6, 2], [-18, 3, 5, 7, -3, 6, 5, 0, -4], [-9, -6, -1, -8, 3, -3, -4, 13, 0], [-19, 0, 2, -6, 9, -1, 4, -1, 2], [-24, -8, -14, -4, -1, -4, 3, -8, -5], [12, -7, -8, -10, -4, 1, 4, 6, 6], [-23, -14, -10, -2, -4, -5, -3, 3, -1], [-2, -2, -2, -1, 0, -5, 1, 6, 3], [-2, -2, -1, -4, -4, -6, 1, -4, 0], [-3, 7, -2, -3, 6, 4, 10, -5, -4], [-25, 14, 8, 5, 4, -1, 0, -3, 0], [9, -1, 1, 6, -10, 1, -6, -7, 3], [-6, -1, -9, -6, -3, 0, -1, 13, 4], [5, -13, -7, -9, -5, 4, -2, 4, -4], [-10, 0, 3, 3, 3, 7, 0, 10, 3], [-38, 4, -7, 0, 6, -3, 0, -3, 3], [-28, 2, 6, -3, 22, 3, 9, 5, -2], [-3, -2, -8, -4, -3, 1, 0, 2, 5], [0, -3, -9, 6, 1, 0, 4, 0, 0], [22, 0, 11, 5, 3, 3, 0, 4, 1], [2, -18, 10, 2, 3, 6, -1, 10, 0], [9, 5, -8, 2, 2, -4, -5, -8, -5], [11, -15, -6, -7, 4, -1, 1, -6, -1], [-7, -6, 2, 10, -10, 3, 0, 8, -5], [21, 6, 10, 7, 2, 2, -6, 0, -1], [-5, 5, -8, 2, -6, -5, -1, 2, 3], [-12, 9, -1, 2, -2, -8, -1, -21, -1], [20, 8, -1, 4, 6, 4, -1, -16, 0], [16, -1, 2, -1, -3, -3, -1, 3, -5], [-1, 4, 2, 5, 6, 4, 6, -4, -1], [-13, -5, -11, 6, -7, -3, -7, -2, -7], [8, 4, -5, -7, -3, 1, 2, 6, 1], [-13, 4, 4, 5, 3, 2, -8, 4, -2], [13, 1, -4, -7, 0, 6, 3, -7, 1], [-5, 7, -2, 0, 1, -1, 4, 5, -6], [-6, -2, 5, 3, -4, -1, -4, 16, 3], [13, 1, 5, -2, 4, 3, 6, 5, 7], [-9, 2, -20, 2, -9, -4, 0, -9, -3], [-15, 8, -15, -4, 0, -5, 1, -4, 4], [21, -15, -4, -9, 4, -2, 0, 5, -3], [9, 6, 11, -12, 4, -1, -4, 14, 0], [0, -16, -11, -7, -4, -3, -2, 2, 5], [-5, 0, 6, -3, 2, -4, -3, 0, 2], [-7, -2, 1, 1, 4, 2, 7, -4, -1], [-20, -3, -6, 4, -10, -3, -4, -6, -3], [0, 6, -14, -2, -2, -2, -1, -6, -5], [-2, 8, 1, -9, 3, -3, 5, 2, 4], [29, -9, 12, 1, 0, 1, -5, 14, 2], [-16, -2, -11, -5, 7, -3, 8, -6, -2], [12, -8, 6, 0, -1, -4, -1, 0, 1], [11, -13, -4, -6, -3, 5, 0, 9, 2], [-32, -10, -2, -2, -5, 0, -9, 1, 1], [-22, 13, 8, 6, -9, -2, -1, 4, 3], [8, -7, 11, 0, 9, 8, 0, 12, 4], [16, -5, 9, -3, 3, 5, -1, 5, -3], [22, 8, 5, 3, 0, -1, 2, -15, 5], [22, 8, -3, 3, -6, -1, -7, -18, 4], [-10, -4, -7, 5, -2, -4, -4, 5, 0], [13, -11, -6, -5, 5, -2, 1, 7, -6], [-4, 10, 8, -5, -2, -1, -3, 5, 0], [-1, 8, 0, 6, 2, 2, 6, -19, -4], [-5, -2, -7, -8, 6, -1, 6, -2, 1], [4, 5, -2, -9, 14, 2, 0, -4, 5], [-25, 6, 1, 7, -2, 2, 3, -10, -1], [18, 4, 5, 1, 5, 5, 2, -6, 1], [-16, -7, -12, -11, 0, -11, 1, -9, -4], [24, -3, -9, -1, -11, -5, -7, -9, 2], [-7, 1, -5, -5, 2, -3, 1, -2, 6], [-1, -1, 6, 11, 8, 8, 8, -3, -4], [3, -12, 6, -14, 5, 1, 0, 20, 2], [-8, -9, -1, 7, -2, -3, 0, 9, -1], [-2, 13, -3, -5, 2, -3, 4, -3, 0], [6, -2, -7, 5, -1, -7, -1, -6, -3], [12, 12, -6, -5, -2, 1, -1, -4, 7], [-2, -13, -7, 1, -13, -4, -1, -2, -3], [-26, 19, 4, -2, 1, -4, 3, -13, 1], [6, -2, -4, -10, 12, 6, 10, 0, 2], [7, 12, 11, 2, 3, 3, 6, 10, -2], [0, 0, 5, -7, 0, -1, -8, 2, 1], [24, -1, 0, 4, -3, 4, 2, 4, -3], [-16, -8, 7, 8, 3, 10, 8, 4, -5], [-23, 11, 9, 3, 12, 4, 1, 3, 1], [-14, -6, 7, 14, 2, 0, -1, 8, 0], [4, -9, 3, 4, -8, -8, -7, 3, -2], [-16, -19, -7, 1, -9, 1, 1, -5, 1], [7, 13, 0, -4, -6, -3, 0, 1, 4], [13, 4, 13, 8, 2, 11, 3, 2, 4], [6, 4, 4, 9, -11, -1, -6, -12, 6], [-3, -17, -5, -1, -3, -7, -4, -2, -4], [-18, 8, 10, 0, 15, 4, 9, 9, 1], [20, -2, 3, 1, 0, 6, 4, -3, -4], [20, 10, 8, 12, -10, 0, -9, -14, 6], [24, -3, 0, 4, -8, 2, -2, 7, -2], [19, -13, -4, -5, -20, -6, -10, 6, 7], [11, 0, 5, 5, 8, -1, 6, -11, -4], [45, -8, 3, 5, -5, 2, -5, 4, 2], [-20, -8, 0, -1, 4, 1, 5, 24, -3], [-10, -9, -4, -4, -7, 10, 0, -9, 2], [-3, -3, 7, -2, 0, -5, -2, 9, -1], [-30, 19, 8, 10, 9, 9, 11, 4, -4], [-33, 14, 9, 2, 12, 5, 4, 0, -2], [3, -1, 5, 8, -3, -3, 2, 7, -9], [2, 0, 5, 4, -6, -4, -5, 8, 0], [16, 1, -3, -14, 0, -8, -8, -14, 4], [-6, 8, 15, 11, 0, 3, 2, 4, 4], [-33, -11, -4, 8, -11, 2, 2, 6, -5], [-1, 1, -15, -10, 10, -1, 10, -3, -3], [32, -6, -8, 0, -6, -6, -4, -17, -10], [-15, -10, 1, -2, 4, 0, -14, 13, 5], [3, 8, 14, 9, 6, 9, 5, -1, -3], [-1, 14, -7, 9, -9, -2, -6, -22, -2], [-1, -5, -11, -14, 1, -11, -5, -1, -3], [8, -18, -11, -7, -1, 2, 7, 5, -8], [29, 6, -7, 3, -17, 2, -1, 0, 3], [21, -21, -8, -12, 1, -3, -1, -3, -4], [-17, -4, -10, 0, -7, 1, 6, 0, 4], [9, 10, -2, 6, -3, -1, 4, -12, 2], [-2, 2, 9, 10, 5, 8, 2, 8, -1], [2, 11, 17, 8, 6, -1, 3, 0, 3], [-28, 25, 13, 4, 11, 8, 4, 9, -3], [-3, 6, 1, -1, 13, -4, -1, 12, -7], [-20, 8, 3, -9, -3, -1, 1, -5, 1], [48, 3, -6, 2, -12, -7, -8, -4, 2], [-19, -12, 1, 6, 4, -3, -6, -1, -3], [-27, -5, 8, 0, -3, 1, 1, -1, 4], [-24, 3, 2, 0, 2, 3, 4, 4, 5], [-8, 17, 5, 6, 8, 2, 9, -15, -7], [-13, 10, -2, 1, 7, 0, 4, -4, 1], [11, 7, 1, 9, -10, -3, -3, -2, -1], [14, 6, -7, -8, 3, 0, -5, -6, 7], [-12, 9, 15, 7, -2, -1, -1, 0, 0], [0, 3, -15, -6, 0, -4, -1, 1, 2], [-1, 2, 16, 7, 2, 8, 9, 5, -4], [-5, -3, -14, 13, -10, 0, -3, -15, -2], [-2, -5, -2, -9, 0, 1, -1, 7, 0], [-26, -13, -8, 0, -11, 2, -3, 1, 2], [2, -8, 7, 9, 10, 12, 4, 11, -2], [-8, 12, 4, 10, 6, 0, 3, -16, -3], [4, 28, 11, 0, 8, 6, 10, 2, 3], [12, 5, -12, -13, -16, -10, -11, 0, 6], [36, 0, -5, -5, 1, -7, 0, 4, 0], [-18, 10, -10, 7, 0, -5, 0, -22, 2], [21, -21, 5, -10, -1, 4, 1, 8, 4], [-3, 5, 18, 1, 2, 2, 3, 5, 3], [15, 9, 2, -6, 17, 3, 4, -5, -1], [40, 7, 2, 7, -9, 3, -4, -13, 2], [-24, 2, -12, 0, 2, -3, -2, 1, -1], [29, -16, 10, 5, -4, 1, -8, 5, 2], [12, -3, 5, -8, 4, -6, -9, -13, 3], [13, -6, -16, -6, -1, 4, 3, 1, 4], [8, -10, -13, 1, -6, 1, -2, -12, 4], [2, -12, 0, -8, -2, 3, -2, 27, 3], [6, 23, 0, 5, -5, -1, 4, -3, 0], [-9, -10, -3, -11, -3, -2, 6, 7, 2], [22, -7, 7, 3, -5, 0, -6, 10, 1], [-15, 4, 3, -12, 20, 9, 13, 14, 4], [6, -3, 14, -3, 5, 10, 2, 0, 0], [20, -2, -8, -4, -3, 0, 14, 0, -4], [-15, 1, -14, -13, -11, -4, -3, -3, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8008_2_a_q();".
// 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_8008_2_a_q(:prec:=9)
    chi := MakeCharacter_8008_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_8008_2_a_q();".
// 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_8008_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-26, -69, 252, -53, -201, 64, 45, -15, -3, 1]>,<5,R![-277, 1357, -2241, 1458, -91, -319, 118, 4, -8, 1]>,<17,R![374, 13409, 1732, -5625, -451, 820, 37, -49, -1, 1]>],Snew);
    return Vf;
end function;