
// Make newform 8000.2.a.bz in Magma, downloaded from the LMFDB on 10 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8000_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_8000_2_a_bz();".
// 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_8000_2_a_bz();".
// 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 := [5, 0, -30, 0, 34, 0, -11, 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, 2, 0, 0, 0, 0, 0, 0], [13, 0, -37, 0, 12, 0, -1, 0], [-1, 0, -32, 0, 18, 0, -2, 0], [0, -71, 0, 83, 0, -24, 0, 2], [-64, 0, 151, 0, -66, 0, 7, 0], [0, 26, 0, -56, 0, 21, 0, -2], [0, -41, 0, 59, 0, -21, 0, 2]];
    Rf_basisdens := [1, 1, 9, 3, 9, 9, 3, 3];
    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_8000_a();"
function MakeCharacter_8000_a()
    N := 8000;
    order := 1;
    char_gens := [2751, 2501, 5377];
    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_8000_a_Hecke(Kf)
    return MakeCharacter_8000_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]];
    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, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 1, 1], [1, 0, 0, 1, 0, 0, 0, 0], [-2, 0, 1, 1, 0, 1, 0, 0], [0, 1, 0, 0, 1, 0, 1, -1], [0, 1, 0, 0, -2, 0, 1, 2], [2, 0, 0, -1, 0, -1, 0, 0], [0, 2, 0, 0, -1, 0, -1, -1], [3, 0, -1, 0, 0, -1, 0, 0], [2, 0, 2, 1, 0, -1, 0, 0], [0, 1, 0, 0, -2, 0, 1, 1], [0, -2, 0, 0, 1, 0, 0, -2], [8, 0, 4, 0, 0, 0, 0, 0], [0, -2, 0, 0, 5, 0, 1, -3], [3, 0, -1, 2, 0, 2, 0, 0], [0, -2, 0, 0, 3, 0, 1, -5], [0, -3, 0, 0, 1, 0, 1, -1], [-3, 0, -6, -1, 0, -2, 0, 0], [0, -1, 0, 0, 0, 0, 0, -2], [0, 1, 0, 0, -2, 0, -1, 4], [1, 0, -3, 0, 0, 0, 0, 0], [-1, 0, 2, 1, 0, -2, 0, 0], [6, 0, 8, -2, 0, -1, 0, 0], [0, -2, 0, 0, 1, 0, 3, 1], [0, -1, 0, 0, -4, 0, 1, 1], [4, 0, 1, 0, 0, 2, 0, 0], [-7, 0, 2, -1, 0, -2, 0, 0], [0, 2, 0, 0, -3, 0, 0, 2], [0, -1, 0, 0, -4, 0, 0, -2], [-12, 0, -3, -1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, -4, -6], [-5, 0, -11, 0, 0, 2, 0, 0], [0, 1, 0, 0, 2, 0, 2, -4], [-2, 0, -9, -1, 0, -1, 0, 0], [0, 0, 0, 0, -4, 0, -3, 2], [0, -1, 0, 0, 4, 0, 1, -5], [8, 0, 7, 1, 0, -1, 0, 0], [0, 2, 0, 0, -5, 0, 3, 7], [4, 0, 4, -1, 0, 1, 0, 0], [0, -3, 0, 0, 11, 0, -1, -3], [-1, 0, -5, 0, 0, 3, 0, 0], [10, 0, 7, -1, 0, -1, 0, 0], [0, 1, 0, 0, 7, 0, 3, -3], [0, -1, 0, 0, 6, 0, 2, -4], [0, -3, 0, 0, -2, 0, 1, 3], [0, 1, 0, 0, 2, 0, -3, 0], [0, 0, -6, 2, 0, -1, 0, 0], [-18, 0, 3, -1, 0, -1, 0, 0], [0, 1, 0, 0, 2, 0, -2, 4], [4, 0, 2, 0, 0, 3, 0, 0], [0, 0, 0, 0, 0, 0, 4, 4], [-4, 0, 6, 2, 0, 2, 0, 0], [0, -1, 0, 0, 6, 0, -3, -1], [2, 0, -10, 2, 0, 2, 0, 0], [0, 2, 0, 0, -2, 0, -2, 6], [4, 0, -5, 1, 0, -1, 0, 0], [-2, 0, 6, -2, 0, -1, 0, 0], [0, -6, 0, 0, 3, 0, 1, 3], [-11, 0, -6, 3, 0, 2, 0, 0], [0, -4, 0, 0, 8, 0, -1, -2], [0, 2, 0, 0, -1, 0, -1, 3], [2, 0, 2, 4, 0, 2, 0, 0], [1, 0, -11, 0, 0, 1, 0, 0], [0, -3, 0, 0, -8, 0, 0, -2], [-3, 0, -11, 0, 0, -3, 0, 0], [0, -2, 0, 0, 8, 0, -1, -2], [10, 0, -4, -2, 0, 1, 0, 0], [-15, 0, -4, 1, 0, -4, 0, 0], [0, -2, 0, 0, 3, 0, -1, 3], [0, -5, 0, 0, 2, 0, 3, -2], [-5, 0, 17, -2, 0, -3, 0, 0], [0, -6, 0, 0, 2, 0, -2, 2], [0, -2, 0, 0, -1, 0, -4, 2], [0, 0, -3, -4, 0, -2, 0, 0], [-5, 0, 2, 5, 0, 2, 0, 0], [13, 0, 7, -2, 0, 4, 0, 0], [2, 0, 8, 0, 0, -1, 0, 0], [0, -4, 0, 0, 8, 0, -4, -8], [14, 0, 1, 0, 0, -4, 0, 0], [0, -3, 0, 0, 16, 0, 4, -2], [-11, 0, -5, -2, 0, -1, 0, 0], [0, -1, 0, 0, -3, 0, 1, 3], [0, 3, 0, 0, 0, 0, -3, -6], [6, 0, -10, -2, 0, -6, 0, 0], [24, 0, 4, 0, 0, 0, 0, 0], [22, 0, -2, 0, 0, 1, 0, 0], [0, 5, 0, 0, -2, 0, 5, -2], [0, 2, 0, 0, 14, 0, 5, 0], [0, 5, 0, 0, 0, 0, 4, 2], [0, -1, 0, 0, 4, 0, 5, -5], [0, -3, 0, 0, -1, 0, -1, 1], [0, -1, 0, 0, -5, 0, 3, -3], [0, -2, 0, 0, 10, 0, 1, -2], [12, 0, 18, 0, 0, -2, 0, 0], [12, 0, 11, 2, 0, 2, 0, 0], [0, -5, 0, 0, 2, 0, 5, -3], [8, 0, -6, -3, 0, -5, 0, 0], [0, 4, 0, 0, 10, 0, 5, -4], [8, 0, 5, -1, 0, -7, 0, 0], [0, -2, 0, 0, 7, 0, -3, -9], [-22, 0, -14, -1, 0, 1, 0, 0], [0, -4, 0, 0, -2, 0, 6, 10], [-8, 0, -2, 2, 0, 2, 0, 0], [0, -4, 0, 0, 3, 0, -3, -5], [25, 0, 19, -2, 0, 3, 0, 0], [0, -1, 0, 0, -4, 0, 0, 6], [0, 0, 8, 5, 0, 1, 0, 0], [0, 4, 0, 0, -9, 0, 1, -1], [-18, 0, -14, -4, 0, -2, 0, 0], [-6, 0, 1, 1, 0, -3, 0, 0], [0, 5, 0, 0, -3, 0, -7, -5], [0, -6, 0, 0, 7, 0, 3, -5], [-10, 0, 0, -1, 0, -3, 0, 0], [0, -1, 0, 0, 2, 0, 7, 8], [0, -6, 0, 0, 19, 0, 1, -5], [17, 0, -1, 2, 0, -5, 0, 0], [0, 5, 0, 0, -1, 0, -5, -7], [18, 0, 13, 0, 0, 4, 0, 0], [19, 0, -4, -1, 0, 4, 0, 0], [12, 0, 14, 2, 0, -6, 0, 0], [0, 8, 0, 0, 3, 0, -4, 8], [0, -7, 0, 0, 12, 0, 0, -2], [8, 0, 14, 4, 0, -2, 0, 0], [8, 0, 10, -2, 0, 7, 0, 0], [0, -1, 0, 0, -2, 0, 6, 0], [0, -1, 0, 0, 2, 0, -9, 1], [7, 0, 2, 1, 0, -2, 0, 0], [0, 3, 0, 0, -6, 0, 2, 0], [0, 4, 0, 0, 7, 0, 1, -1], [0, -3, 0, 0, 10, 0, -6, -12], [-15, 0, 2, -1, 0, -2, 0, 0], [-13, 0, 1, -2, 0, 6, 0, 0], [26, 0, 9, -4, 0, 2, 0, 0], [-12, 0, -14, 2, 0, 2, 0, 0], [0, 2, 0, 0, -1, 0, 10, 14], [9, 0, 30, -1, 0, -2, 0, 0], [-1, 0, 11, -4, 0, -6, 0, 0], [0, 4, 0, 0, -11, 0, 1, 1], [-5, 0, -15, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, -9, 9], [0, -8, 0, 0, 3, 0, 5, -5], [10, 0, 17, -2, 0, 2, 0, 0], [0, -3, 0, 0, 14, 0, -2, -12], [-16, 0, -8, 0, 0, 4, 0, 0], [-3, 0, -8, 1, 0, -4, 0, 0], [0, 4, 0, 0, 0, 0, 4, 8], [0, -11, 0, 0, 0, 0, 3, -8], [-7, 0, -5, 2, 0, 11, 0, 0], [-6, 0, -4, 0, 0, 1, 0, 0], [0, 5, 0, 0, 8, 0, 5, 2], [0, -5, 0, 0, 10, 0, -1, -14], [0, 11, 0, 0, -6, 0, -3, 8], [0, 10, 0, 0, -9, 0, -1, 7], [0, -1, 0, 0, -3, 0, 5, 3], [-6, 0, 8, 4, 0, 3, 0, 0], [22, 0, -8, -6, 0, -4, 0, 0], [-14, 0, 2, -6, 0, -6, 0, 0], [0, 2, 0, 0, 3, 0, -10, 2], [-6, 0, 6, -4, 0, -2, 0, 0], [0, 7, 0, 0, -4, 0, 1, 3], [0, -1, 0, 0, 6, 0, -2, 0], [-8, 0, -11, -5, 0, -3, 0, 0], [0, 8, 0, 0, -3, 0, -5, 5], [0, -2, 0, 0, 0, 0, 12, 8], [-2, 0, 23, -1, 0, 3, 0, 0], [38, 0, 29, 4, 0, -2, 0, 0], [-5, 0, -23, 2, 0, 1, 0, 0], [0, -1, 0, 0, 3, 0, 3, -3], [10, 0, -20, 1, 0, 5, 0, 0], [0, 6, 0, 0, -5, 0, 7, 3], [-9, 0, -10, 5, 0, 2, 0, 0], [0, -7, 0, 0, 13, 0, 1, -13], [-26, 0, -3, 0, 0, -2, 0, 0], [0, 0, 0, 0, -2, 0, -10, -10], [6, 0, 32, -3, 0, -7, 0, 0], [0, -2, 0, 0, 9, 0, -2, 0], [-2, 0, 9, 2, 0, 6, 0, 0], [0, -2, 0, 0, 1, 0, -4, -2], [0, 0, 0, 0, 9, 0, -3, -15], [-3, 0, -4, -7, 0, -8, 0, 0], [-5, 0, -19, -2, 0, -7, 0, 0], [0, 7, 0, 0, -4, 0, 3, 17], [-10, 0, 10, 1, 0, -7, 0, 0], [-11, 0, -22, -5, 0, -2, 0, 0], [0, 0, 0, 0, -11, 0, -2, 6], [6, 0, 0, -8, 0, -4, 0, 0], [0, 7, 0, 0, 4, 0, 0, 10], [-16, 0, -8, 0, 0, 12, 0, 0], [0, -9, 0, 0, 14, 0, 1, 1], [0, 7, 0, 0, -5, 0, -9, 9], [-16, 0, -17, -4, 0, -2, 0, 0], [0, -2, 0, 0, -2, 0, 5, -4], [-3, 0, 15, 2, 0, 7, 0, 0], [6, 0, -16, -4, 0, -3, 0, 0], [-28, 0, -10, 6, 0, -6, 0, 0], [-9, 0, -8, 11, 0, 0, 0, 0], [0, 2, 0, 0, -18, 0, -11, -6], [-10, 0, 0, 0, 0, 0, 0, 0], [0, 6, 0, 0, -16, 0, -8, -4], [-10, 0, -6, 0, 0, -10, 0, 0], [-4, 0, 3, 0, 0, -8, 0, 0], [0, -6, 0, 0, -11, 0, -3, -3], [20, 0, 15, -3, 0, 3, 0, 0], [0, 12, 0, 0, -9, 0, -1, -1], [0, 8, 0, 0, -19, 0, -2, 6], [7, 0, -15, 6, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, -4, -8], [-9, 0, 10, 1, 0, 2, 0, 0], [-12, 0, 9, -2, 0, 0, 0, 0], [0, -2, 0, 0, -9, 0, -12, -2], [0, 3, 0, 0, -2, 0, 1, -8], [0, -6, 0, 0, 1, 0, -11, -11], [20, 0, 10, 4, 0, 2, 0, 0], [0, -7, 0, 0, 6, 0, 7, 1], [26, 0, 19, 4, 0, -2, 0, 0], [0, -2, 0, 0, -16, 0, -3, 12], [6, 0, -31, 5, 0, 5, 0, 0], [-12, 0, -20, 6, 0, -4, 0, 0], [0, -3, 0, 0, 14, 0, 6, 4], [16, 0, 43, 4, 0, 4, 0, 0], [0, 3, 0, 0, 4, 0, 3, 0], [0, 5, 0, 0, -2, 0, 3, 15], [3, 0, 17, 0, 0, -2, 0, 0], [-11, 0, 15, -2, 0, -1, 0, 0], [0, -6, 0, 0, 8, 0, 8, -8], [0, 5, 0, 0, -2, 0, -1, 2], [0, -10, 0, 0, -2, 0, -2, 6], [10, 0, -2, -8, 0, -2, 0, 0], [0, -11, 0, 0, 14, 0, -2, -4], [0, -1, 0, 0, -13, 0, -13, -3], [-30, 0, 2, 2, 0, -5, 0, 0], [0, -2, 0, 0, 6, 0, -5, 8], [0, 4, 0, 0, -17, 0, 1, 7], [12, 0, 38, 1, 0, 5, 0, 0], [4, 0, 6, -6, 0, 2, 0, 0], [0, 0, 0, 0, -7, 0, -11, -3], [0, -2, 0, 0, -8, 0, -4, 4], [0, 7, 0, 0, 8, 0, -9, -1], [0, -13, 0, 0, 29, 0, -3, -17], [0, -7, 0, 0, 4, 0, 3, -12], [24, 0, 28, 0, 0, 5, 0, 0], [-13, 0, -22, 5, 0, -6, 0, 0], [0, -4, 0, 0, 0, 0, -4, -12], [0, -2, 0, 0, 17, 0, 3, -7], [0, -6, 0, 0, -12, 0, -4, 8], [0, -5, 0, 0, 13, 0, 1, -13], [0, 8, 0, 0, -17, 0, 4, 2], [29, 0, -9, 6, 0, 7, 0, 0], [0, 0, -3, -4, 0, 0, 0, 0], [0, -2, 0, 0, 4, 0, 9, 4], [8, 0, 18, 4, 0, 2, 0, 0], [19, 0, -4, -5, 0, 0, 0, 0], [0, 2, 0, 0, -6, 0, 2, 10], [0, 0, -20, -4, 0, -9, 0, 0], [0, -7, 0, 0, 0, 0, -5, -14], [10, 0, 6, 4, 0, 6, 0, 0], [33, 0, 17, 0, 0, 5, 0, 0], [0, 9, 0, 0, -12, 0, -1, 17], [0, 6, 0, 0, -7, 0, -5, 1], [18, 0, -14, 5, 0, 5, 0, 0], [29, 0, 12, 5, 0, 4, 0, 0], [-21, 0, -13, 0, 0, 11, 0, 0], [0, 7, 0, 0, 16, 0, 0, 6], [-16, 0, 20, 8, 0, 13, 0, 0], [-16, 0, -11, -8, 0, 0, 0, 0], [0, -2, 0, 0, -10, 0, -5, 12], [17, 0, -9, 2, 0, -9, 0, 0], [-24, 0, -24, 4, 0, 7, 0, 0], [0, -1, 0, 0, -2, 0, 3, -1], [-4, 0, 17, 3, 0, -7, 0, 0], [0, -9, 0, 0, 9, 0, -3, 7], [7, 0, 5, -6, 0, -3, 0, 0], [0, 0, 0, 0, 19, 0, -3, -5], [0, 13, 0, 0, -4, 0, 7, -6], [30, 0, 34, -3, 0, 9, 0, 0], [34, 0, 22, -2, 0, 2, 0, 0], [0, -5, 0, 0, 19, 0, 7, -15], [0, -2, 0, 0, 0, 0, -9, 8], [0, 3, 0, 0, -9, 0, -9, -11], [0, 5, 0, 0, -10, 0, 3, 5], [36, 0, 10, -3, 0, -1, 0, 0], [0, 11, 0, 0, -16, 0, -7, 13], [0, -2, 0, 0, 1, 0, 1, -15], [-36, 0, -41, 1, 0, -1, 0, 0], [38, 0, 11, -5, 0, 3, 0, 0], [0, -3, 0, 0, -11, 0, -7, -5], [-2, 0, 22, -2, 0, 2, 0, 0], [8, 0, 37, 2, 0, 8, 0, 0], [0, -10, 0, 0, 25, 0, 8, 0], [-27, 0, -14, 3, 0, -10, 0, 0], [0, 17, 0, 0, 1, 0, -7, 7], [8, 0, 0, 0, 0, 12, 0, 0], [-3, 0, 1, 6, 0, -2, 0, 0], [0, 2, 0, 0, -17, 0, 19, 3], [16, 0, 3, 9, 0, -5, 0, 0], [0, -1, 0, 0, 30, 0, 6, 0], [0, -13, 0, 0, 4, 0, 7, 10], [-14, 0, -22, 0, 0, -2, 0, 0], [6, 0, 0, -6, 0, 0, 0, 0], [0, 9, 0, 0, 5, 0, 9, 7], [0, -9, 0, 0, 36, 0, 5, -14], [0, -15, 0, 0, 14, 0, 6, -4], [-11, 0, 29, -4, 0, -3, 0, 0], [0, -3, 0, 0, 6, 0, 5, -4], [6, 0, 6, 0, 0, 3, 0, 0], [0, 18, 0, 0, -23, 0, -3, 9], [-14, 0, -15, 9, 0, 9, 0, 0], [0, -12, 0, 0, 21, 0, -6, 0], [-22, 0, -2, 4, 0, -9, 0, 0], [-14, 0, 2, -6, 0, 10, 0, 0], [0, 8, 0, 0, -8, 0, -7, -10], [0, 6, 0, 0, 7, 0, -6, 0], [42, 0, -7, 0, 0, -2, 0, 0], [0, 3, 0, 0, 10, 0, 10, 0], [0, -12, 0, 0, 11, 0, 7, 7], [30, 0, -14, 0, 0, 6, 0, 0], [-4, 0, 6, 0, 0, -11, 0, 0], [0, -5, 0, 0, -20, 0, -4, 6], [44, 0, 24, 12, 0, 0, 0, 0], [-21, 0, -33, 2, 0, -6, 0, 0], [0, 11, 0, 0, -18, 0, -7, 3], [-39, 0, 4, 5, 0, 0, 0, 0], [28, 0, -4, 6, 0, 13, 0, 0], [0, -13, 0, 0, 35, 0, -5, -15], [0, 3, 0, 0, -18, 0, -7, 25], [0, -1, 0, 0, 6, 0, -9, -26], [31, 0, -3, -6, 0, 1, 0, 0], [-6, 0, -23, 0, 0, 8, 0, 0], [42, 0, 3, 3, 0, -1, 0, 0], [0, 2, 0, 0, 1, 0, -7, -19], [0, 4, 0, 0, -24, 0, -7, 2], [0, 13, 0, 0, -11, 0, 1, 27], [0, -11, 0, 0, 16, 0, 1, -3], [29, 0, -1, 0, 0, -12, 0, 0], [-29, 0, 17, 2, 0, 5, 0, 0], [2, 0, -22, 2, 0, 14, 0, 0], [0, -5, 0, 0, -16, 0, -11, 16], [6, 0, -1, 7, 0, 11, 0, 0], [-50, 0, -21, -1, 0, 3, 0, 0], [9, 0, 27, 0, 0, 12, 0, 0], [0, -9, 0, 0, 3, 0, -1, -11], [-27, 0, -7, 8, 0, 9, 0, 0], [0, -9, 0, 0, 0, 0, 4, 2], [-6, 0, 33, -4, 0, -12, 0, 0], [0, -4, 0, 0, -7, 0, 0, 12], [0, -10, 0, 0, 16, 0, 4, -4], [-4, 0, -23, -1, 0, 5, 0, 0], [0, -3, 0, 0, 10, 0, 10, -4], [-40, 0, 12, 0, 0, -8, 0, 0], [-20, 0, -3, -2, 0, -12, 0, 0], [0, -6, 0, 0, 3, 0, -16, -18], [2, 0, 18, -4, 0, 11, 0, 0], [8, 0, 3, -11, 0, 7, 0, 0], [0, 4, 0, 0, 13, 0, -7, -19], [0, 6, 0, 0, 5, 0, 13, -7], [-24, 0, -11, -5, 0, 1, 0, 0], [0, -6, 0, 0, 13, 0, 7, -3], [-53, 0, 6, -3, 0, 2, 0, 0], [-10, 0, 6, 10, 0, 14, 0, 0], [0, -2, 0, 0, 14, 0, -13, -22], [0, 0, 0, 0, -7, 0, -11, -3], [0, 3, 0, 0, -8, 0, 11, 18], [25, 0, 0, 1, 0, 12, 0, 0], [-35, 0, -16, 1, 0, 4, 0, 0], [0, 1, 0, 0, -4, 0, 11, 28], [18, 0, -4, 3, 0, -11, 0, 0], [0, 5, 0, 0, -7, 0, -11, -5], [0, 3, 0, 0, 29, 0, 9, 7], [0, 6, 0, 0, -18, 0, -7, 6], [-16, 0, -37, 2, 0, 0, 0, 0], [0, 4, 0, 0, -26, 0, -14, 2], [17, 0, -6, 11, 0, 6, 0, 0], [0, -8, 0, 0, -2, 0, -2, 18], [0, 7, 0, 0, -21, 0, 7, 17], [-57, 0, -22, 5, 0, -10, 0, 0], [8, 0, -2, 5, 0, -7, 0, 0], [28, 0, 9, -5, 0, -7, 0, 0], [34, 0, 28, 14, 0, -1, 0, 0], [10, 0, 27, -1, 0, -13, 0, 0], [0, -12, 0, 0, 4, 0, -11, -8], [-24, 0, -25, 1, 0, 11, 0, 0], [0, 5, 0, 0, 3, 0, -9, 1], [0, -8, 0, 0, 26, 0, -11, -26], [0, 3, 0, 0, 4, 0, -8, -18], [48, 0, 17, -1, 0, -3, 0, 0], [0, -6, 0, 0, 11, 0, 12, 0], [0, 6, 0, 0, 31, 0, 10, 4], [-2, 0, 28, 8, 0, 7, 0, 0], [3, 0, 1, -6, 0, 1, 0, 0], [0, 6, 0, 0, -8, 0, -4, 16], [0, -4, 0, 0, 21, 0, -4, 4], [0, 7, 0, 0, -31, 0, 5, 19], [29, 0, 23, -6, 0, 3, 0, 0], [0, -2, 0, 0, 26, 0, 6, -6], [-16, 0, -34, -8, 0, 2, 0, 0], [0, -2, 0, 0, 6, 0, 10, -18], [-39, 0, -19, 14, 0, 0, 0, 0], [14, 0, 5, -10, 0, -10, 0, 0], [-25, 0, -15, 2, 0, 5, 0, 0], [0, -12, 0, 0, -4, 0, 1, -8], [41, 0, 1, 4, 0, -3, 0, 0], [-18, 0, -19, 2, 0, -14, 0, 0], [0, 3, 0, 0, -18, 0, -6, 20], [23, 0, -18, -7, 0, -2, 0, 0], [54, 0, 23, 4, 0, -6, 0, 0], [0, 18, 0, 0, -18, 0, -1, 0], [0, 2, 0, 0, -8, 0, -4, 0], [58, 0, -2, 4, 0, 2, 0, 0], [17, 0, -3, -12, 0, -7, 0, 0], [0, 2, 0, 0, -15, 0, -18, 6], [0, -14, 0, 0, -6, 0, 6, -10], [7, 0, -30, -11, 0, -14, 0, 0], [10, 0, 4, -10, 0, -9, 0, 0], [0, 8, 0, 0, 8, 0, -16, -4], [0, 8, 0, 0, -30, 0, -9, 6], [31, 0, 31, -4, 0, 3, 0, 0], [0, -10, 0, 0, -10, 0, 13, -2], [15, 0, 7, 10, 0, 8, 0, 0], [17, 0, -9, -6, 0, -1, 0, 0], [0, 1, 0, 0, -6, 0, -17, -3], [0, 7, 0, 0, -20, 0, 0, 22], [49, 0, 11, -2, 0, -1, 0, 0], [20, 0, 36, 4, 0, -4, 0, 0], [0, -8, 0, 0, 25, 0, 10, -6], [-60, 0, 4, 3, 0, 3, 0, 0], [0, -8, 0, 0, 30, 0, -2, -6], [0, -10, 0, 0, 27, 0, -9, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8000_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8000_2_a_bz();".
// 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_8000_2_a_bz(:prec:=8)
    chi := MakeCharacter_8000_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_8000_2_a_bz();".
// 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_8000_2_a_bz( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8000_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![5, 0, -40, 0, 69, 0, -16, 0, 1]>,<7,R![1805, 0, -1540, 0, 389, 0, -36, 0, 1]>,<11,R![1280, 0, -9280, 0, 1344, 0, -64, 0, 1]>,<13,R![144, 72, -32, -4, 1]>,<19,R![800000, 0, -128000, 0, 6720, 0, -140, 0, 1]>],Snew);
    return Vf;
end function;