
// Make newform 8022.2.a.r in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8022_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_8022_2_a_r();".
// 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_8022_2_a_r();".
// 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 := [-79, -49, 285, 149, -282, -110, 114, 28, -19, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-238207, -136188, 461973, 163808, -247296, -46396, 49952, 1250, -2973, 229], [-28117, 57434, 341569, 128906, -251022, -94908, 49542, 13902, -2587, -489], [234811, -94928, -558531, 135320, 337626, -71692, -70776, 15958, 4339, -1035], [-119763, -9444, 450947, -1142, -353133, 35234, 86773, -15424, -5743, 1237], [-159203, 75368, 507824, -138938, -362380, 95113, 84083, -23156, -5479, 1517], [-34183, 139568, 129262, -280558, -130262, 164831, 41662, -32398, -3335, 1818], [-17763, 225360, 298399, -699314, -364498, 435548, 120350, -87796, -9633, 5003], [157869, 353760, -440123, -982554, 99738, 574984, 35508, -106280, -5345, 5605]];
    Rf_basisdens := [1, 1, 18602, 18602, 18602, 9301, 9301, 9301, 18602, 18602];
    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_8022_a();"
function MakeCharacter_8022_a()
    N := 8022;
    order := 1;
    char_gens := [2675, 6877, 2311];
    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_8022_a_Hecke(Kf)
    return MakeCharacter_8022_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 := [[1, 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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0, 1], [1, 0, 1, 0, 1, 1, 0, 1, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0, 1], [3, 1, 0, 0, -1, 0, 0, 0, 0, 0], [-1, -1, 0, 1, 1, -1, -1, 1, 1, 0], [3, 1, -1, -1, 0, 1, 1, -1, -1, -1], [2, 1, -1, -1, -2, 1, -1, -1, 0, -2], [2, -2, 0, 1, -1, -1, 0, 0, -1, 1], [0, 1, 0, -1, 2, -1, -1, 1, 1, -2], [2, -1, 0, 1, 0, -1, -2, 1, 0, -1], [2, 1, -1, -1, 1, 1, -1, 1, 1, -2], [-1, -1, 2, 1, 1, -1, 1, 1, 1, 1], [0, -2, 1, 1, -2, -1, 1, -3, 0, 1], [4, -2, -1, 1, -3, 2, 0, 0, -2, 0], [-1, -3, 1, 1, 0, -1, 1, 1, 1, 0], [1, 3, 1, -1, 1, 1, 0, -3, 1, -1], [0, 0, 1, 3, 1, -1, 0, 1, -1, 2], [2, 0, 0, 2, 0, 0, -1, 1, -2, 2], [3, 0, 1, 1, -2, -1, -1, 2, -1, 0], [4, -1, -1, 0, -1, 0, 1, 2, 0, -2], [2, -3, -2, 2, -2, -1, -1, -2, -1, 2], [-1, 0, 0, 0, 0, 2, -1, -1, -1, -1], [2, 1, 1, 1, -1, 0, 1, 0, -3, 1], [-2, 3, -1, 0, 1, 2, -1, 0, 2, -2], [3, 0, 1, 0, 2, 2, 1, 1, 1, -1], [-2, 2, 3, 2, 1, 1, 0, 0, 0, -1], [0, -3, -3, 0, -4, -1, -1, 0, 1, 0], [-4, 3, 1, 1, 3, 0, -1, -1, 2, 0], [0, -1, 2, 1, 0, -1, 0, 2, 1, -2], [-2, 4, 3, 1, 4, -1, -1, -1, 3, 1], [3, -1, -2, -1, -1, -2, 2, -3, 0, 2], [7, -5, 0, 0, -2, -1, 2, 2, -2, 2], [1, 5, 0, -2, 2, 1, 2, 0, 1, 0], [-1, 2, 4, -1, 3, 1, 1, 0, 1, -1], [-5, 0, 1, 0, 0, -2, 0, 1, 1, -2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, -2, -2, -2, -2, 0, 2, -3, -2, 1], [-2, -1, -1, 1, -3, -1, -1, 3, 1, 1], [1, 3, 2, 2, 1, 0, -1, -2, 2, -1], [-2, 3, 2, -3, 5, 1, 2, 1, -1, 1], [2, 1, -1, -2, 2, -1, 1, -1, 2, -1], [7, 0, -2, -2, -1, 0, 0, -2, -2, -1], [3, -2, -4, -1, -5, -1, -1, 0, -1, -3], [2, -5, -3, 0, -4, -3, 2, 1, -1, 0], [-4, 3, 5, 3, 2, -1, -1, 2, 1, -1], [-1, 2, 1, 1, 5, 1, 1, -1, 1, 0], [3, 5, 1, 0, 3, 0, -4, 1, 1, -1], [0, 5, -1, -1, 0, 3, -1, -4, 3, -1], [7, -1, -2, -2, -1, 2, -1, 2, -2, 1], [1, 5, 1, -2, 3, -1, 0, 0, 6, -2], [-1, -1, 1, 0, 0, -1, -1, -2, 0, -3], [3, -4, -3, -1, 2, -1, -2, 3, -1, -1], [-1, 6, 0, 1, 0, 2, -3, 1, 3, -4], [8, -1, 0, 3, -7, -1, 0, 1, -1, -2], [8, 1, -4, 0, -2, -2, -3, 0, 0, -2], [1, 4, 1, 1, 2, -1, -1, -4, -1, 7], [10, -3, 0, 0, -3, 3, 2, 4, -3, -1], [1, 0, 0, -2, 3, 2, 2, -2, -1, 5], [0, 2, 0, 3, -3, 2, -1, -1, -1, 1], [4, 0, -5, 0, -1, 2, -3, 1, -2, -3], [-5, 5, 2, -1, 1, 2, 3, -1, -1, 1], [7, 1, -1, 4, -5, 0, 0, -2, -3, 2], [0, 2, 3, 0, 1, -2, 1, -1, 0, 3], [2, 5, 1, -1, 2, -3, 1, 0, 4, -7], [2, 3, -3, 1, -1, -1, -4, -2, 1, 2], [7, -4, 5, 5, 2, -3, 1, 5, 0, 1], [3, 6, -1, -1, -5, -1, 0, -6, 0, 0], [-2, -5, -3, 2, -6, -1, 0, -2, -3, 1], [3, 1, 0, -1, -2, 1, 0, 0, -3, 1], [0, 3, 0, 0, -2, -2, 0, -1, 4, -2], [3, -6, -4, 1, 2, -5, 1, 1, -1, -1], [-4, -3, -1, 0, 2, 3, 0, 1, -2, -2], [2, -5, -2, 0, -3, 0, 1, -5, -4, 1], [4, -4, -1, 2, -3, -1, 1, 3, -1, 2], [-7, -2, 3, 0, 0, -1, 1, 5, 2, -1], [7, 1, -3, 0, -3, 0, -2, 4, -2, -2], [-4, 3, 4, -2, 4, 5, 1, 0, 2, 4], [1, 6, 7, 0, 4, 3, 0, 2, 2, 0], [2, 0, 1, 3, 3, -2, -1, 2, -3, 4], [0, -4, -5, 1, -1, 1, 3, -2, -3, 4], [-2, -2, 5, -2, 4, 3, -1, 2, 1, -5], [-6, -1, 7, 4, -2, -5, 3, -3, 2, 5], [-6, 3, -1, -2, 4, 4, 2, -3, -3, -1], [-2, -2, -5, -4, 0, 1, 3, -2, 2, -1], [17, 0, -1, -2, 3, 2, 1, 1, 1, 4], [-3, 1, 3, 2, 2, 1, 1, -2, 4, 1], [5, -2, 1, -2, -3, -1, 2, -5, -3, -4], [7, 1, -2, -5, -2, 3, -1, 3, 2, -5], [6, 2, -3, -3, 1, 0, -1, 2, 1, -5], [-5, 3, 8, 2, 3, -1, 3, 2, 0, 0], [0, -5, 1, 1, 3, 0, -1, 3, 5, 0], [-2, -1, -2, 1, 3, 0, 2, 2, 2, -6], [-7, -2, 4, 2, 4, 0, -1, -2, -1, 6], [3, 1, -1, 3, 3, 2, -2, 3, -2, 4], [2, -3, -5, 0, -3, 4, 4, -4, -9, 7], [8, -1, 0, -1, 1, 3, -2, -1, 1, 2], [-1, -1, -4, -4, 2, 5, 1, 0, 3, -3], [-13, -6, 4, 2, 7, -3, 4, 0, -4, 9], [7, -9, 4, 10, -5, -4, 2, 1, -4, 3], [3, 3, 3, 3, 8, -1, -3, 5, 3, 1], [5, -1, -2, 5, -4, -3, 0, 0, -3, 7], [2, -3, 3, 3, 5, -2, -3, 1, 3, -2], [2, -1, -1, 2, -2, -1, -3, 5, 1, -5], [2, 4, 3, 1, -3, 7, -1, 2, -3, 2], [-8, 0, 0, 2, -7, -4, -2, 3, 0, -5], [7, -1, -7, -6, -4, 1, 4, -7, -4, 0], [2, 6, 3, -2, 4, -4, 0, -1, 9, -7], [-13, 5, 4, 2, 6, -1, 0, 8, 6, -2], [2, 1, -1, 4, -9, -2, -2, 1, 2, -4], [-1, 0, -3, -2, 4, 1, 1, 1, 2, -9], [10, 4, 1, -6, 6, 0, 1, -2, 0, -5], [1, -9, -3, -1, -4, 0, 1, 1, 1, -4], [1, 5, 1, 1, 6, -2, 1, -1, 5, -2], [-4, -5, -2, -4, -3, -3, 2, -4, 0, 1], [-5, 9, -6, -2, -3, -1, -1, -8, 1, -1], [-8, 2, 2, -3, 4, 2, 1, 3, -4, 1], [-6, 6, 0, -4, 8, 2, -3, 5, 6, -4], [-5, -6, -1, 3, 2, 0, -5, 0, -2, 5], [-9, 12, 2, -5, 12, 2, 1, 1, 2, 0], [-11, 7, 3, -1, 2, 3, -2, -1, 2, -2], [2, 2, -1, 1, -2, 6, -3, -3, -3, 1], [-2, 5, -2, 1, -2, 1, -2, -2, 3, 4], [-11, -3, 0, 3, 3, -1, 2, 0, -3, 2], [-6, -10, 0, 7, -7, -4, 3, -3, -5, 2], [-4, -2, 1, -3, 3, 5, 3, 4, -3, -4], [3, 0, -4, -1, -6, -2, -3, 3, 1, -10], [-5, 8, 5, 1, 0, 0, 3, 5, 2, 0], [-12, 0, 0, 1, -4, -2, -4, -1, -1, 2], [-1, -5, 1, -1, -5, 4, 5, -4, -9, 6], [4, 0, -3, 1, -8, -1, -5, -3, 7, -3], [7, -4, 1, 2, -1, -3, 1, 2, -7, 2], [4, -11, -1, -4, -5, -2, 2, -1, 0, 2], [-4, -1, 2, 4, -2, -9, 2, -1, 0, 6], [-15, 1, 6, 1, 4, 1, 4, 0, 2, 1], [-10, -2, 7, 1, 0, 4, 5, 0, 0, 4], [-8, 2, 0, 1, 8, -1, 1, 2, 2, 2], [-1, 0, 3, 0, -2, -4, 5, -2, 1, 2], [7, 5, 2, -2, -2, 2, 4, -2, -2, 3], [18, 6, 5, -1, 3, 7, 0, -1, -1, 1], [-7, 0, 2, 3, 10, 0, 2, -3, -2, 5], [10, -16, -5, 2, -7, -3, -1, 1, -8, 6], [-1, -2, -8, -6, 2, 6, 0, 0, -1, -3], [0, 7, 8, 2, -4, 0, -1, 2, 2, 2], [-1, 3, -3, 1, -10, 2, 0, -6, -3, 2], [-2, 9, -3, -3, -2, -1, -4, -3, 6, -13], [-5, 3, -2, -2, -1, -1, 4, -5, -4, 8], [-1, -12, -1, 1, -5, -5, -3, 3, -3, -6], [-2, -5, 2, 0, -5, 2, 2, -10, 0, 5], [0, -10, 1, 6, -6, -2, -3, 4, -2, 7], [1, 9, -1, 3, -5, 0, 0, 1, -7, 4], [-12, -4, 7, 8, -1, 3, 0, 4, -2, 0], [-9, 8, 2, 0, 5, -1, -7, 4, 4, -2], [12, -3, -2, 2, -5, 8, -2, 5, -1, -2], [-14, 1, -1, -6, 6, -3, -3, 1, 2, -3], [2, 4, -2, -5, -1, 0, 1, -5, 1, 0], [-14, 6, -1, 1, -1, -5, 0, 6, 3, 3], [-11, -8, 0, -2, 1, 1, 1, 2, -7, -4], [-8, 6, 2, -7, 11, 6, 0, 1, 4, -5], [3, 10, 3, -4, 0, 6, -6, -2, 2, -5], [-11, 9, -5, -5, 4, 1, 0, -5, -1, -4], [-2, 5, 0, -2, 1, -4, -4, 3, 9, -12], [-8, 1, -7, -2, 6, -3, -6, 0, 3, 1], [-14, 7, -2, -5, 9, 1, -1, -6, 1, 2], [7, -2, -7, 0, 3, -2, 3, 0, -2, 3], [0, -14, -8, -2, -10, -4, 2, -2, -4, 6], [-4, -4, -1, -2, -2, 0, -1, 4, -2, -2], [-8, 11, 2, -4, 3, 7, -2, 0, -4, -4], [-3, 6, 2, -3, 3, 5, 1, -5, 1, -4], [4, -2, -4, -4, 2, 2, 0, 4, 0, -10], [-2, -3, -7, -1, -12, 0, -1, -4, -7, 1], [-4, -8, -2, 1, -7, -4, 4, 2, 3, -5], [5, -10, -7, 0, -2, -6, -2, 4, 0, -1], [-3, -1, -3, -8, -2, 1, 5, -4, -2, 5], [-6, -7, -1, 3, -4, -3, 1, 2, -3, -1], [-1, 0, -1, 10, -3, -6, -3, 0, -2, 10], [-8, 16, -3, -9, 5, 1, -6, 3, 5, -8], [-13, 2, 2, 7, 0, 0, -2, 8, -3, 4], [27, 0, -4, 1, -5, -1, -5, -1, -7, 2], [6, -8, 1, -4, -2, 0, -1, -3, -3, -2], [-13, -3, 1, -2, 1, 2, -1, 2, -3, -9], [-2, -4, 7, 8, -1, -5, 2, 1, -1, 13], [-11, -3, 9, 4, 2, -5, -3, 8, 2, -1], [1, -2, 8, 3, 4, -2, 5, -7, 0, 2], [-13, -6, 9, 10, -2, -1, -3, 9, 0, 3], [-10, 4, 1, -1, 7, 0, 1, -12, -1, 1], [-18, 6, -5, 3, 7, -2, -1, -4, 5, 0], [-18, 1, 3, 2, 0, -5, -4, -2, 1, 3], [11, 2, -2, 0, -6, 4, 4, -4, -1, 5], [7, -5, -1, -6, -2, -3, 1, 2, -2, -1], [-20, 4, -2, -1, 3, 3, -6, 4, -1, 5], [-8, 3, 3, -3, -3, 0, 2, -2, 3, 2], [-4, -1, 2, 3, -5, -2, 1, -12, 5, 5], [-7, 0, 4, -1, 10, -8, 3, -3, 1, 7], [-3, -12, 10, 0, -3, 1, 6, 2, -2, 3], [-11, -3, 3, 4, -8, 5, -2, 3, -2, 1], [6, -6, -1, 10, -10, 0, 1, -4, -12, 3], [14, 6, -5, 6, -5, -1, -9, -3, 2, 0], [7, 10, -3, -5, 7, -1, -2, 4, -1, -4], [0, 8, 2, 5, 6, -1, -10, 7, 9, -9], [6, 2, 0, -3, -11, -3, -1, -1, -5, -3], [-1, -6, 2, 5, -13, -1, -4, -2, 2, -4], [8, -5, 4, 4, -2, -3, 1, 8, 3, 4], [-5, 18, 4, -11, 13, 9, 0, 3, 9, -5], [-8, -1, 2, 12, -10, 1, -3, 2, 0, 6], [-1, -6, -5, -3, -14, -2, 4, -9, -2, 1], [-5, -2, 5, 9, -3, 3, 1, -1, -3, 2], [-15, -11, -3, 13, -11, -8, 0, -5, -6, 6], [13, 1, -2, -3, 2, 3, 6, -1, -3, 2], [-8, -3, 5, -4, -2, 2, 11, -3, -4, 1], [-4, -10, 0, 5, -3, -5, -4, 10, -1, 1], [-9, -5, -2, 0, 6, 3, 6, -6, -2, 2], [3, 7, -9, -2, -1, 6, -3, -7, -2, 2], [1, -10, -3, 1, 1, -8, 4, 8, -5, 1], [-23, -7, -3, 1, -1, -4, -2, 9, 2, -2], [-8, -2, -6, -2, 0, -4, 2, -4, -2, -7], [-3, 1, 3, 6, 0, -3, -4, 3, 0, 1], [9, 3, -5, -5, -3, -4, 2, 3, 2, 4], [6, 15, -2, -5, 3, 0, -5, -3, 0, -4], [-6, 13, 3, 0, 11, 1, -2, 3, 2, -4], [-19, 0, 2, -5, 5, 8, 4, 1, 3, 1], [18, -8, -1, 5, -10, 0, 4, 6, -5, 3], [0, -9, 6, 4, -1, 3, 1, 1, -3, 5], [-2, -2, -8, -3, -1, 1, -2, 2, -5, 3], [-20, -2, -6, -6, 3, -1, -2, 5, 3, -3], [1, 9, -3, 2, -8, 2, -4, 7, 4, -9], [-17, 4, 6, 4, -6, -6, -2, -2, -4, 1], [-17, 15, 4, -5, 15, 4, 1, 1, 7, 1], [-3, -7, -6, -1, -6, -1, 3, -1, -10, -5], [18, -14, -7, 2, -17, 3, 0, -5, -15, 7], [-5, 3, 7, 1, 0, 1, 6, -1, -2, 6], [-11, 7, 7, 5, 10, 2, 3, 1, -1, -4], [15, 5, 4, 0, 9, 0, 0, -3, -7, -1], [9, 4, 1, -2, 8, 4, 0, 0, 2, 1], [8, -8, -1, 0, -4, 6, 0, 2, -5, -2], [9, -8, 0, -2, -3, 3, 5, 7, -5, 7], [-10, -6, 2, -5, 14, 0, 3, -2, 1, 0], [-18, 11, 2, 0, 3, -6, 0, -6, 8, 1], [6, -5, 1, 4, -1, 8, -1, 5, -5, 7], [-5, 11, 2, -2, 0, 3, -5, -7, 4, -14], [-7, 14, -5, -10, 7, 7, -2, -1, -2, -10], [2, 9, 0, 0, -4, 0, 3, -4, 2, 12], [-9, 9, 5, 1, 1, -2, -2, -6, 2, 7], [-2, -8, 4, 2, 5, -8, 8, 8, 5, 1], [-18, -3, 3, 0, 5, -7, 6, -7, 2, 6], [-22, 3, 1, -2, 10, -9, -2, -4, 6, 5], [-5, -9, -4, 9, -1, -6, -1, -5, 5, 3], [3, -14, 1, 8, -7, -1, 3, 4, -2, 6], [6, -6, -1, 3, -2, -4, 2, -7, -8, 6], [3, 6, 12, 3, 1, 3, 4, 1, -1, 5], [-16, 7, 4, 7, -7, 1, -1, -6, -3, 0], [11, 5, -11, -8, -4, 3, 0, -8, -3, -4], [-4, -3, 4, -1, 6, 7, 7, 5, -3, -6], [-21, 6, 0, 6, 6, 4, -12, 1, 6, -10], [3, 5, -6, -7, -3, 3, -6, -1, -2, -5], [3, 10, 2, -11, 9, 7, -1, 3, 2, -4], [1, -5, 4, 4, 7, -5, 4, 7, 4, 5], [-8, 3, 12, -2, 4, 1, 3, 3, 5, -5], [4, -5, -4, -6, 7, -1, -4, 4, 1, -5], [4, -9, -6, -3, -6, -4, 7, -3, -5, 2], [-1, 11, 0, -7, -2, 0, -2, -9, 9, -14], [-6, -1, 8, 8, 10, 5, -2, 3, -2, -2], [-7, 1, 5, 13, 4, -2, -4, -2, 5, 5], [-6, -9, 1, 4, 4, -2, -6, -4, 6, 1], [-13, 1, 4, 3, 0, 4, 2, 5, -1, -2], [0, 17, -5, -4, -5, -2, 2, -5, 3, -4], [-6, 12, 9, 8, 4, 0, 4, 7, 0, 5], [-11, 11, 8, -6, 11, 1, 5, 6, 6, 0], [-1, 4, 5, 0, 11, 8, 7, -3, 3, 2], [12, -14, -5, -2, -5, -3, 0, 6, -6, 2], [-20, 4, -5, 6, -5, -2, -5, 7, 0, 5], [-10, -12, 1, -1, -3, 7, 5, 0, -8, 0], [8, 5, -15, -5, -7, 2, 0, -6, -7, -4], [4, -1, -5, -3, -17, -1, -2, 0, -3, -4], [14, 4, -4, -2, -5, -6, 3, 0, 0, -1], [-13, 4, 6, 11, -3, 5, 0, 3, -1, -3], [21, 8, -6, -8, -5, -2, -5, 3, 9, -5], [-8, 2, 7, 4, 8, 8, -5, 1, 1, -8], [3, 14, 3, -6, 7, 5, 1, -7, 5, -7], [8, 7, 5, 1, -3, 4, 1, 2, -15, 7], [-11, -10, -3, -2, 6, -2, 9, 1, -5, 9], [5, -3, -1, 4, 11, -4, -10, 8, 9, 4], [-4, 23, 0, 2, 1, 4, 3, 1, 10, -3], [8, -2, -2, 9, -6, 0, 1, -6, -9, 7], [-9, -3, -4, 12, -7, 2, -4, 2, -7, 2], [5, 18, -10, -7, 5, 6, -1, 0, 5, -8], [11, -16, -3, 9, -11, -7, 3, -9, -7, 6], [12, -14, -8, -2, 4, -6, 7, 9, -1, 8], [23, -13, -3, -1, -16, 7, 3, -5, -5, 3], [-6, 12, 0, -7, 5, -2, 3, -5, 1, 3], [-3, -7, -6, -9, -5, -2, 8, -10, 5, -6], [20, -7, 4, -1, -5, 0, 7, -6, -9, 5], [-4, 5, -11, -3, 1, 7, -10, 2, 3, -2], [-14, 2, 5, 7, -7, -2, -5, 4, 5, -9], [13, 0, -6, -11, -2, 7, 2, 6, 1, -12], [-22, 10, 5, 1, 3, 1, 3, -3, -4, 3], [-25, -4, 3, 2, 6, -2, 10, -8, 2, 1], [6, 11, 0, 1, 11, -3, -4, 5, 9, -13], [-30, 15, 11, 1, 10, 2, 0, 1, 9, -10], [7, 0, -4, 2, -12, 6, -1, 8, -6, 1], [5, 8, -8, -13, 9, 4, -4, 5, -1, -12], [-30, -4, 0, 5, -12, -6, -5, -9, 0, 3], [-1, -2, 7, 8, 2, 2, 5, 1, -6, 17], [5, -8, 4, 4, 6, -2, 8, -9, -2, 14], [-11, -1, 4, 4, 17, -3, 8, 11, 8, 5], [-5, 12, 2, -1, 9, -3, -11, -1, 12, 6], [-17, 14, -2, -9, 17, 0, -3, 4, 9, -6], [-14, 11, 6, 1, 0, -1, -4, -2, -3, 4], [21, 10, 5, 7, -6, 0, 0, -3, -5, 3], [6, -1, -2, -7, 1, -5, -2, -8, 2, -1], [-8, 14, 8, -4, 0, 1, 2, -10, 8, 5], [12, 16, -2, -6, 0, 0, -6, -12, 8, -2], [1, -6, -5, 4, -5, -5, -1, 0, 2, -6], [25, -6, 2, -1, -4, -1, 3, -7, 0, -8], [-15, -5, -1, -1, 4, 4, 3, 11, 1, -8], [-10, -16, -4, 4, -14, -7, 4, -5, -1, 2], [9, -5, -12, -5, -1, -1, -2, 8, -7, -3], [-8, -1, 6, -3, 8, 3, 6, -6, 1, 0], [2, 6, -5, -12, -16, 0, -3, 0, 8, -11], [-2, 8, -3, 2, 4, -4, 1, 0, -4, 9], [8, 0, -8, -2, 3, 4, -2, 7, -8, 3], [0, -5, 5, 0, 11, -5, 0, -11, 6, 3], [-2, -9, -6, -8, 1, 0, -3, -6, 0, -4], [3, -3, 14, 0, 18, 2, 9, -5, 4, 16], [-10, -21, -8, 9, -3, -12, -3, 3, -2, 8], [10, -12, 1, -1, -6, -8, 12, 2, -4, 5], [-13, -7, 3, 5, 2, -13, -5, 1, 7, 7], [11, 13, 13, 2, 12, -1, -3, 8, 11, -9], [-10, -7, 5, -6, -2, 7, 7, 0, -7, 4], [4, 1, 4, 3, 11, 0, 4, 15, 1, -3], [16, -16, -14, -3, -6, -2, 1, -14, -6, 6], [-1, -3, 3, 7, -3, -2, -2, 4, 1, -1], [2, -10, -9, 3, -4, 2, -6, 2, -7, 7], [-2, -5, 6, -3, 3, -8, -4, 7, 10, -3], [-11, 6, 0, -9, 2, 4, 0, -9, 4, 3], [4, 2, 0, -7, -1, 3, 3, -5, -9, -1], [-8, -2, -2, -1, -8, -2, 4, 3, 1, -14], [-3, -5, 3, 5, -1, -2, 3, -10, 5, 10], [1, -5, 5, 5, 1, 7, 5, 0, 3, -6], [4, 17, -1, 3, 8, -2, -4, 3, 3, -5], [-5, 4, -2, -1, -6, -3, 2, 0, 9, 1], [-22, 8, 13, 11, 4, 1, -4, 4, 5, -4], [38, -12, -8, -8, -8, 1, 2, 4, -12, 2], [11, 13, -1, 9, -14, 5, 0, -1, -6, -2], [18, -24, -9, 9, -8, -6, 0, -8, -10, 17], [-31, 5, -2, 0, 3, -2, 3, -1, 2, 2], [-2, -12, 2, 5, -2, 6, -3, -1, -4, -5], [18, -3, -13, -5, 4, -2, 0, 3, -3, 8], [8, -12, 3, 4, 1, -1, 0, -5, -3, 7], [1, -8, 4, 3, -16, -6, 0, 2, 5, -2], [20, -19, -13, 1, -13, 2, 2, -6, -5, 5], [7, 11, -4, 6, -11, -10, -9, 8, -3, -5], [-16, -5, 10, 1, -1, 6, 9, -4, -3, -3], [40, 4, -5, 2, 5, 11, -1, -2, -10, -1], [7, 13, 9, 10, 4, 0, -5, 8, 8, -6], [-17, -17, 2, 9, -1, 1, 2, 14, -9, 6], [4, 14, 3, -3, -2, -2, 1, 1, 3, 4], [17, -10, -13, -12, -8, 4, 1, -10, -7, -2], [-1, 2, -7, -7, 0, -6, -3, 0, 13, -5], [14, 1, 2, -7, 9, 12, -3, 0, -5, 7], [2, -9, -17, -5, 5, -9, -9, 7, 7, -11], [6, 11, -7, -5, 4, -2, 0, 0, 16, -9], [21, 11, 10, 2, 5, 10, 5, -2, -6, -1], [-12, 24, 15, -3, 16, 7, 5, -1, 1, -4], [-2, 10, 2, 1, -3, 5, -8, 8, 11, -19], [-7, 17, 7, 2, 12, 3, -2, 11, 12, -11], [35, -6, -14, -15, 1, -1, -2, 10, -2, -8], [-10, 7, 7, -1, -2, 5, 1, -6, -5, -3], [-19, 7, 9, 5, 0, -5, -4, 12, 5, -7], [-12, 16, 4, -1, 6, -2, 0, -10, 10, 9], [32, -5, -3, -1, -18, 1, 4, -3, -9, 3], [16, -8, 3, 4, -9, -6, 0, 2, -13, 10], [13, 3, -2, 0, -13, 2, -9, -6, -6, 1], [-11, 6, -3, -7, 0, 6, 3, -10, 5, -5], [9, -7, 13, 4, 6, 9, 1, 7, -11, 6], [25, 4, -3, 4, -8, -6, -2, 4, 0, -1], [33, 2, 7, -4, 12, 2, -3, -1, -6, 3], [15, 10, 6, -2, -9, 12, 9, -5, -5, -3], [-3, -4, -2, -7, -4, 14, 13, -10, -10, 9], [31, -8, -12, -3, -5, -5, 3, -4, -7, 10], [-13, 23, -1, -3, 0, -3, 1, -3, 3, -13], [4, 4, -3, -3, 6, -4, -2, 2, 13, -9], [0, 6, 2, -2, 3, 3, 3, 5, -3, 10], [31, 17, 2, -16, 8, 4, 4, -10, 10, -2], [11, -8, -4, 5, -9, -12, -4, 10, 4, -3], [-6, 3, 0, -9, 4, -4, 3, 3, 1, -1], [3, 1, -1, 5, -3, 2, -12, 3, 6, -18], [-2, -18, 10, 8, -5, 7, 2, 8, -3, -4], [-6, -15, 5, 0, -3, -7, 4, -9, -6, 5], [-14, -2, -2, -5, -5, -8, 6, -9, 4, 3], [1, 2, 7, -3, 16, 10, 8, 5, 1, -7], [-20, 13, -4, -2, -2, 1, 3, -6, 2, 8], [9, -17, -3, 4, 2, -2, -3, 16, -2, -4], [-24, -6, 3, 4, 17, 0, 1, 7, 0, 3], [3, 7, -1, 0, 10, -3, -1, -8, 4, 11], [19, -7, 5, -6, 14, 5, 3, 12, 2, -5], [4, 19, 4, -1, -5, 8, 7, -2, -3, -5], [-20, 19, 10, -5, 6, 6, 7, -3, 6, 0], [22, -5, 10, 11, 10, -3, -1, 11, -3, 4], [22, -14, -8, -1, -8, 0, 1, 4, -17, 8], [-3, -9, 7, 0, 4, 3, 4, -3, -13, 1], [12, -4, -10, -6, -10, -2, -6, 6, 14, -17], [17, 3, -6, -10, -1, -5, -1, -2, -8, 6], [6, -18, -12, 1, -14, -1, 2, -11, -13, 3], [18, -1, -4, -6, -1, -7, 2, -2, 1, -1], [10, -2, 5, 8, -5, -13, -8, 8, 3, -10], [18, 9, -3, 3, 0, -4, -11, 6, 13, -14], [16, -2, -2, -3, 10, 8, -10, 8, -4, -11], [17, -8, -1, -2, -3, 11, 5, -8, -14, 6], [-21, -4, 4, -6, 6, 3, 9, -4, 6, 1], [-2, -8, 3, 1, 2, -16, 6, -1, 12, 2], [12, 3, -1, 8, -8, 1, -9, 3, -10, 8], [19, -15, -1, 4, -11, 1, 1, 10, -7, 4], [25, -5, -7, -2, -3, 10, -4, -8, -6, 0], [-8, 3, -5, 0, -10, -2, -4, -16, -6, 13], [17, 1, 3, 5, 3, 6, 4, -3, -17, 6], [-22, -2, 10, 13, 2, -12, -6, 1, 7, 15], [27, 6, 4, -9, 9, 15, 2, 11, -1, -7], [0, -4, -12, -12, -4, -3, -1, 7, 1, 6], [10, 11, -4, -9, -5, 10, -4, 0, 0, -8], [-3, -11, 5, -5, 13, -1, 4, -3, 5, 9], [2, -3, 0, -2, 1, -2, -7, -1, 12, -3], [26, -5, -7, 3, -7, -2, -14, 2, -7, 6], [-2, 15, 5, -2, -8, 11, 2, 5, 4, 4], [8, 18, -5, -6, 2, -6, -3, -12, 2, -3], [4, -7, -1, 3, -10, -6, 9, -6, -5, -1], [-6, 4, -11, -6, -5, 10, 1, -5, -2, 2], [25, -2, 3, -1, 8, -4, -1, -5, -7, -6], [-20, 10, 1, -2, 12, -2, -4, 11, 24, -13], [-14, 4, -5, 1, 6, -6, 0, 6, 11, -1], [11, -8, 5, 8, 4, -9, -2, 10, -5, -1], [-19, 15, -2, -9, 14, -1, -2, 0, 10, -5], [-28, 9, 6, 8, 11, 2, 1, 1, 7, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8022_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8022_2_a_r();".
// 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_8022_2_a_r(:prec:=10)
    chi := MakeCharacter_8022_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(3067) - 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_8022_2_a_r();".
// 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_8022_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8022_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![36, -10, -321, -69, 486, -98, -180, 76, 8, -8, 1]>,<11,R![252, -8768, -1835, 8662, -805, -2246, 437, 193, -46, -4, 1]>,<13,R![-1636, 7258, -7653, -3052, 8761, -3938, -95, 367, -46, -6, 1]>],Snew);
    return Vf;
end function;