
// Make newform 8008.2.a.x in Magma, downloaded from the LMFDB on 18 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_x();".
// 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_x();".
// 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 := [-64, 0, 1069, -1431, -682, 1484, -4, -536, 80, 79, -17, -4, 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], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [42444, -30863, -130123, 100022, 55928, -71120, -2976, 18608, -1221, -2017, 116, 77], [31376, 37113, -207707, 47918, 180912, -57900, -60424, 18752, 8491, -2453, -416, 113], [-248, -52764, 60651, 48306, -40751, -22080, 8097, 6284, -468, -896, -2, 46], [18376, 144063, -231757, -246082, 259612, 95000, -100224, -5448, 15541, -1703, -816, 163], [-3234, 6817, 25304, -13574, -29113, 9303, 11464, -2821, -1764, 386, 91, -19], [4848, 32399, -68611, -29511, 100626, -7975, -46427, 9646, 7968, -2044, -443, 124], [9208, 16547, -53469, 6980, 51436, -17414, -19166, 7580, 2943, -1221, -154, 65], [-6892, -19017, 108835, -36568, -97584, 43218, 31846, -15052, -4397, 2115, 214, -103], [-22808, -157229, 549981, -193544, -562446, 269800, 203642, -97266, -30103, 13849, 1528, -679]];
    Rf_basisdens := [1, 1, 1, 6100, 6100, 1525, 6100, 610, 3050, 1220, 1220, 6100];
    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, 0, 0], [0, 1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [2, -1, 0, -1, -1, -1, 0, -1, 0, 1, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0], [3, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 2, 0, 1, 1, 0, 0, 1, 0, -1, 1, 0], [0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0], [1, 1, -1, 1, 0, -1, 0, 0, 0, 0, 0, 1], [2, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 1], [0, 1, -1, 1, 1, 0, 0, 1, 1, -1, 0, 1], [1, -2, 1, 0, 0, 1, -1, 0, -2, 1, 1, -1], [3, -3, 1, -1, -1, 0, 0, 0, -1, 1, 0, 0], [4, -2, 1, 0, -1, -1, 0, 0, 0, 1, 0, 1], [1, 2, -1, 0, 0, 0, 0, 0, 0, -1, 1, 1], [0, -1, 1, -1, 1, 0, 0, 1, 0, 1, 0, 0], [2, -1, 0, 0, 1, 0, 0, 1, -1, -1, 1, 0], [-1, 0, 0, 0, 1, 1, -1, 0, 1, 0, -1, 0], [-1, 2, 2, 2, 1, 1, 0, 0, 0, -1, 1, -1], [1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0], [3, 1, 0, 0, -1, 1, 1, 1, 0, 0, 0, 0], [1, -1, 0, -1, 1, 0, 1, 0, -2, 1, 0, 0], [2, -1, -1, -1, -2, -1, 0, 0, -1, 0, -1, 0], [1, -1, 2, 1, -1, 0, -1, -1, -1, 1, 1, 0], [0, 2, 1, 2, 0, 1, 0, 1, 1, 0, 1, 1], [1, 1, -3, -1, -1, -1, -1, -2, 2, 0, -2, 1], [0, 0, -1, 0, -1, 0, 0, -1, -1, 1, -1, -1], [-1, 2, 0, 2, 1, 0, 1, 0, 1, -1, 1, -2], [1, 0, 1, 0, -1, 0, -1, -2, 1, 2, 1, -1], [2, -1, -2, -1, -3, 0, 0, -1, 0, -1, -2, 0], [-5, 5, -1, 1, 3, 0, 0, 1, 0, -1, 1, 0], [-1, 1, -1, -1, -1, 0, -1, 0, 1, 0, 0, 1], [3, 1, -3, -1, 0, -1, 1, 1, 1, -1, -1, 0], [-1, 0, -1, 0, -2, 0, -1, 0, -1, 1, -1, -1], [3, 0, 1, 2, 0, 1, 0, -1, -1, 0, 0, -1], [1, 2, -2, 0, -2, 2, 0, -1, -1, 0, -2, 0], [0, 0, -1, -1, 4, 0, 0, 1, 2, 0, 0, 1], [5, -3, 0, -2, -1, -1, 0, -1, 1, 3, 0, 0], [3, 0, -1, -2, -3, -2, -1, 0, 1, 1, -1, 1], [-3, -1, -1, 0, 0, 1, 0, -2, 0, 0, -1, -1], [1, 1, 3, 1, 1, -1, 1, 1, 1, -2, 3, -1], [3, -1, 3, 1, 0, 0, 2, 1, -1, 2, 3, -3], [4, -1, -2, 1, -1, 0, -1, 0, -2, 1, 1, 0], [1, 0, -5, 0, -2, 0, -1, -2, 1, -1, -3, 1], [5, -1, 1, 1, -1, 0, 2, 2, -3, -1, 2, -1], [-2, 5, -2, 1, 1, 1, 0, 1, 2, -3, -2, 2], [-3, 4, 0, 2, 2, 1, 2, 2, 2, -3, 1, -2], [2, -2, -3, -2, 0, -1, 1, -1, 0, -1, -1, -1], [9, -5, 4, -3, -5, -1, -1, -3, -3, 3, -1, 0], [9, -2, 4, 0, -4, 1, 0, 0, -2, 2, 3, 0], [-5, 0, 0, 0, 0, 2, -1, 1, 1, -1, -2, 0], [-5, 3, -2, -1, 4, -1, 0, 0, 1, -2, 0, 0], [2, -4, 1, 1, 2, -1, -2, 1, 0, 0, 0, 1], [3, -2, 2, -2, -5, -1, -2, -1, -1, 3, 0, 0], [9, -3, -1, -3, -2, 0, -2, -2, -1, 2, -2, 3], [1, -1, -1, -3, 0, -1, -1, -1, -2, -1, -1, 1], [3, 0, 2, 0, -2, 0, 0, 1, -1, -1, 2, 0], [5, -6, 2, 0, -1, 1, -1, -1, 0, 0, 2, 0], [3, -2, -4, -2, -1, -1, 0, 0, 0, 1, -3, 2], [-1, -2, 1, 0, 3, 1, 0, 3, -1, 1, 2, 2], [4, -2, 1, 0, 2, -1, 1, 1, 0, -1, -3, 3], [5, -1, -2, 1, -1, -2, 0, 2, -1, 0, 2, 2], [8, 2, 0, 0, 0, 1, 2, 2, 0, -2, 0, -2], [1, 1, -1, 3, -2, 1, 0, -1, 2, 0, -1, -1], [-4, 0, 1, -2, 2, 0, -2, -1, 1, 0, -1, -1], [6, -3, 5, 2, -4, -1, 2, 2, -4, 0, 2, -1], [6, 0, 3, 0, -5, 1, 2, 0, -2, 3, 2, -3], [-7, -3, -4, -1, 1, -1, -2, -3, 3, -1, -3, 2], [5, -5, 1, -1, -2, -1, -3, -1, -1, 1, 1, 1], [-5, 6, -2, 0, 4, 1, -3, 2, 3, -1, -1, 3], [5, -6, 2, 1, -1, 0, -1, 0, 2, 2, 2, 0], [10, -5, 1, 1, -4, -1, 2, 1, 2, 2, 0, -1], [1, -4, 1, 0, 0, -1, 2, -1, 1, 0, 0, -1], [8, -5, 0, 0, -2, 0, 0, -2, -2, 2, -2, 0], [0, 0, -5, 0, 0, -2, 2, 0, 2, -2, -2, 3], [-1, -1, -2, 3, 1, 0, -1, 3, 3, 0, 1, 2], [1, 4, -2, 2, 2, 0, 1, 3, 0, -3, 2, 2], [3, 3, 1, -1, -2, 0, 1, 0, 2, 1, -2, -1], [-3, 1, 1, 1, -1, 1, 1, 2, 2, -2, -2, 1], [4, 1, 0, -3, 1, -2, 1, 0, 0, -2, 1, 0], [10, -5, -1, -1, 1, 1, 0, -2, -3, 0, -1, -1], [-1, 2, 0, 2, 6, -1, 1, -1, 0, -3, 0, 2], [1, -3, 1, 1, 2, 2, -1, 3, -3, -1, -1, 1], [6, -4, -1, 3, -3, 0, 0, 3, 0, -1, 0, 1], [0, 0, 2, 0, -4, -1, -1, 3, -2, -1, 1, 1], [-3, -3, 0, 1, -1, 0, 2, 2, -1, -1, -2, 0], [-8, 1, 2, 1, 4, 0, 1, 0, 2, -1, 1, 0], [-5, 2, 1, 0, 1, -2, 0, 1, 1, 3, 4, 0], [8, 1, 1, 1, -3, 0, 1, 1, 1, 0, 2, -1], [11, -4, 0, 0, -7, 1, 0, -4, -2, 1, -1, -2], [9, -5, 5, -1, -2, -1, 0, 0, -1, -2, 2, -1], [0, -3, 2, 1, 0, -1, 2, 0, -1, 0, 1, -2], [8, -9, 1, -3, 1, -2, 0, 3, -2, 1, 2, 1], [-8, 1, -5, 3, 1, 0, 1, -1, 5, -2, -2, 1], [7, 2, 6, 0, -1, 0, 0, 0, 0, 0, 3, -2], [-1, 3, -2, -3, 2, -1, -2, 1, -2, -2, -1, 2], [9, 1, 2, 1, -2, 3, -2, 1, -2, 0, 3, -2], [-1, -2, 3, -2, 0, 2, 0, -1, -1, 1, -2, -3], [0, -7, 1, -2, 0, 1, 0, -2, 0, 0, -2, -1], [-5, -3, -4, 1, 0, 1, 0, 0, -1, -2, -4, 2], [4, -3, -3, 3, 2, 1, -2, 2, 1, 0, -1, -1], [8, -4, -2, 0, -2, 0, -1, 2, -1, -3, 2, 4], [11, 0, 1, 2, -1, 0, -1, 2, 0, 0, -1, 3], [11, -9, -1, -3, -1, -1, -2, -2, -7, 3, 0, 1], [4, 3, 2, -1, 2, 0, 0, 0, -1, 2, 5, -4], [-5, -6, 1, 2, 1, -2, -1, -2, 3, 0, -1, 1], [-13, 6, -9, 2, 4, 2, 2, 1, 5, -4, -4, 3], [-9, 0, -1, 0, 1, 0, -3, -1, -1, 4, 2, 3], [0, 2, 3, 2, -4, 1, -1, 1, -1, 1, 3, -1], [-4, -9, 3, -1, 4, 0, -1, 2, 0, 1, 1, 1], [5, -1, 0, -5, -3, -1, 1, -2, 0, 2, 0, -2], [-5, 9, -1, -1, 0, 0, 2, 0, 2, -2, 0, -1], [-13, 8, -7, 3, 4, 1, 1, 0, 0, -3, -2, 1], [6, 0, 2, 0, -7, -2, 0, -2, 2, -2, 0, -2], [-9, 2, -2, -2, 4, 1, 3, -1, 4, -1, -4, 0], [4, 0, -8, 2, 0, 1, 1, 0, 2, -5, -4, 2], [-1, -2, 1, -2, 6, -3, 0, -2, -1, 2, 3, -1], [1, -6, 1, 2, 5, 0, 2, 3, 1, -1, 2, -3], [0, 8, 1, 0, 0, 1, -5, 0, 3, -1, -2, 1], [3, 7, -2, 1, 2, -2, 2, 2, -1, -2, 4, -2], [8, -1, 1, 0, -2, 2, 2, 0, 0, 2, 0, -3], [3, -1, -7, 1, 1, -2, -1, -4, 4, -1, -2, 1], [-4, -2, -1, 0, -2, 1, -1, -3, 0, -1, -3, 1], [-11, 8, -4, 4, 2, 1, 2, 1, 3, -2, 0, 2], [-3, -7, 2, -1, -1, -2, 1, -1, -1, 2, 3, -4], [-6, 0, 0, 0, 3, -2, 1, 0, 1, -4, 0, 0], [-15, 14, -5, 4, 7, 0, 1, -1, 8, -4, 2, 2], [3, 2, -2, 4, -4, 1, -1, -2, 1, -1, -1, 0], [9, 1, 0, 1, -2, -1, -3, 0, 0, 1, 4, 0], [-7, -4, -4, -3, 1, 0, 1, 0, 2, -2, -2, -2], [-1, -5, 2, 3, 1, 2, 1, 2, -3, 4, 2, 2], [-8, 11, -8, 3, 9, -1, 2, -1, 2, -7, 0, 2], [-4, 8, -2, 2, 2, 1, 0, 2, 2, -4, -2, -4], [2, -8, 4, -4, -2, 0, 0, -4, -4, 2, -2, 1], [-1, 4, -4, -1, -1, 3, -2, -1, 4, -3, -5, 2], [0, -1, -3, -1, 1, 1, 0, -1, -2, -3, -6, 3], [-1, -2, -2, 2, 5, -1, 1, -1, 6, -4, 0, 0], [-8, 6, 6, 2, -1, 1, -1, 7, -2, 0, 3, 2], [6, -2, 1, -6, -5, -1, -1, 0, 1, 0, -2, 3], [7, -9, -4, -1, 9, 2, 0, 3, 0, 1, -1, 2], [0, 4, 3, 6, 3, -1, 4, 4, 0, -3, 6, -6], [0, -6, -2, -2, 8, -1, 1, 0, 3, -1, -2, 0], [1, -5, 2, 1, -3, 1, -3, 0, -2, 4, 0, -2], [1, 3, -3, -2, 0, 2, -3, 0, 3, -1, -3, 1], [9, -5, -3, -3, 3, 1, 1, -2, 0, 2, -4, 3], [-8, -8, 5, -2, 1, -1, -2, -5, 3, 1, 1, 1], [-5, 2, 0, 2, 3, 0, -1, 1, 2, -2, 2, 4], [-4, 5, 4, 5, 3, 2, 3, 6, 0, -6, 5, -6], [5, -9, -4, -3, -2, 0, -1, 1, -1, -1, -3, 4], [-5, 6, -1, 1, 5, 0, 3, 3, 1, -2, 3, -1], [-4, -9, 7, -2, 0, 1, 0, 2, -4, 0, -2, 1], [14, -13, 4, 1, -4, 0, -1, 3, -7, 5, 2, -3], [-12, -4, -1, -2, 0, -1, 1, -5, 4, 1, 1, -3], [3, 0, 5, 2, -2, 1, 2, 2, -3, -2, -1, 1], [20, -9, 5, -5, -8, 0, 1, -1, -7, 7, 6, -1], [-15, 6, 2, 2, 8, 0, 0, 7, 1, -2, 4, 0], [3, -11, 3, -1, -5, 0, -1, -1, -3, 5, 1, 1], [4, 3, 4, -1, 0, 0, 2, 0, -5, 2, 1, 0], [2, 5, 3, 5, -1, 5, 2, 2, -1, 1, 3, -1], [-4, -7, 6, -1, -3, -2, 0, 0, 3, 3, 5, -2], [1, 1, 10, 1, -5, 2, 2, 2, -5, 1, 4, -5], [10, 4, 4, 2, -4, -3, 5, -3, 2, 0, 1, -2], [-4, 3, 1, -1, 7, 0, -1, 0, 0, -4, 3, -3], [-6, -2, 3, -2, -1, 1, -1, -3, 0, 4, -3, -1], [-7, 9, 2, -1, 1, 2, 0, 0, -1, -1, 0, -6], [9, 4, -3, 2, -4, -3, -1, -4, 1, 0, -1, -1], [-2, 0, -2, 1, -5, 3, 0, -2, -1, 3, -7, 4], [-4, 0, -5, 1, -2, 1, -4, 0, 3, 0, -3, -1], [5, -2, -3, -6, 3, 0, -1, -2, -1, 4, -3, 3], [-12, -12, 3, -4, 1, 1, -1, -1, -8, 2, -1, -3], [8, 4, 0, -2, 3, -2, -1, -3, 4, -2, -3, -2], [8, -4, 4, 2, 1, 4, 3, 6, -7, -2, 6, 2], [-11, 0, -1, -4, 2, -1, -2, -1, 3, 0, -4, 0], [-11, 2, -4, -2, 10, -1, -1, 5, 4, -1, 2, 3], [2, 0, -4, 0, -3, 0, -3, 6, -1, 0, -2, 6], [-15, 7, 1, 1, 3, 3, 0, 5, 2, -5, 1, -2], [4, 7, 2, 3, 7, 2, 3, 5, 3, -2, 6, 0], [-3, -5, 0, 1, -4, 0, -3, -2, 1, 1, 0, 6], [-9, 10, -2, -2, 9, -1, 2, 0, 2, 1, 1, 0], [8, -3, 1, -1, 5, -2, -1, 0, -2, 0, -1, 5], [-10, 11, -3, 1, -1, 2, 0, 0, -1, -5, -3, 3], [12, -7, -3, -7, -9, -5, -1, -6, 2, 5, -5, 3], [8, -4, -2, 0, -4, 0, 1, -2, -5, 3, 2, -2], [0, -4, 0, -4, -7, -4, -3, -2, -3, 0, -2, 2], [-1, -8, -2, -4, -6, -2, -3, -1, 0, 1, -4, 6], [-6, 1, 5, 1, -4, -1, 0, -1, 0, 2, 8, -7], [3, -7, 7, -4, -6, -3, -2, -5, -3, 4, -2, 1], [24, -6, 11, -4, -7, -1, 0, 1, -1, 5, 5, -1], [7, -7, 5, -3, -4, -2, -2, -2, -3, 6, 4, 1], [-4, -1, 4, -4, -3, 0, -2, 0, -2, 1, 4, 0], [7, 6, -2, 0, -5, 3, 1, 1, 1, -4, -6, -2], [10, -2, -2, -2, -4, -1, 0, -7, 3, 2, -3, -2], [17, -1, 6, 1, -3, 4, 2, 3, 0, -3, 1, -8], [7, 9, -6, 2, 1, 1, 1, -1, 0, -2, -6, 0], [11, 2, 1, -2, -4, 1, -2, -4, -2, 4, 5, -1], [0, 2, 3, 0, 0, 4, -1, 3, 0, -2, -5, 3], [-10, -5, 3, 3, 7, 2, -2, 0, 1, -1, -3, 2], [-17, -7, 1, -1, 0, 3, -2, 3, 2, 0, 3, -1], [3, -7, 2, -1, 2, 2, -1, 4, -4, 1, 4, 2], [-5, 1, 0, -1, 5, 1, 3, -2, -4, 0, -2, 2], [-6, 9, 5, 3, 1, 1, -1, -1, -1, 0, 0, 1], [-2, 2, -2, 2, -10, 1, 1, -1, -2, 0, -1, -2], [-17, 3, -2, -3, -2, -2, -4, 0, 1, 1, 0, 0], [3, -6, -2, -6, -4, -1, -3, 1, -2, -1, 2, -4], [7, -9, -1, -3, 6, -5, 0, -1, 6, 4, 3, 5], [-16, -4, 5, -2, 1, 1, 1, 3, -2, -4, -3, 1], [5, -4, -3, -4, -7, -2, -1, -3, 4, -2, -2, -5], [12, -3, -2, -3, 0, -3, 4, -5, 0, 4, 2, 3], [13, -5, -2, -1, -2, 2, 1, 5, -1, -2, 1, 2], [-3, -13, 1, 0, 3, 2, -1, 1, 2, 4, 2, 3], [4, 8, 8, 4, 6, 1, 3, 4, -1, -3, 2, 0], [-7, 1, -6, -1, -4, -4, 0, -6, 4, 0, -2, 2], [-10, 17, 0, 7, 6, 2, 3, 1, 5, -5, 6, -2], [-8, 18, 1, 8, 1, 0, 0, -2, 2, -5, 2, -3], [8, 7, 5, 1, -1, -1, 1, 1, 1, -2, 6, -5], [-6, 9, -1, 5, -1, 3, -2, 1, 2, -7, 2, -1], [-9, 4, -2, 0, 4, 1, -1, 1, 2, -3, -2, 7], [9, -9, 10, 3, -4, 1, 2, 7, -2, 4, 5, 0], [2, -2, -6, -6, 1, -4, 2, -9, 1, 5, -3, 2], [-5, -7, -7, -3, -2, -1, 1, -4, -2, 1, -2, -5], [3, -2, 2, 4, 5, 3, 3, 3, 0, -4, -2, 2], [6, 9, -4, 1, 5, -1, 3, -6, 4, -2, -1, 0], [-15, 6, -3, 0, 5, 2, 3, 0, 3, -2, -1, -5], [5, -2, 0, -4, -11, -1, 1, -5, -3, 2, -2, 0], [-16, 2, -5, -2, -3, -5, 0, -3, 1, 3, -5, 1], [0, -14, 1, -2, 2, 0, 3, 1, 0, 1, 1, -3], [-8, 7, -2, -1, 7, 1, 1, 0, -4, -4, 3, -2], [-7, 5, 0, 7, -4, 3, 1, 0, 4, -6, 0, -4], [15, -8, -1, 0, -9, -1, -2, -2, -3, 5, -5, -3], [-9, 6, -7, 0, 0, -5, 0, -1, 3, -2, -2, 3], [13, -3, -4, -1, 5, -1, -3, 0, 0, -1, -4, 6], [-7, -3, 4, 3, 0, 2, 2, -1, 0, -4, 1, -6], [-12, -3, 4, 2, 9, 3, 0, 1, -3, -1, 3, 0], [0, -7, -3, -5, -5, 0, 0, 3, 0, 1, -6, 3], [-8, 4, 0, 0, 7, -3, -1, 4, 7, 0, 2, 4], [-10, 11, 0, 0, 5, 4, -2, 1, 1, -1, 3, 0], [12, 8, -9, 0, -3, -3, 3, -2, 5, -4, -2, 1], [-1, -7, 1, -3, 0, -2, -1, -3, 5, 5, -3, 3], [7, 3, -5, 1, -1, 1, -3, 2, 0, -2, 4, 3], [10, -13, -3, -7, -15, -3, -2, -4, -1, 3, -7, 1], [-4, -1, 6, 1, 1, 4, 2, 1, -2, -1, -2, 2], [-9, -2, -5, 0, -6, -3, -2, 0, 4, -4, -1, 7], [3, 0, -7, -8, -2, -2, 2, -3, 1, 4, -6, 3], [-5, -5, -7, 1, -6, -2, -2, -5, -2, -1, -5, 1], [9, -6, 8, 2, 0, -3, -3, 6, 1, 5, 3, 2], [3, 1, -4, 1, -2, 5, -3, 0, 3, -5, 0, -4], [0, 3, 4, 1, -2, 0, -1, 2, -2, 3, 1, 0], [-3, 4, -5, 6, -4, -1, 5, 1, 0, -7, 0, 0], [-8, 10, 0, 2, 5, 3, 0, -2, 0, 1, 2, -6], [-17, 8, 3, -2, -2, 2, 4, -2, -1, -4, 1, -1], [2, -2, 1, -6, -7, -4, 0, -3, -1, 3, 1, 4], [14, -3, 5, -7, -4, -5, -1, -2, -8, 9, 3, -1], [-4, -8, 0, 0, -11, 1, -2, 1, -3, 3, 3, 2], [15, 1, 1, 8, -1, -3, 5, -5, 2, 0, 2, 3], [-11, 1, 6, 6, -5, 0, 1, 1, 0, 2, 8, 2], [1, -2, -8, -2, -3, -2, 0, 1, -7, 7, 2, 2], [-17, 13, -2, -1, 1, 3, -4, -6, 5, -1, -4, -1], [-8, -5, 0, -4, 1, -3, -6, -5, 1, 1, 1, 2], [8, 1, 0, -2, -8, -2, -2, 0, 0, 0, 6, 2], [-6, -10, -2, -6, -3, -8, 0, -3, -3, 5, -1, 2], [2, 2, -6, 1, -7, -5, 4, 1, 0, -5, 2, -2], [-8, -8, 0, -2, -1, 3, -2, 6, -8, 5, 2, 6], [5, -8, 3, -4, -6, -1, -1, -1, 4, 5, 4, -7], [-8, -1, 4, 1, 4, 2, 3, 5, 3, 1, 6, -2], [-7, 0, 9, 2, -5, -1, -1, -7, 2, 4, 0, 0], [-11, -1, 8, 3, 4, 4, -2, -1, 2, 3, -1, 4], [13, 0, 8, 5, 0, 7, 0, 6, -11, 0, 4, 0], [5, 10, -5, 8, 3, 3, 1, 8, 8, -8, 1, 1], [-6, 10, -7, -2, 8, 0, -4, 4, 6, 0, -4, 3], [11, -12, 2, 0, 0, 1, 3, 2, -9, 1, 9, -5], [-9, 12, 4, -1, 1, 1, -3, 1, 5, -2, -3, 0], [8, 15, -4, -1, 6, -1, 1, 2, 2, -1, 3, 2], [12, -1, -2, -1, -9, -5, 1, 0, 2, 2, 1, 6], [-11, 4, -6, 0, 3, 2, 0, -1, 9, -1, -4, 0], [3, -13, -1, 1, 3, 2, -1, -2, -6, 2, 4, -4], [3, 10, -8, -4, -4, -2, 0, -1, -3, 2, 0, 0], [12, -5, 1, 1, 0, 0, -1, 5, -7, 1, 2, 1], [15, -13, 2, -3, -7, -5, -1, -2, -1, 6, 0, 0], [-4, 6, -6, -2, 4, 5, 3, -4, 3, -3, -2, -6], [-4, -10, 3, -2, -6, 4, 0, -2, -8, 2, 2, -4], [1, -16, 0, -3, -7, -4, 3, -3, -5, 4, -7, -2], [-12, 7, 5, 7, 2, 0, -1, 0, 0, -3, -1, 5], [3, 18, -4, 2, 12, 0, -2, 3, 2, -2, 0, 0], [-12, 4, -2, -2, 4, 3, 1, 6, 3, -1, -2, 4], [-4, 17, -3, -3, 7, -2, -1, -1, 11, -2, -8, -1], [8, 6, -6, -2, 2, -3, 0, 1, 3, -9, -5, 4], [-24, 6, -2, 6, 0, 2, -1, 2, 3, -3, -4, 5], [0, 3, 3, -5, -11, -5, -2, -6, -1, 3, 1, -3], [8, -12, 4, -4, 2, -1, 0, 2, -6, 4, 0, 4], [2, 9, 0, 4, 3, 2, 0, -1, 1, -3, 1, -6], [6, -14, 3, 2, -2, -5, 1, -2, -7, 3, 0, 3], [-12, -1, -6, 1, 9, 1, -1, 6, -4, -2, 1, 2], [-15, -4, 0, -2, -13, -5, 1, 0, -1, 0, 1, 4], [4, 16, -8, 8, 2, 1, 8, 4, 0, -10, -2, -2], [19, -14, 6, 3, -9, 0, -2, 3, -8, 3, 7, -4], [3, -1, -5, -1, -8, 0, -1, 3, -5, -1, -1, -3], [3, 6, 1, -2, 4, 5, 2, -2, 2, 2, -1, -5], [-8, -15, 0, -3, -2, -6, 1, 0, 0, 1, -3, 4], [14, -10, 8, 0, -7, 6, -1, -2, -7, 8, 0, 0], [10, -14, -10, -2, 1, -2, 0, 4, 0, -1, -6, 4], [-12, 3, 1, 9, 0, 7, 0, 0, 1, -4, -1, -3], [-5, -3, -2, 3, 0, -1, 1, 1, 1, 3, 5, -7], [15, 2, -2, 4, 1, -3, 4, 0, 2, 3, 3, 6], [10, 4, -1, 0, -2, 3, 1, 1, 0, 5, -3, -3], [-13, 5, 5, 1, 5, 4, 0, 3, 4, -5, -7, -5], [-2, -2, -14, -2, -1, 2, 0, 0, 6, -1, -8, 4], [-39, 3, 2, -1, -1, -5, -5, 1, 5, -2, 1, 6], [4, 3, 0, -1, -1, 4, 1, -2, -2, -4, -5, -2], [8, -2, 3, 0, -13, 6, 0, -6, -4, 5, -6, 0], [-8, 12, -5, 2, 1, 5, -1, 4, -1, -10, 0, -1], [-1, 9, 4, 3, 3, 3, 0, 9, 0, -1, -1, -2], [17, -7, 8, -1, 2, 4, 1, 5, -11, 0, 5, -4], [-29, 6, 9, 4, 7, 6, 4, 6, -2, -3, -3, -3], [1, 1, 4, 1, -4, 1, -2, 3, -6, 2, 7, 0], [8, -6, 2, 2, 1, 0, 3, 2, 5, 0, 0, -2], [0, 13, 6, 9, 8, 6, -1, 2, -3, -9, 5, -2], [-19, -8, -6, 0, 2, -5, 4, -2, -1, -2, -5, -2], [18, 7, -5, 0, 10, -1, 2, 5, 3, -6, 3, 3], [-20, -1, -2, 1, -4, -2, -2, -4, 5, 0, -1, 8], [1, 2, -6, -2, -9, -5, 1, -5, 3, 4, -6, 4], [-4, -6, 1, 6, 0, -3, 3, 0, -1, -1, 2, -5], [-5, 6, -8, 8, 5, 0, 0, -1, 11, -8, -2, 2], [-26, 14, -10, 8, 6, 5, 1, -3, 6, -7, -5, -4], [7, -12, 4, 3, 1, 2, -1, -1, -7, 4, 3, -6], [-17, 10, -2, 4, 0, 2, -1, -4, 11, -7, -3, 3], [-5, 0, -5, -6, -8, 0, -1, 0, 3, -5, -3, -3], [-6, 5, -2, -3, 6, 6, -4, -6, -1, 0, -7, 1], [-28, 1, 4, 3, -1, 0, 3, 0, 0, -4, 1, -4], [13, -11, 1, -3, 4, 0, 9, 1, 3, -1, -7, -1], [4, -6, 9, -2, -5, 1, 3, 0, 1, 2, -2, 1], [-16, 6, 2, 6, 9, 0, 3, 1, 0, 2, 5, 0], [-7, -9, 3, 3, -3, 4, 4, 0, -3, -2, -4, 1], [15, -4, -6, 0, 7, 0, -2, 5, -5, -1, 0, 2], [-13, 1, -6, 3, 3, -1, 2, -4, 5, -3, -2, 0], [-22, 6, -1, 0, -7, 0, 1, 1, 4, -6, 1, -1], [7, 12, -10, 2, 12, -1, 2, 3, 3, -4, 0, 4], [2, 1, -5, 1, -4, -2, -3, -3, -3, 3, -2, -3], [3, -6, 9, 2, -12, -2, 1, 1, 4, -1, 2, -1], [13, -18, 14, 2, -13, -2, -1, 2, -7, 5, 3, -2], [2, -10, 6, 4, -2, 2, -1, -1, 4, 1, -3, -4], [-6, 6, 4, -2, 2, 2, 4, -2, 0, 8, 2, -10], [-8, -8, -12, -2, 2, -8, 3, 0, 7, -3, -6, 6], [11, 5, 1, 5, -3, -1, -2, 3, 4, -5, 1, 1], [-17, 19, -11, -1, 8, 2, 0, -5, 5, -4, 3, 3], [7, -12, 2, -4, -5, -2, 3, -1, -2, 0, 0, -4], [-7, -2, 2, 2, 7, 3, 0, 4, 0, -1, 3, 6], [12, -15, -2, -5, 3, 1, 3, 2, -2, 8, 3, -2], [-11, 10, -12, 6, 11, 0, 2, 6, 12, -13, 1, 2], [22, -9, 0, 3, 0, -4, -2, 4, -7, 2, 7, 10], [33, -1, 3, 1, 2, -2, 0, -3, 0, 2, 9, -5], [3, 6, 4, 4, 15, 1, 4, 8, 0, -3, 9, -8], [11, 6, 0, 0, 6, 0, -1, -3, 2, -5, 0, -4], [-23, 23, 1, 1, 3, 3, 0, 0, 9, -3, 0, -2], [3, -8, 3, -5, 14, -7, 3, 2, 2, 1, 6, 1], [10, -6, 2, -4, 6, 2, -1, 3, -8, 1, 5, -3], [0, -13, -3, 3, 4, 1, 0, 1, 2, -4, 2, 3], [0, -14, 9, -6, -8, -4, -1, -5, 2, 5, 3, -1], [-6, -5, -2, -8, 8, 3, 4, 2, -2, 0, 0, -4], [-2, 2, -1, 0, 0, 1, 4, -2, 4, 2, -6, 4], [-13, 7, -7, -1, -1, -4, 1, 2, 4, -4, 0, 9], [-12, 5, -4, 5, 2, 3, -1, 2, 2, -3, -5, 4], [-14, -2, 7, 2, 9, 3, 4, -1, 1, -1, 5, 5], [-9, 14, 6, 6, 6, 2, -4, 3, -1, 0, 10, 0], [-13, 0, -2, 2, 8, 6, -1, 6, -1, -8, -1, 0], [10, -6, -6, -4, -11, -3, -4, -6, -6, -1, -2, 0], [14, -1, -5, -3, 7, 2, -3, 8, -6, -2, -1, 5], [-12, 15, -2, 11, 1, 7, 3, 8, -2, -8, 1, 2], [-17, -2, -4, -8, -3, -3, -5, -5, 5, 0, -4, 0], [12, 19, 0, 3, 2, 4, 3, 1, -1, -3, 2, -4], [-1, 4, 0, 4, -1, 0, -3, 1, -4, 1, 0, 8], [-22, 5, -7, 3, 8, 1, -2, -6, 3, -2, -7, 3], [23, 4, 2, -4, 0, -1, 1, -9, 5, 1, -6, -2], [16, 4, 8, -2, 0, 1, 1, 1, 2, 1, -5, 5], [-8, 3, -8, -1, -3, -3, -7, -8, 0, -1, -1, 0], [-25, 10, 6, -2, 1, 1, -1, -1, -4, 0, -2, -6], [5, -7, 17, -3, -15, -4, -1, 2, -2, 8, 2, 5], [18, -24, 0, -8, -10, -5, -5, -11, 0, 2, -5, 4], [-13, 18, -3, 4, 8, 0, 6, 3, 5, -8, 0, 1], [25, 0, 6, -6, -11, -6, -5, 2, -7, 4, 3, 0], [5, -19, 8, -3, -9, 1, -2, 2, -1, 5, 10, -4], [-3, -7, -4, -3, -7, -1, 0, -4, 1, -1, -6, 0], [-15, -8, 8, -4, -9, 0, -3, 5, -8, 6, 2, 0], [11, -3, 3, -1, 9, 1, 2, 6, -5, 2, 2, -9], [-9, -8, -1, -2, 3, -8, 2, 2, -2, 7, 7, 7], [23, -2, 13, 0, -8, -2, 4, -5, -5, 4, 2, -5], [21, 1, 1, 5, 6, -5, 2, 0, 4, -2, 6, -1], [-11, -3, -13, -3, -1, 0, -8, -3, 4, 0, -5, 7], [-3, 17, -4, 7, 4, 0, 0, 10, 4, 0, 2, 8], [2, -9, -3, -1, -19, 1, -1, 5, -3, 0, 4, -1], [16, -8, 1, -2, -3, 2, 0, 0, 0, 5, -12, 2], [31, -3, 2, -7, -6, 2, -3, 1, -3, -3, 1, -2], [-36, 6, -6, -2, 7, -4, 1, -4, 5, -6, 0, -6], [5, -6, -3, -10, 2, 0, 1, 1, 2, 2, -4, 1], [-5, -7, 5, -1, 1, -5, 1, -8, -2, 6, -2, 7], [4, -1, -9, 1, -6, -5, 3, -7, 8, -3, -6, -5], [-6, -1, 3, 3, 7, 5, -4, 5, 6, 1, -2, 3], [6, -1, 14, 4, 1, 0, 0, 3, -3, 1, 7, -8], [11, -11, 5, 1, -17, 1, 0, 2, 3, 1, 2, -5], [15, -1, 4, 3, -1, 2, 1, 9, -5, -6, -1, 0], [-5, 0, 1, 0, -1, -5, 0, 1, 1, 1, -2, 7], [-38, 7, 5, 9, -5, 3, 0, 2, 1, 0, 1, -3], [-18, 16, -1, 6, 5, -1, 3, 4, 5, -2, 10, -7], [-16, 7, -3, 1, -4, -2, -2, 0, 5, -2, 1, 5], [-2, 13, -13, 1, 7, 5, -1, 0, 0, -6, -5, 0], [-6, 7, -2, 11, -9, 5, 0, 3, -2, -1, 6, 0], [-5, 2, 4, 0, 9, -1, 0, 3, -1, -1, -2, 6], [-9, 2, -8, -4, -9, -3, 1, -10, 3, 7, -5, 0], [-10, -9, -2, -9, -10, -4, -3, 0, 0, -1, -1, 2], [22, -1, 2, 3, 2, 0, 2, -5, 2, 4, -4, 4], [-11, 4, -5, -6, 6, -4, -4, -4, 6, -6, -1, 1], [23, -2, -6, -2, 2, -4, 0, -3, 5, -1, -8, -2], [1, -22, 2, -4, 2, -1, -2, -3, -5, -3, -6, 2], [41, 9, 8, -1, 4, -4, 3, 5, -5, -1, 3, -4], [9, 7, 3, 5, -5, 8, 2, -3, 0, -3, -3, -2], [-13, 2, 3, 4, 4, 3, -1, -2, 9, -3, -5, 3], [13, 3, -3, 1, 7, 1, 1, -4, -6, -8, 0, 4], [7, 18, 3, 2, 6, 0, -2, 4, -6, -2, 9, -3], [6, -4, -2, -5, -9, 0, 6, -5, 0, 9, -6, -6], [-26, 14, -6, 4, 3, 4, 5, 2, 1, -5, -4, -2], [4, 17, -3, 7, 1, 6, 2, 9, 0, -5, 0, 9], [-2, -5, 5, 5, -1, 0, -5, 2, 4, -1, 1, 7], [-9, 7, -5, -3, -3, -4, 3, 1, 7, 0, -3, 7], [40, -9, 2, -5, 7, -3, -4, -2, -3, 5, -5, 6]];
    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_x();".
// 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_x(:prec:=12)
    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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-64, 0, 1069, -1431, -682, 1484, -4, -536, 80, 79, -17, -4, 1]>,<5,R![-92, 340, 1063, -2287, -3024, 3604, 816, -1310, 38, 157, -19, -6, 1]>,<17,R![-3460, 32928, 175719, -38441, -323954, -12522, 89876, -13894, -4150, 1031, 7, -16, 1]>],Snew);
    return Vf;
end function;