
// Make newform 8008.2.a.k in Magma, downloaded from the LMFDB on 10 May 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_k();".
// 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_k();".
// 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 := [-4, 5, 29, 11, -13, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-5, 0, 1, 0, 0, 0], [0, 19, 12, -12, -1, 1], [-10, 7, 23, -11, -2, 1], [-6, 16, 22, -12, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1];
    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, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1], [1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 1], [1, 0, 1, 0, -1, 1], [2, -2, 0, -1, -1, 0], [2, 0, 0, -1, 0, -1], [4, -2, 0, 0, -1, 1], [-1, 0, -1, 1, 1, 1], [0, -2, 0, -1, 0, -1], [2, 1, 0, 1, 1, 0], [1, -2, 1, -1, 0, 0], [0, 3, 0, 1, -1, 0], [0, 2, 0, 1, -1, -2], [-3, -1, 1, -2, -1, -2], [-5, -1, -1, -1, 0, -2], [-1, -2, 1, 0, 0, 0], [1, 2, -1, -1, 2, -2], [4, -2, 0, 0, 1, -2], [3, 4, -1, 0, 1, 1], [9, -1, -1, 0, 0, 1], [5, 2, -1, 0, -1, -2], [1, 4, -1, 2, 1, 3], [8, 0, 0, 0, 0, 0], [2, -3, 0, 1, -1, 2], [-3, 0, 1, -1, -1, 0], [3, 0, 1, 0, -2, -2], [0, 2, 0, 0, 1, -4], [2, 0, -2, 1, 0, -1], [10, -2, 0, 0, -1, 1], [2, -2, 0, 0, 0, 2], [7, -4, -1, 1, 1, 0], [-3, 6, -1, 0, 4, -1], [2, -2, 0, 0, 3, -3], [3, -4, 1, -1, 1, 0], [10, -3, 2, -2, -1, -1], [-1, 4, 1, 0, -1, 1], [0, -2, -2, 1, 1, -2], [-5, -2, -1, -2, -3, -2], [6, 0, 2, 0, -3, 3], [-1, 1, -3, 1, 1, 1], [2, -1, 2, 1, -1, 2], [0, 0, 0, 1, 3, 3], [-14, 3, 0, 3, -1, 0], [7, 0, 3, 1, -3, 1], [-7, 1, -1, 0, 3, 4], [-2, 0, -2, -2, 0, 5], [1, 2, -1, -1, 0, -2], [5, 4, -1, 4, 4, 1], [-1, 0, 3, -2, 0, -1], [-6, 4, 0, 2, 2, -1], [-2, 6, -2, 3, 2, -1], [2, -2, -2, 0, -1, 2], [-6, 2, -2, 1, 0, 1], [8, 1, 0, 4, 1, 1], [2, -2, 0, 1, 2, -1], [11, 5, 1, 1, 3, 3], [-13, -4, 1, 1, 2, 0], [7, 4, -3, -1, -2, 0], [-5, 5, 1, 2, -3, 2], [4, 0, 0, 3, 3, 1], [-10, -2, -2, -1, -3, 0], [1, 6, 3, 0, 3, 4], [-8, 0, 0, -1, 0, 6], [10, 4, -2, -1, 0, -1], [-5, 0, 1, 1, -3, 6], [-4, 2, -2, -2, 4, -8], [-1, -2, 5, 0, 1, -6], [-2, 0, 0, -3, -4, 3], [5, 3, -1, 0, 0, -3], [11, 6, -1, 1, 2, 4], [-9, 1, -1, 3, 0, -2], [7, 2, 1, 4, 3, 0], [8, 0, -2, -1, -2, 4], [1, 5, -3, 0, -1, 4], [-4, -4, 0, -1, -2, 3], [4, 6, -2, 4, 2, 4], [-1, -3, 1, 0, -3, -4], [7, -2, 1, 2, 4, -3], [1, 0, 1, 5, 4, 2], [-4, -2, 2, 4, -3, -1], [11, 8, -1, 1, 0, -2], [6, 0, -2, 0, -1, -5], [5, 2, 3, -3, -3, -3], [-2, 4, -2, 6, 1, 4], [1, -4, -3, 0, 2, 1], [-1, 6, -1, 0, 5, 0], [4, -3, 0, 0, 0, 0], [-4, 8, -4, 3, 5, 1], [11, 2, -1, 2, -1, 0], [-10, 0, -2, -6, -2, 2], [11, -4, 5, -3, -1, -6], [-10, -2, 2, -4, -4, 0], [29, 0, 1, 0, -1, -2], [12, -2, 0, -3, 3, -2], [12, -2, 2, -2, -2, -2], [12, -4, 2, 0, -4, 10], [-14, 4, 2, 0, 0, -2], [11, 0, -1, 3, 3, -4], [-17, 8, -1, 4, 4, -3], [-7, 8, -3, 2, 4, 5], [-9, 4, 1, -1, -5, 0], [15, 1, 1, -2, -4, -3], [20, 2, 0, -3, 0, 1], [-1, 0, -1, 0, 0, 5], [2, -6, -2, 2, -2, 2], [0, 11, -4, 2, 3, 5], [1, 4, 3, -4, 0, -5], [4, 0, 2, -4, 0, -6], [1, -6, 1, -2, -2, 3], [-21, 4, -1, 4, -1, 2], [-6, 3, 2, -3, -2, -3], [-4, 7, 4, -1, 2, 1], [16, -2, 2, 4, 1, -1], [-1, 0, 1, 2, -5, 8], [7, 11, 1, 1, -2, 2], [18, 3, 0, -1, 1, 2], [-16, 4, 0, 2, -6, 5], [-9, 2, 3, 2, 4, 1], [-3, -3, -3, -2, 3, 6], [-6, 1, 0, -3, -1, 2], [-1, 6, 1, 3, 3, -5], [-7, 2, -1, -2, -2, 1], [-11, -4, -3, -5, -1, -3], [-11, 11, -5, 6, 4, 3], [5, 4, -1, 3, -2, -1], [16, 2, 4, 1, -6, 7], [17, -6, -1, 0, 2, -7], [-11, -8, 1, -1, 4, 4], [-17, -4, -5, 1, -3, -1], [-3, 8, -3, -1, 9, -4], [4, 8, -4, 1, 0, -3], [4, 4, 0, -3, -4, -1], [1, -15, 1, 0, -1, -4], [5, -5, -1, 8, 5, 6], [6, 6, -4, 4, 5, 1], [-12, 0, -6, 2, 2, 0], [-3, 4, -3, -4, 1, 5], [-10, -3, -2, -1, -5, -4], [-13, -8, 1, -2, -1, -4], [13, -2, 5, 1, -6, 8], [2, 10, 0, 0, -1, -3], [-12, 8, -6, 3, 7, -4], [1, -8, 3, 1, 2, 2], [6, 6, -2, -1, 5, 1], [-11, 8, -1, 6, 7, -2], [6, -2, 4, -4, 2, 8], [-10, 1, -6, -1, -1, 0], [12, -8, 2, 0, -1, -7], [-8, -2, -4, -2, 4, -12], [9, 2, -3, -1, 2, 0], [-16, 6, -2, 1, 6, 1], [-11, 2, -1, 2, 8, -7], [9, -1, 3, -1, -5, -9], [6, -5, -6, 0, 1, -5], [-6, -10, 2, -3, -5, -1], [8, -4, 4, -9, -4, -7], [-7, -5, 1, 1, -4, 2], [0, -10, 4, 3, -8, 7], [-21, 10, -3, 8, 0, -1], [-6, -8, 0, -8, -6, -5], [1, 14, 1, 0, -5, 4], [18, 2, 0, 9, 5, 2], [-12, 2, -2, -1, 0, -5], [-5, 8, 1, -2, 3, -11], [-22, 0, -2, -6, -1, -6], [-12, -2, 0, -1, -3, 10], [2, 10, -2, 2, 2, -2], [24, -4, 2, 0, 0, 6], [4, -6, -4, -4, -9, -2], [17, -11, -1, -7, -5, 1], [3, -13, 1, -8, -2, 1], [22, -6, 0, 2, 1, -5], [-24, -6, 0, -4, -6, -4], [4, -10, -4, 0, 3, -3], [-2, 0, 4, -5, -6, 5], [5, 4, -1, 8, 7, 7], [-10, -2, 2, -2, -3, -2], [10, 6, -2, -3, 4, -3], [2, -8, -4, -5, 2, -2], [15, 0, 3, 2, -4, -7], [-2, 7, 6, 3, -4, 1], [-2, 4, 0, 2, 6, 2], [-7, -8, 5, -1, -2, 10], [1, 6, 3, -3, -2, 6], [-2, -8, -2, 4, 1, 6], [26, 12, -2, 4, 2, 0], [11, 2, -7, 3, 5, -9], [9, -2, 3, -4, -6, -1], [-9, 8, 3, 3, 7, -3], [11, -1, 3, 0, 5, 0], [8, -12, 4, -6, -10, -8], [-4, 6, -4, 1, 5, 0], [-21, 1, 1, 3, -1, 5], [11, -7, 5, -1, -4, 2], [-27, 4, 3, -3, 0, -4], [13, -10, 1, -7, -6, -2], [-20, 1, 4, 5, 9, -2], [-3, 8, 1, -1, 2, 2], [5, 4, -1, 0, -1, 7], [-10, -18, -2, -3, -3, -5], [9, 0, 5, -4, -7, -8], [-1, 10, -3, 5, 11, 1], [23, -4, 1, 0, -3, 4], [18, 0, 2, 2, 4, 0], [-10, -8, -2, -4, -5, -4], [7, -12, -1, -4, -3, 3], [27, -4, 3, -2, 2, 1], [9, 2, -3, -2, 5, 0], [-11, 2, 7, 2, 5, -4], [20, 4, 8, -1, 2, 1], [6, 12, 6, -1, 7, 7], [9, -6, 3, -5, -2, -3], [17, 9, 3, 3, 3, -1], [-10, 2, 2, 4, -6, 6], [23, -11, 1, -1, -5, -3], [-22, 10, -8, 7, 5, 8], [15, 0, 3, 1, -6, 4], [30, 0, 6, -2, 4, 8], [17, -10, 1, -2, -1, -2], [-15, 2, -1, 0, 2, -6], [-3, 6, 5, -2, -4, -1], [-17, 0, 5, -4, -8, 11], [25, -6, 3, 1, -4, -12], [-5, 10, 3, 1, -4, 12], [8, 6, -6, -1, -4, -9], [-17, 0, -3, 0, 8, 3], [-21, -9, 1, -5, -4, -2], [6, -8, -4, -6, 2, -6], [-14, -2, -4, 5, 2, -3], [30, 8, -2, 1, 8, -1], [12, 8, -2, -4, 8, -9], [-18, -3, -6, -8, 1, -5], [-1, -10, -7, -2, 3, 2], [-15, 0, 1, -9, -3, 0], [-20, 8, 0, 0, 1, -1], [1, -5, 3, 0, -6, 7], [-9, -12, 1, 0, 1, 0], [-18, 10, -6, 1, 2, 11], [-4, 3, -4, 4, 7, -5], [16, 10, -6, -1, -1, 0], [18, 10, 0, -4, -6, 0], [-5, -8, -3, 2, 6, 9], [-5, -6, -1, -5, -10, 2], [5, -14, -3, -8, 0, -5], [-27, -14, 5, -7, -4, 2], [6, 6, 4, -4, -4, 6], [21, -12, 1, -7, -3, 0], [-2, 10, -6, -1, 0, 1], [14, 5, -2, 9, 10, -3], [16, -2, 2, -2, 4, 14], [-11, 0, -5, 0, -7, -1], [-14, 0, 0, -11, -4, -9], [-16, 13, 0, 1, -3, -6], [-3, 12, -3, 0, -5, 5], [-2, -6, 0, 4, -3, -7], [-11, -6, 1, -12, -12, -1], [-35, -5, 3, 1, -5, 5], [26, -11, 8, -3, -15, 8], [-17, 0, 5, 7, -2, -2], [13, -4, 3, 4, -3, 15], [6, 1, 6, -4, -1, -11], [5, 9, 3, 0, -7, 6], [-4, -12, -8, -2, 1, 1], [-11, 17, -5, 6, 12, -1], [-10, 8, 8, -5, -8, 5], [2, 2, 2, -4, 2, -6], [8, -4, 4, 7, -3, 6], [2, 12, -6, 9, 4, 14], [6, 14, -10, 7, 7, -1], [1, 9, 3, -3, -7, -5], [18, -15, -2, -10, -5, 3], [-22, -4, -2, 0, 3, 5], [34, -8, 6, -5, 0, 0], [30, 1, 6, 3, 5, -2], [-10, 12, -8, 8, 8, -4], [24, -4, 4, 1, -2, 3], [25, -4, 1, -3, 6, -2], [7, 6, 5, 4, -2, 0], [15, 10, -3, -2, 0, 3], [6, 22, -2, 5, 6, 15], [-9, 3, 1, 0, -5, -6], [-21, 14, 1, 0, -7, 6], [14, 0, 6, -4, 3, 4], [-16, -7, -4, -1, 3, -10], [-32, -10, 0, 2, -7, -3], [6, 3, 2, 4, -1, 1], [19, -4, 7, 2, 1, 0], [-2, 0, -2, 0, 12, 2], [-12, -12, 0, -10, -1, -12], [3, 0, 3, 4, 7, -9], [-11, 14, -3, 5, 2, 2], [-10, 8, 0, 2, 12, -2], [-13, -15, -3, 2, -12, 1], [34, -4, 0, 4, -1, 1], [8, 2, 4, 1, 0, 9], [-10, -4, -2, 6, -4, 4], [-2, 6, 2, 2, -6, 12], [-36, 7, 0, -5, 1, -12], [0, 10, 2, 4, 2, -6], [13, 18, 5, 3, -2, 8], [-41, -6, -3, 4, -5, 4], [-7, 10, -9, 5, 3, 1], [-15, 2, -11, -3, 7, -3], [22, -6, 8, 4, -9, 17], [2, -6, -2, -10, -1, 6], [-6, -16, 6, -4, -6, -6], [42, 2, 0, 1, 0, 5], [8, 0, -6, 3, -2, -9], [-6, 4, 6, 3, 8, -3], [-8, 4, 2, 5, 9, -8], [-10, -4, 0, -3, 6, -5], [-26, -2, -10, 1, 3, -7], [-28, 10, -8, 9, 11, 4], [1, 20, 1, 1, -4, 8], [-16, 10, 8, -3, -3, -5], [3, 4, 3, 8, 2, 7], [9, -4, 13, 0, -2, -1], [1, 6, -5, -1, 9, -11], [3, 27, 3, 2, -3, 6], [-6, 1, 2, -5, 6, 3], [-25, -7, 1, 1, 1, 5], [-1, -4, -1, 10, -1, 8], [-9, 15, 3, -1, 6, -4], [10, -8, 4, -2, -1, -5], [-8, -10, 0, 4, 3, -6], [25, 0, 1, 13, 8, -4], [20, -11, -8, 3, 3, 2], [-1, -12, 11, -5, -5, -10], [17, 2, -3, 13, 8, 12], [-19, 4, 5, 0, -11, 7], [10, -12, 0, -9, -2, 13], [-23, -8, -3, -8, -13, -6], [26, 6, -4, -1, 5, 10], [-3, -18, 5, -10, -2, -7], [31, -6, 1, -1, 2, 15], [-17, 6, -1, 3, 2, -2], [15, 6, -11, -2, -5, -4], [-21, 20, -1, 8, -1, -4], [-5, 8, -1, 0, -6, -7], [-5, -22, 7, -5, 1, -19], [25, 16, -1, 4, 1, 8], [-13, -8, 1, -7, 3, -6], [-18, 12, -2, 3, -1, 10], [11, 20, 1, 4, 1, 12], [16, 4, 4, 6, 10, 0], [-17, 4, -3, 1, 13, -3], [-27, 11, 3, -4, -3, -10], [-6, -6, 2, -7, -5, 18], [-12, 10, 8, 2, 4, -6], [-14, 8, -4, -5, 8, 9], [-2, 2, 2, -8, -6, -2], [8, -2, -4, 4, 6, 16], [-9, 10, -3, 4, 3, -10], [6, -8, 6, 6, -8, 8], [53, -6, 5, -2, -6, 3], [-3, 10, -1, 10, 8, 8], [28, 3, 4, -1, -7, -12], [-12, 2, -8, -3, -9, 3], [39, 10, -1, -1, 3, -7], [10, -2, -8, 5, 2, -3], [32, 1, -4, 2, 2, 4], [27, 16, -3, -4, 0, -10], [-29, -12, 7, 2, 0, 1], [14, 16, 2, 0, -4, 8], [4, 12, -2, 10, 16, -4], [14, -12, 2, 0, -7, -1], [1, -8, 9, -6, -1, -13], [34, 6, 4, 3, 0, 11], [-1, -4, 11, -7, -6, 6], [18, 0, 4, 13, 6, 3], [37, -3, 3, 1, -6, 8], [15, 2, -11, 11, 8, -2], [0, -14, 12, -6, -7, -4], [-37, 2, 3, -4, -4, 13], [-18, -7, 6, 10, -5, -1], [18, 14, 2, 11, 9, 5], [-1, 12, 5, -2, 7, 7], [-32, 6, -4, 7, 13, 10], [13, -14, -9, -6, -4, 1], [-16, 20, 4, 2, -1, 8], [12, -12, -8, -6, -12, 2], [13, -2, 3, 4, 1, -10], [16, -2, -10, 0, -2, 14], [9, -6, 9, -10, -9, -2], [11, 12, -1, 1, -4, -12], [21, 16, -9, -1, 6, 7], [4, -20, 2, -6, -5, -1], [13, -16, -3, -12, -5, -2], [-27, 3, -11, 1, 12, -10], [16, -16, -2, -10, -8, 3], [-2, -11, 6, -3, -12, -5], [-2, 6, 0, -16, -2, -8], [-14, 0, -6, 5, -7, -3], [19, -4, 7, -2, -6, 11], [14, 6, 4, 13, 0, 11], [-21, -4, 3, -5, 5, 16], [5, 2, 5, 0, 8, 17], [44, -13, 0, 5, 4, -11], [-24, -6, -4, -8, -8, -12], [10, -5, 2, -2, -7, -7], [35, -2, 3, 8, 0, -13], [-5, -24, 3, -7, -4, -10], [-26, 4, 4, -7, -10, 7], [10, -4, -2, -4, 3, 10], [13, -28, 3, -3, -8, 0], [12, 2, -6, -6, 5, 7], [-16, -2, -8, -1, 9, 3], [-21, 6, 1, 10, 0, 5], [11, -4, 3, 1, 4, -14], [-25, -28, 5, -1, -9, -10], [12, 20, 2, 4, 4, 14], [-10, -6, 2, -3, 7, -20], [-23, -4, -7, -3, -6, 0], [-17, 0, -3, -1, 2, -7], [20, -14, -4, 11, 7, 5], [-9, -24, 1, -6, 2, -5], [-20, -4, -6, -5, 4, 10], [21, -4, -5, 11, 8, 10], [6, 22, -2, 6, 7, 2], [-27, -2, 7, -7, -6, -6], [28, -1, 8, -2, 9, 11], [8, -12, -4, -4, 0, 17], [25, -12, 3, -10, 3, 10], [12, -12, 4, -4, -7, 4], [-9, -4, 17, -3, -5, 9], [12, -8, 2, 1, -10, 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_k();".
// 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_k(:prec:=6)
    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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -5, 29, -11, -13, 1, 1]>,<5,R![2, 27, 85, 9, -19, -1, 1]>,<17,R![-1792, 200, 1225, -20, -71, 0, 1]>],Snew);
    return Vf;
end function;