
// Make newform 8967.2.a.y in Magma, downloaded from the LMFDB on 02 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8967_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_8967_2_a_y();".
// 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_8967_2_a_y();".
// 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 := [-17, 10, 31, -2, -11, 0, 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], [-4, 0, 1, 0, 0, 0], [9, 0, -8, 0, 1, 0], [-9, 21, 8, -10, -1, 1], [-23, 32, 22, -18, -3, 2]];
    Rf_basisdens := [1, 1, 1, 2, 2, 2];
    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_8967_a();"
function MakeCharacter_8967_a()
    N := 8967;
    order := 1;
    char_gens := [2990, 5491, 3235];
    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_8967_a_Hecke(Kf)
    return MakeCharacter_8967_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, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, 0], [-2, 1, -1, -1, 0, 0], [-1, 0, 0, -1, -1, 0], [-2, 0, -1, 1, 0, 1], [0, 0, 2, 2, -2, 2], [0, 1, 1, -1, 0, 0], [-2, 0, -1, 1, 2, -1], [0, 0, -2, 0, 0, -2], [0, -2, 0, 2, -2, 2], [1, 2, -2, -1, 1, -2], [0, 0, 0, -2, 0, 0], [0, -2, 0, 0, 2, 0], [0, 2, -1, -1, -2, -1], [2, 1, -1, -1, 2, -2], [1, 0, 0, 0, 0, 0], [0, -3, -2, 2, 0, -1], [-2, 0, 1, 3, 0, 1], [-5, -2, 0, -1, -1, 0], [2, -1, 2, 0, -2, 1], [4, -2, 4, 4, -2, 4], [-2, 0, 3, 1, -2, -1], [-2, 4, 0, 0, 0, -2], [2, 2, 1, 3, -2, 3], [2, -2, 2, 4, -4, 4], [0, -2, 0, 0, 0, -2], [3, 0, 0, 3, -1, 2], [-1, 2, -2, -3, 3, 2], [0, -2, -2, -2, 2, -4], [2, 2, -4, -2, 4, -4], [9, -4, 2, 3, -1, 2], [-4, 3, -6, -2, 4, -3], [1, -2, 2, 3, 1, -2], [-8, 3, -2, -2, 0, 1], [-2, -4, 0, 4, 0, 4], [2, 4, 2, 0, 0, -2], [6, 6, -2, 0, 0, -2], [2, -4, 1, -5, 0, -5], [-8, 0, -4, 4, 2, -2], [2, 2, 2, -2, 4, 0], [-4, 1, 1, -1, 2, -2], [-2, 2, 4, 4, 0, 2], [1, -4, 4, 5, -1, 0], [0, -6, 0, 4, 4, -2], [8, 0, 0, 2, 0, 0], [2, 1, -4, -6, 2, -3], [8, 1, -1, 5, -4, 2], [7, 2, 0, 3, 3, -2], [10, -4, 5, 7, -6, 5], [-14, 4, -4, -2, 4, -2], [7, -2, 0, -1, -5, 2], [-4, 4, -1, -1, -4, 1], [-20, 0, 2, 0, 0, 0], [6, 0, 4, -2, 2, 0], [6, -4, 4, 0, -4, -4], [0, 0, 4, 4, 0, 4], [-2, -2, -2, 2, 0, -4], [-4, 4, -3, 1, -2, 3], [-10, -2, 4, -6, 6, -2], [-2, 4, -2, -2, 0, -6], [-2, -5, 6, 0, -2, -3], [-4, -3, -5, 1, 0, -2], [-12, 2, 0, 2, -2, 6], [8, 8, -2, 0, 4, -4], [8, 7, 0, 0, -4, -1], [0, -6, 2, -4, -2, 4], [-2, 4, 2, 0, -6, 6], [6, 8, -4, -4, 4, -4], [-15, 8, 0, -7, -1, 0], [-8, -8, 1, 1, 0, -1], [4, 4, -2, -6, -2, 2], [-8, 12, -8, -2, 6, -4], [12, 0, -4, -4, 4, -8], [-14, 1, -1, -9, 4, 4], [-2, 0, 5, -1, -4, -1], [2, 4, 0, 0, -4, 0], [-4, -2, -5, 3, -4, 1], [0, -2, 2, 4, 2, 8], [14, -4, 7, 5, -8, 3], [-2, 0, 6, -2, 0, -2], [4, 2, 8, 0, 0, 2], [-2, -2, 0, 0, 0, -6], [-10, -4, 4, -2, 2, 4], [-4, -4, -2, 6, 4, 6], [-3, 0, 6, -5, 3, 2], [-12, 2, 6, 8, -6, 8], [-16, 0, -2, -4, 4, -4], [14, -8, 6, 4, -4, 6], [0, 3, -1, 1, 6, -2], [-8, -8, 2, -2, 2, 0], [-10, 8, 0, -6, -2, 8], [8, 4, -2, 6, -2, 8], [8, 5, 2, -2, -8, -1], [-20, -2, 8, 4, -2, 0], [2, 0, 5, -1, -4, 7], [-2, -8, -3, 3, 0, 3], [2, -7, 0, -6, -2, 1], [-8, 8, 6, 0, 2, -2], [10, -5, 4, 10, 2, 3], [2, -8, -5, -7, 6, -5], [10, -6, -4, 2, 2, 2], [5, 8, 2, -1, -9, 6], [18, 10, -2, 4, 0, 0], [-6, 2, 4, 8, -4, -2], [4, -8, 3, -1, 0, -3], [-16, -2, 1, 5, -4, 3], [8, 3, -5, -7, 6, -10], [7, 12, -4, -1, -1, -4], [-6, 4, -4, -2, 6, 4], [-6, 4, -4, -8, -4, -6], [0, 4, -3, -7, -4, 1], [0, -2, 2, -6, -6, 0], [6, -7, -4, 2, -6, 1], [-2, -10, -7, 3, 2, -9], [-4, 0, 2, 8, 0, 2], [0, -5, 3, 1, -10, 6], [-4, -8, 1, 9, -4, 9], [-6, -2, 0, -10, -2, -10], [0, 0, 4, -6, -2, 0], [-2, -4, 0, -4, 0, -4], [8, 2, 1, 5, -8, 7], [12, -4, 6, 6, -12, 6], [2, 4, 2, -4, 12, -2], [-6, 0, -5, -3, 0, -7], [-18, -2, 2, 2, -8, 4], [0, 8, 0, 0, -8, -4], [8, 10, -2, 10, -6, 4], [3, -4, -8, 3, 3, -6], [-10, 5, -10, -8, 14, -9], [22, 1, -1, 7, -6, -2], [2, -4, 0, -10, 10, 4], [7, -2, 4, 11, -5, 8], [2, 10, -11, -5, 8, -7], [-6, -4, -4, -12, 4, -4], [6, 0, -6, 2, 4, 2], [0, 0, 2, 4, -8, 6], [-17, -4, -4, -5, -7, 4], [5, 0, 10, -1, -5, -2], [-12, 5, 0, -4, -4, 1], [-2, 2, -1, -7, -4, 7], [-4, 12, -4, 2, 4, 4], [-4, -8, 0, -4, 6, -10], [-3, 0, 8, 13, 1, 6], [-2, 0, 4, 6, -2, 8], [3, -6, 8, 15, -1, 12], [1, 2, 2, 3, -11, 2], [-26, 14, -4, -14, 10, -6], [2, 0, 4, -2, 8, 2], [16, 0, 0, 2, -6, 0], [-3, 12, 4, 1, -1, 0], [-8, -15, -2, 2, 0, 3], [14, -4, -9, -3, 8, -9], [0, 0, -2, -4, -4, -2], [-6, -8, 0, 2, -8, -2], [-16, 4, 4, -4, 4, -4], [1, -10, -6, -5, 1, -2], [-15, -4, 4, -3, 5, 4], [4, 4, 1, 1, 6, -1], [-14, -3, -1, 7, -2, 6], [-6, 6, -7, -5, 8, -11], [26, -10, 12, 2, -14, 2], [18, -6, 0, 2, -4, 4], [-22, 4, -4, -4, 8, 4], [-18, -4, -3, -9, 8, -7], [10, 0, 6, -4, 0, -6], [-10, 8, -4, -12, 8, 0], [-12, -18, 6, 4, 2, 8], [10, -8, 5, 3, -2, 5], [4, -1, -11, -9, 18, -12], [-7, -4, 0, -11, -3, -10], [4, 3, -11, -13, 2, -12], [-6, 0, -4, 8, 8, 8], [-2, 5, -6, 4, -2, 3], [11, -6, 4, 7, -5, -8], [14, 10, 4, -2, 2, 6], [-8, 12, -5, 7, 2, 9], [-14, 1, -10, -4, 6, -5], [14, -12, 4, 4, -4, -4], [-10, 3, 0, 6, -2, -5], [-6, -5, -5, -1, 8, 10], [-1, -6, -12, 3, 3, 4], [-22, -8, 10, 2, 8, -2], [-2, 4, -4, -10, 6, 8], [-12, -4, -9, -1, 12, -13], [1, -10, 8, 5, -3, 2], [18, -10, 16, 10, -2, 14], [6, -12, 2, 2, -4, 2], [-12, 7, 3, 9, -8, 8], [-4, -8, -2, 8, 2, -6], [-8, -4, 2, -6, 2, -4], [10, 0, -2, -8, 0, 6], [9, 2, 0, 1, -15, 4], [6, -5, -5, 7, 2, -8], [-21, -6, 0, -5, -1, -8], [26, -4, 12, -4, -4, -2], [-16, 0, -14, -8, 14, -10], [-8, -4, -14, 4, 4, -8], [-16, 2, -2, -2, 2, -6], [-1, 2, 16, -1, -5, 12], [8, 0, 6, -4, -4, 2], [-8, -6, -6, 12, -6, 0], [0, 18, 2, 8, 6, 0], [-12, -2, -8, -4, 12, -10], [-31, 10, -12, 5, 5, -4], [28, -3, 8, 16, -8, 13], [-2, 7, 5, 9, -4, 12], [-2, 2, 5, 3, 0, 9], [-14, -3, -6, -4, 14, -9], [-11, -2, -12, -15, 21, -18], [9, 8, 6, -1, -5, -10], [28, -6, 10, 10, -6, 8], [-2, 0, 1, 3, -4, 1], [-16, -1, 1, -5, -10, 4], [22, -10, 0, 8, -4, -6], [-2, -8, -12, 6, -2, -4], [17, 4, 8, 1, 7, 4], [10, -2, 20, 2, -8, 8], [-10, -12, 9, 3, 4, -1], [13, -2, 0, 1, 1, 10], [12, -2, -2, -2, -2, 4], [-24, 0, -1, -13, 2, -3], [-28, -8, -6, -2, -2, -6], [-10, 1, 1, -11, 4, -6], [4, 14, 8, 6, -14, 6], [16, 6, 5, -7, -8, 3], [-24, 8, 6, 10, -10, 4], [16, 8, 4, 2, -4, 8], [-18, -6, 2, -16, 8, 6], [-8, 12, -14, -8, 6, 6], [10, 0, 2, -12, 8, 2], [6, -5, 8, 2, -10, -5], [-16, 10, -9, -13, 0, -7], [-26, 8, -12, 10, 6, 0], [26, 11, -1, -1, -8, 2], [-23, 0, -4, 5, -3, 12], [-8, 2, 9, 13, -6, 9], [-12, -8, 3, -5, 8, -3], [14, -2, 14, 12, -22, 16], [2, 8, 1, -9, 12, 5], [14, -7, 6, -8, -2, 3], [-32, 12, -4, -10, 4, 4], [26, -4, 4, -8, 8, 4], [16, 2, -1, -13, 2, -5], [26, -2, 10, 12, -10, 8], [16, 16, -8, -4, -4, 4], [-2, 6, -6, -8, 4, 2], [-8, -16, 4, 10, 12, -4], [-28, 8, -8, -8, 2, 6], [-12, -6, 2, -16, 2, 4], [18, -4, -12, -8, 8, -4], [8, 0, -1, -21, 8, -23], [0, 0, 6, -12, 10, -2], [-18, 0, 2, 2, -4, 6], [-40, -1, -1, -3, -4, 0], [8, 4, 14, 12, -22, 10], [-8, -18, 8, -4, 0, -6], [-34, 0, -7, 3, -2, 5], [-14, 24, 8, -8, -4, 8], [20, 4, 4, 16, -12, 16], [-8, 12, 8, -8, -10, 10], [-18, -16, 8, 4, 0, 2], [-3, 4, 0, 9, -7, 6], [-10, -18, 5, -1, 4, 5], [0, 8, 2, -6, -6, 6], [11, 0, -2, 9, 5, -6], [-8, -4, -10, -8, 4, -12], [-12, 0, 6, -18, 14, -10], [3, 0, 6, -7, -3, -2], [-4, -6, -4, -2, 10, -2], [14, -12, 12, 6, 2, 8], [-13, -16, 0, -1, 11, -8], [-28, 0, -12, 6, 8, 8], [-4, -4, 22, 16, -6, 14], [0, -12, 12, 16, -12, 12], [-12, 16, -9, -5, 20, -3], [-13, 2, -12, -13, 27, -4], [2, 8, 0, 12, 4, 8], [-4, 2, 8, 0, -2, 8], [26, 1, 5, 9, 12, -10], [8, 10, 8, 0, -8, 10], [-22, -5, 11, -9, 10, -8], [28, -4, -4, 18, -6, 8], [48, 0, 4, 2, -8, 0], [-10, 26, -10, 0, 12, -2], [38, -10, 2, 14, -12, 4], [23, 14, 8, -9, -5, -8], [-38, 0, 2, -12, 8, 2], [-8, 14, 5, -15, 12, -1], [-6, -10, -9, 1, 14, -3], [-12, -10, 14, 10, -12, 12], [-7, -6, -4, 5, -11, 0], [20, -11, 15, 1, -8, -2], [5, -16, -8, 13, -3, -4], [17, 4, 10, 3, -13, 6], [-2, -6, -6, -20, 12, -12], [6, -2, 9, 23, -2, 3], [6, 0, -6, 4, 20, -8], [22, 9, 4, 2, -14, 13], [-22, 8, -8, -12, 16, -6], [32, -4, 4, 2, 4, -2], [-12, -10, -6, -14, 14, -8], [6, 13, 9, 17, -6, 20], [20, -3, 16, -12, 4, 1], [1, 4, 8, 5, 1, -4], [4, -14, 6, 6, 2, 14], [5, -10, 4, 17, 13, 6], [24, -4, -3, 9, 8, 7], [-10, -18, 12, 4, 8, 2], [-28, 4, -9, 7, 4, 13], [-13, 2, 4, 7, -1, 8], [-4, -6, 5, 13, -14, -3], [0, -2, -10, -2, -6, 0], [20, 12, -2, 10, 0, -2], [-13, -6, -8, -9, 7, 4], [2, 16, -18, -8, 2, 2], [0, -5, 1, 7, -4, 14], [-5, -4, 16, 11, -9, 14], [-34, 8, -17, -15, 10, -13], [4, -2, -12, 8, -8, 10], [-8, 12, 12, 14, -10, 16], [-10, -10, 7, -3, 4, 7], [-12, 4, -4, 10, 4, -8], [-30, -4, 7, 13, -18, 7], [10, -8, -2, 6, -12, -2], [16, -16, 16, 14, -16, 16], [-4, -11, -12, 12, -4, -3], [6, -5, -15, 1, 4, -8], [-24, -6, 7, 7, -6, -5], [-30, 0, -8, 4, -4, -2], [-15, 8, 0, -23, -1, 12], [16, -7, 0, 12, 12, -7], [18, 8, -12, 2, 12, -18], [42, 1, -2, -4, 14, 3], [-40, 21, -9, -15, 14, -8], [-2, 4, -8, -8, 0, -2], [23, 10, 16, -9, 7, 0], [-8, 2, -6, 0, -6, -4], [12, 3, -4, 4, 4, 3], [27, -4, 4, -9, -9, -6], [20, -10, 11, 11, 4, 17], [-52, 16, -8, -10, 8, -6], [18, 5, -2, 8, 2, -5], [-29, -14, 8, 3, -1, 8], [-16, -5, 1, 7, -2, -4], [16, -2, 14, 8, -22, 0], [-10, 3, 6, -12, 6, -11], [-34, 5, 3, 3, -6, 6], [23, 2, -2, -3, -1, 2], [38, -9, 4, 14, -14, 7], [0, 14, -2, -20, -14, 0], [-2, 12, -19, 3, -2, -7], [2, 2, 10, 16, -8, 0], [26, 4, 20, 8, 0, 20], [-15, -8, -8, -7, 27, -16], [-12, 4, 6, 2, 0, 22], [-10, 0, 1, -5, -12, 1], [-26, 8, -11, 11, 2, -3], [22, -3, 5, 21, -4, 22], [-9, 14, -8, -9, -13, -6], [-17, 14, -4, 3, 7, 0], [18, -5, 11, -1, 2, -8], [-18, -4, -14, 12, -4, 8], [10, -16, -6, -16, 4, -18], [-8, -12, 0, -10, 14, 0], [18, 8, -3, 15, 0, -13], [-20, 4, -12, -10, -4, -8], [-17, 24, 4, -13, -5, 0], [26, -16, 1, -5, -4, -11], [-22, 13, 8, -14, 10, -7], [-22, 18, 4, -6, 8, -4], [0, -4, -4, -14, 12, -14], [4, 20, -22, -16, 16, -18], [30, 4, -8, -16, 0, -6], [-8, 7, 9, -5, 0, 2], [-8, 21, -1, -11, 2, 0], [24, -6, 28, 2, -2, 6], [23, -12, -14, -15, 5, -6], [20, 0, 6, 0, 14, 6], [-28, 12, -19, -7, 16, -7], [-18, 36, 3, -3, 6, -1], [6, 7, -16, -2, 2, -9], [30, -8, 4, 0, -16, 4], [40, -6, 10, 6, -10, 12], [-20, 5, -19, -5, 2, -10], [-2, 8, -4, -6, 2, -8], [-30, -10, -6, -10, 20, -20], [18, 12, -8, -10, 26, 0], [20, -14, 8, 16, -8, 18], [-11, 0, -8, 17, 5, 8], [17, -18, 4, 5, 1, -4], [-14, 2, 2, 8, -18, 4], [-50, -9, -8, 2, 10, -9], [-8, 2, -10, -10, 2, -22], [-16, 12, -10, -8, -10, -10], [24, 1, -6, -10, 16, -13], [33, 8, 0, 17, -21, 4], [34, 12, 6, -8, 4, -14], [-5, 0, 6, -15, 1, -6], [-25, 2, -8, -5, 7, 6], [-14, 2, -1, -11, -16, 11], [-4, 14, -4, 0, 12, -6], [0, -4, -3, -23, 6, -13], [-42, -14, 1, -1, -6, 11], [-6, 10, -18, 4, 4, -16], [-25, 0, -4, -1, 11, -4], [-31, -2, 4, 5, -7, 0], [22, -2, 8, 10, -2, 2], [6, 4, -12, 14, -8, 6], [9, 6, 12, 1, -27, 6], [8, -8, 3, -1, 14, -7], [-8, -8, -1, 11, 4, -15], [14, -18, -4, -2, -2, -2], [-8, -18, 10, 0, 2, 8], [26, 14, 7, 1, 12, -5], [-12, -16, -2, 2, 4, 14], [4, -12, 16, 12, -28, 20], [16, 24, -1, 7, 2, 9], [18, 16, -16, 6, -10, 4], [-29, 14, -4, 7, -5, 4], [-18, 26, -4, -4, -4, -2], [-4, -14, -20, 4, 0, -2], [24, 0, -1, 23, -16, -3], [7, -14, 8, -1, -1, 12], [-32, -6, 17, 9, -6, 21], [-52, 27, 1, -21, 2, -4], [-12, 12, -16, -2, 16, -10], [-22, -7, 4, 10, 2, 1], [4, -7, 7, -3, 8, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8967_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8967_2_a_y();".
// 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_8967_2_a_y(:prec:=6)
    chi := MakeCharacter_8967_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_8967_2_a_y();".
// 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_8967_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8967_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-17, -10, 31, 2, -11, 0, 1]>,<5,R![-144, 80, 144, -28, -23, 2, 1]>,<11,R![4, 8, -68, -110, -5, 8, 1]>],Snew);
    return Vf;
end function;