
// Make newform 8281.2.a.by in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8281_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_8281_2_a_by();".
// 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_8281_2_a_by();".
// 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 := [-2, -6, 7, 8, -5, -2, 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], [-2, -1, 1, 0, 0, 0], [1, -3, -1, 1, 0, 0], [2, 2, -4, -1, 1, 0], [-2, 4, 6, -4, -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_8281_a();"
function MakeCharacter_8281_a()
    N := 8281;
    order := 1;
    char_gens := [1522, 3382];
    v := [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_8281_a_Hecke(Kf)
    return MakeCharacter_8281_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, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 1, 0, -1, 0], [0, 0, 0, 0, 0, 0], [0, -1, 1, 0, 1, 1], [0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 1], [0, 0, -2, 0, 0, -1], [-2, -2, -2, 0, 0, -2], [0, -2, -1, -1, 1, 2], [-3, 0, -1, -1, 0, -2], [-3, 0, -1, -2, -1, -3], [5, 0, -1, 2, -1, 1], [-1, 2, -3, 1, -1, -2], [3, 0, 1, -3, 2, 2], [-3, 0, 0, 0, -2, 2], [1, -2, -3, 1, 0, 2], [2, 3, 0, 1, 1, 2], [-5, 2, 3, 1, 1, -1], [0, -2, -2, -2, 2, 0], [5, 2, 1, 2, -1, -1], [-4, 0, 4, 0, 0, 2], [5, -2, 1, 1, 2, 0], [6, 0, 0, 0, 2, -3], [5, -4, -3, -1, 2, -1], [-2, 1, -6, -3, -2, 0], [-3, -3, -2, 1, 2, -3], [-6, 5, -1, 4, -2, -2], [-3, 6, 1, 1, -1, -3], [-1, 0, 2, -2, 2, 2], [-1, 2, -5, -1, 1, -2], [-1, -5, 4, 1, -1, -1], [-5, -3, 0, -4, 4, 6], [-1, 5, 4, -5, -1, -3], [-1, 3, 2, 0, -2, -2], [-5, 8, 1, -3, 1, 3], [1, -4, -2, -6, 4, 1], [4, -4, -8, 0, 0, -2], [3, 2, 3, 3, 0, 4], [-5, -1, 2, 3, 0, 1], [-7, -5, 2, -1, 3, 2], [-6, 5, 6, 1, -1, 2], [-4, 3, -5, -2, -2, -2], [-3, -4, -5, 6, -1, 1], [2, 3, 3, 2, -5, 5], [-3, 5, -6, 5, -6, -5], [-3, -1, 0, 1, 5, -2], [3, -2, 1, -1, -4, -5], [-11, 2, -3, 3, 2, 0], [-2, 4, -2, -10, -2, 2], [-6, 11, 5, 0, 0, 0], [-3, -6, 5, 5, -5, 3], [-10, 4, 4, -1, 5, -2], [-2, 0, 0, 8, -2, -2], [-9, 2, 8, 2, -2, 3], [-3, -1, 2, 7, 3, -4], [-11, -5, -6, 3, -1, -7], [-5, -4, 1, 7, -2, -2], [7, -6, 6, 0, 2, 6], [-5, 2, 3, 8, -5, -1], [10, 4, 6, 2, -6, -2], [5, -10, -7, 0, 7, 1], [-3, 2, 3, 5, -2, 3], [-5, 11, 0, 1, -3, -1], [-10, 6, 6, -4, -8, -2], [-14, -2, 4, 3, -2, 0], [6, -3, -7, 2, 5, 3], [8, -10, -1, -1, 3, 6], [10, -10, -2, 0, 4, -2], [-7, 4, -5, 1, 0, -6], [7, 0, 3, 0, 3, 12], [5, -7, -2, -3, 0, 4], [-2, -12, 4, -4, 2, 6], [-3, 1, 1, -10, 0, -6], [-8, 1, -5, -6, 5, 3], [-9, 0, -1, 1, 0, -5], [0, 5, 6, 9, -1, 4], [8, -10, 6, -4, 8, 10], [1, -1, 4, 4, -4, 0], [11, 8, 7, -8, 5, -2], [-17, 5, -8, -5, 0, -3], [13, 6, 1, 2, -3, 3], [-5, 5, 4, 11, -10, 2], [-9, -2, 6, -4, 7, 9], [-1, -3, -4, 3, -11, -9], [-16, 8, -4, -4, -8, 0], [-18, -6, 0, -6, 4, 0], [1, -16, 3, -4, 3, 1], [1, -6, 7, -6, 5, 6], [1, 10, 5, 1, -5, -11], [-11, 1, 0, -1, 4, -5], [-10, 6, -10, -6, 4, -1], [-3, -5, 0, 9, 0, 6], [9, -12, 1, 7, 3, 4], [9, -9, -4, -1, 10, 4], [2, -4, -4, -8, 9, 2], [4, 4, 4, -1, -1, -5], [-6, -3, -2, 1, 5, -2], [4, 0, 12, 4, 2, -2], [-15, 5, 2, 0, 4, 2], [11, 5, 4, -9, -1, -4], [-12, 9, 11, 3, 1, -1], [-10, 4, 2, 6, -8, 6], [-8, -4, -2, -2, 0, 8], [12, -1, 1, 0, 2, 0], [12, -10, 2, -1, -1, 7], [21, 0, 9, -1, 2, 11], [8, 4, -6, 3, -11, 2], [-9, -2, 3, -5, 11, 4], [-12, 1, -6, 5, 0, -8], [-5, -5, -6, -9, 2, 7], [6, -3, -3, 5, 3, 1], [-16, -4, 4, -5, 2, 2], [-16, -2, 6, 8, -10, 10], [-2, 5, 5, 2, 9, 3], [-1, 1, -13, 4, 2, -4], [-3, 0, 3, -11, 0, 2], [-14, -4, -8, 4, 2, -4], [-5, -4, 1, -9, -7, -4], [-8, 8, 4, 12, -14, -2], [-7, 10, 5, 8, -13, -1], [-1, 8, 4, -4, 0, 8], [-2, 6, 12, 2, -7, 0], [-7, 6, -3, -1, -9, -7], [0, -16, -14, 2, 2, -1], [-10, 2, -10, 16, -8, -8], [1, 3, 0, -8, 8, -8], [13, -17, -2, 3, 6, 9], [3, 9, -2, 13, -4, -5], [-2, -16, -12, -3, 5, -4], [10, -4, 6, -6, 0, -2], [-9, -7, -4, -1, 0, 4], [-3, 1, -4, -5, -3, -18], [20, 10, 16, -2, -6, 6], [21, -6, 3, -5, 4, 7], [-13, 2, -9, -5, 8, -4], [5, -8, -5, -7, -4, -2], [0, 4, 0, 2, 4, -4], [-30, 14, 10, 0, 0, 0], [5, -2, -6, -8, 8, -2], [-12, 4, 6, 8, -4, 9], [19, -2, 7, -5, -1, 11], [5, -1, -6, 3, 5, -6], [7, 13, -4, 3, 6, -6], [0, 1, -4, -5, 4, -2], [8, 10, 12, 0, -8, -5], [0, 0, 16, 3, -1, 5], [-3, 8, 12, 0, 2, 3], [6, 0, -8, -12, 10, -10], [-28, 16, 4, -4, -6, -8], [0, -8, 0, -9, 2, -6], [1, 16, 6, -6, 1, -1], [-7, 7, 0, 9, 1, -2], [10, 4, 14, -2, 4, 16], [4, -6, -10, -4, -2, 6], [-12, -13, -11, 0, 8, 6], [-20, 4, 6, -2, 10, 2], [28, 2, 2, 3, -9, -3], [9, -12, 0, -4, -5, 1], [7, 10, 19, 7, -6, -6], [-8, -1, -11, 10, 1, 3], [-11, 17, 10, 1, -5, 0], [5, -2, -3, -1, -9, -7], [-17, -11, -12, -5, 8, -7], [-18, 6, 0, 1, 6, 0], [-9, 4, 7, -9, 4, -5], [-12, 7, -1, -2, 12, 0], [13, -2, -16, 10, -11, -1], [-1, -11, -20, -1, 12, 6], [21, -8, 12, -4, -3, 3], [9, 7, 4, 7, -4, -6], [-22, -2, -2, 1, -5, 4], [5, -12, 3, -11, -1, 14], [2, 13, -3, 3, -11, -11], [-2, 6, 8, 2, 3, -8], [1, 8, 12, 4, -3, 11], [6, 16, -10, -4, 6, -4], [-16, -1, 9, -1, 17, -5], [23, -29, -2, -1, 7, 5], [5, -7, -8, -9, 11, -1], [19, 0, 5, 5, 3, 1], [0, -6, -8, -14, 13, 8], [-33, 16, 2, -2, -8, -8], [4, 0, -10, -10, 0, -3], [6, 5, -11, -6, -13, 5], [-5, -8, 0, 0, 9, -3], [4, 4, 2, 10, -6, -4], [5, 10, 7, -3, 4, 1], [7, -16, 17, 2, -3, 13], [-6, 8, 4, -12, 0, 11], [9, -6, 12, 10, 9, 1], [19, -9, 14, -3, 0, 12], [2, 3, 19, -6, 2, 6], [-30, 18, 18, -8, 2, -3], [6, 3, -9, -6, -10, -4], [-3, 13, 2, 13, -2, 3], [6, -3, -3, -1, -11, -9], [-35, 21, -1, -4, -8, -10], [5, -4, 7, 4, -11, 1], [3, -3, -16, -1, 6, 5], [-12, -13, 5, -15, 1, 1], [-4, -14, 6, -4, -5, 4], [-12, 12, 4, 13, -15, 5], [14, -15, 20, -5, 14, 14], [-5, -4, -7, 13, -22, -8], [15, -18, -14, 0, 13, 1], [8, -14, 10, -4, 2, 6], [-3, -15, -16, 7, -5, 6], [-10, 13, -7, -1, -1, -17], [-8, -8, -4, 4, -14, -6], [14, 2, -5, -1, -1, 10], [-21, 7, -16, 1, -11, -11], [-23, -3, -10, -7, 3, -16], [3, 6, 3, 9, 8, 4], [15, 6, 5, 19, -6, 1], [-16, 2, 2, -1, 5, -11], [15, 1, 0, -9, 7, 8], [-31, -12, 2, -10, 10, 3], [-1, 13, -8, -13, -4, 0], [-14, -7, 5, 6, -13, 5], [-18, 10, -17, -5, -1, -6], [-23, 22, -3, 15, 4, -16], [-5, 8, -3, -4, -11, 5], [25, -23, 10, -11, 0, 16], [2, 24, -8, 12, -5, -10], [0, -4, 9, -9, 7, 12], [17, -12, -4, 8, -9, -9], [-15, 4, 13, 11, -19, 2], [11, 1, -6, -3, 9, 13], [-7, -16, -1, 1, -7, -15], [22, -5, 6, 11, -9, -8], [-39, 9, -4, 7, -8, -9], [-15, -3, 2, 3, -18, 6], [10, -7, -10, -11, 3, -12], [-6, 8, 12, 0, 10, -10], [-1, 10, -1, 29, -6, -8], [7, -10, -15, 5, -6, -9], [18, 0, 14, -11, -2, 2], [-4, -2, 4, 6, 6, 8], [-5, -11, -6, -7, 20, -3], [-18, 11, -5, -8, -3, -1], [23, -7, -20, -13, 10, -7], [1, 1, -18, -7, -5, -21], [24, -2, 14, 7, 2, 0], [13, 18, -7, -12, 7, -9], [14, 0, 6, -2, 18, 10], [18, -6, -4, -8, 16, 2], [-12, -6, 0, -18, 6, 9], [0, -16, -12, 8, -6, -10], [-3, 10, -7, -7, 1, -24], [-5, 13, 12, 11, 6, -2], [13, 10, -5, -21, 2, -7], [21, -2, 13, 9, -7, -7], [13, -4, -4, 16, -11, -3], [-23, -9, 4, 1, -4, 1], [9, -18, 1, 5, -5, -1], [2, 7, 5, -28, 2, 4], [29, 12, -13, -3, 2, -6], [11, -9, -6, -5, 10, 5], [24, -4, -26, 7, -15, 4], [-3, -13, -22, -9, 17, -14], [5, 7, 6, 15, -11, 14], [-20, 12, -24, -9, -3, -13], [-15, 22, 8, 2, -13, -11], [21, -10, 1, -28, -1, 5], [22, 4, 20, -12, -6, 10], [-21, 6, -5, -9, -13, -15], [10, -32, -8, 1, 3, 6], [-23, -18, -19, 3, 11, -15], [-1, -6, -6, -4, -6, -10], [-7, 6, -28, -6, 0, -3], [-25, 13, 4, -5, -4, 2], [5, -8, -1, 2, 15, -15], [13, -13, -12, 13, 13, 12], [0, -6, 12, -6, 18, 22], [20, 2, -8, 8, 0, 9], [-37, 8, -15, -5, -5, -1], [-19, 16, 3, 12, -21, -9], [-9, 2, -9, 10, -23, -11], [-7, 9, -6, 9, 8, 1], [-17, 27, 6, -9, -1, -5], [-1, -9, 6, -11, -2, -2], [-14, 6, -10, -12, 11, 12], [-13, -4, -3, 10, 5, 11], [-3, 12, 11, 13, 0, 0], [-22, 28, 8, -12, 6, 0], [5, 1, -11, -6, 18, 0], [4, -1, -3, 13, 5, -13], [-24, -10, 6, 20, 0, 6], [-14, 17, -20, 3, -12, -20], [21, -5, -28, -3, -9, 0], [-15, -5, 8, 13, -9, 11], [-37, 6, -1, -4, 3, -1], [-13, 0, -21, 23, -14, -11], [38, -12, 8, 12, 0, 2], [-19, 12, 8, -24, -2, -5], [36, 2, 2, 24, -3, 2], [23, -2, 12, -14, 0, 1], [2, -18, -22, 4, 17, 2], [-6, 24, 12, 2, -4, 17], [16, 9, 16, -13, -3, 14], [-9, 3, 2, -8, 18, -2], [18, 14, 0, 18, -4, 8], [-3, -11, 12, 3, 7, -6], [-11, 19, -4, -3, -1, -2], [-4, 6, -2, 1, 20, 12], [-15, 17, 4, -1, 13, -4], [-27, -2, 12, -18, 5, 1], [-1, -2, -13, 7, -3, -9], [-2, -27, 0, 7, -1, 4], [-19, 27, 10, 15, -26, -9], [-3, -6, -7, 1, -15, 9], [11, 14, 0, 2, 4, -2], [12, -15, 15, 4, 6, 12], [11, 3, 6, -7, -10, 2], [33, 0, -4, -8, 10, -7], [15, -10, -9, -19, -10, -4], [-22, 12, 2, 0, 0, 6], [-17, -10, -5, 11, -2, 0], [20, -15, 8, -21, -7, 8], [-1, -7, 4, 11, -10, 7], [-21, -3, 4, 3, 10, -18], [11, -2, -6, 8, -5, -1], [-28, -1, 21, -12, -1, -3], [-16, -22, -10, -4, 0, -8], [11, 0, 8, -12, 18, 15], [-42, -13, 7, 12, -9, 5], [-26, -26, 6, -8, 4, -1], [0, 15, -15, 4, 8, 2], [2, -13, -4, -3, 15, 8], [8, -10, -26, -13, 4, -6], [-16, -12, 13, 3, 9, 6], [-34, 12, 0, -12, -2, -2], [5, -12, -29, -9, -2, -6], [11, -15, -24, 3, 2, 10], [18, 6, -4, 20, -2, -4], [14, -16, -2, -11, 18, 8], [18, 22, -8, 2, -8, 4], [28, -8, 2, -1, 9, 6], [-31, 5, -8, 3, -14, -13], [-13, -3, -8, -13, 12, -10], [-2, -27, -8, 23, -7, -4], [-18, -10, -14, 1, 5, -14], [0, -16, -4, -8, 28, -4], [-2, -28, 14, -10, 10, 22], [-34, 10, 8, -6, -12, -6], [-12, 33, 4, 19, -18, -4], [36, -10, -2, -4, -14, 2], [-13, 6, 11, 5, 0, -14], [-7, -1, -8, 5, 2, -9], [13, -6, -27, 9, 10, -2], [-22, -22, -20, 22, -4, -8], [24, -24, -4, 7, 8, -6], [-9, -21, 4, -3, 9, 7], [-5, -10, -22, 8, -4, -4], [23, 12, 18, -2, -6, 19], [-12, -2, -4, 10, -20, 3], [-6, 20, 10, 10, -8, 0], [48, -10, 5, -7, -1, 12], [4, -4, 0, 26, -16, -4], [48, -5, -3, -1, 15, -3], [-14, -11, 0, -1, 15, 10], [-36, 46, 12, 14, -10, -14], [45, 8, 7, 5, -7, -7], [-14, 24, 10, -10, 10, 4], [1, -17, 16, 29, -5, 2], [24, -4, 20, 8, 14, -2], [2, 1, 25, 6, 15, -7], [-21, 23, -3, 4, -12, -26], [-32, 14, 28, -10, 16, 2], [-7, 3, 28, 1, 16, 11], [-12, 19, -15, -2, -13, -19], [-23, 17, 27, 12, -6, 8], [-26, -14, -2, 4, -2, 14], [-6, -18, -9, -25, 21, 8], [13, -20, 15, -5, 14, 4], [-16, -18, -2, -24, 6, -6], [-9, -18, 9, -7, 16, 25], [11, -36, -2, -18, 11, 13], [-47, 0, -8, 8, -3, -5], [-17, 36, 7, 9, -10, -15], [18, -20, 6, -19, 16, 10], [11, 0, -7, 7, -7, 1], [-9, 6, 23, -9, 23, 21], [-1, 26, -11, -3, -2, -7], [0, -10, -6, -14, -14, 6], [-17, 18, -25, -5, 19, -3], [-1, 15, -2, 7, 6, 7], [-19, 36, 3, 3, -15, -15], [2, -8, -20, 16, 16, 2], [1, 9, 2, 18, 0, 0], [22, 20, 14, 1, -7, -1], [21, -19, -22, -7, 6, -8], [-3, 3, -10, -9, -10, -5], [31, -7, 4, 23, 10, 16], [32, -6, 0, 6, 18, 4], [-19, 18, 23, -9, -8, -17], [-26, -12, -12, 16, -2, -12], [9, 28, 5, -9, -1, -28], [13, -5, -12, 2, 24, -4], [-1, 14, 19, -1, 10, -16], [29, 8, 27, -7, 9, 10], [-12, 6, 6, 28, -12, 8], [44, -2, 24, 3, 5, 12], [16, 4, 10, -23, -1, 16], [-25, 21, 4, 23, -14, -5], [14, -18, 12, -2, -6, -6], [-17, -4, 13, -24, 15, 11], [26, -10, -12, -22, 4, -8], [20, -28, 24, -4, 12, 36], [31, 16, 34, 2, -8, -5], [-22, -11, 8, -9, 15, 4], [-21, 3, 28, -19, -9, -4], [4, -20, -2, -2, 0, 6], [31, -8, -14, -18, -8, -4], [-4, -19, 20, 3, 1, 12], [8, 8, -4, -6, 14, 0], [2, -8, -26, -14, 13, -12], [7, 6, 7, -17, -16, -8], [21, -15, 8, 15, -18, -11], [-17, -16, -8, 0, 18, 18], [8, 14, 22, 1, -27, -3], [13, 30, 33, -3, -5, 5], [1, -5, 4, -15, -14, -3], [8, -2, -14, 4, 10, -2], [34, -46, -10, -7, 23, 10], [36, -8, 4, 8, -20, 0], [17, -5, 0, 12, 16, 24], [15, 16, 21, 17, -24, -8], [-21, -19, 4, -1, 6, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8281_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8281_2_a_by();".
// 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_8281_2_a_by(:prec:=6)
    chi := MakeCharacter_8281_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_8281_2_a_by();".
// 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_8281_2_a_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8281_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 8, -4, -12, 0, 4, 1]>,<3,R![4, -20, 25, 2, -11, 0, 1]>,<5,R![-59, -128, -2, 50, -2, -6, 1]>,<11,R![16, -72, 85, -16, -17, 4, 1]>,<17,R![-491, -224, 167, 60, -21, -4, 1]>],Snew);
    return Vf;
end function;