
// Make newform 5054.2.a.bb in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5054_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_5054_2_a_bb();".
// 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_5054_2_a_bb();".
// 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 := [-1, -3, 3, 7, -3, -3, 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], [1, 4, -2, -3, 1, 0], [1, 1, 4, -2, -3, 1], [-1, -1, 5, -2, -3, 1], [-3, -2, 9, 0, -4, 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_5054_a();"
function MakeCharacter_5054_a()
    N := 5054;
    order := 1;
    char_gens := [1445, 1807];
    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_5054_a_Hecke(Kf)
    return MakeCharacter_5054_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, 1, 0, 0], [0, -1, -1, 0, -1, -2], [-1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 1], [2, -1, 0, -1, 0, 0], [-1, 1, 3, -2, 1, 2], [0, 0, 0, 0, 0, 0], [-2, 2, 0, 2, 0, 2], [0, 1, -2, 1, 0, 0], [0, 2, 3, -1, -1, 2], [-2, 2, -3, 1, 1, 0], [2, 0, 2, -1, 1, 0], [-1, 2, 0, -2, 4, 2], [2, -3, -1, 2, -1, 2], [4, -2, -3, -4, -1, -3], [4, -4, 0, 1, -3, -3], [-8, 4, 4, 2, -2, 2], [6, -4, -3, -3, -2, -5], [6, -1, -2, -4, 4, 1], [-6, 2, 4, -1, -3, 3], [-2, -3, -2, -3, 2, -4], [-4, 0, -2, -3, 3, -4], [-1, 2, 2, -4, 2, 6], [-8, 7, 2, 3, 1, 5], [0, -2, 2, -3, 2, 3], [4, 2, 7, -2, -1, -3], [3, -3, 4, 4, 0, 1], [-8, 4, 4, 1, 0, 3], [-2, 9, 5, 0, 0, 4], [-2, -5, -4, 3, 2, 0], [-6, -3, 0, -4, 2, 0], [-2, -1, -9, -1, -1, -2], [-8, 0, 3, -5, -2, 3], [2, -5, 1, -1, 3, 7], [8, -7, -9, -2, -1, -8], [-2, -3, -2, 4, -4, -9], [2, -5, -5, 1, -5, -6], [0, 10, 7, 1, -1, 2], [-2, -2, 2, -3, 2, 7], [0, -1, -2, 3, -7, -1], [0, 5, -5, 3, 3, 3], [-10, 3, 1, 5, -7, -3], [-11, 4, 6, 0, 2, 2], [-6, -6, -1, 7, -5, -10], [-4, -2, -2, -6, 2, 8], [-3, -7, -6, 7, -8, -8], [2, -2, 0, -5, 6, 7], [-8, 2, -1, 4, 0, 0], [-12, 0, 3, 3, 1, -2], [6, 1, 0, 0, 0, -6], [2, 6, 7, -8, 1, 3], [-2, -5, -11, 3, 3, -6], [-2, 1, -6, 1, -2, 1], [6, 1, -7, -6, 1, -9], [4, 0, 2, -1, -2, 5], [-6, 2, 6, 6, 0, -6], [-4, -3, -1, -4, 1, -6], [8, 1, 3, -9, 7, -1], [-8, 4, -3, 2, 3, -6], [2, -3, -6, -4, -1, -9], [12, -10, 3, 4, -1, 1], [-4, 0, 5, 7, -10, -8], [6, -6, -4, 6, -2, -6], [-6, -5, -12, 1, -4, -1], [-8, 0, 0, 4, 6, 8], [-14, 6, 3, 5, -6, 6], [-6, -3, -7, 3, -11, -4], [10, 0, 2, -10, 1, 5], [0, 2, -2, 0, 4, -8], [-6, 10, 3, 5, -6, -1], [-6, 5, -2, 4, 6, -5], [-8, 0, 7, -3, 3, 6], [-4, 6, 11, 1, 7, 12], [0, 0, 11, 6, -3, 9], [6, -6, -2, 1, -10, -7], [-4, 3, 2, 10, -10, -15], [-10, -3, 0, 10, 2, 9], [-8, 8, 4, 0, 7, 16], [-4, -15, -5, 8, -2, -10], [7, -6, -6, 1, -8, -9], [2, -1, 8, 4, 6, 11], [0, -4, 0, 12, -8, -10], [10, -14, -1, 4, -3, -3], [-18, 6, 9, -7, 7, 2], [2, 3, 13, 1, 5, 4], [-16, 4, 5, 13, 1, 1], [2, -7, 0, 10, 1, 0], [-16, -1, 7, 12, 3, 8], [-6, 0, -10, 3, 2, 1], [8, -10, -4, 3, 3, -8], [-12, 8, -2, 0, 6, 12], [-12, 9, 9, 5, -1, 13], [-19, 6, -6, 1, -4, -7], [-6, 12, -11, -2, 0, -3], [10, -4, -5, -3, 5, -4], [-16, -1, 3, -1, -1, -1], [0, -12, -13, 3, -11, -17], [0, 0, -25, 0, 1, -17], [-10, 2, 1, 8, -9, -13], [-19, 1, 8, 8, 0, -3], [-12, 11, 14, 6, 0, -1], [-6, 9, 2, 1, -1, -2], [-5, -11, -1, -11, 3, -3], [-14, 1, -3, 6, -5, 5], [-6, 4, 2, 10, 1, 8], [13, -7, -4, -6, 4, -3], [18, -10, 3, -3, 2, 4], [-8, 0, -10, -7, -8, -1], [16, 1, -8, -5, 1, 4], [6, -7, 2, -16, 2, 17], [16, -15, -5, -13, -7, -3], [6, -5, 3, 9, -4, -5], [-17, 1, -1, -3, 3, 5], [12, -19, 5, 3, -13, -17], [2, 7, 4, -1, 7, -3], [-16, 12, 21, 6, 1, 0], [-4, 10, -7, 3, -1, 8], [14, 9, 9, -9, 3, 17], [-1, 1, 1, 1, 5, 5], [6, -3, 14, -1, 6, 20], [-7, -6, -13, 0, -3, -23], [6, 7, -2, -8, 2, -9], [19, -8, -12, -7, -2, -15], [-17, -10, -9, 6, 1, -1], [-6, 7, -10, -6, 4, 9], [-8, 12, 2, -8, 2, 2], [-18, 2, -11, -1, 3, -18], [-20, 3, 21, 12, -1, 20], [-24, 10, -5, -2, 7, -1], [16, -1, -15, -16, 2, -17], [0, 5, 25, 4, 9, 30], [-4, 4, 15, 7, -3, -8], [0, 4, 0, 7, -6, 3], [6, 5, 17, -2, 13, 9], [-13, -15, -11, 3, 5, -1], [16, -19, -8, -3, -2, 2], [16, -18, -4, -7, 7, 7], [14, -10, 0, -5, 6, -7], [24, -14, -11, 8, -10, -5], [26, -1, 1, -5, -3, -5], [10, 5, -6, -3, -6, -14], [-2, 2, 1, 17, -11, -20], [20, 1, -12, -6, -1, -9], [-10, -2, -4, 6, -12, 10], [16, 11, -5, -7, 9, 13], [4, 9, -8, -10, 0, 7], [-6, -3, 11, 6, -5, -13], [-8, 8, 16, 9, 3, 9], [10, -2, -28, 4, 0, -16], [-18, 23, 18, 12, 4, 9], [0, -9, 13, 15, 4, 14], [2, 6, -12, -7, -1, 0], [12, 1, -1, 0, -1, 20], [-8, 15, 14, 3, -1, -2], [-14, 1, 2, 26, -6, 5], [0, -2, -24, -1, 4, -11], [-2, -6, -9, -3, -8, 0], [-4, 8, 0, 1, 4, 0], [-12, 8, 15, 7, -1, 16], [18, 11, 11, 4, 5, 16], [10, -7, 1, 1, -4, 23], [2, -11, -6, -6, -10, -13], [-4, -14, 6, 20, -8, -11], [-12, -10, -20, 1, -10, -14], [2, 3, 1, -16, 19, 0], [14, 2, -16, -16, 10, -8], [-28, 18, 12, 1, 10, 7], [8, 3, -11, 12, 11, -6], [-10, 16, -3, 5, -7, 2], [-14, 19, 12, -5, -6, 17], [8, 7, 4, 6, 12, 9], [38, -4, -5, -13, 3, 0], [34, -7, -1, -8, -3, 1], [8, 5, 2, 4, 2, -21], [-6, 6, 5, 2, -10, -3], [-24, 3, 1, -9, 10, 5], [-12, -1, -11, -6, -5, -6], [28, 3, 5, -16, 13, 14], [-20, -3, 28, 11, 0, 12], [-6, -5, -5, 15, -1, -21], [19, 4, 9, -2, 7, 11], [-6, 23, 6, -11, 10, 12], [-4, 5, -4, -12, 7, 23], [6, 1, -10, 7, -10, -12], [32, -6, 3, 0, -1, -15], [16, 9, 22, -1, -4, 28], [-9, 21, 8, -5, 6, 8], [-12, -2, -12, 6, 1, -1], [14, -10, 6, -12, -8, -6], [-8, 1, -12, 4, 7, 10], [2, 7, 3, -1, -5, -11], [-12, -2, -7, -3, 1, 31], [8, 2, 2, 7, -4, 23], [28, -6, -6, -12, 14, -9], [-13, -12, -24, 3, -14, -13], [-8, 12, -6, 5, -5, -10], [6, 3, 22, -4, 0, 11], [4, 11, 21, 9, 12, 25], [26, -26, -4, -7, -6, -15], [2, -9, 9, 6, -11, -4], [-8, -2, 2, 14, -12, -6], [-2, 3, -10, -6, 6, -25], [6, -19, -13, -7, -3, -22], [8, -4, 23, -5, -16, 13], [0, -17, -15, 15, -1, -15], [-18, -11, -20, 10, -8, -3], [-16, 0, 1, 2, 7, 4], [-6, 23, -9, -2, 4, 7], [8, -6, 3, 4, -14, -9], [-12, 18, -12, 3, -5, 6], [4, 16, 9, -11, 9, 14], [2, -5, 15, -2, -11, 8], [2, -17, -9, 0, -14, -23], [-14, 22, -4, 0, 4, 6], [17, 0, 9, 8, -5, -13], [8, 1, 14, 9, -8, 4], [-7, -24, -15, 10, -1, -15], [2, 5, 4, 8, -8, -7], [-30, 3, 8, 5, -4, 4], [22, 1, -7, 4, 1, -10], [24, 8, 11, -15, 15, 26], [-22, -13, -4, 3, -11, -10], [-8, 13, 9, 6, 1, 2], [8, -16, 1, -11, -4, -9], [-10, 12, 20, -9, 8, 15], [-2, 14, 20, 0, 19, 13], [18, 2, 11, -11, 11, -12], [-2, -8, -9, 1, -15, -2], [10, 5, 18, 5, 0, 4], [2, -30, -30, 0, -2, -18], [18, 11, 25, -6, 4, 18], [-14, 14, 16, 16, -18, -4], [-34, 3, -6, 12, -6, -22], [-10, 3, -5, 7, 18, 7], [10, 0, -18, 17, -10, -9], [-12, 3, -12, -12, 1, 6], [26, -1, 12, -18, 10, 25], [-28, -3, -23, 7, 2, -8], [-6, 15, -11, -4, 3, 10], [6, 25, 6, 0, -14, -6], [-32, 16, 13, 0, 13, 18], [8, -18, -13, -2, 7, -17], [0, -6, 7, 0, 11, 25], [-33, 12, 19, 13, -9, 16], [6, -3, -11, 17, 5, -13], [-12, 19, 7, 10, -3, -4], [10, -1, -18, 7, 5, -23], [22, 11, -5, -1, 3, 8], [-18, 5, -5, -8, -1, -6], [18, -28, 7, 12, -9, -23], [-43, 19, 15, -9, 15, 7], [14, -1, 2, 8, -18, -9], [-12, 13, 9, 19, 8, 21], [14, -3, -10, -15, 0, 4], [-1, 7, 18, -5, 0, 30], [-12, 7, -7, -11, 5, -25], [-4, -6, -1, -17, 6, 33], [20, -10, -1, 15, -5, -10], [-25, -5, 30, -4, -2, 5], [10, -24, -21, 3, -3, -28], [24, 1, -6, 15, -11, -4], [16, -21, -6, -3, 0, 2], [42, -8, -11, 8, -13, -13], [-2, 24, -3, -10, 12, 16], [2, -25, 3, 13, 1, -6], [26, -20, -15, -3, 1, -28], [8, 14, -7, -1, 15, 13], [11, -14, 13, 7, -23, -6], [40, -7, -17, -27, -1, -17], [14, 19, 12, -11, 8, 20], [-9, 7, -7, 11, 9, 9], [-4, -7, -39, 6, -6, -23], [-22, -7, 4, 7, -16, -12], [29, -15, -18, -19, 12, -22], [-28, 25, 10, 3, 22, 16], [-14, 21, 1, 10, 11, 25], [-24, 13, 4, 10, 4, 1], [20, -8, -1, 11, -12, -12], [-18, -1, 12, 6, 8, 23], [22, 7, 6, -15, 4, 26], [-8, 9, 0, -14, 30, 15], [4, -1, -21, -7, 11, -27], [14, -8, 7, -13, 3, -14], [-25, 7, -2, 8, -22, -5], [-36, 12, 0, 6, 20, 24], [22, -7, 21, -13, -16, 9], [0, -2, -1, -3, 9, 6], [16, 4, 25, -4, 17, 15], [-15, 11, 1, -24, 11, 8], [-8, -6, 1, -3, -21, -4], [-5, 4, -18, 10, -6, 12], [-22, -1, 3, -15, -2, 0], [3, -6, -16, -12, 4, -22], [4, 2, 30, 4, 2, 42], [4, 7, 27, 14, 7, 10], [0, 9, 32, -12, 14, 1], [4, -26, -32, 6, -6, -24], [1, 5, 23, 3, -13, 23], [10, -29, 1, -15, 14, 6], [-30, -3, 13, 8, -19, -5], [-22, 10, -3, -3, -9, -6], [-30, 2, -10, 17, 10, -1], [22, 8, 22, 6, 9, 1], [-18, 10, 7, 2, 0, 24], [-18, 3, -1, -21, 3, -10], [-2, -4, 3, -25, 28, 22], [0, 20, -1, -26, 9, 23], [26, -16, 3, 19, -13, 2], [0, 24, 11, -6, 7, 39], [-12, -18, 4, 26, -20, -42], [-12, 1, 22, -7, -10, 6], [-10, 1, 40, -6, -9, 17], [40, -9, 14, 7, -9, -11], [2, -35, -26, 0, 2, -25], [-19, -15, -13, -5, -5, -15], [8, 14, -6, -4, 3, 5], [-12, 20, 20, 16, -10, 8], [20, -1, 26, 5, -1, 17], [-23, -9, -3, -1, -15, -5], [-26, 1, -3, 2, 16, 11], [-6, -9, -19, 16, 11, -5], [-2, -17, -20, 12, -22, -27], [34, 1, -9, -11, 1, -3], [21, -8, 5, -27, 1, -20], [-16, -3, -17, 12, -19, -2], [2, -12, 2, -9, 5, -14], [-12, 2, -8, 33, 0, 10], [-14, -6, -23, 20, 5, -1], [-28, 24, 11, 5, -7, -10], [-12, 14, 16, 12, -2, 46], [26, -29, -26, -12, 6, -19], [-22, 5, -8, -1, -2, -18], [36, 0, -23, -27, 17, -12], [20, -33, -31, -6, 8, -21], [3, -21, 0, -3, -14, -32], [-26, -13, -2, -9, -10, -8], [22, 0, 17, -4, 6, -15], [-45, -6, 8, 17, -14, -9], [-32, 16, -4, -8, 6, -12], [24, -1, 3, 9, 5, 3], [-18, -15, -5, 25, -14, -15], [42, -23, -23, -18, 5, -12], [-2, 7, -20, 1, 6, -8], [6, 9, -9, -3, -1, 27], [-24, -1, 17, -8, 8, 17], [-46, 23, 7, 22, 3, -6], [35, -28, -11, -20, 7, -5], [22, -4, -11, -13, -7, 4], [30, -19, -8, -15, -18, -6], [32, -20, -17, -11, 17, -11], [-8, -6, -23, 10, -26, -11], [4, -10, 10, 11, 16, 9], [-8, -14, -10, 4, -4, 2], [8, 5, 2, -35, 16, 18], [8, -35, -32, -4, 11, -29], [-12, -6, 13, 23, -1, 12], [-8, 4, 36, -16, 5, 26], [-10, 19, -9, 14, 18, 9], [0, 42, 17, 17, 9, 40], [16, -25, -13, -12, 9, -12], [0, -10, -20, -10, -14, -27], [28, -10, -12, -10, -8, -36], [16, 0, 25, -12, -1, 32], [-11, 13, 26, 7, 4, 34], [-9, 2, 1, -23, 5, -22], [-26, 10, 25, -22, 11, 19], [14, -12, 33, -7, -7, 28], [-26, 15, 6, 17, -7, -7], [-34, 20, 14, 17, -19, -5], [-24, 12, -19, 1, -15, -16], [-16, 3, 20, 13, 0, -20], [-2, -13, -20, 3, 0, 2], [-6, 16, -34, -14, 4, -14], [30, 1, 10, 3, 16, -8], [-33, 13, 24, 20, -4, 19], [-2, -21, 27, -1, 1, 1], [-8, 33, 9, 9, 0, 34], [24, -21, -15, -37, -18, -21], [-26, 2, 3, -3, -12, 20], [-36, 24, 17, 14, 25, 33], [-11, -12, 16, 5, 14, 23], [8, -4, 7, 20, 9, 3], [-42, 15, -6, 2, -11, 30], [-8, -2, 0, -17, -3, 9], [16, -18, -4, 3, 10, -29], [-24, 18, 16, 27, 6, 49], [12, -17, -2, 4, -12, 5], [-20, 32, 33, -4, 23, 28], [-12, 12, 33, -1, 1, 10], [-6, 5, -25, -6, -22, -36], [-18, 1, 12, -4, -18, 29], [-8, -17, -21, -6, 0, -17], [-6, -9, 0, -1, 11, 16], [18, -1, -10, -20, -2, -41], [2, 7, -17, 24, 2, 7], [-2, -21, -10, -16, 4, 5], [-6, -34, -4, -5, 3, -12], [4, 15, -10, 17, 5, 8], [26, -46, -23, -20, 3, -15], [10, 8, 10, -15, 2, 9], [-10, 7, -3, -9, 10, 6], [18, -15, -20, 41, -28, -4], [-33, 23, 14, 19, -16, -10], [0, -21, -43, -6, -15, -34], [-2, -8, 8, -10, -8, 10], [0, 32, 18, -19, 6, 39], [24, 0, -1, -3, -10, 6], [-37, 12, 4, 36, -20, -28], [0, 12, 34, 2, 3, 13], [-14, 34, -9, 5, 10, 32], [46, -41, -30, -23, -8, -14], [12, 5, 4, -40, 15, -4], [-16, -18, -31, 7, -7, 3], [22, 1, -12, -13, -7, -35], [-36, 20, -7, -4, 13, -17], [-46, 5, 13, -8, -19, 18], [-4, -1, -7, -8, -9, -22], [2, -12, -4, 34, 0, -14], [-26, 17, -11, 10, -1, 14], [-38, -22, -40, 11, -8, -39], [30, 1, 2, -12, 12, 17], [-38, 41, 13, 16, 17, 38], [-12, 12, 34, -29, -2, 24], [12, -3, -15, -17, -7, -47], [-6, -4, -12, -5, 18, 11], [40, -29, -10, 11, -19, -9], [-32, 19, 14, -15, 16, 6], [-47, 30, 32, 13, 10, 33], [-6, 1, 2, -6, 28, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5054_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5054_2_a_bb();".
// 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_5054_2_a_bb(:prec:=6)
    chi := MakeCharacter_5054_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_5054_2_a_bb();".
// 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_5054_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5054_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, -9, 3, 11, -3, -3, 1]>,<5,R![8, 0, -42, -47, -12, 3, 1]>,<13,R![8, -12, -12, 17, 3, -6, 1]>],Snew);
    return Vf;
end function;