
// Make newform 5054.2.a.z in Magma, downloaded from the LMFDB on 25 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_z();".
// 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_z();".
// 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 := [-16, 16, 41, -4, -13, 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], [-44, -55, 24, 3, -4, 1], [4, -15, -24, 7, 4, -1], [172, -75, -112, 31, 12, -3], [18, 65, 2, -31, -2, 3]];
    Rf_basisdens := [1, 1, 40, 10, 40, 10];
    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, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 1], [-2, 0, 0, 0, -1, 1], [-2, -2, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 1], [-3, -1, -2, -1, 1, -2], [-2, -1, -2, 1, 2, 0], [-1, -2, 0, -1, 1, -1], [0, 2, 2, 1, -1, 2], [-3, 1, -1, -1, 1, -2], [0, 1, -2, 1, 2, -2], [1, 2, 0, 1, 1, 1], [-2, -1, -2, -3, 2, 0], [-6, -1, -2, -4, 3, -3], [0, 0, 3, 3, -3, 1], [-4, 2, 0, 0, 0, 0], [-4, -1, -2, 0, 1, 2], [-11, -1, -3, 1, -1, 0], [2, -4, -2, -4, 2, 0], [2, 1, 8, -2, 1, -2], [-6, -1, -8, 0, -1, -3], [0, 2, -4, 3, 3, 0], [8, -3, 8, 0, 0, 0], [-7, -1, -10, 2, 2, -1], [-4, 1, 1, 4, 0, -1], [0, 0, -1, 4, 0, 0], [-5, -3, 3, -1, -3, 4], [4, 0, 2, 1, 2, 2], [6, 3, 0, -3, -1, 2], [2, 3, 2, 5, -6, 0], [0, -2, 7, 2, 2, -2], [-5, 3, 8, 2, -4, 3], [6, -3, 2, 1, 3, 1], [0, 0, -4, 2, 4, -4], [10, 0, 0, 2, -4, 2], [4, -2, 2, 0, -1, 2], [1, -5, 5, -7, -1, -4], [-6, -1, 0, -3, 4, 0], [-5, 5, 0, 0, -4, -1], [-6, -2, 4, -4, -4, 0], [14, 2, 9, 2, -4, -2], [10, -1, 4, -4, 0, -2], [-9, -1, -6, 0, 2, 1], [2, -5, 2, -5, 2, -4], [-8, 1, 0, 1, 0, -4], [2, 5, -4, -2, -1, 2], [-1, -3, -1, -4, -6, 1], [7, 3, 5, 1, -9, 2], [-10, 2, -2, 1, -2, -2], [8, 2, 4, 6, -4, 4], [0, 4, -4, 2, 1, 4], [-11, -1, 2, 0, 6, -1], [-10, -3, -2, 0, 1, -1], [-2, -1, -6, 1, -2, 4], [-4, 7, 0, -1, 3, 0], [1, -3, 0, -3, 3, -8], [-2, 3, 12, 4, 0, -2], [0, 1, 10, 5, 2, -4], [6, 2, 4, 6, 0, 2], [-6, -3, -8, 1, -4, -2], [-2, 8, -6, 1, -3, 0], [5, 0, 8, -1, 1, -3], [-4, -6, 2, -6, -2, 2], [-6, 3, -5, -6, -6, 3], [-6, 0, -4, -6, 4, -6], [0, -1, 6, 0, -1, -1], [-6, 1, -4, -1, -3, 3], [-13, -1, -12, 0, 0, -1], [-4, 7, -12, 2, 0, 0], [-7, 3, 3, 2, 0, 5], [2, 0, 3, 1, 1, 3], [12, 1, 0, -5, 0, 4], [-6, 4, 7, 0, 2, 0], [18, 4, 12, 3, -8, 4], [2, -2, 10, 8, 6, -6], [-12, 9, -8, -2, -3, 0], [-6, 5, -6, 4, -2, 4], [-1, 6, 0, 1, -3, -1], [-6, 12, -8, 0, -4, 2], [10, -5, 2, -6, -1, 4], [2, -7, -2, -3, 2, -4], [11, -1, -1, -5, 1, -4], [2, -4, -12, -9, 3, 1], [0, 5, 4, -1, 3, 0], [-14, -4, -12, 0, -4, 0], [-6, -6, 3, -3, 5, 3], [0, 1, 2, -1, -6, 2], [14, -6, 0, -2, 0, -2], [-11, 7, -2, -4, 2, -5], [6, -2, 6, -6, -2, 8], [9, 3, 10, -6, -2, -1], [12, 6, 16, -6, -4, -4], [8, 2, 0, -1, -1, 3], [-12, 3, 12, 2, -1, -1], [2, -3, 12, 4, -4, -2], [-7, 1, -2, -1, 3, -4], [-9, 9, 6, 8, -2, 1], [3, -3, 4, 0, 0, 5], [-6, -6, 2, -6, -6, 0], [7, -2, -3, -2, 4, 2], [-14, 12, 6, 6, -2, 4], [-8, -1, 4, 2, 9, -3], [2, 1, 0, 5, -4, -2], [-16, -1, 8, -2, 3, -4], [-15, -5, -8, 4, 8, -7], [-16, -9, -2, 3, 6, -4], [0, -2, -8, -6, 0, -4], [-4, -3, 2, -9, -3, 6], [-24, -8, -5, 3, 1, -5], [-10, 1, -20, 6, 0, 6], [2, -6, 12, 2, -4, 2], [0, -2, 1, 8, 2, 2], [2, -6, -14, -4, -2, -4], [-4, -2, 14, -1, -6, 6], [-9, -2, 1, 2, 2, -2], [1, -9, 0, -8, 0, -3], [14, 1, 22, -7, -2, -2], [-18, -1, -15, 4, 0, 1], [6, 8, -20, 5, -1, 1], [9, -7, 15, -3, 1, -2], [-6, 1, 8, 1, 4, -6], [11, 8, -9, 0, -8, 4], [19, 1, -4, 5, -3, 0], [16, 3, 18, -5, 6, -2], [22, -11, 4, -3, 4, 2], [2, 3, 12, 4, -3, 7], [-18, 6, -2, 2, -2, 0], [2, 0, -3, -9, 11, -3], [-8, -2, 2, 0, 2, -2], [18, -2, -8, 1, 9, -5], [-2, 7, 2, -1, -6, 2], [2, 9, 20, 3, -1, 6], [0, -9, -10, 1, 9, 1], [0, 10, -4, 7, -4, -4], [2, -5, -16, 5, -9, 9], [6, 11, -5, 6, -8, -1], [-16, 2, -8, -6, 0, 0], [31, 7, -2, 5, -13, 10], [-5, -9, 2, -8, 2, 1], [-10, -6, -3, -5, 11, 5], [-6, -4, 18, -8, -5, 0], [4, 1, -12, -1, -8, 4], [-4, -1, 10, 0, 7, 3], [-4, -3, -8, -5, -1, 7], [8, 5, 24, 3, -8, 0], [3, -1, 0, 8, -12, 3], [9, 13, 23, 0, -12, 1], [-2, -4, 4, -1, -5, 6], [-16, -2, -29, 1, 1, -3], [20, -4, 12, -6, -4, 0], [-5, 7, -17, -5, 1, 0], [-5, 1, -15, 3, -5, -2], [-18, -8, -5, 5, -3, -1], [-4, -9, -30, -2, 15, -9], [-34, -5, -12, 4, 4, -4], [10, 6, -16, 1, -4, 4], [2, -2, 10, -2, 6, 4], [13, -1, 8, 3, -1, 8], [30, -6, 7, 9, 9, 3], [30, -7, 8, 7, 4, 2], [-5, 8, -18, -1, 3, 5], [28, 5, 18, -2, -2, 2], [-4, 2, -2, -2, 2, 10], [-6, -1, 4, 8, 7, -6], [5, 11, 8, 1, -13, 4], [-8, -6, 14, 2, 11, -9], [20, -6, 23, -1, 5, 1], [-2, -2, -6, 2, -2, -6], [-1, -5, -3, 5, 3, 6], [2, 7, 9, -4, -10, 5], [0, -8, -18, -4, 6, -6], [-12, 0, -14, -3, -13, -1], [6, -6, 3, -3, 5, -9], [27, 11, 8, 1, -11, 0], [-10, 0, 0, 3, 4, 2], [0, -2, -4, -3, -3, 9], [-7, -7, -11, -9, 3, 0], [-6, -7, -2, -7, -6, 4], [12, -6, 15, -10, 0, 0], [24, 5, 28, -5, 1, 4], [6, -10, 0, -2, 4, -2], [-4, -5, 10, -3, 9, 2], [-6, 3, 2, 3, 7, -4], [-4, -2, -20, 2, 4, -4], [-19, -7, -24, 3, 7, 0], [36, 0, 2, 7, 6, 2], [22, 4, -16, 3, -7, 2], [6, -2, 4, 8, 0, -2], [-11, -5, 1, 9, 3, -4], [-6, -3, 0, -4, 9, -18], [31, -9, 26, 4, -2, -3], [16, -1, -4, -2, -4, 2], [-16, -2, -37, 2, 6, -2], [-6, -7, -13, 0, 12, -5], [-24, -7, -6, -2, 3, -1], [-12, -10, -8, -9, 12, -4], [5, -5, 2, 1, 5, 2], [-14, -3, -4, 1, 4, -10], [22, 1, 20, 10, 3, 1], [0, 1, 16, -1, 7, 3], [8, 2, 10, -9, -14, 2], [10, 2, -12, -1, 11, -6], [14, 6, 6, 6, -2, 12], [28, 10, -1, 1, -15, 9], [-27, -4, -3, -2, -2, -8], [-2, 7, 4, 0, -4, -2], [13, -12, 2, -3, -3, 3], [-14, 8, -14, -6, 1, -8], [-31, -5, -17, -9, 9, -2], [16, 8, -18, 0, 2, -2], [-10, 3, 0, -4, 12, 6], [4, 6, 14, -10, 5, -15], [-2, 10, -24, 8, -8, 2], [-24, -8, 0, 3, 11, -5], [-20, 12, 12, 6, 8, -4], [-42, 5, -33, -2, 0, 3], [-12, 16, -4, 2, 4, 6], [-10, -4, 0, -11, 8, 2], [6, -4, 23, -8, -4, 2], [2, 12, 19, 7, -7, 3], [8, -8, -2, -2, -2, -6], [-7, 2, 4, -1, 7, 13], [12, 9, 8, 0, -1, -1], [-2, 8, 5, -3, 3, 9], [10, -16, -2, -3, 2, 0], [13, 1, 1, -3, 15, -4], [-5, -3, -4, 0, 8, -11], [8, -2, 16, 10, -16, 8], [-5, 15, 12, -4, 12, -13], [-10, 8, 9, 1, 3, -1], [16, 12, 14, 0, 10, -6], [38, 5, 24, 3, -12, 2], [16, -11, 10, -9, -5, -1], [29, 3, 0, -6, -12, 11], [-2, -6, -10, 0, -14, 12], [0, 4, 10, -5, -18, -2], [-11, 11, -6, 12, -2, -9], [-26, 13, -10, 6, -2, 8], [10, 7, 18, 3, -10, 0], [-21, 2, -9, -10, 6, 4], [-20, -12, -2, 3, -7, -3], [-4, -5, -2, 0, 6, 2], [4, -3, -4, 0, -12, 4], [-22, 1, -16, -3, 4, -2], [13, -3, 12, 6, 12, -7], [-29, 3, -2, -16, -10, -7], [3, -2, 5, 4, -10, 22], [-24, -3, -6, 9, 13, -2], [29, 9, 6, -6, 6, 3], [-12, -5, 4, -4, 3, -5], [-6, 3, -30, 3, -6, 10], [-4, 2, -23, -1, -5, -5], [7, 16, -21, 14, 0, 4], [-26, 3, -8, -6, 16, 2], [4, -8, 0, -5, -4, 6], [-4, -2, 20, 11, -8, 2], [-7, 9, 4, 2, 20, 1], [17, 15, 17, 1, -13, 8], [14, 14, 2, 10, -10, 14], [-10, -14, 2, 0, -2, 2], [24, -4, 18, -14, -1, 7], [2, -9, 20, -8, 0, 14], [-2, 14, -23, 13, 3, 3], [-10, 12, 28, 7, -7, -1], [-31, -11, -21, 5, 5, -10], [13, -19, 14, 2, 6, -3], [-2, -6, -22, 4, -13, 12], [-22, 0, 3, 1, 13, 7], [26, 15, -6, 6, -17, 13], [0, 2, -9, 1, -3, -7], [20, 0, 24, 16, 5, -4], [12, -4, -20, -4, 4, 4], [-3, 5, 20, -4, 4, -11], [-13, 10, -8, 17, -3, 11], [-7, -19, -13, -4, 12, 5], [-14, 5, -14, 11, 2, 12], [-32, 14, 4, 4, 4, -4], [-3, -3, -16, 8, -4, 5], [38, 11, 8, -1, -16, 2], [12, -2, 8, 8, -5, 3], [-16, -6, -42, -1, 2, -6], [-3, -5, 15, -7, 13, -14], [-31, -2, -15, -8, 2, 0], [-23, -7, -19, 8, 10, 7], [14, 7, -4, 13, -4, 6], [6, -3, 26, 7, -14, 6], [-4, 14, -20, 15, -7, 5], [24, 12, 2, 5, -18, 10], [-4, 3, -12, -3, -15, -4], [2, 2, 2, 12, -6, 0], [-31, 5, -10, 3, 5, -2], [12, 0, -8, 0, 7, -12], [-20, 14, -12, 10, 8, 8], [-6, 9, 8, 4, 0, 16], [-8, 6, -32, 4, -8, 0], [-30, 5, -10, 5, 2, 0], [26, -6, 30, 4, -3, 4], [12, 12, 31, 2, -4, 4], [-16, -14, 9, -9, 7, -9], [16, 12, -10, -4, -2, 2], [-18, -12, -20, -10, 0, -14], [5, -23, -17, -6, 8, -3], [-3, -13, -27, 1, 7, -4], [12, 18, 14, 7, -9, 2], [-2, -2, 5, -1, 15, -7], [48, 5, 8, 3, -1, 4], [-2, -6, 6, 1, -2, -12], [-13, -3, 9, 17, -9, 0], [1, -18, -12, -13, 5, 5], [2, 0, 0, 10, -20, 2], [-23, -5, -6, 10, 2, -5], [-18, -10, -10, 0, 9, -9], [-8, 15, -16, 16, 4, -4], [-39, 3, 4, 0, 0, -7], [4, 7, 26, -5, -17, 7], [11, 0, 26, 7, 9, 1], [-26, 12, 0, 5, 0, 6], [12, -2, 18, 9, 11, -5], [14, 0, 22, 8, -5, -8], [-9, 11, -23, 7, -13, -6], [5, 6, 2, 23, -7, 3], [2, -5, 14, 0, 1, 7], [19, 1, -11, -1, 11, -10], [2, 8, 8, -2, -12, 6], [-47, 9, -13, 3, 1, -2], [26, 0, 2, 5, -5, -3], [-2, -5, 1, -2, 6, 1], [4, 16, -6, -6, -10, 8], [-6, -3, -6, 17, -14, 4], [-34, -4, -20, -6, 0, 2], [7, -11, -4, -12, -20, -1], [20, -14, -4, 3, -8, 8], [8, -2, 16, 3, -17, 7], [6, -6, 2, 12, 3, -12], [-44, -13, -20, 7, 1, -7], [0, 11, 20, 14, -8, 8], [20, -8, 26, -8, -2, 8], [-13, -23, -12, 4, 16, -7], [14, -10, -8, 6, -16, 12], [-8, 20, 8, -4, -4, -8], [-13, 9, -14, 3, -15, -4], [4, -2, 2, 4, -22, 2], [-4, -19, 16, -19, 7, 4], [-41, -9, -11, 1, 3, -10], [-34, -5, -18, 10, 2, 0], [-36, 8, 4, 10, -3, 5], [16, -3, 4, -14, -12, 4], [-19, 20, -9, -2, 2, -10], [-3, -15, -14, -3, -9, 20], [16, 12, 14, 4, 10, -14], [-47, 3, -11, -10, 2, -1], [24, -11, 12, -14, -3, -8], [-4, -6, 14, 10, 6, -6], [-36, -8, -46, 6, 6, 6], [-7, -26, 5, 4, -10, 4], [1, -3, 3, -1, -7, 6], [-4, -8, 7, -6, 14, 6], [-42, 19, 2, 3, -5, 5], [-16, 2, 0, -14, 8, -4], [-30, -4, -19, -3, 7, 5], [-44, -12, -10, 12, 10, -10], [11, 5, -2, -3, 21, -12], [12, 7, -6, -7, 2, -4], [-16, 4, -8, -8, 20, 0], [-28, -14, -18, -3, 13, 5], [60, 4, 0, -2, -12, 0], [-34, -15, -20, -2, 28, -2], [-42, 8, -20, 8, -8, 2], [-20, -8, -4, -6, -12, -2], [30, 2, 0, 7, -16, -10], [-48, -12, -20, 9, 1, -7], [-20, -8, -12, -4, 28, 4], [28, 2, -5, -1, -3, -7], [30, -8, 10, -11, 7, 13], [-22, 20, 2, 13, -15, -4], [-24, -20, -16, 0, 27, -5], [-12, -16, 4, -10, 3, 3], [-23, -12, 10, -1, -7, 7], [7, -11, -6, 8, 10, -11], [-14, 1, -19, 6, 6, 9], [-8, -15, 14, -6, -6, -2], [-26, -1, -2, -13, -6, -4], [1, 1, 41, -5, -13, 0], [-6, -15, 6, 5, 11, -11], [9, 7, 8, 0, 8, 5], [14, 4, -10, -6, -6, 0], [-26, -26, 6, 1, 13, -17], [-2, 11, 8, 15, 5, -2], [6, -28, 22, -4, -2, -8], [30, 3, 8, 1, -20, 6], [-11, -3, -12, 12, -4, -11], [-4, -4, 5, -4, -6, -2], [-10, -5, -22, -21, 18, -8], [34, 0, 10, 1, 2, -2], [5, -3, -13, 9, -11, 14], [-8, -12, 7, 4, -2, -6], [-12, 0, -4, -6, -11, 8], [-2, 5, 13, -4, -14, -3], [-26, 6, 4, -6, 4, -10], [-24, 10, -16, 13, 7, 7], [-24, -8, -20, 8, 5, -12], [14, -13, 4, -25, 8, -2], [-3, -19, 9, 8, 0, -7], [-5, 1, -6, 23, -7, 8], [-24, -2, -46, 2, -2, -6], [24, 1, 6, 9, -10, 10], [20, 10, 2, -1, -3, -7], [5, -21, -15, 0, 8, 1], [31, 17, 5, 11, -25, 18], [-22, -2, -13, -1, 5, 7], [5, 4, 4, 3, 11, -7], [-10, -11, 18, -21, 10, -18], [25, 5, 24, -6, -4, 5], [10, 31, -10, 5, -14, 16], [12, -5, 6, -1, -19, 6], [-8, -11, -18, -23, 14, -2], [8, 20, 39, 2, -16, 12], [30, -8, 6, -17, 11, 4], [-21, -3, -12, -6, -4, -13], [4, 8, -4, -3, 4, 8], [36, 2, 10, -6, -5, 10], [-26, 4, -30, -7, 3, 8], [-28, -10, 26, -8, -6, 6], [2, 7, 24, -9, -5, -2], [-10, 1, 0, -9, 8, 6], [16, -2, 36, 10, -20, 12]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5054_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-16, 16, 41, -4, -13, 0, 1]>,<5,R![-61, -7, 67, -6, -15, 1, 1]>,<13,R![-109, -525, -250, 73, 70, 15, 1]>],Snew);
    return Vf;
end function;