
// Make newform 3549.2.a.z in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3549_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_3549_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_3549_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 := [-728, -252, 224, 55, -25, -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], [-10, -1, 1, 0, 0, 0], [0, -10, -1, 1, 0, 0], [76, 18, -17, -2, 1, 0], [-92, 72, 39, -17, -3, 1]];
    Rf_basisdens := [1, 1, 2, 2, 4, 4];
    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_3549_a();"
function MakeCharacter_3549_a()
    N := 3549;
    order := 1;
    char_gens := [1184, 1522, 3382];
    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_3549_a_Hecke(Kf)
    return MakeCharacter_3549_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, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, -1, 0, 0, -1, -1], [0, 0, 0, 0, 0, 0], [-3, 1, -1, 1, 2, 1], [-2, 0, 0, 0, 0, 1], [-2, 0, -1, 0, 0, -1], [0, -1, 0, 1, -3, 0], [-1, -1, -1, 1, 0, 1], [5, -1, 2, -1, -2, 0], [-1, 1, -2, 0, 4, 1], [-1, 0, 1, -2, -1, -1], [-4, -2, 3, -1, 0, -1], [1, -1, 4, -1, -4, 0], [9, -1, 4, 0, 0, -2], [-6, 0, -3, -1, 2, 1], [-2, 0, -1, 2, -2, 1], [-1, -1, 3, -2, 0, 0], [-3, 1, 3, 1, -2, 1], [-3, 1, 2, -2, 1, -1], [-1, -1, 0, 0, 6, -1], [-3, 3, 0, 0, -4, 2], [-3, -1, -1, -1, 4, 0], [-4, 2, 6, 0, 0, 0], [2, 2, -3, 1, 2, 2], [3, 1, 0, 0, -9, 1], [6, 0, 6, -2, -3, -1], [0, -2, 5, -1, 1, 1], [0, -3, -4, 1, 5, 0], [-5, 1, 7, -1, -6, -2], [6, 2, -5, 1, 3, 2], [-9, -3, -3, 1, 6, -1], [-5, 2, 3, 2, -3, -2], [4, 1, -1, 2, 3, 3], [-1, 1, -2, 0, 4, -1], [-8, -1, -9, -2, 11, 0], [-6, 0, -8, 2, 2, 1], [-7, 3, -6, 0, 0, 2], [-10, 0, 5, 1, -3, 2], [7, 1, 5, 1, 0, -2], [-11, -2, -3, 0, 3, 0], [8, -1, 12, 1, -9, 0], [-2, 2, 2, -2, -4, 1], [-6, 2, -1, 3, 4, 1], [-18, 2, -6, -1, 4, 3], [2, -2, 1, 3, 4, -1], [13, -1, 1, 1, -4, -1], [3, -1, -3, -1, -2, -2], [-9, -2, 4, -1, -3, 0], [8, -1, 6, -5, -5, -1], [-9, -1, -5, 1, 8, -2], [-13, 1, -3, -3, 6, 1], [-14, 0, -3, -1, 4, 0], [2, -2, -10, 0, 1, 0], [3, -1, 8, -4, -8, -4], [14, -2, 6, 0, -6, -2], [1, -3, -9, -1, 1, 1], [-3, -1, -2, -4, 4, 0], [6, -4, 5, -1, 2, -6], [-14, 0, -4, 0, -4, -1], [-5, 1, -3, -1, -8, -1], [-5, 5, -5, 1, 8, 4], [12, 0, 0, -2, -10, 1], [11, 1, -2, 4, -5, 2], [-5, -3, -7, -4, 14, -1], [0, -3, 5, -2, 7, -3], [11, -5, -2, -1, -2, -3], [-9, 5, -3, 3, 8, 2], [13, 1, 2, 2, 2, 3], [7, -3, -13, 0, 8, 1], [-1, -3, 3, -1, 4, -3], [-2, 4, 10, -1, -2, -2], [19, 2, 13, 2, -9, 2], [-10, 0, 2, 2, 8, -2], [-8, 2, -11, -1, -5, 3], [-2, 2, -5, 3, -4, 4], [0, -1, -3, 0, 11, 3], [1, 5, -9, 1, -6, 3], [-11, -1, 2, 2, -10, 4], [6, 0, -11, -1, -10, -1], [-8, -2, -6, 4, -5, 1], [4, 2, 6, 4, -16, 0], [9, -1, 4, -4, 2, -1], [-6, 1, -5, 4, 11, 1], [1, 1, 4, 3, -8, 1], [-3, -1, -11, 1, -3, 4], [-16, 4, -14, 0, 18, 3], [-6, -2, -12, 2, 11, -1], [1, -3, -5, 1, -8, -5], [-2, -4, -9, -1, 12, -3], [-12, 4, -13, 3, 9, 2], [-2, 2, 6, 1, -10, 5], [-1, 8, -6, -1, 7, -1], [14, 0, 23, 1, -8, 2], [-16, 4, -16, 0, 10, -1], [-6, 0, -11, -1, 6, 1], [-1, 5, -5, 5, -4, 5], [6, -2, 10, -1, -2, -5], [4, 4, 2, 0, 1, 1], [2, 2, 3, 2, -2, -2], [8, -6, 13, -5, -14, -4], [6, -4, 4, 0, 1, -4], [3, 4, 9, 0, -3, 0], [4, 2, -11, -3, -4, -2], [-17, 3, -6, -2, 0, 0], [13, 1, 8, -2, -4, -1], [-18, -3, -3, 0, 1, -4], [2, 4, 6, -2, 0, -4], [15, 1, -2, -2, 8, 3], [-9, 2, -4, -7, 1, 1], [-8, 4, -6, -2, 0, -1], [-7, 3, 2, 4, -6, 4], [11, 1, 8, 0, -1, 0], [-10, 2, -27, 2, 14, 2], [-1, 3, 19, -1, -16, -1], [-5, -1, 10, 2, -2, 2], [-7, -6, 4, -1, 13, 1], [-4, 1, 14, -3, -19, -5], [3, 3, 16, 2, -8, 0], [15, 1, 2, 0, -11, -3], [-8, 0, 16, 0, 4, 1], [-8, -4, -8, 0, 0, -6], [-8, 6, -10, 2, 6, -4], [-21, 1, 5, 0, 2, 3], [-7, -2, -14, -5, 5, 0], [-8, 4, 1, -5, -2, -4], [-5, 3, -5, 1, -4, 7], [-11, -3, 8, -2, 6, 3], [-7, 1, 7, -5, -14, -3], [3, 2, 1, 0, 19, -1], [-4, -2, 0, 2, 1, 6], [-17, -2, -8, 1, -3, -7], [9, 1, 8, 4, -6, 1], [-4, -2, -12, 6, 0, -2], [22, 0, 5, -7, -8, -2], [7, -1, -2, 2, -4, -6], [11, -3, 7, -1, 10, -3], [-4, 0, 0, -3, 12, 3], [-18, 4, -9, 1, 4, 6], [18, 2, 12, -3, 4, -1], [24, 6, 10, -1, -8, 2], [-16, 2, -15, 0, 24, -1], [2, -2, -16, 0, -6, 4], [-14, 2, -22, 2, 24, 3], [9, -7, 19, -5, -18, -6], [7, -7, 7, -1, 8, -6], [-22, 0, -1, 3, 0, -6], [5, -1, 21, -5, -17, -4], [5, -7, 0, 2, -19, -4], [-2, 8, 11, -3, 2, -4], [-24, 8, -11, 5, 23, 6], [19, 3, 9, -1, -14, -3], [-11, 1, -15, -2, -6, -6], [-5, 1, 1, 0, 16, 1], [-12, -2, -7, 4, -2, -7], [-7, 3, 0, -6, 18, 4], [14, -4, 3, -3, -4, -8], [-9, -1, 2, -4, -12, -8], [2, -4, 15, -2, 0, -7], [7, -3, 9, 10, 2, 5], [11, 9, -2, 1, -4, 6], [-33, -3, 0, 2, 18, -2], [29, -7, 19, -1, -5, -6], [-2, -2, 18, -2, 10, -4], [6, 5, -12, 3, 11, 6], [26, 2, 12, 4, -14, -1], [4, -1, -2, -5, 3, 2], [24, 4, 24, 2, -20, 1], [-4, 2, 4, 12, -4, 5], [3, -5, -3, -3, 4, -2], [-16, -7, -16, 3, 13, -1], [-11, -3, -12, 5, 22, -1], [-16, -2, -9, -1, 16, 0], [6, -6, -6, -2, -14, -7], [-1, 0, -7, -8, 9, -2], [6, -8, -6, 2, -7, -2], [-22, -8, -4, 0, 18, -5], [14, -4, 12, 6, 4, 5], [12, -2, 18, 0, -17, 0], [-7, -3, -6, -2, -16, -3], [12, 1, -2, 5, -3, 0], [2, -4, 1, -7, 8, -7], [-11, 1, -12, 1, 6, 11], [18, -4, 12, 0, -12, -3], [-15, 1, 0, 2, 11, 6], [21, 7, 22, -6, 0, -1], [11, -7, 15, -5, -25, -6], [13, -3, 24, 0, -16, 1], [27, -1, -11, 3, 20, 4], [-17, 5, -9, 6, 4, 7], [-11, 1, 15, -2, -10, -4], [5, 1, 22, 2, 2, -7], [-11, -3, -9, 3, -6, -1], [-19, -1, -27, 5, 14, 5], [12, 4, -7, 2, -4, 4], [34, -4, 23, 1, -17, -2], [-2, -2, 17, -5, -10, -2], [-18, -6, -27, -1, 8, -3], [4, -1, 3, 4, 13, -6], [28, -8, 17, 1, -16, -4], [12, -6, -6, 0, 16, -2], [0, -4, -14, -2, 8, -2], [0, -4, 20, -4, -14, 3], [17, 3, -5, -1, 16, 3], [4, -12, 0, -4, 6, -9], [1, -1, 1, 1, 10, -4], [14, -2, 13, -1, -31, -6], [-15, -1, 10, -8, -8, 1], [-4, -2, 0, -1, 10, -2], [-7, -7, 13, -3, 0, -2], [-14, 6, -21, -5, 13, -2], [28, 2, 10, 2, -24, -1], [8, 2, 16, 4, -22, 2], [-16, -6, -15, -1, -4, -9], [-22, 4, -6, 2, -8, 1], [-17, 3, -14, 4, 4, -5], [-18, 5, 11, 4, -9, 5], [27, 0, 6, 5, -15, 1], [-25, 7, -21, 7, 27, 5], [-9, 7, -18, 2, 6, 2], [-37, 0, -27, -4, 13, 1], [7, 2, 30, 1, -3, 3], [15, -5, 4, 4, -32, 0], [-8, 8, -12, -1, -2, -3], [14, 4, 7, 3, -10, 2], [-4, 2, 5, 7, -2, 8], [-33, 3, -15, 3, 4, -1], [-18, -1, 11, -6, 5, -1], [-22, 6, -2, 5, 26, 5], [15, -7, -2, 2, -8, -5], [26, 0, 28, 2, -19, -2], [-32, 9, -11, 2, 5, 5], [3, 3, -19, 3, 6, -3], [-5, -3, -13, -3, -2, -5], [6, -4, 10, -4, 10, -4], [34, -4, -2, 6, 11, 6], [-14, -2, -3, -3, 17, 7], [-10, 0, 1, 3, 6, -2], [-7, 1, -1, -5, 2, -1], [-4, -2, 12, -4, -2, 3], [20, -6, 15, 1, -16, 0], [-11, -1, 20, -2, -8, -4], [31, -3, 33, -11, -20, -5], [14, 12, 10, 2, -16, 6], [-7, -1, -24, 8, -4, 0], [-17, 1, 5, 5, 22, 1], [25, 2, 20, -5, 1, 0], [21, -3, 20, -2, -28, 0], [27, -5, 1, -7, -7, -7], [24, 2, -11, -3, -24, 0], [-33, 0, -14, 5, 11, 4], [2, 0, 12, 8, 2, 6], [32, -6, 6, 0, -26, -4], [8, 6, -5, 2, -12, 3], [-44, 6, -1, 4, 28, 3], [-19, 3, -27, -5, 2, 2], [-4, 6, 14, 8, -18, 10], [27, 5, 10, 0, -12, 1], [-5, -13, -1, 0, 22, -6], [1, -7, -10, 4, 8, 0], [10, 0, -13, 7, 10, 2], [28, 8, 14, -6, -16, 3], [3, -5, 9, 5, 20, -1], [-21, 3, -21, -5, -2, 0], [-2, 11, -13, 0, 17, 7], [-39, 5, -18, 0, 42, 3], [-18, 12, -6, 4, 19, 8], [21, 14, 7, 0, -9, 0], [-25, -5, -28, 6, 6, 0], [-19, 5, -16, -4, 7, -3], [2, -8, 30, -8, -14, -5], [-8, -8, -10, 10, 12, 0], [-21, 1, -27, 1, 36, 0], [-30, 6, 15, 5, 20, 5], [21, -9, 37, -9, -37, -10], [24, -2, 18, 6, -17, -3], [4, 1, 6, -1, -5, -11], [-3, -3, -2, 8, 14, 0], [22, 2, -6, -8, -12, 1], [10, -6, -7, 4, -14, -4], [-15, -3, -18, 6, 8, 8], [10, 0, 17, -1, -20, 5], [8, 0, -10, -14, 10, -10], [-18, 2, -6, 0, -23, 0], [5, -1, 1, -3, -7, 2], [6, -8, 16, 0, 3, -8], [26, 6, -15, 3, -2, 5], [-12, 0, -8, 4, -2, -8], [-13, -9, 12, -6, 15, -3], [-23, -7, -29, -7, 32, -6], [-3, -11, 5, -4, 6, -4], [-18, 2, 8, 4, -2, 0], [33, 1, -9, 4, -8, 0], [39, -3, 14, 6, -22, 0], [0, 0, -10, -4, -2, -1], [-36, 2, -21, 1, -2, 11], [19, -5, 2, 8, 4, 1], [-27, 3, 2, 8, 8, -5], [6, 2, -16, -4, -4, 2], [-9, 0, -29, 0, 31, 4], [-15, -2, 5, 0, 33, 0], [-12, 4, -3, -6, 12, 6], [-34, 12, 4, -2, -13, 0], [-8, 2, -4, 3, -18, -4], [-4, -3, -17, -12, 19, -6], [-17, -5, 6, -4, -4, -5], [-8, -1, -22, -3, -7, -6], [3, 5, 34, -4, 15, 1], [-11, 3, 6, -6, 20, -2], [6, 4, 23, 4, 2, 9], [14, 8, 3, 9, -21, 7], [9, -11, 6, -4, -4, -6], [-5, -7, -2, -2, 25, 7], [6, 12, 11, 5, -32, 4], [-49, 5, -10, 0, 10, 4], [-18, -1, -17, 4, 3, 10], [8, -2, 0, -8, -4, -2], [11, -7, -23, -7, 3, 5], [20, -6, -1, 5, 4, 2], [-5, -7, -28, -3, 10, 4], [-10, 0, -37, -1, 22, -6], [34, 0, -4, 1, 4, -4], [37, -1, 35, -5, -47, -1], [7, -9, -1, 7, -30, -3], [-4, 4, 6, -4, -9, -1], [-30, 6, -39, 1, 24, 5], [4, 10, 15, 1, -2, 8], [1, 1, 3, -5, 11, 0], [-12, 0, -7, -1, -6, -4], [-22, 10, -4, -1, -4, -1], [27, -17, -7, 1, 2, -1], [5, 1, 25, -9, -32, -7], [27, -7, 4, 6, 12, -3], [18, 0, 4, 0, 2, 3], [8, 6, -17, 8, 8, 1], [29, -19, 7, -1, -8, -10], [14, -6, 19, 9, -2, 7], [11, 1, -25, 1, 30, 6], [3, 12, 2, 5, -9, 11], [-3, 0, -41, 0, -1, -3], [-25, -3, -24, -2, 6, -8], [14, -2, -2, 1, 8, 1], [21, 11, 12, 1, -24, 0], [-2, -4, -28, 4, 0, 3], [-27, -1, -8, 0, 2, -6], [-16, 10, 1, -3, -17, 3], [-6, -4, 14, 6, -16, -4], [0, -10, 3, 3, -20, 0], [-9, 1, 17, 3, -28, -12], [-11, 11, -19, 1, -3, 8], [22, -2, -14, 7, -8, 3], [23, 7, 29, 5, -24, -6], [-6, 2, -7, -1, 1, 1], [34, 4, 22, -8, -24, -3], [18, 6, -5, -3, 14, 7], [-2, 14, 20, 2, -4, 2], [-3, 17, 14, 13, -20, 17], [19, 7, 36, -11, -2, -5], [-5, -1, 1, 5, -2, 6], [-11, -7, -21, -1, 6, 9], [0, -3, 5, -2, -25, 6], [17, 0, 2, 3, -3, 4], [-14, 4, 0, 0, -8, -8], [38, -4, 26, 5, -18, -2], [-10, 0, 13, 3, 20, 3], [-4, 20, 1, 1, -17, 8], [10, -6, 17, -4, -28, -2], [-16, -12, -9, 6, 16, 0], [-47, 3, 8, 8, 16, 2], [15, 3, -1, 10, -12, -4], [0, -8, 11, 5, -15, 4], [-21, 11, -17, -9, 4, -5], [-11, -12, -2, -7, 7, 0], [-35, 7, 6, -2, -28, -4], [29, 5, 4, 4, 1, -6], [-18, 6, 0, 4, -6, -5], [18, -4, 1, 1, -20, 0], [8, 10, -16, -2, 18, -5], [4, 12, 15, -5, -30, -2], [-28, -13, -9, -4, 25, -2], [-12, -4, -22, 15, 36, 3], [-53, 4, -6, -5, 3, 2], [-16, 10, -18, 6, 6, -6], [-18, 2, -21, -5, 44, 8], [11, -3, -19, 7, 15, 5], [8, 14, 9, 9, -22, 6], [20, 0, 58, -1, -20, -1], [-33, 7, -52, 4, 24, 3], [-37, -1, 18, 0, 29, 2], [-14, 8, 10, 4, -4, -1], [-11, -9, -26, 12, -4, 3], [-46, 16, -7, 1, 6, 6], [-28, 4, 7, 1, 2, 13], [1, 3, -15, 13, 7, 4], [-15, -5, 27, 1, -6, 0], [10, 8, 9, 3, 8, 2], [-16, 6, 9, 0, 14, 7], [-30, -2, -3, -9, 11, -2], [-11, -3, -43, 3, 36, 0], [-25, 2, 7, 2, 5, -8], [5, -15, -9, -1, -14, -1], [12, -6, 21, -3, -22, 0], [20, -2, -6, -8, -22, -9], [23, -9, -9, -3, -14, -12], [13, 5, -19, 1, 25, 7], [-14, 0, 3, 1, 1, -1], [33, 1, 37, 5, -38, -2], [12, 4, -31, 7, -8, 3], [0, 4, -26, 2, 24, 12], [12, -9, 15, -2, -29, -15], [26, -10, 42, -9, -30, -5], [17, 5, -11, -5, 7, -4], [-4, -2, -12, -5, 6, 0], [34, -2, 32, -6, -16, -3], [-49, -1, -15, -3, 26, 3], [-3, -7, -2, -8, 16, -2], [-30, 8, -2, 4, -16, 13], [36, -4, 9, 7, -34, -2], [7, 14, -19, 2, 3, 5], [-37, 3, 4, -2, 16, 10], [-26, -3, -30, -1, -1, -13], [-21, 1, -18, 10, 20, 1], [9, 5, -4, 10, 4, -1], [7, -3, 7, -9, -4, 9], [-13, 19, -11, 7, 7, 3], [-28, -8, -12, 0, 4, 1], [14, 0, 10, 8, -10, -2], [-13, 15, -16, 0, 8, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3549_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3549_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_3549_2_a_z(:prec:=6)
    chi := MakeCharacter_3549_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_3549_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_3549_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3549_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -2, -1, 1]>,<5,R![-728, 252, 224, -55, -25, 3, 1]>],Snew);
    return Vf;
end function;