
// Make newform 3549.2.a.bf in Magma, downloaded from the LMFDB on 23 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_bf();".
// 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_bf();".
// 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, 6, 12, -41, -18, 37, 8, -11, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 2, -3, 13, 5, -8, -1, 1, 0], [-8, 1, -14, -11, 27, 7, -10, -1, 1], [-1, -3, 15, 24, -22, -15, 9, 2, -1], [1, -18, -11, 27, 7, -10, -1, 1, 0], [9, -3, -65, 4, 52, -1, -13, 0, 1], [9, -1, -68, -21, 66, 14, -21, -2, 2], [4, 17, -62, -30, 65, 15, -21, -2, 2]];
    Rf_basisdens := [1, 1, 4, 4, 4, 4, 4, 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_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, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 1, -1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, -1, 0, 1, 1, 0, 1, 1], [-1, 0, -1, -1, 0, -1, -1, 1, 0], [1, 1, -2, 1, 1, -2, 0, 0, 0], [2, 0, 0, 1, 1, 1, -2, -1, -1], [-1, 2, 0, 0, 0, 1, 1, 0, -2], [1, 3, -1, 1, 1, 2, -3, 0, -2], [-2, 0, -2, 0, -1, 0, 0, 0, 0], [4, 2, 0, 2, 0, 2, 0, 2, 0], [-1, 2, -1, -1, 4, 0, 3, 1, 0], [1, 2, 1, 2, -1, -1, 1, 0, -1], [-1, -2, 3, 0, -3, 1, -1, 0, 1], [-5, 4, -5, -1, 2, -2, 3, -1, 2], [3, 2, 1, -3, -2, 0, -1, -1, 2], [-1, -2, -3, -1, -1, -4, -1, -3, 2], [2, -2, 4, -5, -3, 1, 0, -1, 1], [1, 3, -3, 0, 0, 3, -2, -3, 1], [-3, 1, -1, -1, 3, 2, 0, -2, 4], [2, -1, 2, -1, -3, 1, -5, 2, 1], [-2, 3, -2, 0, -3, 0, -1, -1, 0], [2, -2, -1, -3, -3, -1, 1, -2, 1], [4, -1, 3, -3, -3, -3, -3, 0, 4], [9, -3, 0, -3, -3, 6, -5, 1, 2], [0, 4, -4, -1, 1, 1, 3, 3, -1], [-1, 0, 1, 2, -3, -3, 3, 4, -5], [-4, -2, 2, -3, 1, -5, 6, -1, 1], [6, 3, 4, -2, 3, 4, 3, 1, 0], [1, 3, 1, 1, 1, 4, -2, -2, 0], [3, -2, -2, -2, 4, -1, 1, 2, -2], [2, -3, 0, 3, -2, -3, 3, -2, -5], [11, 0, 1, -2, 3, 7, -3, 0, 1], [3, 1, 3, -3, 1, -2, -4, 4, -4], [-3, -1, -1, -1, 1, -2, 6, -6, 4], [2, 5, 2, 5, 0, 5, -5, 6, -5], [-1, -4, 0, -5, -2, -6, 4, -2, 4], [7, 1, 5, 1, -3, -6, 4, 0, -2], [4, -7, 2, -2, -7, -4, -3, -3, 0], [12, 1, 5, -2, -7, 2, -6, -2, -2], [1, -2, 2, 0, -2, 3, -7, 2, 2], [0, -1, 2, 1, 6, -1, 3, 6, -3], [3, -5, -3, 2, -2, 1, 3, -1, -5], [4, -7, 0, -5, -4, -1, -5, -4, 5], [-8, 2, -5, -2, 3, -8, 4, -2, 1], [-2, -4, -2, -5, -3, -9, 4, 1, 1], [0, -5, 0, 0, 1, 2, -1, -5, 2], [1, 0, -1, -5, -2, -2, 7, -5, 4], [6, -3, 1, -3, -2, -1, -8, 1, 3], [2, -3, 0, -2, -3, 4, -3, 5, 0], [8, -2, -4, 3, 1, -3, 4, 5, -3], [3, 1, 1, -4, 0, -1, -4, -3, 5], [8, 5, -5, 1, 2, 7, -3, -2, -1], [2, -1, -2, -3, 3, -3, -1, -2, 5], [-11, -3, 1, -2, 2, -4, -2, -3, -1], [1, -1, -1, -2, 2, 8, -4, -3, 5], [5, -4, 1, 0, -1, -3, -5, 2, -3], [0, -1, 3, -1, 1, -3, 5, 6, -2], [-9, 4, -7, 8, 4, -3, 3, -2, -1], [6, -1, 6, 2, -1, 6, 0, 7, -2], [14, 5, 2, -4, 3, 10, -1, 1, 0], [-8, 7, -2, 3, -4, 1, 1, 4, -5], [9, -4, 1, 3, 4, 4, -5, -3, 4], [8, -6, 0, 3, -3, 5, -2, 3, 3], [-4, 1, 4, -4, -1, 11, -9, 3, -4], [9, 7, -5, -5, -1, 6, -4, 0, 2], [-12, -3, -2, -2, 9, -4, 1, -7, 4], [-1, -4, -4, -3, 4, -4, 9, -3, 2], [-6, -1, -1, -1, 4, -5, 13, 6, -5], [-3, 7, 1, 9, 1, -5, 2, 4, -8], [-6, -3, 2, 1, 0, -5, 0, 0, 3], [6, -7, 0, -3, 0, 3, 9, -2, -1], [6, -2, 8, -4, 0, 12, -6, 0, 0], [3, 10, -3, 4, -1, 1, -7, 4, -5], [-11, 1, -3, 6, 0, -13, 12, 5, -3], [13, -6, -1, 4, -7, -1, -7, -6, 3], [2, 4, -10, 3, 3, -3, 0, 3, 1], [9, 2, -1, 6, 5, 5, 3, 2, 3], [14, -11, 10, -5, -6, 10, -5, -4, -1], [-12, 1, -7, -1, 2, 3, 8, -3, 3], [-12, -8, -4, 0, 0, 2, -4, 0, 6], [0, 4, 2, 1, -7, 2, -2, -5, 3], [4, -3, 0, 1, 5, -3, 11, -2, 7], [15, -9, 3, -1, -3, 14, -12, 0, 0], [-5, -4, -7, 9, 4, -5, -1, 1, -4], [2, 4, -3, -3, 1, 13, 2, 5, 5], [22, 3, -2, 4, -5, 8, -23, -3, 0], [6, 6, -8, -3, -1, 1, 2, 3, -3], [17, -12, 7, -3, -4, 6, -7, -5, -4], [11, -13, 3, 1, -1, -4, -6, 0, -2], [-2, -2, -7, -1, 5, -19, 2, 3, 7], [-1, 5, -7, 2, 4, 7, -6, 7, -3], [9, -11, 5, -6, -2, -3, -8, -5, 7], [0, 10, -1, 9, 1, 7, -6, 1, -1], [3, -7, 8, -5, -3, -16, 8, -2, 4], [7, 13, -1, 9, 1, 7, -2, 4, -8], [3, -4, -3, 3, 6, -9, 3, 1, 0], [4, -5, -4, -4, -1, -2, 5, -11, 6], [1, -1, 7, 0, -2, -13, 10, 3, -3], [3, 5, -1, 2, -6, -1, 4, 3, -5], [4, -4, 2, 5, -3, 7, -8, 5, -1], [14, 1, -2, -1, -8, 11, -11, -2, 5], [1, -1, -3, -6, -2, 7, 4, -3, 5], [-13, 2, -7, 2, 5, 9, 1, 0, 9], [-5, 5, -3, 5, -1, -6, 2, 6, 2], [-7, 5, -1, -7, -5, -12, 10, -4, 0], [-16, 0, 2, -2, -4, -8, 20, 0, 0], [-1, 4, 1, -8, -1, 1, -4, -4, 1], [2, 3, 9, -1, -5, -1, 9, 0, 2], [-4, -3, 0, -5, -2, -13, 13, -6, -1], [-5, -6, 5, 8, -5, -11, -5, -2, -5], [9, -4, 7, 5, -4, 8, 3, 3, -4], [21, -5, 11, 0, -6, 9, -14, -1, -5], [3, 8, 0, 8, -4, 15, 3, -2, -10], [-3, 6, -3, 2, 5, 3, -11, 0, -3], [3, -1, -3, -2, -2, 3, 8, 5, 1], [4, -2, -3, 4, 2, 2, 10, -6, 2], [-5, -4, 5, 0, -1, -13, 5, 6, 3], [-22, 6, -14, 0, 4, 9, 4, 0, 4], [-10, -1, 2, -1, 4, -7, 5, 6, 7], [7, 9, -7, 1, 6, 10, -12, -2, -6], [-5, 4, -8, 7, 3, 6, -11, 5, 1], [14, -4, 2, -1, -3, 27, -18, -5, -1], [8, -11, 16, -3, -6, -4, 1, -4, 5], [-3, 1, -1, 2, 0, -5, 2, -5, -1], [3, -1, 7, 3, 5, 5, 0, 0, 2], [-5, -1, -1, 7, 9, 12, -6, -2, 8], [10, 2, -6, 11, 3, -5, 4, 1, 1], [4, 7, 5, 4, -7, 8, 13, 7, -6], [2, 0, 0, -3, -5, -9, 8, -3, -13], [-11, -4, 3, 1, -6, -14, -6, -3, -2], [3, 7, 2, -4, 2, -13, 5, -2, 1], [-16, -10, 8, 8, -4, -14, 6, 2, -10], [7, 3, 5, 6, -4, 5, 14, 1, -1], [2, -14, 9, 1, -8, 5, -8, -7, -2], [-7, 7, -5, -4, -5, -1, -2, -3, 3], [-8, 7, -4, -3, 2, -7, 11, -4, 7], [-1, 2, 9, 5, -6, 6, -6, 1, 2], [0, -3, 4, -5, -12, 11, -11, -4, -1], [4, 0, 0, 4, -4, -4, 4, -8, -6], [-6, -1, -6, -4, -2, -18, -7, 1, 6], [0, 6, -2, 8, -2, -6, 8, 8, -6], [-16, -3, 2, 5, 0, -3, -13, 4, -5], [-8, 9, 0, -8, 3, 2, -11, 3, 2], [14, 5, 9, 5, 2, 21, -8, 9, -7], [2, 20, -4, 0, 0, -1, 6, 2, 4], [-13, -4, 5, -10, 0, 1, -5, 2, 1], [-1, 5, 1, 4, 12, 15, 3, 3, -5], [17, 3, 2, 3, 5, 10, -13, -5, -6], [3, -3, 3, -1, 5, 10, -6, 0, -4], [4, -2, 0, -5, 9, 5, 8, 1, -5], [-14, -1, -6, 2, 1, -6, -1, 5, -2], [22, -20, 7, -12, -8, 2, -11, -5, 10], [19, -10, 5, -1, 0, 10, -15, 11, -4], [-11, 0, -9, 6, 5, 5, 11, 0, -11], [-11, 3, -5, -10, 10, 7, -2, 1, 9], [21, 2, 0, -13, -6, 4, -6, -2, 10], [15, 4, -2, -14, -5, 5, -8, 3, 1], [12, -4, -4, -3, -7, 5, -20, 3, -7], [7, -10, -1, -2, 3, -15, 1, 0, 1], [15, 8, 1, 0, -5, -1, -5, 0, -11], [2, 17, 0, 7, 0, 19, -11, 6, -3], [14, 0, 10, 4, -16, 10, -10, -2, -8], [22, -2, -4, -5, 9, 7, -14, 5, 3], [0, -9, 12, 13, -8, -1, 11, 4, -9], [19, -11, 9, -1, -7, 11, -14, -8, -8], [9, -2, -1, -6, 2, -11, -5, 2, 7], [6, 5, -5, -2, -5, 0, 0, 2, 4], [-12, 1, 4, -6, 3, -18, 15, -3, 8], [14, -4, -1, -4, -2, 0, -6, -10, 12], [13, -6, -1, -12, 7, 17, -8, 0, 9], [12, -13, -1, -5, -7, 1, -7, -2, 10], [14, -7, 1, -2, 3, 10, 2, 4, 2], [6, -9, 2, -9, -8, -15, 15, -2, -1], [-2, 10, -4, 7, -7, 1, -2, -1, -9], [0, -4, 0, -4, -4, -10, 13, 2, 6], [-6, 11, 4, 5, 4, 1, 19, 4, 3], [-24, -20, -6, -1, 5, -15, 10, 3, 9], [11, 1, 5, -5, -1, 0, -4, 10, 2], [7, 3, 1, -13, -5, 3, -4, -2, 0], [17, -8, -2, -8, -9, 11, 0, -9, 11], [-11, 21, -2, 4, 4, -7, 7, 0, 5], [-4, 7, -11, 8, 3, -4, 12, 4, 10], [-4, -8, -5, -8, 7, -10, 0, -4, 1], [5, 12, -1, 6, -7, 3, -18, -8, 1], [-15, 22, -17, 1, 8, -9, 7, 5, 0], [-13, 15, -17, -5, 5, -2, 2, -4, 4], [-8, 11, -14, -2, 1, 10, -5, -11, 4], [4, -20, 13, -4, -12, -14, -9, -7, 4], [28, 3, -4, 8, 5, 10, -7, 1, -8], [2, -2, 10, 4, -1, -20, -2, -2, 0], [28, -4, 5, 0, 0, 6, -1, 5, 8], [-6, 11, -13, 7, 8, 21, -1, -2, -1], [-11, 3, 1, 0, -6, -5, 14, 1, -1], [-7, -4, -13, 8, 13, -11, -16, -2, 1], [-12, 0, -8, -3, 8, -1, 4, 7, 5], [-11, -8, 1, 13, -2, 6, 7, -1, -4], [-1, -1, 1, 3, -7, 2, 6, -6, 8], [13, 11, -9, 8, 4, 1, -8, -11, 9], [0, -6, -14, -6, 6, -6, -12, 12, 8], [8, -10, -10, 8, -6, -6, -8, -8, -6], [2, 5, 0, 11, 8, 19, -11, 4, -7], [8, 3, 1, 3, -6, -21, -3, -6, 5], [1, -17, 22, 4, -9, -5, 10, 7, -8], [-11, -8, -1, 11, 7, -14, 3, 7, -4], [9, 7, 5, -3, -5, 4, -18, 4, -6], [27, -3, 9, 3, -3, 4, 6, 2, -2], [2, -17, -2, -11, -6, -1, -10, -2, 7], [11, -8, 1, -5, -9, 10, -11, -15, -2], [-9, 5, 3, 7, -3, 6, -12, 10, -4], [2, -7, -4, 13, -6, -21, -7, -12, 1], [8, 6, -4, 2, 10, -8, 18, -2, 8], [-5, 20, -5, -3, 6, 0, 7, 5, 2], [-12, -13, 6, -7, -2, 1, 17, 8, -7], [-7, -8, 6, 3, -8, -20, 27, 3, -10], [1, -13, 6, -6, -6, 5, 0, -5, -11], [11, 3, 3, -14, -2, 7, -16, -5, 13], [24, -1, 12, 11, -18, 11, -16, -4, -9], [-21, 11, -15, 1, 5, 0, 1, -4, 10], [6, 10, 4, 2, 0, -16, 12, 10, -6], [11, -7, 5, -4, -2, -3, -22, 3, 5], [27, -14, -3, 0, -11, 3, -19, -6, 7], [5, -5, 10, 4, -11, 13, -8, -3, -2], [-21, -6, 0, -2, 5, -25, 25, 2, 7], [-5, 2, 0, 8, 15, 15, -1, 8, -7], [-6, -1, 6, -4, -9, -2, -12, 3, 8], [-5, 10, 0, 15, 0, -8, 14, 4, -6], [-17, 5, -2, 4, 9, 1, 12, -1, 6], [18, -19, 13, -12, -12, 14, -25, -7, 13], [16, 4, -2, 11, 9, -15, 18, 7, 3], [-11, -6, -7, -10, 13, -9, -7, -10, 5], [-8, 2, -4, 13, 9, -5, 4, -5, 1], [-20, -6, 0, -3, 5, -25, 22, -3, 7], [0, -16, -6, 12, -10, -18, 8, -6, -2], [-2, 2, -10, -7, 9, -7, -20, 5, 3], [28, 7, -8, 7, 9, 11, 1, 6, -3], [7, -11, 9, 6, -18, 1, 2, 1, 3], [16, 8, -9, 6, -2, 20, -7, 1, 2], [37, 3, 2, -14, -9, 11, 2, -7, 2], [-4, 10, -6, -6, 2, -9, -6, 6, 0], [21, -17, 7, -1, 3, 8, -13, -6, 4], [-23, 13, 3, 14, 9, 13, -8, 1, 1], [-4, -20, -6, 6, 0, -24, 14, -2, 4], [-21, 5, -11, -1, 13, 6, -7, 4, 12], [36, -2, 4, 1, -7, 13, -32, 1, 1], [15, -4, -7, 13, 6, -12, -3, 1, -14], [-29, -16, 8, -17, 4, -22, 28, -8, 10], [13, 5, -5, -2, 12, 20, -18, 5, 7], [11, -6, 4, -14, 5, 15, -23, -8, -1], [3, 3, 6, -11, -7, 8, 20, -6, 10], [-2, -6, 8, -6, -6, 16, -6, -8, 10], [21, 5, 11, 5, 4, 12, -14, 6, -2], [-19, -2, -8, -6, 5, -15, 8, -5, 11], [-16, -26, 13, 8, -7, -16, 14, 6, -5], [-5, -5, 8, 9, -8, 20, 2, -8, -1], [-8, 19, -19, -2, 7, -12, -9, 5, -4], [-3, -7, 7, -1, -5, -6, -12, -10, 10], [-10, 9, -14, 4, 7, -6, -11, 7, -8], [-13, -3, -12, -4, 4, -11, -16, -15, 9], [-9, -7, -9, 2, -2, 7, -2, 7, 5], [0, -1, -8, 3, -4, -1, -19, 4, -7], [5, 1, -5, -8, -3, -7, 12, -7, 5], [17, -13, 8, -9, 0, 10, -20, 2, -7], [-7, -16, 15, -10, -7, -19, 15, 4, -5], [15, 10, -6, 1, 10, 16, -9, 13, -6], [18, 6, 4, -5, -9, 37, -8, -1, 1], [6, 1, 6, 3, -12, -3, -7, 6, -13], [7, 6, 5, -2, -3, -1, -11, -8, 1], [11, -11, -9, 0, -2, -21, 12, -13, 13], [-6, 13, -4, 9, 2, 11, 13, 0, 11], [20, 0, -10, -14, 2, 4, -2, -6, 4], [-2, -13, 12, -8, -11, 0, -13, -3, 4], [6, -11, 4, -1, -12, -30, 7, -2, -7], [-26, 11, -2, 7, -2, -11, 7, -8, -1], [-2, -2, 6, 9, -11, -9, -5, 9, -9], [-16, 2, -8, 17, 11, -23, 15, 1, -7], [-7, -6, -3, -12, -5, 3, 9, 8, 1], [6, 3, -4, -3, 0, -11, -11, -2, -5], [23, -1, 13, 4, 0, 1, -18, 5, -1], [-13, 16, 3, 6, 1, -15, -7, 16, -1], [-13, 10, -1, 1, 0, -10, 15, 11, -6], [-1, 2, 8, 5, -9, 0, 23, 1, 3], [-7, 1, 3, -12, -11, -5, 0, 3, 5], [-18, -12, 14, 6, 0, 16, 3, -6, -8], [4, 24, -6, 3, 7, 25, 4, 7, 5], [7, 2, 9, 7, -4, 26, -23, 13, -2], [-26, 3, 3, 0, 7, 0, 17, -3, 0], [7, -19, 1, -16, -4, 3, 2, 9, 1], [-12, 7, 2, -12, -5, -16, 13, -15, 14], [46, -7, 4, -12, -5, 2, -23, 3, 4], [-12, -4, 10, -12, -2, -12, -20, -4, 12], [-20, 4, 0, 5, 7, -25, -1, -3, -11], [22, -1, 2, 1, 1, 25, -17, 2, -3], [-14, -13, -10, -7, -10, -17, -3, -2, 15], [-8, 3, 4, -4, -6, 12, 3, 9, -14], [27, 11, 7, -5, 3, 24, -8, -2, -4], [37, 0, -1, 6, 1, -1, 6, 8, -7], [4, 18, 0, 16, -10, 10, -16, 8, -14], [30, -11, 0, -11, -4, 35, -19, 4, 5], [-12, -3, 2, 9, -8, -21, 13, 2, -13], [18, 4, 2, 13, 4, 3, -18, 9, -13], [24, 2, 8, -1, 17, 15, 2, -1, 9], [29, -11, 7, 0, -26, 2, -14, -5, 1], [5, -9, 6, -4, -14, 17, -23, -14, 5], [-3, -5, -3, -1, -15, 6, 0, 2, 2], [-2, 8, -18, -3, 15, 1, -14, -19, 9], [-12, -21, 17, -1, -11, -25, 15, -12, 6], [7, -10, 3, 0, -7, 11, -17, -8, 15], [-5, 8, -17, -7, 22, 16, -27, -7, -4], [-13, 6, -11, -20, 5, -5, 7, -6, 3], [19, -12, 15, -6, -3, 17, -7, -6, -1], [-13, -1, -19, -2, 2, 1, -8, -5, -1], [1, 5, -7, -12, 0, 15, 16, 3, 1], [-39, -10, -3, 4, 5, -33, 27, -10, 1], [1, 14, 14, 1, -4, 4, 23, 3, -4], [8, 12, 0, -8, -12, 24, 2, -14, 4], [18, -2, 6, 12, -10, 12, 2, -4, -6], [5, 12, -17, -3, 20, 6, 8, -3, -12], [2, 13, -10, -10, 15, -3, 9, -1, 2], [-4, 4, 16, -9, 4, 5, 20, 9, -9], [-4, 18, -2, 2, 12, 30, 2, 6, 4], [-2, -2, -6, 2, -10, 12, -18, -6, 12], [20, 0, 18, -1, -29, 13, -22, -7, -13], [-22, 13, -12, 10, 23, -2, 21, 13, -8], [23, 26, 1, 1, -2, 12, 1, 7, -8], [-1, -1, -7, 0, 8, 3, -12, 5, -7], [-24, 13, -10, 17, 8, -19, 19, 8, 1], [16, -1, 15, 9, -17, -25, -1, -6, 2], [-4, -3, 0, -13, 8, 15, -15, -12, 11], [-13, -6, -14, 14, 8, -25, 9, -6, 2], [-16, -24, 14, 1, -5, -17, 16, 13, -7], [23, 5, -5, -1, 13, 10, -18, 4, 0], [8, 9, 0, 1, 16, 7, 21, 8, -9], [-3, 0, 9, -11, -16, -4, -1, 13, 8], [-18, 6, 1, -4, -12, 6, 22, -12, 0], [-19, 2, 19, -1, -4, 10, 23, -1, 10], [-21, 4, 7, -12, 15, -18, 33, 4, 3], [-1, -4, 5, -31, -8, 4, -4, -7, 8], [10, 8, -10, 6, -2, 16, 24, 0, 16], [9, -11, -2, 12, 10, 21, -16, 5, -1], [-20, 5, -12, 3, 12, -5, 1, -8, 13], [-5, -6, -3, 4, -11, -13, 19, 16, -9], [-10, 0, -9, 1, -3, 11, -8, -7, -7], [19, 0, 3, -10, -3, 31, -37, -14, 13], [11, -4, 13, 12, 5, 1, -31, 2, -3], [0, -8, -12, 4, -10, 10, 6, 0, 14], [-9, 7, -12, 13, 1, 2, 5, 1, -14], [16, -9, 0, 2, 3, -12, 17, -1, 14], [-6, -15, 16, -3, -10, 9, 1, -4, -11], [-18, -3, -2, 5, -4, -5, -23, -10, 9], [-32, 5, -4, -16, -1, -6, -1, -5, 20], [-2, -23, -2, -8, 1, 7, 1, 3, 22], [10, 5, -3, 0, 13, 32, -22, 2, 4], [-3, 12, 3, 1, 10, -12, 3, 11, 0], [-18, -6, 14, -4, 14, 0, 28, 20, -12], [11, -9, 17, 9, 3, 14, -10, 2, -4], [-19, 22, -4, 17, 4, 4, -5, 1, -12], [2, -4, 6, 3, 7, 19, 13, 3, 1], [2, -6, 2, -11, -1, 9, -14, 25, -7], [7, -9, 17, 29, -9, -2, 10, 2, -2], [-13, -12, -11, 23, 8, 10, -11, 3, -4], [36, -23, 12, -11, -2, 3, -12, -6, 11], [-38, 1, -22, 15, 32, -5, 16, 0, -3], [-23, -14, 16, -12, -3, -17, 24, -13, 3], [17, 7, -7, 17, 1, 22, 0, -8, -8], [-9, 0, 5, 1, -6, -6, 18, -1, 12], [5, 9, -10, 13, 5, 14, 9, 3, -14], [2, -8, -3, 11, -6, -31, 14, -11, 0], [-4, -1, 17, -11, -2, -7, -2, -3, -7], [11, -15, 1, -9, -17, 20, -36, -14, 16], [0, 17, -20, 16, 27, 4, 16, 9, -22], [-13, 13, 13, -2, 8, -5, 19, -5, -7], [7, -14, -17, 6, -7, -13, 1, -22, 7], [-19, -5, -2, -5, -3, -8, -2, 0, -6], [-6, -5, -4, -5, 2, -17, -19, -4, 11], [20, -6, 0, 7, 10, 27, 6, 13, -3], [-24, -7, 8, 15, -10, -23, 5, 6, -13], [29, 10, 10, -18, -5, 37, -37, 0, 5], [19, 15, -1, 13, -1, -8, -4, -6, -8], [-4, 11, 8, 9, 6, 1, 1, 2, 9], [3, 10, 5, 25, 8, -2, 21, 11, -12], [-4, 25, -12, 25, -2, 19, -8, -2, -9], [33, -27, 11, -14, -20, 7, -24, -5, 15], [4, 20, -20, 0, 6, -4, 2, 2, 10], [-11, -3, 5, -2, -4, 3, -12, -1, -7], [16, -1, -8, 9, 10, 1, -7, -6, 17], [17, 15, -9, 29, 1, -2, 4, 6, -4], [8, -14, 7, 4, -17, -16, -2, 2, -15], [16, -10, -7, -17, -1, 1, -2, 5, 13], [-7, -15, 11, 13, -9, -16, 26, 12, 4], [-2, 15, -9, 1, 3, 9, 19, 0, 16], [29, -27, 20, 3, -9, 12, -8, -2, 2], [1, -14, -5, 7, 2, 29, -29, 5, 8], [-10, 12, 2, -11, -9, -11, 1, -7, 17], [9, -4, 6, 20, -21, 7, 1, -12, -9], [1, -10, -7, -5, -6, -26, 21, -19, 12], [-29, 9, -5, -15, 9, -34, 10, 4, 14], [0, -11, -6, -10, -21, -8, -15, -17, 8], [10, -9, 18, 9, 0, 49, -9, 2, -7], [-16, 18, -10, -1, 1, -1, 36, 1, 1], [11, -12, 22, -24, -15, 1, 3, -12, 1], [18, 4, -7, 1, 9, 5, 2, 11, 9], [13, 11, -16, 9, 16, 14, -32, 8, 11], [9, 6, -10, 7, 11, 16, 15, -7, -7], [-37, 12, -15, 10, 7, 7, 11, -10, 9], [-33, -10, -9, 0, 17, -2, 23, -10, 11], [41, -12, 3, 3, 6, -20, 1, -7, 24], [-50, -2, 10, 4, -4, -6, 8, 2, 4], [7, -26, -1, -8, 3, -15, 5, 2, -3], [5, -10, 3, -6, -4, 15, -5, 16, -1], [-2, -7, -10, -3, -10, -11, -25, 2, -11], [14, 7, -2, 11, 10, 19, -24, 22, -7], [6, -8, 17, 5, -3, -23, -10, 11, -15], [24, 0, 8, 13, -3, 11, -16, 1, 3], [-24, -6, 11, -11, 2, -21, 10, 9, -8], [-2, -11, -9, -16, -1, -6, 12, -22, 24], [-17, 11, -9, 10, -10, -11, 2, 3, 3], [-18, 8, -10, -6, 4, -26, 26, 4, 2], [-15, -2, 15, 6, -1, 3, 19, 12, -27], [-22, 2, 7, 14, 6, 2, 16, -2, -18], [3, -3, -23, 5, 5, -16, -14, -10, 12], [-5, -5, -9, -21, 11, -30, 10, -6, 24], [-36, 2, 11, -4, 14, -24, 39, 13, -6], [-24, -10, 16, -10, 2, 14, 4, 4, -16], [20, -23, 0, 11, -2, 1, -5, -2, 3], [-27, 21, -31, 2, 14, -16, -10, -11, 3], [5, -6, -13, 4, 5, -7, -39, -12, 13]];
    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_bf();".
// 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_bf(:prec:=9)
    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_bf();".
// 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_bf( : 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, 6, 12, -41, -18, 37, 8, -11, -1, 1]>,<5,R![-169, -131, 255, 249, -83, -141, -20, 22, 9, 1]>],Snew);
    return Vf;
end function;