
// Make newform 3549.2.a.ba in Magma, downloaded from the LMFDB on 19 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_ba();".
// 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_ba();".
// 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 := [61, 64, -96, -72, 57, 22, -13, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-29, 16, 64, -56, -27, 21, 3, -2], [8, 4, -7, -1, 1, 0, 0, 0], [16, -81, -64, 69, 27, -21, -3, 2], [-55, 68, 77, -69, -27, 21, 3, -2], [-55, 172, 129, -160, -40, 34, 3, -2], [-200, -112, 202, 80, -58, -17, 5, 1]];
    Rf_basisdens := [1, 1, 13, 1, 13, 13, 13, 13];
    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], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 0, 0, -1, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, -1, -2, 0, 1], [-3, 1, -2, -1, 2, 0, 0, -1], [0, -1, -1, 0, 0, -1, -1, 0], [2, 0, 2, 2, -2, 2, 0, 1], [-2, 2, -2, -1, 1, 0, -1, 1], [0, 0, 1, 0, 0, 0, 2, -1], [2, 0, 1, 1, -1, -3, 0, 1], [0, 3, -4, -2, 1, -1, 0, -2], [1, -1, 2, 0, 1, -1, 0, 3], [0, 0, 2, 2, 2, 3, 0, -1], [-6, 0, 0, 1, 1, 1, -1, 1], [2, -1, 1, 1, -3, -1, 0, 2], [3, -2, 1, 0, -3, 1, 0, 0], [-4, 0, -3, -1, 0, 0, -1, 0], [1, -1, -2, 3, -1, 0, 0, -1], [5, 0, -3, 1, 0, -1, 0, -2], [-7, 2, 0, 3, 1, 2, 0, -1], [0, -1, 2, 1, -2, -3, 0, 4], [-2, 2, 2, 1, -1, 2, -1, 1], [-3, 3, -3, -6, 0, 0, 0, -1], [0, 3, -1, 3, -1, 1, -1, 4], [-4, 3, -3, -2, 2, 0, 2, -3], [1, 0, -1, 0, 1, 1, -4, 0], [2, 1, 1, 0, 0, 0, 0, 6], [4, 0, -1, -1, 0, -1, -3, 0], [-3, -2, 1, -1, -2, -2, 1, 1], [-5, 4, -4, -1, 0, -1, -2, 3], [-4, 4, -1, -4, 3, 2, 0, -1], [-2, 1, 3, 5, 2, 2, -1, 4], [2, -4, 2, 2, 2, 0, 0, 0], [-7, 0, 0, 3, 1, -1, 0, -2], [-6, 2, 1, -2, 3, 1, 2, -3], [-8, 3, -4, -5, 4, -1, 0, -2], [-9, 1, 1, 4, 2, 6, -1, -3], [6, -6, -2, 0, -2, -2, 1, -3], [-5, -3, 0, 2, -1, 0, 1, -3], [4, -3, -1, -2, 0, -3, 1, 2], [1, 2, -2, 0, 5, -1, 0, -1], [-6, 4, -1, 3, 0, 0, 5, -2], [-1, 0, 1, 5, -2, -5, 3, -5], [6, -5, 0, -6, 1, -1, 0, -2], [-3, 5, 0, 3, 0, 0, 2, -1], [-3, -1, 2, -6, -3, -1, 0, 3], [2, 0, -1, -5, 2, 3, -2, 1], [4, -4, 4, 0, -2, 1, -3, 5], [-14, 2, 0, -2, 2, 2, 0, -6], [2, -6, 1, 0, -4, -1, 3, 2], [-2, -1, 2, 0, 1, -2, 2, -2], [-8, 2, -2, -5, 5, 0, 3, -6], [2, -5, 5, 2, -6, -2, 6, -1], [4, -5, 0, 0, -1, -6, 0, 0], [-12, -6, 1, -4, 3, 3, 1, -2], [4, -4, 2, 0, -2, -2, -5, 2], [-7, 3, -4, -1, 1, -2, 1, 0], [0, 2, -5, -4, 5, 2, 0, -1], [-3, 0, -1, -2, -2, -1, -2, 2], [-3, 11, -3, -3, 5, 1, -1, 1], [5, -8, 1, 2, -1, -2, 3, -1], [-9, 1, -3, -2, -2, 0, 2, 0], [-2, 5, -8, -10, 6, 0, -2, -4], [-7, 0, 5, 8, -7, 1, 2, 4], [9, -4, 3, 1, -2, 3, 1, 0], [4, -4, 10, 2, -2, 4, -1, 5], [-1, -1, 1, 3, -5, -7, 1, 1], [-7, -1, 5, -1, -4, 0, 4, 0], [2, -2, 0, -2, 0, 0, -4, -2], [1, -9, 4, 2, -1, 4, -4, 3], [6, -1, 0, 8, -1, -1, 1, 0], [0, -2, 2, -2, -2, 0, -4, 2], [4, -2, -6, -4, -2, 0, -2, -4], [2, 2, -8, -4, 6, 6, -7, -2], [5, 3, -4, 0, 7, 0, 3, -2], [-1, -1, 5, 2, 1, 6, 0, 6], [-5, -1, -4, -6, 0, 4, 3, -8], [-9, 6, -5, 3, 8, 4, 1, -1], [0, -4, 0, -4, -5, -3, -4, 4], [-7, 1, -1, -10, 6, -1, 3, -3], [-8, 3, -5, 1, 7, 5, 2, 0], [-6, -2, 7, 2, 3, 10, -2, 3], [12, -2, 2, -2, -4, -1, -1, 3], [-4, 1, 4, -4, 3, -3, 3, -1], [5, -12, 6, 6, -5, -1, -4, 3], [-1, 3, 5, 7, -4, 10, -2, 0], [11, 6, 3, -1, -4, 2, 0, 2], [-6, -9, -4, -1, 2, -2, -2, -6], [-10, -1, -6, 2, 1, -8, 1, -5], [1, -2, 7, 0, 3, 1, -2, -2], [4, 0, 2, 0, -8, -6, -1, 5], [-5, 0, 0, 4, -2, -2, -4, 7], [-13, 3, -3, -5, 7, 6, 1, -7], [-16, 0, -3, 3, 3, 5, -2, -3], [2, 3, 8, 7, -4, 2, 3, 4], [1, -2, 10, 5, 1, 7, 1, 0], [7, -2, -6, -7, 0, -8, 2, 3], [-5, -1, 5, 3, 5, 4, -2, 8], [5, 3, -5, -8, 9, 4, -4, -2], [5, 0, 9, -4, -7, -2, 3, 1], [8, -5, 7, 4, -4, 3, -5, 6], [11, -1, 7, 3, -1, 0, -6, 14], [0, 6, -10, -4, -1, -2, 2, -8], [-7, 3, -14, -8, 3, 1, 0, -1], [1, -10, 7, -1, 1, 2, -1, 7], [-4, 4, 0, -12, 6, -1, 5, -7], [-11, 1, 3, 4, -4, 0, -1, -2], [-8, -3, 8, 1, 0, 3, 5, 1], [8, 0, 1, -2, 0, -2, -2, 7], [4, -1, -5, -5, -4, -6, -3, 0], [-12, -6, -2, 4, -6, -4, 4, -10], [-9, -2, 16, -2, -6, 0, 2, 5], [-16, 11, -7, -14, 8, 3, -4, -4], [-1, 5, -8, -7, 8, -4, 0, -5], [2, -5, 0, 1, -8, -2, 0, 0], [-12, 6, 0, 2, -2, 3, 1, -7], [-6, -3, 13, 2, -6, 4, -1, -2], [0, -2, 1, 8, -8, -3, 1, -2], [4, 1, 0, -2, -1, 9, -3, 2], [2, -5, 2, 3, 0, 2, 0, 2], [1, -10, 4, -1, -9, -6, -3, 8], [-18, -4, 1, -3, -6, -1, 2, -1], [-4, -8, 6, 0, -4, -4, 1, 5], [7, 8, 1, 4, 0, 4, 2, 6], [8, 0, 4, 1, 5, 2, -4, 12], [-6, -4, 7, -6, -3, -2, 0, -1], [10, -10, 2, 1, -4, -4, 5, -1], [-19, 0, -5, -1, -8, -6, 4, 0], [7, -3, 6, 7, -8, -3, 0, 7], [-2, -3, 3, -7, 3, -4, 7, 0], [-10, 5, -14, -6, 1, -9, 8, -10], [-5, 1, 0, 2, -7, 3, -4, 7], [0, 4, 2, 4, 2, 14, -4, 2], [-3, -3, 4, -3, -2, -4, 1, 6], [5, -1, -5, 1, 7, -3, -7, 5], [-22, -2, 4, 1, -5, 2, 0, 2], [16, 0, 0, -6, 10, 1, 1, -2], [-23, 9, 1, -4, 12, 5, 2, -10], [-16, 6, -8, -8, 10, 0, 2, -2], [-3, -4, -4, 18, 2, 2, -5, 0], [-14, 9, 1, 7, 3, 5, 0, -9], [4, -16, 12, 4, -2, -6, 1, 2], [-5, -2, -2, 2, 4, -5, -1, -2], [23, -3, 6, 3, -13, -8, 0, 5], [26, -9, 0, 3, -8, -16, 3, 4], [25, -2, 0, -9, 1, 1, 1, 6], [7, 0, 2, 1, -15, -10, 3, 6], [-5, 18, -10, -9, 6, -2, -2, -5], [4, 7, -6, -9, 6, 6, -3, 6], [7, 0, 5, 1, 0, 7, 0, -2], [0, 4, -8, 4, -8, -2, -4, -4], [20, -4, -5, -5, 0, -15, 3, 2], [-10, 0, -4, 2, 4, -1, -3, 5], [-3, -2, 1, -9, 4, -5, 2, 4], [10, -3, 3, 7, -4, 7, -3, -2], [-4, -18, 6, 10, -12, -8, 5, -4], [11, 0, -4, -7, 7, -6, -4, 5], [7, -1, 7, 2, 0, -9, 3, -1], [12, -2, 8, 2, -4, 1, 4, -4], [-6, -4, 5, -4, 5, 5, 2, 5], [-3, 9, -9, -3, -3, 4, -3, -3], [-9, 8, 5, -8, 10, 1, 1, -5], [-4, 15, -10, -11, 12, 7, -2, 2], [2, 13, -5, -11, -1, 6, 1, -4], [-20, 0, -4, 14, 8, 4, -1, -2], [-8, 4, -1, 8, 3, 5, 0, -9], [3, 2, 7, -2, -7, -6, 2, 7], [-25, -1, -4, 9, 10, 1, -2, -1], [-23, 5, 0, 7, 11, -2, -2, -5], [-16, 12, -14, -6, 4, -3, 4, -6], [16, -4, -1, -1, -7, 3, -2, 9], [-8, 3, 0, 3, -8, 1, 5, -1], [0, 12, 8, 16, 6, 18, -1, 4], [-14, -2, -10, 1, 7, 3, -4, -8], [5, 0, -1, -6, -10, -3, -3, 7], [-14, 4, -2, 12, 2, 14, -4, -4], [-5, 9, 0, 8, -1, -2, 6, -3], [-13, -6, 12, 4, -8, 0, 8, -1], [-10, 7, -6, -7, 2, -4, -2, 6], [-7, 4, -8, 2, -2, 10, 2, -2], [-9, 16, -2, -1, 11, 8, 8, -13], [-5, -14, 14, 1, -3, 4, 3, 2], [10, 3, -6, 1, 4, 2, 7, 6], [2, -5, -10, -6, -1, -7, 6, -12], [16, -14, -2, -6, -16, -20, -2, 0], [-2, 14, -3, 7, 7, 13, 2, -7], [-12, -2, -6, 9, -1, -1, 5, -9], [-1, -4, -4, -13, -3, -5, -4, 6], [-15, 11, -10, -13, 10, -1, 0, 1], [8, 2, 2, 0, -4, 2, -8, -2], [-1, 7, -4, -14, 3, 1, -6, -3], [-2, -3, 11, -8, 4, 7, 0, 1], [8, 6, -6, 2, -2, -8, 0, -8], [22, 0, 7, 12, -10, 2, 2, 11], [-9, -7, 0, 4, -7, -3, -6, -4], [-8, 0, 3, 2, 6, 8, -5, 4], [20, 9, -4, 1, -2, -4, 0, 6], [-13, -5, 6, -7, 9, 14, -3, -4], [-14, -10, 8, 2, -16, -8, 10, 2], [26, -4, -1, 4, 6, -7, -9, 16], [-7, -13, 8, 0, -5, -4, -1, 9], [-7, 1, 10, -5, -8, 2, -5, 10], [6, -8, 8, -14, -4, 0, 1, -2], [-20, -6, 9, 3, 8, 17, -4, -5], [-2, 4, 4, 6, -8, -12, 9, 5], [-11, -4, -7, 0, 0, -9, -11, 5], [4, 0, 10, -2, 0, 8, 0, -2], [-8, 10, -3, 3, 8, 9, 12, -10], [-5, 6, -5, 14, -1, -8, -3, 3], [26, -6, 12, 12, -14, -4, 0, 16], [-13, -9, 8, 6, -3, 1, -6, -7], [3, -6, -4, 8, 12, 5, -1, 0], [-8, 4, -2, 3, 2, -2, 3, -17], [10, -7, 2, -10, -11, -15, 4, 4], [13, -2, 7, 5, -4, 6, -4, -3], [10, -15, 10, 4, -3, -1, 5, -11], [-4, 1, -9, 7, 7, 7, 0, -21], [2, 4, -12, -8, 4, -18, 3, -4], [-5, 7, -16, -3, 4, -8, -6, 5], [9, 4, -5, -4, 2, -17, 1, -5], [0, 10, 1, 11, -6, -6, 6, -1], [16, 1, 8, 3, -12, -2, 4, 4], [3, 7, -9, -13, 7, -10, -5, 4], [12, 12, -8, -9, 9, 2, -3, 8], [0, 1, -9, -8, 14, 0, -3, 8], [7, -17, 11, 0, 0, 14, 6, -2], [-1, 7, 17, 12, 4, 11, 1, 7], [6, -2, -6, -6, -10, -6, 7, -2], [-22, -2, 0, 0, 6, 14, 2, -4], [2, -12, 15, 7, 2, 18, -4, -3], [6, 12, -8, -4, 4, 0, 3, -10], [-13, 6, 2, -11, 9, -5, 3, 8], [-7, -4, -2, -12, -4, 3, -1, 4], [23, -13, 11, 3, -11, 1, 3, 9], [-8, 3, -2, 2, 0, 6, -14, 0], [-10, -4, -12, -14, 6, -3, -2, 4], [1, -1, 1, 15, 7, 6, 5, 3], [-4, -11, -3, -3, -13, -9, 10, -5], [-5, 12, -1, 13, 16, 13, -13, 3], [-3, 23, -9, -18, 14, 6, -6, 2], [8, -14, -2, 1, -10, -6, -4, 2], [-14, 4, 7, -4, 4, -4, -7, -4], [13, -7, 3, 17, 1, 0, 7, -5], [14, 14, -9, 6, 1, 15, -1, -4], [11, -7, -4, 8, 3, 7, -2, -11], [-22, 0, -9, -8, 1, 1, 10, -9], [-8, 7, 1, -2, 16, 7, -3, -6], [-14, -6, -4, 8, -2, 10, 2, -8], [5, 4, 12, 12, -4, 3, 7, -2], [-11, 0, -16, -3, -11, -8, -7, 4], [8, 0, -9, -2, 15, 2, -7, 7], [3, 17, -11, -1, 5, -2, -3, -7], [-23, -8, 7, 6, -13, -14, 6, 4], [26, -4, 4, -6, -2, 4, -6, 10], [-5, 3, 3, 11, 1, 11, 3, 5], [-14, -10, 10, 17, -3, 12, -2, -2], [-10, -4, -7, -11, -1, -9, 6, -5], [24, -3, 9, -1, -1, -4, 7, 10], [16, -5, 7, 6, -2, -4, 5, -12], [-19, 19, 1, 3, 14, 7, 7, -11], [-13, 0, 5, -13, 6, 3, -4, -3], [-29, 9, -9, 1, 6, 2, -2, -14], [-9, 6, -3, 3, 18, 19, 5, -9], [-9, -3, 17, 2, -12, 1, 7, 1], [13, -11, 17, -1, -16, -2, 2, 2], [-10, -4, 8, 13, 11, 11, -6, 4], [-14, -14, -2, 2, -8, -5, 6, -3], [22, 0, 14, 14, -6, 6, 1, 10], [-17, 20, -7, 4, -1, -5, 4, -8], [5, -1, 8, 16, 8, 4, -5, -2], [-20, -1, -3, 13, -5, -2, -3, 6], [6, -3, 3, -11, 3, -9, 5, 12], [25, -21, 6, -6, -13, -12, 5, -14], [1, 9, 7, 6, 0, 13, -1, 9], [15, 8, 4, 1, 4, 2, -8, 15], [-11, 4, -14, -3, 18, -8, -2, -7], [12, -2, 8, -19, -5, 4, 0, 4], [-33, 9, -1, -1, -1, -4, 7, -5], [-27, 14, -3, -1, -2, 10, -4, -2], [-4, -9, 2, 9, 2, 8, -8, 4], [1, -7, 8, -2, -6, 0, -1, -2], [-16, 18, -16, -9, 11, 10, 1, -9], [24, -14, 2, 10, -16, -11, -5, 9], [17, 2, -1, 9, 0, -3, -1, 14], [36, 6, 6, -6, -5, -7, 5, 7], [-1, 13, 4, -10, -1, 2, 9, -2], [-3, -15, -3, -6, -2, 0, 2, 3], [4, -9, 5, -8, 6, 13, -7, -2], [15, -19, -7, -4, -14, -12, 7, -9], [6, 4, -5, -13, 1, 5, 0, 9], [-14, 22, -6, -13, 5, -1, -3, -7], [11, 0, 18, 2, -12, -4, 7, 2], [-3, 7, -11, 15, -7, -10, -4, 3], [19, -3, 5, -8, -8, -4, 6, -8], [-16, 4, -8, -6, -12, -12, 9, 4], [13, -9, 1, -1, 11, 2, -9, 3], [11, 1, -4, 9, 12, 10, -4, -5], [-3, 4, 1, 3, -6, 3, 10, -11], [-16, 28, -17, -17, 17, 5, -2, 5], [11, -10, 20, 26, -8, 14, 3, -6], [-4, -4, 20, 8, -4, 4, 2, -6], [-32, 8, -9, -13, 12, -1, 1, -8], [31, 2, 2, -10, 11, 11, 0, 1], [12, -5, -3, 8, -2, -5, 9, 0], [-11, 11, 6, 0, 9, 0, 11, -17], [11, -4, -3, -8, -11, -2, 6, -8], [9, 16, -10, -2, 0, 6, -2, -4], [23, -1, 3, -10, 0, -2, -13, 1], [21, -4, 0, 5, -15, 8, -9, 2], [12, 10, -14, 20, 6, 1, -6, 1], [-12, 3, 4, -8, -9, 5, -10, 0], [20, -11, 17, -3, -15, 5, -4, 17], [-7, 7, 14, -4, 1, -4, -7, 13], [-16, 10, -16, -17, 3, -7, -3, -11], [4, 16, -5, 7, 0, 8, 5, 4], [20, 6, 1, 2, -13, -11, 3, -4], [-22, -5, -7, 16, 0, -4, 7, -9], [-24, 10, 5, -4, -5, -2, -2, 1], [21, -4, 0, -1, -7, -13, -8, 21], [1, 8, 3, 9, 2, -14, 0, -1], [1, -16, 15, 28, -11, 1, 6, 10], [-1, -16, 11, 10, -13, -12, 4, 0], [8, -12, 11, 0, -3, -8, -7, -1], [-11, 2, 0, -7, 7, 9, -10, -5], [16, -2, -13, -7, 4, -5, -8, 1], [-32, 1, -7, 6, 2, 19, -3, 2], [6, -2, -4, -20, 12, -2, -6, -3], [24, -12, 9, 2, -8, -18, -8, 21], [-10, 3, -11, 14, 16, 7, 2, -16], [-14, -9, 1, -3, -5, 3, 3, -8], [16, -4, 0, 30, 0, 6, 2, -6], [-19, 4, 8, -5, -7, 9, -8, 1], [-18, -6, -18, -5, 7, -15, -1, -9], [6, 0, -17, -8, 0, -8, 8, -15], [14, -5, 0, -5, -6, 5, 4, 12], [-2, -14, 6, 9, -4, -2, -5, 9], [-30, -6, -19, -4, 7, -18, 10, -19], [2, 14, -9, 23, 12, 6, 0, -11], [-36, 7, -7, -2, 22, 22, 1, -7], [-2, -11, 3, -16, -7, -18, 6, -5], [-9, 12, 3, 7, 0, -7, 2, -4], [-9, 5, 6, 23, 8, 13, -8, -1], [-20, 2, 8, 18, -2, 5, 3, -13], [-1, 5, 1, 12, -2, -8, 7, -8], [-5, -14, 17, -7, -14, -4, -2, -2], [20, -3, -2, -1, 10, 3, 4, -8], [-12, 5, -16, -13, 8, -12, 2, -8], [-13, -5, 20, 0, -5, 8, -1, -4], [45, 4, 5, -9, -6, -14, -2, 0], [-12, -5, -15, 9, 12, 2, -7, -10], [-9, 21, -8, 20, -1, 0, 0, -3], [25, -8, -6, 2, 4, 2, -2, -6], [10, -4, -4, 6, -12, 0, -7, -6], [-20, -4, 2, 15, -5, 13, 5, 3], [21, -18, 18, -12, 8, 19, -3, 6], [12, 14, -20, -6, 12, -8, -3, 4], [-5, -8, -2, 3, -1, 2, -11, 4], [0, -12, -8, -3, -2, -2, -12, -4], [27, -12, 25, 8, -9, 12, -6, 17], [8, 0, -2, 4, -10, 5, -4, -2], [-7, -13, 15, -10, -18, -5, -3, 7], [-28, 14, 5, 8, 11, 30, 2, -7], [-12, 0, -11, 3, 6, 3, 7, 4], [-20, 0, 8, 8, -4, 26, -4, -8], [4, -2, -3, -1, 0, 4, 4, -3], [-17, 4, 2, 24, -2, 14, -4, -6], [-12, -2, 10, -2, 2, -2, 4, 4], [-25, -3, -2, -6, -25, -12, -2, 17], [-22, 8, -6, -6, -10, -8, 8, 0], [12, -15, 9, 4, -2, -16, 3, 7], [-22, 9, 0, -7, 4, -7, -5, 11], [15, 19, -16, -8, 11, 15, -2, 8], [-8, -18, 8, -19, -9, -21, 3, 11], [16, -5, -3, -18, 0, 5, -3, 6], [1, -9, 18, 5, 4, 18, -8, 7], [14, 2, -8, -2, 18, 20, -7, -12], [-13, 8, -20, -23, 11, -7, 0, -4], [15, 1, 0, -1, -12, -10, 9, -4], [30, 2, 0, 14, -1, 3, -15, 11], [-41, 2, -11, -1, 26, 20, -2, -18], [2, -6, 2, 16, -12, -4, 4, -16], [-26, 6, -16, 13, -1, -12, 11, -9], [-28, 12, 6, -1, 29, 27, -3, -5], [16, -12, -8, -3, -17, -23, 0, -8], [-3, -9, -14, -12, -9, -10, -10, 9], [-12, 12, -12, -5, -5, 1, -4, -12], [-37, 12, -3, -2, -5, -2, 5, -5], [20, -16, 12, -12, -5, 5, 4, -8], [10, 7, -15, -7, 10, -9, 1, 4], [-14, 13, -7, 3, 3, -19, -2, -3], [-2, -24, 23, 14, -11, -4, 0, -3], [-8, 19, -1, -21, 23, 17, -4, -1], [-23, 20, 11, 3, 4, 15, 0, 8], [12, 10, 18, 15, -1, 14, -3, 13], [-33, 3, -7, 17, 13, -8, -7, -3], [-23, 10, -11, -13, 4, 13, 0, -18], [10, 6, 13, -11, 3, 9, 10, 5], [11, -21, 18, 23, -6, 10, -4, 1], [6, 0, 18, 8, -2, -3, -6, 6], [-27, -4, 2, -6, -14, -12, 10, 5], [2, -1, -15, -5, 2, 1, -3, -8], [-3, -4, -7, 7, -17, -4, -7, 11], [-37, -9, 10, 6, 1, 4, 2, 1], [-21, 18, 0, -5, 7, -3, 6, -7], [12, 4, -2, 12, 3, -5, 1, -1], [7, -25, 15, -3, -19, -21, -5, 1], [19, 10, 0, -6, -8, -19, 11, 4], [-4, -17, -9, -3, 11, 1, -2, -4], [2, 12, -14, -22, 18, -4, -2, -11], [8, 8, 4, 8, 8, 20, 2, -6], [7, 0, 11, -9, 4, 3, -2, 4], [-28, -11, 14, 4, -5, 5, 1, -10], [14, 6, -12, 2, -6, -18, 7, 4], [4, 2, 16, 20, -8, -10, 6, 6], [-9, -17, 2, -26, -7, -12, 10, 5], [-4, -14, -6, 14, -6, 2, 4, -2], [-15, 10, -9, 7, 12, -16, 7, -7], [-6, -11, 27, -6, -6, 9, -3, 1], [-4, -4, 4, 16, 5, -4, 4, 8], [7, 4, -5, 2, -1, -20, 14, -6], [-51, 9, -9, -7, 25, 24, 1, -9], [30, -20, -1, 11, 2, -11, 10, -3], [3, -12, -19, -15, -13, -26, 7, -9], [26, -30, 18, 10, -6, 4, 2, 4], [18, 23, -3, 13, 8, 22, -1, -6], [-15, 7, 27, 15, -1, 7, 7, 15], [13, -13, 17, 4, -28, -10, -5, 17]];
    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_ba();".
// 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_ba(:prec:=8)
    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_ba();".
// 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_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3549_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![61, -64, -96, 72, 57, -22, -13, 2, 1]>,<5,R![-143, 40, 312, 70, -177, -104, -7, 6, 1]>],Snew);
    return Vf;
end function;