
// Make newform 3549.2.a.bg in Magma, downloaded from the LMFDB on 24 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_bg();".
// 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_bg();".
// 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 := [281, -1041, -3811, 4632, 7806, -6785, -6393, 4553, 2522, -1575, -510, 290, 51, -27, -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, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [17099693, -53360949, -300569102, 106936663, 552269516, -106742226, -381510549, 71234831, 119482005, -24302208, -17154925, 3734994, 946938, -207205, -4840], [-38588056, 286694529, 627931543, -587154214, -1079863813, 489339396, 685631166, -223516985, -202723757, 56338585, 28878896, -7125065, -1761470, 348794, 26527], [5587509, 8811570, -128107759, -95988421, 277030943, 125754442, -212298805, -53203852, 75838990, 9178285, -13591371, -610919, 1175340, 8311, -38567], [-41386850, -106534419, 140774277, 164127904, -230681231, -38850820, 183706814, -19347335, -69250381, 9988723, 12934898, -1536175, -1156542, 79358, 39159], [-20411763, -9188801, 212384073, -51129407, -317954439, 102192067, 171606601, -65837724, -39286903, 18918087, 3036617, -2498265, 139769, 123497, -21755], [9987024, -68658319, -174934433, 77250998, 306145379, 7522680, -205828878, -33989261, 71541267, 13880105, -13912376, -2156565, 1435826, 116866, -60853], [-27937020, 93954739, 438736369, -157471248, -757335097, 91388378, 502314938, -34531577, -163544127, 9206243, 27645810, -1328053, -2302936, 73174, 73663], [17466621, -190640833, -837484336, 22579661, 1600220662, 275576846, -1110194795, -181845153, 369211365, 44708658, -63240615, -4786396, 5359798, 186673, -177180], [58774419, -101829125, -732374977, 114455160, 1256849458, 12720921, -827597654, -33848081, 265785957, 10040241, -44424222, -1127106, 3702933, 43404, -121202], [-37166976, 49042155, 422840940, 101476522, -725804406, -271189129, 497057283, 173650765, -173841121, -47456984, 33124120, 5917076, -3276487, -276767, 131583], [64260237, -112246899, -1004383910, 58512023, 1825407274, 142922354, -1283390115, -126820423, 446409499, 38749648, -82118737, -5103648, 7666606, 246133, -286506], [-101420669, 189938318, 1321185415, 24798435, -2310023707, -443158712, 1581453993, 310055960, -538705514, -84491865, 97315541, 10255723, -8935738, -463195, 328859]];
    Rf_basisdens := [1, 1, 1, 6033052, 6033052, 6033052, 6033052, 3016526, 6033052, 6033052, 6033052, 3016526, 3016526, 6033052, 6033052];
    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, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 1, 1, 0, 1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, -1, 0, 1, 0, 0, -1, 1, -1, 1, -1, -1, 0, 0], [1, 0, 0, 0, 0, -1, 1, 0, -1, 1, 0, 1, 0, 0, 1], [1, 0, -1, 0, 1, 0, 1, 0, -2, 0, 1, 1, 0, 0, 1], [0, 1, -1, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1], [-1, 0, 0, 1, -1, 1, -1, 0, 1, 0, 0, 0, -1, -1, 0], [-1, 0, -1, 2, 0, -1, -1, 1, 0, 1, 0, 0, 0, 0, 0], [2, -2, 0, 1, -1, 0, 0, 1, 0, 2, -1, -1, 0, 1, 0], [2, -1, -1, 2, 0, 0, 0, -1, 0, -1, 1, -1, -1, 0, 1], [1, 0, -1, 0, 1, -3, 0, -1, 0, -2, 0, -1, -1, -1, 0], [1, 0, 0, -1, -1, -1, 1, -1, -3, -1, -1, 0, -1, 0, 1], [1, 0, -1, 1, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 0], [-1, -2, -1, 2, 1, 3, 0, 1, 2, 2, 0, 1, 1, 1, 0], [1, 0, 2, 0, -2, 1, -1, -2, 0, 1, 0, -1, 0, 0, -1], [-1, 2, 0, -1, 1, -1, -1, 1, -2, -1, 1, 0, 0, -1, 1], [2, 2, 0, -3, 1, 0, 0, -1, -1, -2, 1, 1, 1, 0, 0], [1, 1, -1, -2, 1, 0, -1, 0, 1, -3, 0, 0, 0, 0, 0], [-1, 1, 1, -2, 1, 4, 0, 1, 2, 1, -1, 1, 2, 1, 0], [-2, -1, 2, 0, 0, 1, 2, -2, 1, -1, -1, 0, -2, -1, 2], [2, 3, 2, 0, -2, -1, 0, 0, 0, -1, -1, 2, 1, 0, 0], [2, 0, 1, 1, 0, -1, 0, 0, -1, 0, -1, 0, -2, 0, 0], [0, -1, 1, -1, 0, 1, 1, -1, 1, -3, -1, 1, -1, 0, 0], [-1, -1, 2, -3, -1, 1, -1, -2, 4, -1, -1, 0, 0, 0, -2], [0, 1, -1, 0, 2, -3, 1, 2, -1, -1, 2, 2, -1, 1, 1], [-1, -2, 1, 0, -1, 3, -1, 0, 1, 2, 1, 0, -1, 0, 0], [2, 1, -1, -1, -1, 0, -2, 0, 1, -1, 0, -2, -2, 0, 1], [6, 3, 0, -2, 0, -3, -2, 2, -4, -1, 1, 0, 1, 0, -2], [-5, -1, 1, 3, 0, 6, 1, 1, 6, 3, -1, 1, 1, 2, 0], [3, -4, -2, 1, 1, 1, 1, 0, 3, 2, 0, -2, 1, 1, 0], [-2, 0, -1, 1, -1, -1, 0, 0, 1, 4, -1, 0, 1, 0, -1], [1, -1, 0, 1, 0, 3, 0, 1, 3, 3, 0, -1, -1, 1, -1], [3, -1, 1, 2, 1, 0, 2, 1, -2, -1, -1, 3, 2, 3, 2], [1, 6, 0, -2, -1, -6, -2, 2, -2, 0, 0, 2, 1, -1, -3], [-2, 3, 0, 1, -1, -3, -2, 1, 1, -1, 0, -1, 2, 0, -2], [-3, 2, 0, -2, 0, 4, -2, 0, -2, 2, 0, -2, 0, 1, -2], [5, 5, -1, -1, 0, 0, -3, 1, 0, 1, 1, -1, 1, -2, -2], [8, 3, 0, -2, 0, -7, 0, 2, -4, 1, 1, 2, 1, -2, 0], [4, 0, 0, 0, 1, -2, 2, 0, 0, 2, 2, 2, 1, 0, 1], [1, -3, 3, 1, -1, -3, -1, -1, -5, 1, -1, -1, 0, 1, -3], [-2, -3, 2, 1, -1, -1, 0, -1, 1, 1, -2, 1, 0, 0, 0], [3, 0, 0, -1, 0, -3, 0, -1, 1, -2, -1, 1, 0, -1, -1], [8, -1, -1, 3, 0, -1, -1, -2, -3, 2, 0, -3, -3, -1, 1], [-4, 2, -1, 0, 1, 8, -3, 2, 2, 2, 1, 2, 1, 0, 0], [2, -4, 2, 5, -1, 0, 0, -1, -1, 2, -1, 1, -1, 0, -2], [4, 1, 2, 0, 0, -1, 0, 2, 2, 3, -3, 2, 3, 2, 0], [-3, -3, 2, 2, -1, -3, 0, 0, 2, -1, -1, 0, 0, 1, 1], [-2, 4, 2, 1, 0, -6, -4, 1, -5, 2, 1, -1, 2, 0, -3], [-6, -1, 2, -2, -2, 5, 4, -2, 4, -1, -3, 2, -1, 0, 2], [8, 0, 2, -1, 1, 0, 2, -1, -3, -2, -1, 1, 1, 0, 0], [-3, 3, 1, -1, 0, 0, 0, 2, -1, -3, 0, 2, 3, 3, 0], [-2, 2, -1, 2, -1, 3, -1, 1, -7, -1, 2, -3, 2, 3, 0], [-2, 1, 0, -4, 0, 1, -2, 0, 1, -1, 3, 0, 2, 1, 0], [3, 1, 0, -1, 1, 2, 1, -1, 2, 0, 2, 2, 0, 0, 1], [3, 7, 0, -2, 0, 0, -2, -1, -10, 2, 2, 0, 1, -3, -1], [-1, 2, -3, -1, 2, -1, 0, 4, 1, 4, 3, 2, 0, 1, 0], [4, -5, -1, -3, 2, 1, 3, -3, 1, -3, 1, -1, 1, 2, 0], [5, 2, 1, -2, -1, 5, -2, -3, 1, 0, 2, -3, 0, 0, 0], [6, 4, -3, -4, 0, -6, -2, 1, -8, 0, 4, 1, 4, 0, 1], [6, -5, 2, 4, 0, 1, -2, 0, 2, 5, 1, 0, 1, 0, 2], [4, -1, -2, -3, 1, 1, 2, -3, -1, -5, 0, -1, -2, 2, 0], [-3, 4, -5, 1, 2, 1, -1, 1, -2, -2, 2, 1, -2, -2, 0], [2, 4, -1, -3, 0, -2, -3, 0, 3, -3, 3, -3, -2, -1, -1], [10, 2, -2, 0, 1, -3, -1, 2, 1, 1, 2, 3, 1, -3, 2], [-1, 7, 0, -6, 0, 2, -1, 0, 6, -2, 1, 1, -1, -4, -3], [0, 5, 0, 0, -4, -3, -2, 4, -4, 3, -1, 0, 3, 2, -4], [9, 5, 0, -2, -2, -1, 0, 3, 0, 2, 0, 1, 4, 3, 0], [-4, -2, -4, 3, 1, -7, 1, -2, 3, -2, -2, -1, -2, 1, 1], [3, 3, -2, -4, 2, 2, -3, 0, -9, 0, 3, -1, 1, -2, -3], [8, -6, 0, 3, 0, 0, 1, 3, -1, 3, 1, 0, 2, 3, 0], [-2, 2, 5, -1, -3, -1, 0, -4, -3, 0, -1, -2, -3, -4, -1], [6, 0, 0, 4, 0, 2, 0, -2, -4, 4, 0, -4, 0, 0, 0], [3, 2, 3, 0, -3, -3, 1, 0, 1, 2, -1, 2, 3, 0, -2], [1, -1, 1, 1, 0, 4, -2, 0, 5, 5, -2, -4, -1, 1, -4], [-7, 5, -2, 1, 0, -3, 4, -1, 5, -5, -2, 3, -3, -3, 3], [2, 4, -2, 1, 1, -4, 2, 3, -9, 0, 1, 1, 1, 2, 0], [3, -2, 5, -3, -4, 9, 0, 0, 5, -2, -1, -2, 0, -1, -2], [-6, 5, 3, -3, 0, 3, -1, -2, -6, -6, 0, -1, -1, -1, 1], [-4, 5, 3, 0, -1, 2, 3, 2, 5, 3, -2, 4, 2, 1, 2], [8, 0, 0, 4, 0, 0, 4, 0, -2, 0, -4, 2, -2, 0, 4], [-6, -2, -1, 1, 2, 6, -1, 2, 8, 3, 1, -1, 2, 7, 3], [8, -2, 3, 0, -2, -1, 2, -3, -1, -2, 0, 1, 1, 1, -1], [-15, -3, -4, 2, 0, 14, 1, -2, 14, 0, 3, -3, -1, 2, -1], [-5, -2, 0, -5, 1, 1, 4, -2, -3, -3, 0, 3, 3, 3, 1], [6, 3, 3, -1, 0, -10, 0, 3, -7, 0, 3, 1, 2, 4, -2], [-10, 3, 0, -3, 3, 5, -1, 2, 8, -1, 1, 2, 2, -1, 2], [6, 2, -2, 3, 1, -10, 2, 1, -7, 0, -3, -3, -1, 2, 0], [3, 0, 7, -6, -1, -1, 0, -3, 2, 2, 0, 1, 1, -3, -2], [7, 1, -5, 3, 0, 6, -1, 1, -2, 5, 1, 3, 1, -6, 0], [6, -5, 2, 1, -1, 0, 5, 0, -3, 1, -5, 3, 5, 5, -1], [-1, -2, 0, -3, 2, 0, 0, -7, 3, -2, -1, -3, -4, -5, 1], [-1, -5, -5, 3, 2, -12, 0, 4, -3, 5, 2, -4, 1, 5, 0], [4, 13, 1, -5, 0, 2, -2, 1, -1, -6, 1, 5, 2, -4, 0], [-3, -8, 2, -2, 0, 4, 0, -1, 0, 1, 1, -5, -3, 1, 0], [-7, -5, 2, 6, 2, 6, -1, -2, 3, 4, 1, -3, -5, 2, 1], [-5, -10, 5, 2, -1, 7, 0, -5, 3, 0, -2, -5, -3, 3, 0], [-8, -3, -4, 7, 3, 5, -3, 6, 10, 1, 3, 2, -2, 1, 0], [-9, 1, 0, -7, 3, -2, -1, -3, 5, -6, 2, -2, 0, 0, 1], [5, 7, 7, -9, 0, -8, 0, -2, -9, -5, 4, 2, 1, -3, 2], [-6, -8, 2, 0, 4, 10, -3, -3, 5, 0, 1, -5, -4, -3, -2], [10, 6, -3, 3, 1, 3, -2, 0, -11, -2, 3, 0, -1, 0, 1], [5, -3, 9, -5, -2, -4, 0, 0, -11, -1, -4, 2, -1, 3, 0], [-7, -6, 1, 1, 2, 7, 0, 4, -3, -2, 1, 0, 2, 5, -2], [9, 1, 1, -2, -1, -14, 0, 1, -2, -3, -3, 1, 0, 5, 0], [7, 4, 8, -2, -3, 2, -2, -4, 4, 0, -2, 2, -3, -7, 1], [4, 2, -2, 2, -4, 0, -4, 4, 4, 4, 0, 0, 0, 2, 2], [-5, -1, 6, -7, -4, 6, 0, -7, 5, -1, -2, -1, -3, -3, -3], [-16, 2, -6, -5, 2, 5, -3, -2, 9, -4, 6, -4, -2, 2, -1], [-6, 1, 2, -1, 1, 1, 2, -5, -7, -3, -2, -1, -4, -2, -2], [-5, 10, -3, -5, -2, 7, 0, 2, -1, -4, 3, 0, 4, -1, -2], [7, -2, 1, 3, -4, -5, 1, 5, -2, 6, -4, 3, 6, 2, 0], [-5, -1, 4, -1, 1, 2, 1, 3, -5, 2, -2, 0, 0, 2, 1], [3, -4, 2, 3, -1, 3, 1, -4, -1, -2, 0, 2, -5, -3, 0], [-1, 0, 3, 5, 0, 3, -2, 0, -5, 0, -3, 0, 0, -5, 0], [-7, 2, -4, -1, 2, -2, 2, 1, -3, -4, 3, -1, 2, -1, 1], [4, 0, 2, 0, 3, -8, 0, -1, 0, 3, -1, 1, 0, 2, 1], [-5, -8, -1, 7, -2, 5, 0, -2, 5, 6, 1, -2, -2, 3, 0], [10, 2, 5, -5, -2, 2, -2, -3, -3, -1, -4, 2, 4, -4, -3], [6, 9, 0, -1, -3, -9, 0, 3, -7, 1, 0, 7, 4, -2, 0], [-9, 3, 1, 3, -2, 0, 0, 4, -6, -1, -4, 4, 0, 4, 0], [-5, -4, -2, 4, 2, 1, -3, -1, -4, 8, 1, -3, 0, -1, 0], [-12, 4, 3, -5, 2, 6, 5, -2, 11, -3, 1, 3, 0, 1, 1], [-10, -4, -4, 5, -4, 5, 1, 1, 6, 1, -3, -2, -4, 5, -2], [-1, 3, -1, 3, -2, -2, 2, 4, 3, 1, -4, 2, -1, 3, 2], [5, -3, 0, 2, -2, -12, 1, 2, -11, -2, -5, 1, 3, 6, 1], [7, -1, 1, -4, -3, 2, 2, -3, 4, -9, -3, -1, 0, 5, -2], [2, -6, -4, 3, 1, -2, 4, -1, 3, -2, 1, 1, -1, 2, 4], [-14, 1, 6, -2, -3, 5, -2, 3, 14, 0, -4, 1, -1, -4, -1], [0, 2, 4, -6, -4, -2, -1, 3, 3, 2, -3, -1, 2, -1, -4], [-3, -4, -7, 0, 7, 10, 2, -1, 0, -2, 4, -1, -3, -1, 2], [-11, -1, -8, 1, 3, 11, 2, 1, 7, -5, -2, -1, -2, 1, 4], [-4, -8, -2, 6, 2, 2, 4, 2, 2, 6, -2, 2, 2, 4, 0], [17, 1, -5, 3, 4, -2, 0, 0, -3, 1, 2, -2, -1, -1, 0], [-2, -8, 3, 3, 2, -8, -1, 1, 9, -2, 0, 1, 0, 2, 2], [-1, -5, -5, 6, 3, -10, 0, -5, -5, -5, 3, -1, -7, -2, 4], [-8, 1, 1, -3, 0, 1, 1, 0, 1, -4, 2, -1, -1, -1, -5], [7, -9, -6, 6, 1, -4, 2, 0, 4, 1, 1, -4, -4, 3, 5], [4, -3, 1, -5, 2, -1, -5, -6, -1, -6, 4, -3, -1, -1, -1], [6, 8, 5, 6, -5, -2, -3, 5, -6, 11, -2, -2, 3, -1, -1], [-6, -1, 1, -1, 2, 9, 1, 0, 10, -6, 2, 1, 2, 2, 1], [-4, 2, 2, 4, 2, 4, 0, -4, -2, -4, 2, 0, -2, -2, 2], [12, -3, 2, -1, -1, -3, 2, -1, 1, -1, -4, -3, -2, -2, -6], [12, -11, 4, 0, -2, -3, 2, -4, -2, -3, -5, 0, -5, -2, 0], [-10, -3, -9, 3, 4, -12, 4, 0, 1, -3, 0, 2, -1, 4, 2], [16, 2, 3, -4, -3, 2, 1, 1, 3, 5, 0, 2, 3, 1, 3], [-5, -6, -3, 5, 0, -13, 3, -1, 7, 0, 2, 1, -5, 1, 4], [-11, -2, -6, -4, 3, 9, 1, -1, 3, -4, 3, -1, 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-6, 0, -5, -9, -4], [-18, -3, -6, 6, 4, 13, -8, -6, 9, -5, 5, -6, -4, -1, 0], [18, -7, 0, 3, -7, -7, -4, -7, -11, 3, -2, -1, 0, -6, 0], [-8, -5, 4, -1, -7, 17, 0, -3, 8, 13, 2, 3, 3, 3, 0], [9, 15, -11, -10, 1, -4, 4, 3, -4, -9, 3, 5, 2, -5, 6], [29, 15, -2, -5, 2, -12, 2, 3, -5, 1, 0, 9, 1, -5, 3], [-12, -8, -4, 4, 0, -8, -2, 0, 10, -8, -2, -8, -4, 4, -4], [-14, -16, -1, -9, 5, 5, 8, -14, 7, -16, 1, -2, -5, 0, 3], [4, -4, -3, 5, 7, -5, 8, 6, 1, 6, -1, 2, 3, 0, 1], [0, -6, 5, -4, 5, -6, 1, 1, -11, -3, -2, 2, 4, 8, -3], [-6, -22, -8, 10, 4, 16, 3, 9, 7, 16, -1, 1, 2, 13, -4], [-1, 5, 1, -3, -2, -16, -10, 4, -6, -5, 6, 4, -3, -7, -6], [7, -4, -3, 1, 3, 1, -2, 14, -17, 4, 1, 4, 7, 3, 1], [-11, -1, 3, -6, 3, -12, 2, -3, -2, -5, -5, -1, 10, 9, -2], [-8, 11, -1, -7, -3, -4, 2, 0, -7, -9, -4, 4, 2, 2, -1], [-10, -11, 7, 2, -7, 3, 0, -11, 3, 3, -3, -5, -4, -3, 0], [1, -2, 13, 1, -8, -7, -6, -12, -5, -6, -5, -4, -4, -1, -4], [-5, -6, -4, 4, 6, 1, -3, 0, -4, -3, 6, -5, 0, -6, 1], [-1, 0, 1, 6, 1, -1, -1, -2, 7, -8, 1, -2, -7, 0, 4], [7, 6, -1, -5, 4, -5, 2, -2, 11, 2, -1, -2, -6, -11, 6], [-3, -11, 3, 8, -3, -2, -5, -4, -15, 5, 0, -4, 0, 10, 0], [-15, 9, -1, 3, 4, 8, 0, -2, -3, 1, 2, 6, -1, -5, -2], [7, 0, -1, 1, 0, -15, 2, 2, 7, 6, -3, 2, 4, 7, 2], [-9, 10, -5, -4, 6, -17, 6, 0, 4, -3, 3, 6, 9, 7, -1], [16, -2, -2, 2, 4, 2, 0, -2, 12, 12, 4, -6, 0, -6, -4], [0, 0, -6, 1, -1, -6, 1, 2, 0, -4, 6, 8, -3, -3, 4], [1, 7, 6, 0, -5, 2, -4, -2, -4, -3, 3, 8, 2, -3, 5], [8, 1, -10, 4, 2, 1, -4, -4, -9, -5, -1, -12, -5, -2, 2], [12, 20, 4, -4, -4, -8, -14, 8, -19, -4, 8, 4, -3, -7, -6], [-2, 0, -15, 4, -1, 8, 1, -1, 0, -1, 0, -8, -5, -1, 1], [-8, 12, -8, -5, 3, -14, 5, 0, 8, -6, 2, 8, -1, -9, 10], [-8, -6, 4, 7, -1, 22, -6, 5, 19, 6, -1, -1, 7, 2, 0], [6, 15, 4, -7, -7, 5, -7, 0, 2, -1, 3, -8, 2, -3, -12], [-21, -2, -3, 2, -1, 5, 4, 1, 16, -6, -4, -1, 1, 3, 0], [3, 13, -3, -3, -4, -14, -6, 0, -9, -1, 2, -2, -3, -9, -6], [-4, 3, 4, 0, 4, -5, 5, 7, -9, -1, -8, 1, 5, 1, 2], [-8, -3, 5, 9, 6, 11, 7, 11, -9, 5, -5, 11, 9, 8, 2], [-8, -15, -5, 7, -4, 15, -3, -8, -1, 0, 4, -11, -3, -3, -3], [11, 14, 10, -5, -1, -15, 3, -2, -9, -10, -4, 6, -1, -5, 8], [4, 0, -6, -1, 9, -12, -2, 3, -17, 4, 9, -1, -1, 0, -2], [-3, -9, -5, 8, 5, 2, 4, 9, -20, 3, -1, -1, 8, 15, 6], [20, 7, 4, -11, -7, -1, -4, 1, -1, 3, 2, 7, 2, -4, -4], [-1, 6, 0, 6, 0, 9, -3, 0, 12, -1, 0, -1, 0, 4, -1], [14, -7, 11, 0, 1, -2, -4, 2, -6, 10, -8, 0, 3, 6, -2], [5, 8, 1, -6, -1, -21, 2, 9, -14, 8, -2, 7, 9, -7, -4], [-15, -2, -4, 11, 3, 13, -1, 1, 0, -7, -1, -6, -1, 4, 1], [-5, -2, -3, 10, 3, 4, 2, 9, 6, 12, 4, 5, 3, 5, 4], [-8, 5, -3, 0, 4, 8, -8, 1, -2, -7, 3, -5, 1, 0, -1], [-21, 0, 6, -7, 1, 8, -2, 4, 17, 7, 2, 6, 6, 3, -10], [12, -11, -5, -1, 2, -3, 1, -10, -3, 2, 2, -5, -7, -7, 5], [3, 2, 0, -11, -1, -11, -5, -3, -11, -13, 7, -4, -5, -2, 5], [12, 3, 5, -7, 0, 8, 12, 7, -3, -4, -5, 5, 4, 8, 0], [3, 4, 7, -9, -4, -5, 4, 0, 17, -2, 1, 4, 2, -7, -10], [-23, -2, -1, 4, 1, -1, -3, -2, 15, 0, 7, 0, 3, 4, 2], [12, 2, 0, 12, -4, 2, -4, 4, -2, 4, 2, 4, -4, -6, 2], [-7, 3, 4, 2, -2, -1, -8, 10, -6, 3, 1, 2, 1, -1, -4], [-16, -1, -18, 4, 4, 9, 2, 6, 8, -1, 7, -4, -3, 8, 0], [-6, -3, 8, 8, -10, 11, -5, 5, 13, 15, -8, -5, -3, 1, -2], [-2, 13, 1, -15, 2, -13, 1, 4, -17, -4, -2, -5, 3, 7, -3], [8, 3, 0, -2, 6, -1, -4, -6, 14, -9, 1, -8, -9, -6, 0], [-8, 1, 1, 5, -4, -9, -8, -1, -1, 4, 5, -4, 1, -6, -5], [-10, -3, 9, 4, -5, -2, 0, 7, 4, -1, -11, -1, 2, -4, -4], [11, 2, -1, 0, -5, 5, -2, -3, -16, -4, -4, 1, 5, 3, 0], [-10, -12, 0, -6, 2, 12, 0, -2, 8, 0, 6, 6, 0, 8, 2], [-3, -5, 10, -10, 4, -12, -3, 2, -18, 0, 3, 7, 7, 2, -5], [3, 3, 9, -2, -3, -15, 3, 8, 8, 1, -6, 9, 1, 2, 3], [-8, 1, 15, -9, -7, -1, -6, -6, -7, 3, 0, -2, -4, -4, 1], [10, 0, 8, 9, -3, -22, 4, 1, -3, 0, -7, 3, -1, 0, 8], [9, 8, 4, -4, -3, -22, -6, 2, -6, 0, 0, 16, 5, -9, -5], [-1, -19, -2, 4, -3, 7, 12, -8, 14, -1, -15, -2, -4, 5, 3], [16, 12, 15, -1, -13, -6, -2, 0, 7, 6, -7, 6, 3, 2, -1], [-8, 2, 2, -1, 0, -4, -3, -1, 23, -7, -1, 0, 2, 5, -6], [7, 2, 2, 3, -5, -4, -6, 1, 11, 4, -3, -5, 3, -1, -4], [9, -11, 9, -9, -2, -4, 1, -7, -16, -7, -5, -1, -5, -6, 0], [-1, -9, 1, -2, 4, -8, 3, -5, 8, 0, -9, -4, -3, -10, -10], [47, -5, 1, 0, -3, 5, -5, 3, -6, 0, -3, -6, -2, -2, -1], [6, -4, 5, 1, 0, -10, 5, 7, -3, -6, 0, 3, -4, 2, 8], [12, 3, 2, 5, -1, -18, -3, 5, -23, -4, 0, -2, -4, 1, 5], [-19, -3, 3, 5, -10, 20, -5, 5, 6, 11, -9, -3, -5, -2, -6], [-16, 6, 3, -4, 0, 2, 2, 1, -2, -4, -2, 9, 2, -2, -5], [1, 1, 1, 3, -2, 13, -2, 8, -7, 13, -2, -4, 5, 9, -4], [-15, 2, -7, 3, -6, 7, -4, 0, 15, -10, 3, -6, -2, 3, 4], [21, -2, -8, 5, 3, -4, 5, 5, 14, 11, 3, -1, 4, -2, -6], [18, 9, 6, 1, 1, -11, -6, 7, -11, -1, -2, 9, 4, -2, -4], [-8, -2, 6, 8, -2, 2, 8, 4, 15, 14, -10, 12, 7, -3, 2], [28, 13, -4, -1, -1, -19, -2, 7, -17, 7, 4, 5, 12, 0, -10], [7, -9, -6, 11, 5, -7, -6, 0, -15, 0, 1, -8, 3, 5, -2], [17, 2, 8, 4, 1, -4, 2, 10, 0, 12, -8, 10, 11, 5, -3], [-4, 3, -7, 9, -2, -11, 1, 4, 3, -2, 0, 3, -1, 7, 7], [-5, 19, 0, -6, -3, -9, -8, 10, -22, 5, 5, 2, 8, -1, -3], [8, 6, -3, -3, 5, -18, 8, -4, 3, 2, 7, -4, -1, 6, 3], [-13, -7, -5, 9, 2, 0, -6, 0, 3, 9, 2, 8, -3, -9, 2], [-2, 6, -6, 5, 5, -10, 3, 8, 4, 6, 4, 4, 5, 15, 4], [-3, -1, -3, 7, 4, -8, 8, -4, -7, -1, 0, -4, 1, 1, 4], [-12, 3, 13, -7, -2, -3, -3, -8, 15, 2, -4, -1, -5, 3, -5], [15, -15, 7, -2, 3, -12, 6, -13, -2, 7, -11, -5, -2, 3, 0], [-8, 0, 3, -1, -4, 0, -6, -6, 12, 0, 1, -14, -6, -1, -6], [36, -9, -3, 15, -6, 6, 2, 1, 5, 6, 5, -1, -6, 2, 2], [-27, 1, 7, -6, -1, -18, 2, -5, 10, -13, -1, -5, -2, 11, 4], [30, 1, 7, -3, -2, -21, 5, -1, -19, 3, -5, 5, 5, -2, -2], [-5, -6, -4, 4, -2, 6, -4, -1, -2, 7, 1, -9, -5, 7, 4], [-7, -8, -4, 17, -5, -3, 0, 2, 3, 1, 2, -3, -1, 9, 3], [2, 17, 3, -3, 7, -21, 4, 10, -11, 7, 6, 8, 14, 8, -7], [-25, -13, -8, -1, 11, -1, 0, 2, 11, 0, 9, -2, -5, -1, 2], [-5, 4, 4, -14, -2, 15, -1, 4, 18, -13, 0, 7, 6, -2, 1], [-11, -15, -1, 17, 0, 6, -3, -3, -4, -3, -1, 1, -9, -4, 0], [-12, 17, 4, 0, -10, 17, -14, 2, 8, 5, 3, -10, -3, -2, -10], [-10, 14, -3, -11, -1, -1, 2, 8, -11, -2, 7, 6, 7, -4, -1], [0, -12, -12, 11, 3, -10, -4, -7, -25, -6, 13, -7, -15, 0, 8], [-7, 11, 0, -9, 1, 5, -8, 0, 21, -6, 1, -4, -5, 3, -4], [26, -9, 7, 7, -6, 2, 4, -9, 5, -10, -7, 1, -2, -2, 2], [-7, 5, -8, 2, 6, -4, -9, 5, 20, -3, 8, -7, -7, 3, 4], [13, -4, 4, 3, -1, -11, 2, -1, 18, 2, -1, 3, -5, 4, 0], [7, -16, -3, 12, -3, -13, 3, -2, 3, 2, 1, -8, -5, 6, 0], [-27, -2, 6, 3, -3, 27, 7, 5, 16, 9, -7, 6, 7, 16, 5], [-9, -2, -5, 6, 3, 27, 0, -5, 12, 0, -2, -7, -5, 3, 6], [-2, 12, 2, -2, -10, 20, 4, 0, 10, -2, -2, 0, -4, -6, 2], [3, 1, 8, -17, -2, 1, 4, -3, -1, 5, -2, -3, 5, -9, -3], [9, -8, -4, 12, -6, -11, 1, 2, -29, 3, -4, -3, -5, -1, 1], [-2, -7, -6, 14, -6, 17, 7, 9, 9, 5, -4, 1, 5, 7, -2], [0, -18, 4, -1, 4, -4, 1, -3, -5, 13, -5, -2, 2, 1, 0], [-2, -9, 9, -3, -2, -6, 6, -1, -11, 6, -3, 1, -6, -6, -4], [12, 10, -2, -5, 3, -10, -10, -3, -23, -8, 9, -5, 1, 2, -4], [8, 12, 11, 3, -10, 2, -3, -1, -5, 4, -10, 1, 4, -6, -2], [6, 5, -1, 2, -7, -8, 2, 9, -14, -5, 5, 7, 4, 8, 4], [-7, 25, -3, -1, -2, -6, 4, 8, 13, 1, 0, 6, 7, -3, -2], [-12, -15, 1, 4, -2, -14, 4, -3, -20, -3, -5, 1, -3, 10, -1], [1, 8, 11, -3, -8, 3, -8, 0, 13, -4, -11, -2, 6, -3, -14], [22, -6, 4, 4, -3, 14, -8, 3, 4, 11, 3, -3, 2, -4, -7], [25, 1, 5, 6, -9, -18, 4, -5, 0, 1, -1, 5, -4, -5, -2], [-13, -1, 3, -10, -3, 18, -6, -3, 10, -3, -1, -7, 2, -9, -10], [-16, -2, -19, 18, 5, -3, -10, 7, 12, 6, 6, -5, -5, -2, -4], [18, -2, 0, -1, 5, 16, 3, -6, 2, -6, 0, -4, -5, 3, 8], [2, -17, -6, 3, 11, 3, 0, -9, 15, -7, 0, -11, -4, 0, -2], [-5, 9, 6, -5, -11, -6, -1, -11, 2, -16, -8, -4, -6, -8, -5], [11, 14, -7, -9, -4, -25, 2, -6, -7, -10, -3, -4, 4, -1, -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_bg();".
// 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_bg(:prec:=15)
    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_bg();".
// 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_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3549_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-281, -1041, 3811, 4632, -7806, -6785, 6393, 4553, -2522, -1575, 510, 290, -51, -27, 2, 1]>,<5,R![397312, 402944, -675648, -576600, 504860, 306312, -206759, -74303, 47961, 7829, -6081, -169, 380, -22, -9, 1]>],Snew);
    return Vf;
end function;