
// Make newform 273.2.bj.d in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_273_bj();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_273_bj_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_273_2_bj_d();".
// 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_273_2_bj_d();".
// 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 := [256, 0, -688, 0, 1273, 0, -1196, 0, 807, 0, -310, 0, 85, 0, -11, 0, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [122770133, 0, 19033136, 0, -34658945, 0, 48900157, 0, -19993643, 0, 6775609, 0, -905544, 0, 93708, 0], [1373874032, 0, -2550992573, 0, 2135288524, 0, -1648811299, 0, 647847966, 0, -192872281, 0, 25335623, 0, -2439541, 0], [0, -635028480, 0, -344398261, 0, -2653944980, 0, 2049303605, 0, -1065436466, 0, 239720495, 0, -31489585, 0, 2051459], [-7631273072, 0, 14007293741, 0, -15396256332, 0, 8479881587, 0, -3074557294, 0, 653830489, 0, -79437735, 0, 5499477, 0], [-1532777472, 0, 6790029511, 0, -6405865572, 0, 4946433897, 0, -1943543898, 0, 578616843, 0, -76006869, 0, 7318623, 0], [0, 510925824, 0, -2550992573, 0, 2135288524, 0, -1648811299, 0, 647847966, 0, -192872281, 0, 25335623, 0, -2439541], [0, 1111287280, 0, -2162804917, 0, 657335740, 0, -1749011307, 0, 844736062, 0, -407614369, 0, 58008863, 0, -6901309], [-1111287280, -2985621268, 2162804917, -1621479872, -657335740, 2952681140, 1749011307, -3851063448, -844736062, 1703307836, 407614369, -577230868, -58008863, 78057212, 6901309, -7983216], [-2540113920, 26115780688, -1377593044, -48250376215, -10615779920, 47423070356, 8197214420, -30334459849, -4261745864, 11536837802, 958881980, -3041434107, -125958340, 390591909, 8205836, -34417935], [-2540113920, -26115780688, -1377593044, 48250376215, -10615779920, -47423070356, 8197214420, 30334459849, -4261745864, -11536837802, 958881980, 3041434107, -125958340, -390591909, 8205836, 34417935], [1111287280, -2985621268, -2162804917, -1621479872, 657335740, 2952681140, -1749011307, -3851063448, 844736062, 1703307836, -407614369, -577230868, 58008863, 78057212, -6901309, -7983216], [141925895, 0, 240477056, 0, -437903720, 0, 551180115, 0, -252612728, 0, 85607464, 0, -11648434, 0, 1183968, 0], [0, -7631273072, 0, 14007293741, 0, -15396256332, 0, 8479881587, 0, -3074557294, 0, 653830489, 0, -79437735, 0, 5499477], [0, 141925895, 0, 240477056, 0, -437903720, 0, 551180115, 0, -252612728, 0, 85607464, 0, -11648434, 0, 1183968]];
    Rf_basisdens := [1, 1, 53934263, 862948208, 1725896416, 862948208, 862948208, 862948208, 1725896416, 1725896416, 6903585664, 6903585664, 1725896416, 53934263, 862948208, 53934263];
    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_273_bj();"
function MakeCharacter_273_bj()
    N := 273;
    order := 6;
    char_gens := [92, 157, 106];
    v := [6, 4, 3];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_273_bj_Hecke();"
function MakeCharacter_273_bj_Hecke(Kf)
    N := 273;
    order := 6;
    char_gens := [92, 157, 106];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 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, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1, -1], [0, 1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, -1, -1, 0, 0, 0, -1, -1], [0, 0, -1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 1, -1, 0, 0, 1, 0], [3, 0, 0, -3, 0, -1, -1, 0, 0, 0, -1, -1, 0, -1, 0, 0], [1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, -2, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -1, 4, 0, -3, 3, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 2, -2, 0, 0, 2, 0], [0, 0, 3, 1, 0, 4, -3, 0, 0, 2, 0, 0, -2, 0, 0, 0], [0, -1, 0, 0, 1, 0, 0, 0, -1, 2, 0, 0, 2, 0, 0, 1], [4, 0, 0, -4, 0, 3, -1, 0, 0, 0, 2, 2, 0, 3, 0, 0], [0, -7, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 1, 0, 0, -2], [0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 3, 3], [0, 0, 0, 0, -3, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0], [-1, 0, 0, 1, 0, 1, 4, 0, 0, 0, 5, 5, 0, 1, 0, 0], [0, 0, 0, 0, 3, 0, 0, -2, 0, -4, -4, 4, -4, 0, -1, -1], [0, 3, 0, 0, 0, 0, 0, 3, -3, 0, -2, 2, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -2, -2], [0, 0, -2, 4, 0, -3, 2, 0, 0, 3, 0, 0, -3, 0, 0, 0], [-3, 0, 0, 3, 0, -2, -2, 0, 0, 0, -2, -2, 0, -2, 0, 0], [5, 0, 0, -5, 0, 3, -1, 0, 0, 0, 1, 1, 0, 3, 0, 0], [0, -1, 0, 0, -3, 0, 0, 0, 3, 3, 0, 0, 3, 0, 0, 2], [5, 0, 7, 0, 0, 0, 0, 0, 0, 3, 3, 3, -3, -1, 0, 0], [-3, 0, -1, 0, 0, 0, 0, 0, 0, -1, -1, -1, 1, 4, 0, 0], [0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0], [0, 4, 0, 0, -8, 0, 0, 0, 8, 4, 0, 0, 4, 0, 0, 1], [-5, 0, -2, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, 1, 0, 0], [0, 4, 0, 0, 0, 0, 0, 4, 0, 0, -4, 4, 0, 0, -4, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, 2], [0, 0, 2, 4, 0, 1, -2, 0, 0, 3, 0, 0, -3, 0, 0, 0], [0, 7, 0, 0, 0, 0, 0, 7, -3, 0, 4, -4, 0, 0, 3, 0], [0, 0, 0, 0, -2, 0, 0, 1, 0, -2, -2, 2, -2, 0, 3, 3], [5, 0, 0, -5, 0, -4, 2, 0, 0, 0, -1, -1, 0, -4, 0, 0], [0, 0, 1, 2, 0, 2, -1, 0, 0, -3, 0, 0, 3, 0, 0, 0], [-6, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [2, 0, 0, -2, 0, -1, -2, 0, 0, 0, 1, 1, 0, -1, 0, 0], [0, 1, 0, 0, 5, 0, 0, 0, -5, -3, 0, 0, -3, 0, 0, 4], [0, 0, 0, 0, -1, 0, 0, 2, 0, -2, -2, 2, -2, 0, -4, -4], [0, 0, 3, 0, 0, 0, -3, 0, 0, 1, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, -3, -3, -3, 3, -6, 0, 0], [0, 0, 0, 0, 9, 0, 0, -2, 0, -4, -4, 4, -4, 0, 0, 0], [0, -2, 0, 0, -4, 0, 0, 0, 4, 2, 0, 0, 2, 0, 0, 2], [0, -5, 0, 0, 0, 0, 0, -5, -6, 0, 5, -5, 0, 0, 5, 0], [2, 0, 0, -2, 0, 1, -3, 0, 0, 0, 4, 4, 0, 1, 0, 0], [0, 0, 0, 0, 6, 0, 0, -1, 0, 2, 2, -2, 2, 0, 2, 2], [0, 1, 0, 0, 5, 0, 0, 0, -5, 2, 0, 0, 2, 0, 0, 5], [2, 0, -6, 0, 0, 0, 0, 0, 0, -5, -5, -5, 5, 3, 0, 0], [2, 0, 0, -2, 0, -4, 7, 0, 0, 0, 1, 1, 0, -4, 0, 0], [0, 0, -1, -4, 0, -9, 1, 0, 0, -2, 0, 0, 2, 0, 0, 0], [0, 0, -3, 13, 0, 0, 3, 0, 0, 2, 0, 0, -2, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, -6, 3, 0, -4, 4, 0, 0, 3, 0], [0, 0, -1, -9, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, -4, 0, 8, 8, -8, 8, 0, 3, 3], [0, 0, 5, -1, 0, 2, -5, 0, 0, -1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, 2, 0, 2, 2, -2, 2, 0, -2, -2], [0, 0, 0, 0, 6, 0, 0, 5, 0, -7, -7, 7, -7, 0, -10, -10], [0, 0, 3, -10, 0, 2, -3, 0, 0, 3, 0, 0, -3, 0, 0, 0], [3, 0, 0, -3, 0, -3, 5, 0, 0, 0, 0, 0, 0, -3, 0, 0], [0, -1, 0, 0, 0, 0, 0, -1, 4, 0, -4, 4, 0, 0, -2, 0], [0, 9, 0, 0, 0, 0, 0, 9, 1, 0, 5, -5, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 8, -8, -1, 0, 0], [0, 0, -3, -5, 0, 0, 3, 0, 0, -2, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, -9, 0, 0, -5, 0, 2, 2, -2, 2, 0, 0, 0], [0, -6, 0, 0, 5, 0, 0, 0, -5, 4, 0, 0, 4, 0, 0, 2], [0, -3, 0, 0, 0, 0, 0, -3, -9, 0, 6, -6, 0, 0, 4, 0], [0, 0, 0, 2, 0, -5, 0, 0, 0, -10, 0, 0, 10, 0, 0, 0], [-3, 0, 0, 3, 0, 4, -8, 0, 0, 0, 2, 2, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 0, -9, -9, 9, -9, 0, -6, -6], [0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 4, -4, 0, 0, 13, 0], [0, 0, 5, 6, 0, 0, -5, 0, 0, -5, 0, 0, 5, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 3, -5, 0, -7, 7, 0, 0, -3, 0], [0, -1, 0, 0, 0, 0, 0, -1, 8, 0, -4, 4, 0, 0, -13, 0], [0, 4, 0, 0, 8, 0, 0, 0, -8, -5, 0, 0, -5, 0, 0, -7], [1, 0, -8, 0, 0, 0, 0, 0, 0, -1, -1, -1, 1, 0, 0, 0], [0, 0, 0, 0, -2, 0, 0, 15, 0, 7, 7, -7, 7, 0, 5, 5], [0, -2, 0, 0, -1, 0, 0, 0, 1, 4, 0, 0, 4, 0, 0, 12], [3, 0, -13, 0, 0, 0, 0, 0, 0, -10, -10, -10, 10, 3, 0, 0], [-12, 0, 0, 12, 0, -2, 7, 0, 0, 0, 7, 7, 0, -2, 0, 0], [-24, 0, 0, 24, 0, -4, 5, 0, 0, 0, 3, 3, 0, -4, 0, 0], [0, 0, 0, 0, -9, 0, 0, -6, 0, 4, 4, -4, 4, 0, -1, -1], [0, -1, 0, 0, 0, 0, 0, -1, 5, 0, -13, 13, 0, 0, -7, 0], [0, 0, 0, 0, -5, 0, 0, -2, 0, -6, -6, 6, -6, 0, 3, 3], [0, 0, 0, 0, 0, 0, 0, 5, 0, 5, 5, -5, 5, 0, 4, 4], [-8, 0, 0, 8, 0, 3, 8, 0, 0, 0, 5, 5, 0, 3, 0, 0], [0, -3, 0, 0, 12, 0, 0, 0, -12, -4, 0, 0, -4, 0, 0, -2], [0, -17, 0, 0, -4, 0, 0, 0, 4, 1, 0, 0, 1, 0, 0, -9], [4, 0, -1, 0, 0, 0, 0, 0, 0, -3, -3, -3, 3, 8, 0, 0], [0, -4, 0, 0, 0, 0, 0, -4, 4, 0, -13, 13, 0, 0, -8, 0], [-14, 0, 3, 0, 0, 0, 0, 0, 0, 4, 4, 4, -4, 3, 0, 0], [0, -3, 0, 0, 0, 0, 0, -3, 16, 0, -10, 10, 0, 0, 1, 0], [0, 0, -7, -14, 0, -9, 7, 0, 0, -2, 0, 0, 2, 0, 0, 0], [-2, 0, 0, 2, 0, -6, -6, 0, 0, 0, -3, -3, 0, -6, 0, 0], [0, -3, 0, 0, 0, 0, 0, -3, -6, 0, 5, -5, 0, 0, 13, 0], [19, 0, -5, 0, 0, 0, 0, 0, 0, -4, -4, -4, 4, 4, 0, 0], [0, -5, 0, 0, -8, 0, 0, 0, 8, 2, 0, 0, 2, 0, 0, 3], [0, 0, 9, 24, 0, -2, -9, 0, 0, 3, 0, 0, -3, 0, 0, 0], [4, 0, 0, -4, 0, 7, -10, 0, 0, 0, -9, -9, 0, 7, 0, 0], [0, 0, -4, 22, 0, 1, 4, 0, 0, 4, 0, 0, -4, 0, 0, 0], [0, 12, 0, 0, -9, 0, 0, 0, 9, -9, 0, 0, -9, 0, 0, -1], [0, 0, 0, 0, 11, 0, 0, -5, 0, -6, -6, 6, -6, 0, 6, 6], [0, -3, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0, 0, 0, 13, 0], [0, 0, 12, -11, 0, 9, -12, 0, 0, 4, 0, 0, -4, 0, 0, 0], [-20, 0, 2, 0, 0, 0, 0, 0, 0, -1, -1, -1, 1, -10, 0, 0], [-5, 0, 0, 5, 0, -5, -3, 0, 0, 0, -4, -4, 0, -5, 0, 0], [0, -2, 0, 0, 6, 0, 0, 0, -6, 8, 0, 0, 8, 0, 0, -1], [0, 0, 0, 0, 9, 0, 0, 16, 0, 4, 4, -4, 4, 0, -2, -2], [0, 6, 0, 0, -1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 8], [0, 0, 0, 0, 6, 0, 0, -2, 0, -6, -6, 6, -6, 0, -3, -3], [0, 0, -5, -4, 0, -1, 5, 0, 0, -10, 0, 0, 10, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, -11, 0, 5, 5, -5, 5, 0, -8, -8], [0, 0, -2, -25, 0, 7, 2, 0, 0, -8, 0, 0, 8, 0, 0, 0], [7, 0, 0, -7, 0, 3, -6, 0, 0, 0, -10, -10, 0, 3, 0, 0], [12, 0, -2, 0, 0, 0, 0, 0, 0, 6, 6, 6, -6, -12, 0, 0], [0, -12, 0, 0, 12, 0, 0, 0, -12, -5, 0, 0, -5, 0, 0, -12], [9, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, -2, -6, 0, 0], [-13, 0, 0, 13, 0, 8, -13, 0, 0, 0, -4, -4, 0, 8, 0, 0], [0, -15, 0, 0, -8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, -6], [0, -7, 0, 0, 0, 0, 0, -7, 8, 0, -17, 17, 0, 0, -10, 0], [-16, 0, -4, 0, 0, 0, 0, 0, 0, -10, -10, -10, 10, -6, 0, 0], [0, -2, 0, 0, 0, 0, 0, -2, 7, 0, 6, -6, 0, 0, -2, 0], [8, 0, 0, -8, 0, -1, 9, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 8, 0, 0, 0, 0, 0, 0, 10, 10, 10, -10, 7, 0, 0], [0, -11, 0, 0, 0, 0, 0, -11, -6, 0, -9, 9, 0, 0, 1, 0], [0, 19, 0, 0, -21, 0, 0, 0, 21, 5, 0, 0, 5, 0, 0, 13], [0, 0, 0, 0, 2, 0, 0, 7, 0, -2, -2, 2, -2, 0, 8, 8], [31, 0, 0, -31, 0, 0, -6, 0, 0, 0, -6, -6, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0], [0, 9, 0, 0, 0, 0, 0, 9, -8, 0, 8, -8, 0, 0, 6, 0], [0, 0, 0, 0, 18, 0, 0, 14, 0, -5, -5, 5, -5, 0, -5, -5], [0, -2, 0, 0, 15, 0, 0, 0, -15, -12, 0, 0, -12, 0, 0, -24], [0, 10, 0, 0, -10, 0, 0, 0, 10, -6, 0, 0, -6, 0, 0, 5], [-1, 0, 17, 0, 0, 0, 0, 0, 0, 9, 9, 9, -9, 3, 0, 0], [0, 0, 14, 5, 0, 7, -14, 0, 0, 6, 0, 0, -6, 0, 0, 0], [0, 0, 0, 0, 7, 0, 0, 7, 0, 4, 4, -4, 4, 0, -4, -4], [0, 23, 0, 0, 0, 0, 0, 23, 8, 0, 6, -6, 0, 0, 2, 0], [0, 0, 9, -10, 0, 7, -9, 0, 0, 10, 0, 0, -10, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, -12, 0, -8, -8, 8, -8, 0, -5, -5], [0, 0, -11, -11, 0, 2, 11, 0, 0, 1, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, 16, 0, 16, 16, -16, 16, 0, 4, 4], [0, 0, 0, 0, -5, 0, 0, -9, 0, -3, -3, 3, -3, 0, 9, 9], [0, 0, -5, -1, 0, -7, 5, 0, 0, 3, 0, 0, -3, 0, 0, 0], [11, 0, 0, -11, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, -13, -4, 0, -6, 6, 0, 0, 1, 0], [0, -2, 0, 0, 0, 0, 0, -2, -7, 0, 6, -6, 0, 0, -9, 0], [-15, 0, 13, 0, 0, 0, 0, 0, 0, 15, 15, 15, -15, -5, 0, 0], [-11, 0, 9, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, 14, 0, 0], [10, 0, 0, -10, 0, -13, 8, 0, 0, 0, 7, 7, 0, -13, 0, 0], [0, 0, -11, -1, 0, -13, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [12, 0, 10, 0, 0, 0, 0, 0, 0, 3, 3, 3, -3, -11, 0, 0], [0, 0, 0, 0, 0, -1, 10, 0, 0, 0, -2, -2, 0, -1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, -10, 0, 2, -2, 0, 0, -7, 0], [-6, 0, -2, 0, 0, 0, 0, 0, 0, -12, -12, -12, 12, 9, 0, 0], [0, 9, 0, 0, 15, 0, 0, 0, -15, -10, 0, 0, -10, 0, 0, -11], [0, -3, 0, 0, 0, 0, 0, -3, -2, 0, 6, -6, 0, 0, -3, 0], [-10, 0, -20, 0, 0, 0, 0, 0, 0, -2, -2, -2, 2, 4, 0, 0], [0, 0, 0, 0, -4, 0, 0, -22, 0, -5, -5, 5, -5, 0, -2, -2], [21, 0, 0, -21, 0, -4, -6, 0, 0, 0, 7, 7, 0, -4, 0, 0], [0, 11, 0, 0, 7, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 13], [0, -6, 0, 0, 5, 0, 0, 0, -5, -10, 0, 0, -10, 0, 0, 2], [0, 0, 6, 10, 0, -2, -6, 0, 0, -9, 0, 0, 9, 0, 0, 0], [0, 0, -10, -10, 0, 1, 10, 0, 0, 2, 0, 0, -2, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_273_bj_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_273_2_bj_d();".
// 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_273_2_bj_d(:prec:=16)
    chi := MakeCharacter_273_bj();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_273_2_bj_d();".
// 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_273_2_bj_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_273_bj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![16, 0, -92, 0, 381, 0, -755, 0, 1089, 0, -398, 0, 107, 0, -12, 0, 1]>],Snew);
    return Vf;
end function;