
// Make newform 185.2.b.a in Magma, downloaded from the LMFDB on 21 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_185_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_185_b_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_185_2_b_a();".
// 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_185_2_b_a();".
// 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 := [16, 0, 368, 0, 2672, 0, 7004, 0, 8280, 0, 5049, 0, 1684, 0, 306, 0, 28, 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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-132264, 0, -1758744, 0, -4791660, 0, -5036892, 0, -2540885, 0, -669460, 0, -93358, 0, -6496, 0, -177, 0], [232640, 0, 1639712, 0, 5455192, 0, 7366768, 0, 4838164, 0, 1691115, 0, 317307, 0, 29689, 0, 1077, 0], [115088, 0, 1600528, 0, 5455192, 0, 7366768, 0, 4838164, 0, 1691115, 0, 317307, 0, 29689, 0, 1077, 0], [352608, 0, 3538656, 0, 11144072, 0, 14780768, 0, 9562402, 0, 3278489, 0, 602481, 0, 55331, 0, 1977, 0], [-489696, 0, -5668704, 0, -16305424, 0, -20472248, 0, -13225414, 0, -4649903, 0, -882923, 0, -83557, 0, -3059, 0], [-235536, 164112, -3770352, 90652, -11916532, -681806, -14863300, -1192840, -9197135, -784896, -3070197, -254770, -555680, -43333, -50597, -3684, -1799, -123], [-223360, 0, -2930100, 0, -8890098, 0, -10927927, 0, -6674643, 0, -2199697, 0, -393548, 0, -35498, 0, -1253, 0], [0, -75200, 0, -1738752, 0, -5799814, 0, -7117720, 0, -4218614, 0, -1336033, 0, -229539, 0, -19979, 0, -685], [471072, -101392, 7540704, -4522936, 23833064, -14180304, 29726600, -17081180, 18394270, -10268734, 6140394, -3357773, 1111360, -599299, 101194, -54071, 3598, -1911], [-504560, 804848, -4744224, 15707472, -10800384, 52937480, -10533844, 69344372, -5343484, 44784396, -1515087, 15519345, -240271, 2897505, -19737, 270255, -649, 9783], [0, -1536448, 0, -17429088, 0, -49849508, 0, -60022300, 0, -36568528, 0, -12115545, 0, -2184753, 0, -198583, 0, -7055], [0, -1046496, 0, -13764504, 0, -43013004, 0, -54719876, 0, -34626342, 0, -11795705, 0, -2171369, 0, -200327, 0, -7193], [824496, 3131312, 13616608, 38240352, 47776128, 113365504, 63652892, 140493076, 41121384, 87707656, 14147931, 29652659, 2617063, 5433799, 242029, 499901, 8701, 17917], [-917648, -3131312, -12790656, -38240352, -39299872, -113365504, -49839476, -140493076, -31968484, -87707656, -11118035, -29652659, -2089559, -5433799, -196141, -499901, -7137, -17917], [504560, 2898352, 4744224, 39228240, 10800384, 120025648, 10533844, 148444476, 5343484, 91470624, 1515087, 30450629, 240271, 5501165, 19737, 500307, 649, 17775], [917648, -3131312, 12790656, -38240352, 39299872, -113365504, 49839476, -140493076, 31968484, -87707656, 11118035, -29652659, 2089559, -5433799, 196141, -499901, 7137, -17917]];
    Rf_basisdens := [1, 1, 19592, 39184, 39184, 39184, 39184, 19592, 9796, 19592, 39184, 78368, 39184, 39184, 78368, 78368, 78368, 78368];
    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_185_b();"
function MakeCharacter_185_b()
    N := 185;
    order := 2;
    char_gens := [112, 76];
    v := [1, 2];
    // 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_185_b_Hecke();"
function MakeCharacter_185_b_Hecke( : Kf := false)
    N := 185;
    order := 2;
    char_gens := [112, 76];
    char_values := [[-1, 0, 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, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, -1, -1, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 1, 1, 0, 0, 1], [-1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, -1, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0], [-1, 1, 1, 1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 2, -1, 0, 0, -1], [0, 0, -2, -1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, -2, 0, 1, 0, 0], [1, 0, -1, -1, -1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, -1, 0, 2], [2, 0, 1, -1, -3, -1, 0, 0, -3, 0, 0, 0, 0, 0, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -3, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1], [1, 1, -3, -1, 0, 0, 0, 3, 0, 1, -1, 3, -1, 0, 1, -2, 1, 1], [0, -1, 2, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 3, 0, 2, 2, 0], [-2, 0, -1, 2, 0, 0, 0, 1, 0, -4, 2, 0, -1, -2, -2, -3, -3, -2], [-1, 0, 2, 2, -2, -1, 4, 0, 0, 0, 0, 0, 0, 0, -2, -1, 0, -1], [1, 0, 1, -1, 1, -4, -2, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 3, 0, 0, 0, 0, -3, 0, 2, 0, 0, 0, 3, 0, 3, 3, 0], [0, 0, -1, -2, 2, 2, -1, 0, 3, 0, 0, 0, 0, 0, 2, 1, 0, 1], [0, 0, 3, 0, 0, 0, 0, -3, 0, 4, 0, -2, -1, 0, 0, 3, 1, 0], [1, 0, 0, 2, 1, -1, 4, 0, 1, 0, 0, 0, 0, 0, -2, -1, 0, -1], [0, 0, 2, 0, 0, 0, 0, -2, 0, 2, 0, -2, -1, -7, 0, 4, 0, 2], [2, 0, 0, 0, -3, -2, -2, 0, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0], [2, 2, -1, -2, 0, 0, 0, 2, 0, -2, -1, -3, 1, 0, 2, -1, -2, 0], [-5, 0, -2, 1, 5, -3, -3, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -3, -3, 1, 0, 0, 0, 1, 0, -3, -1, 1, 1, -2, -1, -2, -3, 1], [0, -1, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, -1, 5, 0, 3, 1, 2], [-2, 0, 0, -2, 1, 0, -2, 0, 1, 0, 0, 0, 0, 0, 4, 2, 0, 2], [-2, -1, -2, 2, 0, 0, 0, 1, 0, -9, 1, 2, 2, -6, -2, -7, -2, -5], [-3, -2, 1, 3, 0, 0, 0, 0, 0, -4, 4, 0, 1, -7, -3, -2, -2, -3], [1, 0, 3, -4, -2, -4, -1, 0, -3, 0, 0, 0, 0, 0, -1, 3, 0, -4], [-1, 2, -1, 1, 0, 0, 0, 0, 0, 2, 0, 0, -2, -2, -1, -2, -2, -1], [-1, 0, 1, 1, 2, -6, 0, 0, -2, 0, 0, 0, 0, 0, -3, 2, 0, -5], [-3, 0, -4, 1, -1, 5, -1, 0, 3, 0, 0, 0, 0, 0, -2, -4, 0, 2], [-5, 0, -5, 1, 6, 6, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 3], [3, -8, -6, -3, 0, 0, 0, 5, 0, -4, -4, 2, 1, -4, 3, -5, -1, 1], [3, -2, -4, -3, 0, 0, 0, 2, 0, 4, -5, -1, -5, 0, 3, 2, -2, 6], [-1, 0, 0, 1, 0, 0, 0, -2, 0, -4, -1, 1, 5, 4, -1, -4, 0, -4], [1, -2, -4, -1, 0, 0, 0, 3, 0, 0, -2, 2, -1, -4, 1, -3, -1, 1], [2, 0, -2, -1, 2, 6, 3, 0, 3, 0, 0, 0, 0, 0, 2, 2, 0, 0], [0, 0, -1, 0, -1, 3, -3, 0, 1, 0, 0, 0, 0, 0, 1, 2, 0, -1], [-5, 0, 2, 4, -4, 1, 0, 0, 2, 0, 0, 0, 0, 0, -4, -5, 0, 1], [1, 3, 2, -1, 0, 0, 0, -3, 0, 5, -2, -4, -2, 4, 1, 3, -1, 1], [2, 0, 1, -2, 0, 0, 0, 1, 0, 0, 0, 2, -1, 2, 2, 1, 5, 0], [3, 0, 3, -4, -3, 2, 3, 0, -2, 0, 0, 0, 0, 0, 1, 1, 0, 0], [0, 0, -1, 0, -2, -2, -3, 0, -5, 0, 0, 0, 0, 0, -2, 1, 0, -3], [-5, 0, -3, 5, 0, 0, 0, 0, 0, -6, 2, 6, 1, 5, -5, -6, -2, -3], [0, -1, 0, 0, 0, 0, 0, -1, 0, -3, -1, -3, -1, 6, 0, -1, -3, -1], [5, 0, 6, -3, -3, -4, 2, 0, -2, 0, 0, 0, 0, 0, 2, 6, 0, -4], [-5, 0, 7, 5, 0, 0, 0, -8, 0, -2, 4, 0, 6, -6, -5, 2, 2, -5], [5, 0, 5, 4, -7, -2, 3, 0, -4, 0, 0, 0, 0, 0, -3, -1, 0, -2], [-7, 0, -3, 5, -1, 2, 0, 0, -1, 0, 0, 0, 0, 0, -1, -3, 0, 2], [-5, 0, -5, 2, 2, 0, -5, 0, -1, 0, 0, 0, 0, 0, -1, 1, 0, -2], [4, 2, -1, -4, 0, 0, 0, 0, 0, 0, -5, -6, 0, 2, 4, 1, -3, 2], [0, 6, 2, 0, 0, 0, 0, 0, 0, 6, 2, 2, -5, 1, 0, 0, 4, -2], [-3, 0, 5, 1, 3, -7, 2, 0, -4, 0, 0, 0, 0, 0, -2, 1, 0, -3], [-10, 0, 1, 8, 2, -8, 5, 0, -1, 0, 0, 0, 0, 0, -4, -1, 0, -3], [-2, 0, -3, 2, 0, 0, 0, -4, 0, -4, -5, 2, 2, -4, -2, -3, -3, 0], [9, 0, 3, -3, -5, 6, 0, 0, 1, 0, 0, 0, 0, 0, 5, -1, 0, 6], [6, -5, 2, -6, 0, 0, 0, 1, 0, -3, -3, -9, 5, -8, 6, 5, -1, 3], [-4, 0, 2, 4, 0, 0, 0, 2, 0, -2, 8, -2, -6, -10, -4, 2, -4, 0], [2, 7, 6, -2, 0, 0, 0, -5, 0, 12, -1, -4, -6, 15, 2, 10, 4, 4], [-2, 0, 3, -1, 2, 1, -6, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 3], [1, 3, 5, -1, 0, 0, 0, -1, 0, -5, 3, -3, 3, -2, 1, 4, 3, -1], [3, -4, 3, -3, 0, 0, 0, -2, 0, 2, -2, -4, 0, 4, 3, 0, 2, -3], [4, 0, 4, -1, 4, -6, 1, 0, 3, 0, 0, 0, 0, 0, 2, 2, 0, 0], [-3, 0, 8, 3, 0, 0, 0, -7, 0, 2, 4, -2, 3, -10, -3, 5, 3, -3], [2, -6, -1, -2, 0, 0, 0, 4, 0, 4, 1, 1, -1, 8, 2, 3, 2, 4], [-1, 0, 1, -1, 7, 3, -2, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, -1], [3, 5, 0, -3, 0, 0, 0, 5, 0, 5, 2, -2, 0, 0, 3, -1, 1, -1], [-9, 0, -2, -3, 0, 0, -1, 0, 7, 0, 0, 0, 0, 0, 3, -1, 0, 4], [0, 4, 1, 0, 0, 0, 0, 3, 0, -4, 4, 2, 4, 3, 0, -5, 3, -6], [-3, -8, 2, 3, 0, 0, 0, -3, 0, -2, 2, 2, 7, -4, -3, 1, 1, -1], [-3, 0, -2, -1, -2, 0, 9, 0, 1, 0, 0, 0, 0, 0, 1, -1, 0, 2], [2, -6, -9, -2, 0, 0, 0, 6, 0, 0, -5, 5, -7, -4, 2, -7, -2, 2], [2, 0, -4, -2, -4, 6, 2, 0, -2, 0, 0, 0, 0, 0, -4, -2, 0, -2], [-4, 0, 3, 4, 0, 0, 0, -1, 0, -2, 6, 0, 3, -6, -4, -1, -1, -4], [7, 0, 1, -9, -2, 0, 2, 0, -4, 0, 0, 0, 0, 0, -1, 3, 0, -4], [4, 0, 6, 6, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2], [2, 0, -1, -2, 8, 0, -7, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1], [5, 0, -1, -1, 4, -2, -2, 0, -4, 0, 0, 0, 0, 0, 3, 5, 0, -2], [3, 0, 12, 2, -3, -3, 8, 0, -1, 0, 0, 0, 0, 0, -4, -1, 0, -3], [-6, 8, 7, 6, 0, 0, 0, -11, 0, 2, 2, 2, 9, 4, -6, -1, 3, -8], [5, 0, -1, 6, -3, -2, 1, 0, -4, 0, 0, 0, 0, 0, -7, -5, 0, -2], [-8, -3, -2, 8, 0, 0, 0, -1, 0, -6, 5, 6, 2, -17, -8, -10, -4, -8], [12, 0, -4, -2, 3, -4, -8, 0, -3, 0, 0, 0, 0, 0, 4, 2, 0, 2], [-1, 11, 7, 1, 0, 0, 0, -5, 0, 7, 3, -3, -1, 6, -1, 12, 3, 5], [-1, 0, 3, 1, -6, 4, 0, 0, 2, 0, 0, 0, 0, 0, 5, -1, 0, 6], [-4, 3, 2, 4, 0, 0, 0, -1, 0, -1, 5, 7, -7, 8, -4, -3, 5, -5], [4, 3, -2, -4, 0, 0, 0, -1, 0, 5, -7, -5, -1, 20, 4, 5, -3, 7], [-8, 0, 0, -7, 4, -6, -3, 0, -3, 0, 0, 0, 0, 0, 0, 2, 0, -2], [1, -7, -3, -1, 0, 0, 0, -1, 0, 3, -5, 3, 3, -10, 1, -4, 1, -1], [6, 0, -6, -6, 4, 6, -4, 0, 10, 0, 0, 0, 0, 0, 12, 4, 0, 8], [-4, 0, -1, 11, 3, -3, -2, 0, -1, 0, 0, 0, 0, 0, -7, -6, 0, -1], [-3, 1, 1, 3, 0, 0, 0, -3, 0, -5, 1, -1, 11, -12, -3, -4, -3, -5], [1, 0, 2, 1, -6, 2, -1, 0, -5, 0, 0, 0, 0, 0, 1, 1, 0, 0], [-4, 0, 3, 4, -5, -5, 5, 0, -7, 0, 0, 0, 0, 0, -11, -2, 0, -9], [5, -3, 1, -5, 0, 0, 0, 1, 0, 5, -3, -7, 1, -2, 5, 4, -1, 3], [-7, 0, 1, -5, 3, -4, -8, 0, -3, 0, 0, 0, 0, 0, 1, 7, 0, -6], [2, -7, -6, -2, 0, 0, 0, 1, 0, 3, -7, 3, 1, 6, 2, -5, -1, 1], [0, 8, 1, 0, 0, 0, 0, 2, 0, 8, 3, 2, 0, 8, 0, 9, 3, 8], [-3, 10, -6, 3, 0, 0, 0, 4, 0, -6, 1, 7, 1, -10, -3, -8, -2, -2], [-23, 0, -7, 5, 11, 6, 2, 0, 9, 0, 0, 0, 0, 0, -7, -5, 0, -2], [6, 0, -3, -2, -4, 4, 9, 0, 3, 0, 0, 0, 0, 0, -6, -5, 0, -1], [-11, 7, -7, 11, 0, 0, 0, 3, 0, -15, 7, 14, -8, -4, -11, -14, -4, -7], [3, 2, -4, -3, 0, 0, 0, 2, 0, -4, -5, -3, -1, 14, 3, -2, -4, 2], [-4, 4, -1, 4, 0, 0, 0, -1, 0, 4, 2, 4, -3, -4, -4, 3, -1, 4], [-3, 0, 2, 1, 4, -4, 3, 0, -5, 0, 0, 0, 0, 0, 1, 1, 0, 0], [13, 0, 5, -5, -7, 3, 6, 0, -4, 0, 0, 0, 0, 0, 2, 7, 0, -5], [-3, -9, 3, 3, 0, 0, 0, -3, 0, -5, 3, 3, 5, 6, -3, 0, 3, -3], [3, 2, 6, -3, 0, 0, 0, -7, 0, -4, -4, -12, 5, 0, 3, 7, -3, 1], [-1, -2, -12, 1, 0, 0, 0, -1, 0, 0, -12, 2, 1, 2, -1, -9, -11, 3], [-7, 0, -2, 1, -6, -4, 1, 0, -9, 0, 0, 0, 0, 0, -1, -1, 0, 0], [3, 0, 1, -4, 0, 8, 3, 0, 5, 0, 0, 0, 0, 0, 9, 1, 0, 8], [11, 0, -6, 1, -7, -1, -1, 0, -5, 0, 0, 0, 0, 0, 8, 4, 0, 4], [2, 2, 3, -2, 0, 0, 0, 0, 0, 4, 1, -9, -7, -4, 2, 3, -4, 0], [-1, 6, 4, 1, 0, 0, 0, 2, 0, 14, 7, 3, 3, 10, -1, 10, 6, 6], [4, -7, -1, -4, 0, 0, 0, 1, 0, 3, -4, 4, 6, 10, 4, 2, 7, 3], [0, 0, 1, 0, 8, -6, -5, 0, -7, 0, 0, 0, 0, 0, 2, -1, 0, 3], [-10, 0, -12, -2, 3, 10, 0, 0, 9, 0, 0, 0, 0, 0, -4, -4, 0, 0], [6, 6, -7, -6, 0, 0, 0, 9, 0, 6, -4, 0, -3, 12, 6, -1, -1, 6], [5, 8, -6, -5, 0, 0, 0, 3, 0, 4, -8, 2, -9, 4, 5, -1, 1, 5], [5, -8, -12, -5, 0, 0, 0, 14, 0, -8, -3, 5, 3, -6, 5, -2, -2, 10], [0, 0, 8, 2, -8, 2, 4, 0, 0, 0, 0, 0, 0, 0, -8, -4, 0, -4], [-2, 0, 6, 2, -6, -6, 0, 0, -8, 0, 0, 0, 0, 0, -4, 0, 0, -4], [-1, 0, 1, 5, -6, -2, 6, 0, 6, 0, 0, 0, 0, 0, -5, -3, 0, -2], [10, 0, 9, 4, -2, -4, -1, 0, -3, 0, 0, 0, 0, 0, 6, 5, 0, 1], [-1, 6, 6, 1, 0, 0, 0, 0, 0, 0, 7, -3, 7, -4, -1, -4, 2, -10], [-4, 4, -1, 4, 0, 0, 0, -3, 0, 4, 0, 0, -5, 2, -4, -1, -5, 0], [-7, 0, 6, -1, -4, 2, -3, 0, 7, 0, 0, 0, 0, 0, 7, 1, 0, 6], [-2, 3, 14, 2, 0, 0, 0, -7, 0, 2, 9, 4, 4, -7, -2, 8, 16, -6], [-13, 0, 6, 3, 12, -6, 7, 0, -1, 0, 0, 0, 0, 0, 3, 1, 0, 2], [10, -3, 3, -10, 0, 0, 0, 3, 0, 11, -4, -12, -8, -10, 10, 8, 1, 5], [-5, 0, 0, -1, 10, -2, 3, 0, -1, 0, 0, 0, 0, 0, -5, -3, 0, -2], [-7, 0, -3, -7, -3, -2, -2, 0, -3, 0, 0, 0, 0, 0, 7, 9, 0, -2], [7, 0, 12, -7, 0, 0, 0, -1, 0, 8, 4, -12, 1, 12, 7, 17, 7, 5], [-11, 9, 5, 11, 0, 0, 0, -7, 0, -4, 9, 6, -2, -5, -11, -14, 0, -19], [-12, 7, 14, 12, 0, 0, 0, -11, 0, 1, 15, -2, -2, -8, -12, 5, 0, -9], [11, 0, 7, -5, -3, -2, -10, 0, -7, 0, 0, 0, 0, 0, 1, 9, 0, -8], [0, 0, 1, 2, 8, 0, 3, 0, 1, 0, 0, 0, 0, 0, 2, -5, 0, 7], [4, 0, 5, -4, -8, 6, -5, 0, 7, 0, 0, 0, 0, 0, 0, 1, 0, -1], [5, 5, 0, -5, 0, 0, 0, 4, 0, -2, -1, -8, 1, 29, 5, -3, -3, -3], [-2, 2, -7, 2, 0, 0, 0, 6, 0, -4, 1, 1, -1, 6, -2, 1, -8, 8], [-2, 0, -4, 2, -2, 6, 12, 0, 4, 0, 0, 0, 0, 0, -6, -8, 0, 2], [-6, 0, 0, 6, 0, 6, -2, 0, -2, 0, 0, 0, 0, 0, 4, -4, 0, 8], [-4, 2, 9, 4, 0, 0, 0, -4, 0, 6, 9, -1, -11, 4, -4, 11, 4, 2], [-1, -6, -12, 1, 0, 0, 0, 10, 0, -8, -1, 17, 9, 2, -1, -18, 4, -6], [12, 0, -11, -7, 2, 21, -6, 0, 12, 0, 0, 0, 0, 0, 7, -4, 0, 11], [1, 4, 4, -1, 0, 0, 0, -1, 0, -8, 2, -6, 6, 1, 1, -1, -1, -5], [-8, 2, 8, 8, 0, 0, 0, -10, 0, 0, 6, 4, 0, -6, -8, 4, 4, -4], [10, 0, 0, -4, -5, 5, -4, 0, 8, 0, 0, 0, 0, 0, 9, 7, 0, 2], [6, -1, 4, -6, 0, 0, 0, -5, 0, 5, -7, -1, 5, 2, 6, -1, 9, -5], [-3, 11, -1, 3, 0, 0, 0, -1, 0, -2, 1, -4, -10, 15, -3, 0, -8, 1], [-6, 4, 1, 6, 0, 0, 0, -10, 0, 18, -3, 1, 1, -2, -6, 3, -4, 2], [4, 0, -5, -1, -8, -1, -4, 0, -4, 0, 0, 0, 0, 0, -7, -2, 0, -5], [11, 0, -8, -6, -5, 5, -8, 0, 1, 0, 0, 0, 0, 0, -4, -3, 0, -1], [-17, 0, 7, 5, -1, -14, 9, 0, -5, 0, 0, 0, 0, 0, -6, -3, 0, -3], [-2, -8, -3, 2, 0, 0, 0, 5, 0, -8, 4, 2, -5, -20, -2, -1, -3, 2], [-1, 0, 13, -3, 1, -9, -10, 0, -8, 0, 0, 0, 0, 0, 2, 7, 0, -5], [-5, 7, -3, 5, 0, 0, 0, 1, 0, 7, 3, 9, -13, 18, -5, -2, 1, 1], [6, 0, -1, -6, 0, 0, 0, -5, 0, 2, -12, 0, -1, 12, 6, 7, 5, 8], [-2, -5, -6, 2, 0, 0, 0, 1, 0, -1, -3, -3, -9, 2, -2, -5, -11, 1], [-2, 0, 4, 0, 2, -8, 0, 0, -10, 0, 0, 0, 0, 0, -10, 2, 0, -12], [-2, -7, -4, 2, 0, 0, 0, 3, 0, -5, 1, -1, 3, -20, -2, -7, -7, -3], [-7, 16, 7, 7, 0, 0, 0, -6, 0, 20, 8, 2, -3, 11, -7, 8, 2, 1], [12, 0, 4, 7, 9, -8, 5, 0, -2, 0, 0, 0, 0, 0, -2, -4, 0, 2], [6, -11, -17, -6, 0, 0, 0, 7, 0, -3, -16, 6, 4, -2, 6, -8, -5, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_185_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_185_2_b_a();".
// 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_185_2_b_a(:prec:=18)
    chi := MakeCharacter_185_b();
    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_185_2_b_a();".
// 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_185_2_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_185_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;