
// Make newform 1183.2.e.g in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1183_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1183_e_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_1183_2_e_g();".
// 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_1183_2_e_g();".
// 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, 1, 8, 1, 47, 17, 55, -11, 33, -2, 7, -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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2119374, 5624108, -14058754, 8363982, -31119861, 16008568, -16878368, 2684563, -3846008, 682506, -478424, -29696], [-2229034, 21668998, -14803002, 8704710, -34289666, 16328192, -18206928, 1631143, -4188508, 495146, -498559, -73788], [-6198231, -133032, -450388, -1984215, -2183844, -4702740, 2259680, -3276280, 406680, -747485, 153752, -109660], [572347, 646135, 2423530, 22831649, 12171095, 21902645, -1556185, 14090715, -131145, 2921453, -329655, 439315], [-6707921, 12588213, -44235433, 27901335, -63065800, 58565408, -46353032, 24340355, -9944902, 5617249, -1732988, 566698], [3392561, -2155246, 22259302, -14729214, 12172746, -33035270, 20386532, -19018596, 4066147, -4336084, 1011662, -572347], [6345807, -4177460, 41650219, -27474213, 26529336, -61367800, 38513384, -34760912, 7725614, -7924683, 1869572, -1035034], [566698, 7274619, -8054629, 44802131, -1266529, 72699666, -27397018, 40119354, -5639321, 8811506, -1650363, 1166290], [-1035561, -15484451, 33752077, -96392052, 11027874, -158252610, 66981583, -90433689, 13501144, -19974443, 3882058, -2686072], [1, 8, 1, 47, 17, 55, -11, 33, -2, 7, -1, 1]];
    Rf_basisdens := [1, 1, 3318773, 3318773, 3318773, 3318773, 3318773, 3318773, 3318773, 3318773, 3318773, 1];
    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_1183_e();"
function MakeCharacter_1183_e()
    N := 1183;
    order := 3;
    char_gens := [339, 1016];
    v := [1, 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_1183_e_Hecke();"
function MakeCharacter_1183_e_Hecke( : Kf := false)
    N := 1183;
    order := 3;
    char_gens := [339, 1016];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [1, 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, 1, 0, 0, 0, 0, 0, -1], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, 1, 1, 0, -1, 0, 0, -1, 0, 0, 0, 0], [0, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0, 0], [1, 2, 0, 0, 2, 2, 0, -1, 2, 1, -1, -2], [-1, -2, 0, 0, -1, -2, 0, 1, -1, -3, 1, -1], [-1, 0, 0, -1, 2, 2, 1, 0, 0, 0, 1, 1], [0, 2, 2, 0, 0, 0, -1, -3, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, 0, -2, 0, -3, -1, 0], [-2, 0, -1, -1, 0, 0, 2, 0, 0, 1, 2, 1], [3, 0, 0, 1, 0, 2, -2, 0, 0, 0, -2, -1], [-2, -4, 0, 0, 2, -4, 0, 2, 2, 0, 2, 3], [0, -1, -3, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 1, -1, 1, 0, 0, -2, -1, 2, 0, 0, 0], [2, 6, 0, 0, 0, 6, 0, -2, 0, 2, -2, -5], [0, 1, 2, 2, 0, 0, -3, 3, 2, 0, 0, 0], [2, 0, -1, 3, 2, 0, -2, 0, 0, 1, -2, -3], [0, 5, 0, -1, 0, 0, 0, 4, 0, 0, 0, 0], [2, -1, 0, 0, -1, -1, 0, -2, -1, 2, 0, -3], [4, 0, 1, -1, 3, 0, -1, 0, 0, -1, -1, 1], [-4, -8, 0, 0, -1, -8, 0, 4, -1, 0, 4, 5], [-8, 0, 0, -4, 1, -2, 3, 0, 0, 0, 3, 4], [0, -3, -1, 4, 0, 0, -1, 7, 1, 0, 0, 0], [4, -6, 0, 0, -1, -6, 0, -4, -1, -1, -2, 1], [8, 1, 0, 0, -3, 1, 0, -8, -3, 2, 1, 1], [0, -5, -1, -4, 0, 0, 1, 0, -1, 0, 0, 0], [1, 0, 0, -5, -2, -8, 2, 0, 0, 0, 2, 5], [-1, 0, 2, -3, -1, -8, 0, 0, 0, -2, 0, 3], [4, 2, 0, 0, 1, 2, 0, -4, 1, 2, 2, 1], [0, -1, -1, 1, 0, 0, 1, 5, 4, 0, 0, 0], [-4, 0, 0, 0, -3, -6, 0, 0, 0, 0, 0, 0], [4, -6, 0, 0, -5, -6, 0, -4, -5, -3, 2, 0], [0, -6, 3, 6, 0, 0, 0, -2, -2, 0, 0, 0], [0, 1, 2, 2, 0, 0, 2, 0, -5, 0, 0, 0], [-3, -4, 0, 0, -2, -4, 0, 3, -2, -2, -3, 1], [-2, 0, 2, 3, -2, 0, -4, 0, 0, -2, -4, -3], [6, -4, 0, 0, 0, -4, 0, -6, 0, 1, -3, -2], [0, 7, 3, -8, 0, 0, 0, -3, 4, 0, 0, 0], [4, 0, 2, 1, -3, -3, 2, 0, 0, -2, 2, -1], [2, -6, 0, 0, 1, -6, 0, -2, 1, -3, 1, 7], [0, -2, 1, -3, 0, 0, -3, 0, 4, 0, 0, 0], [4, 0, -5, -1, -1, 4, 5, 0, 0, 5, 5, 1], [0, 6, -3, -3, 0, 0, 4, -2, -3, 0, 0, 0], [6, 0, 4, 6, -4, 2, 1, 0, 0, -4, 1, -6], [-2, 0, 5, 1, 1, -6, 3, 0, 0, -5, 3, -1], [0, -7, -4, 2, 0, 0, 7, 5, -3, 0, 0, 0], [2, -10, 0, 0, -4, -10, 0, -2, -4, -3, -1, 5], [-10, -1, 0, 0, 4, -1, 0, 10, 4, -1, -2, -1], [-8, 0, -1, 7, -4, 0, -1, 0, 0, 1, -1, -7], [0, 6, 1, -1, 0, 0, -2, -9, 0, 0, 0, 0], [4, 0, -8, 3, 3, 7, -1, 0, 0, 8, -1, -3], [10, 2, 0, 0, 2, 2, 0, -10, 2, 1, -7, 0], [0, -10, -8, 1, 0, 0, -2, -1, -5, 0, 0, 0], [0, -9, -1, -3, 0, 0, 2, 13, 2, 0, 0, 0], [5, -3, 0, 0, 5, -3, 0, -5, 5, -4, 0, 7], [0, 0, 3, 6, 0, 0, 2, 0, 7, 0, 0, 0], [5, 0, 1, 5, 6, 7, -2, 0, 0, -1, -2, -5], [0, 2, 6, -7, 0, 0, 1, 5, 1, 0, 0, 0], [9, 0, 3, -1, -4, -6, 0, 0, 0, -3, 0, 1], [2, 0, 7, 8, -3, 0, -2, 0, 0, -7, -2, -8], [0, 8, 5, -2, 0, 0, -3, 6, 9, 0, 0, 0], [5, 1, 0, 0, -1, 1, 0, -5, -1, 6, -1, 1], [-7, 0, 0, 0, -8, 0, 0, 7, -8, -2, -2, 7], [5, -4, 0, 0, 2, -4, 0, -5, 2, 0, -2, 8], [-3, 0, -5, 8, -3, 6, 0, 0, 0, 5, 0, -8], [0, 2, 3, 0, 0, 0, 4, -1, 5, 0, 0, 0], [-12, 0, 5, -4, -1, 5, 4, 0, 0, -5, 4, 4], [0, -8, -6, -5, 0, 0, -7, 7, -9, 0, 0, 0], [6, 4, 0, 0, 4, 4, 0, -6, 4, 1, -6, 1], [0, -9, -1, 7, 0, 0, 3, 9, -3, 0, 0, 0], [6, -3, 0, 0, -1, -3, 0, -6, -1, 0, 9, 0], [-1, 0, 4, 0, -1, -4, -6, 0, 0, -4, -6, 0], [-5, 4, 0, 0, -3, 4, 0, 5, -3, -2, 8, 0], [0, 5, 4, 0, 0, 0, 6, -12, 4, 0, 0, 0], [-17, -5, 0, 0, -2, -5, 0, 17, -2, -5, -1, -6], [11, 5, 0, 0, -3, 5, 0, -11, -3, 6, 1, 0], [0, 5, 3, 2, 0, 0, -2, 7, 3, 0, 0, 0], [-13, 0, -8, -1, 2, 2, 1, 0, 0, 8, 1, 1], [-1, 0, -9, -1, 2, 10, -8, 0, 0, 9, -8, 1], [0, -1, -3, 9, 0, 0, 4, -1, 2, 0, 0, 0], [6, 0, 1, 7, 1, 1, -14, 0, 0, -1, -14, -7], [0, -9, 0, 0, 0, -9, 0, 0, 0, 1, 8, 11], [-4, 6, 0, 0, -2, 6, 0, 4, -2, -5, 1, 7], [-2, 0, 6, 0, 5, 2, 2, 0, 0, -6, 2, 0], [8, 7, 0, 0, -3, 7, 0, -8, -3, 2, -3, -4], [10, 0, -5, -3, 4, -6, -1, 0, 0, 5, -1, 3], [-5, 0, -5, -3, -5, -7, 5, 0, 0, 5, 5, 3], [10, 9, 0, 0, 7, 9, 0, -10, 7, 8, -1, -9], [0, -1, 2, 12, 0, 0, 2, 10, 0, 0, 0, 0], [0, -1, 5, 12, 0, 0, 0, -4, 7, 0, 0, 0], [-13, 0, -3, 0, -5, -2, 10, 0, 0, 3, 10, 0], [-18, -1, 0, 0, 11, -1, 0, 18, 11, -1, -2, 3], [-7, 0, 6, 1, 2, -5, -11, 0, 0, -6, -11, -1], [12, -1, 0, 0, -3, -1, 0, -12, -3, 0, 0, -11], [0, -9, -6, -1, 0, 0, 1, 5, -10, 0, 0, 0], [9, 10, 0, 0, -4, 10, 0, -9, -4, 5, -2, -10], [0, -12, 0, 0, -3, -12, 0, 0, -3, -1, 11, 4], [5, 0, 2, -1, -6, -10, -8, 0, 0, -2, -8, 1], [0, 6, -4, -2, 0, 0, -2, -7, -3, 0, 0, 0], [0, -14, -7, 4, 0, 0, 0, 6, -8, 0, 0, 0], [2, 5, 0, 0, -10, 5, 0, -2, -10, -4, -2, 1], [0, -1, 0, 5, 0, 0, 1, -3, 3, 0, 0, 0], [0, -3, -3, 0, 0, 0, -3, -2, 1, 0, 0, 0], [5, 0, -4, 6, 5, 3, -2, 0, 0, 4, -2, -6], [-5, 18, 0, 0, 7, 18, 0, 5, 7, 11, 8, 0], [0, 11, 1, -6, 0, 0, 10, 5, 5, 0, 0, 0], [-4, 0, 10, -15, -11, -16, 0, 0, 0, -10, 0, 15], [-2, -4, 0, 0, 5, -4, 0, 2, 5, 12, -2, 5], [0, -4, 1, -2, 0, 0, 2, 6, -4, 0, 0, 0], [-6, 0, -2, -4, 1, 13, -6, 0, 0, 2, -6, 4], [0, 5, -1, -3, 0, 0, -4, 3, 3, 0, 0, 0], [15, 0, -12, -3, 8, 0, -5, 0, 0, 12, -5, 3], [0, 17, -2, -4, 0, 0, 3, -18, -1, 0, 0, 0], [-10, 0, -4, 2, -2, -10, -4, 0, 0, 4, -4, -2], [0, 1, -4, 3, 0, 0, -5, 20, 6, 0, 0, 0], [-1, -3, 0, 0, 6, -3, 0, 1, 6, 6, 4, 2], [3, 0, -2, 8, 1, 13, 0, 0, 0, 2, 0, -8], [0, -10, -10, 0, 0, 0, -9, 18, -3, 0, 0, 0], [-2, 0, 2, 0, 4, 4, -3, 0, 0, -2, -3, 0], [7, -17, 0, 0, -10, -17, 0, -7, -10, -7, 8, 4], [0, -2, 5, -8, 0, 0, 5, 4, -12, 0, 0, 0], [-20, 17, 0, 0, 5, 17, 0, 20, 5, 4, 1, -5], [-22, 0, 2, -12, 3, -19, 1, 0, 0, -2, 1, 12], [-2, -14, 0, 0, 0, -14, 0, 2, 0, -6, 5, 22], [20, 8, 0, 0, -2, 8, 0, -20, -2, -6, -11, -10], [-5, 0, -9, -9, 2, -6, 8, 0, 0, 9, 8, 9], [6, 7, 0, 0, -3, 7, 0, -6, -3, -5, 4, -9], [0, 5, 9, -19, 0, 0, 2, -2, 0, 0, 0, 0], [3, 0, 3, -1, 9, -1, 3, 0, 0, -3, 3, 1], [6, 7, 0, 0, 2, 7, 0, -6, 2, 5, -4, 8], [-8, 0, -8, -4, 13, 6, 0, 0, 0, 8, 0, 4], [-4, 16, 0, 0, 7, 16, 0, 4, 7, 19, -2, 0], [2, 0, -6, -10, -4, 1, -3, 0, 0, 6, -3, 10], [0, -6, -11, 3, 0, 0, -2, 8, 1, 0, 0, 0], [0, 7, -5, -12, 0, 0, 2, -10, 5, 0, 0, 0], [-8, 0, 8, -16, -6, -18, 11, 0, 0, -8, 11, 16], [0, -10, 2, 3, 0, 0, 2, 4, 0, 0, 0, 0], [-5, 0, 7, -13, -7, -21, -7, 0, 0, -7, -7, 13], [-14, -12, 0, 0, -4, -12, 0, 14, -4, -1, 17, 11], [0, -3, 1, 5, 0, 0, 1, 9, -7, 0, 0, 0], [-3, 0, 6, -11, -1, -10, -3, 0, 0, -6, -3, 11], [0, 14, -1, -12, 0, 0, 12, -2, -2, 0, 0, 0], [-11, 0, -3, -10, -2, -10, -4, 0, 0, 3, -4, 10], [-2, 0, -6, -7, -5, 7, 11, 0, 0, 6, 11, 7], [0, -7, 2, -1, 0, 0, -5, 11, -6, 0, 0, 0], [-5, 1, 0, 0, 14, 1, 0, 5, 14, -8, -6, 12], [-17, -10, 0, 0, 1, -10, 0, 17, 1, 2, 5, -8], [-14, -17, 0, 0, -9, -17, 0, 14, -9, -6, 2, 4], [-7, 0, -18, -2, 9, 15, -2, 0, 0, 18, -2, 2], [-13, 0, -3, -5, -1, -4, 3, 0, 0, 3, 3, 5], [21, 10, 0, 0, -6, 10, 0, -21, -6, 6, 0, -10], [0, 10, 6, 7, 0, 0, -1, -8, -5, 0, 0, 0], [13, 0, 7, 10, -2, -2, -16, 0, 0, -7, -16, -10], [-10, 11, 0, 0, 3, 11, 0, 10, 3, -6, -2, 5], [-4, -9, 0, 0, -7, -9, 0, 4, -7, -10, -5, 0], [15, 0, 7, 10, -5, 10, 0, 0, 0, -7, 0, -10], [0, -10, -7, 23, 0, 0, 2, 16, -5, 0, 0, 0], [-3, 8, 0, 0, 6, 8, 0, 3, 6, 0, 3, 4], [-19, 0, 9, -3, 7, -2, 1, 0, 0, -9, 1, 3], [0, 0, 10, -1, -5, -15, 0, 0, 0, -10, 0, 1], [4, -12, 0, 0, 0, -12, 0, -4, 0, -10, -1, -11], [0, -10, -7, 5, 0, 0, 5, 16, -11, 0, 0, 0], [0, -8, 5, 18, 0, 0, -10, -5, 4, 0, 0, 0], [0, 12, -9, -21, 0, 0, 14, -8, 3, 0, 0, 0], [0, 2, -6, -1, 0, 0, -4, -12, -9, 0, 0, 0], [3, 0, 0, -20, 1, -13, 0, 0, 0, 0, 0, 20], [23, 15, 0, 0, 9, 15, 0, -23, 9, 3, -2, -16], [0, 10, 4, -2, 0, 0, 10, -19, 2, 0, 0, 0], [22, 0, -17, 5, -1, 21, -5, 0, 0, 17, -5, -5], [5, 11, 0, 0, 1, 11, 0, -5, 1, 3, 10, -7], [0, -8, -3, 19, 0, 0, -4, 14, -11, 0, 0, 0], [0, -20, -7, 14, 0, 0, -12, 23, -7, 0, 0, 0], [16, 0, -4, -1, -13, 1, 8, 0, 0, 4, 8, 1], [-5, 0, -6, 5, 13, -8, 6, 0, 0, 6, 6, -5], [0, -1, -11, 8, 0, 0, -16, 31, -5, 0, 0, 0], [-9, 0, 2, 8, 10, 17, 1, 0, 0, -2, 1, -8], [-6, 18, 0, 0, 10, 18, 0, 6, 10, -4, -1, -8], [-11, -13, 0, 0, -5, -13, 0, 11, -5, -6, 6, 17], [-32, 0, 7, -5, 4, -5, 11, 0, 0, -7, 11, 5], [-15, 0, 3, -20, -3, 2, 7, 0, 0, -3, 7, 20], [14, -6, 0, 0, -3, -6, 0, -14, -3, -6, -8, -10], [0, 15, 7, 1, 0, 0, 2, -20, 4, 0, 0, 0], [0, 0, 9, 2, 0, 0, -9, 21, 13, 0, 0, 0], [0, 0, 7, 8, 0, 0, -13, 2, -6, 0, 0, 0], [0, -2, -7, -19, 0, 0, 12, 13, 1, 0, 0, 0], [0, -8, 0, 0, -1, -8, 0, 0, -1, 3, -10, -5], [0, -15, -5, 10, 0, 0, -13, 7, -14, 0, 0, 0], [-17, -8, 0, 0, -6, -8, 0, 17, -6, 5, 0, 17], [3, 0, -1, 9, -6, 3, -3, 0, 0, 1, -3, -9], [-8, -11, 0, 0, 9, -11, 0, 8, 9, 2, -8, 14], [-2, 17, 0, 0, 3, 17, 0, 2, 3, 13, -1, -7], [0, -6, 4, 1, 0, 0, -6, 15, 8, 0, 0, 0], [0, -5, 5, -7, 0, 0, -11, 8, -14, 0, 0, 0], [0, -26, -10, 5, 0, 0, 4, -17, -8, 0, 0, 0], [-20, -22, 0, 0, -5, -22, 0, 20, -5, -13, -1, 15], [-24, 0, 3, 9, 6, -9, -6, 0, 0, -3, -6, -9], [11, -10, 0, 0, -4, -10, 0, -11, -4, -4, 10, -14], [0, -3, -13, -1, 0, 0, 6, 6, 13, 0, 0, 0], [-14, 0, -10, -3, 4, -8, 13, 0, 0, 10, 13, 3], [-7, 4, 0, 0, 0, 4, 0, 7, 0, -1, 5, 2], [0, 6, -9, -6, 0, 0, -13, 2, -1, 0, 0, 0], [10, 0, 1, 18, 1, 3, 8, 0, 0, -1, 8, -18], [0, 14, 14, 0, 0, 0, -8, -22, -7, 0, 0, 0], [4, -4, 0, 0, 8, -4, 0, -4, 8, -17, 10, -2], [-22, -2, 0, 0, -7, -2, 0, 22, -7, 7, 7, 17], [6, 0, 7, -5, -4, 11, 0, 0, 0, -7, 0, 5], [0, 9, -5, -1, 0, 0, -1, -7, -12, 0, 0, 0], [-25, -10, 0, 0, 0, -10, 0, 25, 0, 9, 12, 2], [0, 0, -16, -13, -5, -5, 0, 0, 0, 16, 0, 13], [7, 0, 0, 3, -7, 9, -14, 0, 0, 0, -14, -3], [5, 7, 0, 0, -8, 7, 0, -5, -8, -5, -11, 0], [0, 0, 5, 0, 0, 0, -8, -4, 1, 0, 0, 0], [14, 13, 0, 0, 1, 13, 0, -14, 1, 12, -7, 7], [0, 17, 7, 1, 0, 0, 17, -10, 5, 0, 0, 0], [0, -5, -8, 16, 0, 0, -16, 4, 6, 0, 0, 0], [8, 6, 0, 0, 9, 6, 0, -8, 9, 20, 1, 8], [20, 0, 2, 1, 0, 3, -5, 0, 0, -2, -5, -1], [-13, 22, 0, 0, 8, 22, 0, 13, 8, -3, -6, -20], [32, 0, 4, 7, -8, 13, 1, 0, 0, -4, 1, -7], [0, 5, 0, 0, 10, 5, 0, 0, 10, 3, 3, 18], [-26, -16, 0, 0, 2, -16, 0, 26, 2, 1, 5, 7], [14, 0, 17, 16, -8, -4, 1, 0, 0, -17, 1, -16], [-6, 0, -9, 23, 5, 28, -12, 0, 0, 9, -12, -23], [2, -6, 0, 0, -1, -6, 0, -2, -1, -7, -2, 15], [0, -3, -8, -17, 0, 0, 1, 20, -15, 0, 0, 0], [7, -3, 0, 0, -15, -3, 0, -7, -15, -9, -13, -6], [13, 8, 0, 0, 0, 8, 0, -13, 0, 11, -7, -19], [0, -7, -17, 1, 0, 0, 5, -2, 0, 0, 0, 0], [0, -28, -6, 23, 0, 0, -13, -14, -1, 0, 0, 0], [-2, 0, 9, 3, -12, 6, 3, 0, 0, -9, 3, -3], [0, -13, -3, 6, 0, 0, 9, -6, 4, 0, 0, 0], [20, 0, -15, 7, 2, 8, -17, 0, 0, 15, -17, -7], [0, 11, 2, -8, 0, 0, 2, 13, -1, 0, 0, 0], [-23, -2, 0, 0, 1, -2, 0, 23, 1, 10, -1, 3], [-12, 6, 0, 0, 2, 6, 0, 12, 2, 7, 0, -20], [-20, 0, -7, 16, -2, 4, -4, 0, 0, 7, -4, -16], [-10, 0, 6, 19, -1, 16, -5, 0, 0, -6, -5, -19], [0, 3, 3, -12, 0, 0, 14, -26, 5, 0, 0, 0], [1, 5, 0, 0, 4, 5, 0, -1, 4, -7, 0, -12], [0, 4, 9, -8, 0, 0, 8, -33, 14, 0, 0, 0], [5, 10, 0, 0, 9, 10, 0, -5, 9, 8, -12, 5], [16, 0, 6, -12, -4, -4, -1, 0, 0, -6, -1, 12], [23, -4, 0, 0, -9, -4, 0, -23, -9, 6, 13, 6], [-20, 0, 11, -6, -7, 4, 10, 0, 0, -11, 10, 6], [0, 0, -3, -2, 0, 0, -16, -5, 4, 0, 0, 0], [0, 19, 19, -18, 0, 0, 9, -7, -7, 0, 0, 0], [-14, 0, 8, -7, -15, -15, 2, 0, 0, -8, 2, 7], [0, 0, 11, -13, -13, -1, -2, 0, 0, -11, -2, 13], [-4, 3, 0, 0, 4, 3, 0, 4, 4, -6, 4, 19], [0, 3, 9, -11, 0, 0, 4, 2, -3, 0, 0, 0], [-14, 0, 11, -14, 9, 12, 7, 0, 0, -11, 7, 14], [0, -6, -15, -8, 0, 0, 11, -8, 1, 0, 0, 0], [-6, 2, 0, 0, 9, 2, 0, 6, 9, 7, -3, -20], [0, 2, 8, -9, 0, 0, 13, -27, 8, 0, 0, 0], [0, -16, -11, 4, 0, 0, -1, -35, -11, 0, 0, 0], [-12, 0, -3, 12, 18, 18, -9, 0, 0, 3, -9, -12], [11, 12, 0, 0, 7, 12, 0, -11, 7, 4, 11, 11], [0, 19, 9, 7, 0, 0, 4, -20, 10, 0, 0, 0], [-21, 0, 2, 3, 16, 17, 9, 0, 0, -2, 9, -3], [0, 0, -5, 5, 0, 0, -5, 45, -3, 0, 0, 0], [22, 0, -11, -15, 21, 5, 9, 0, 0, 11, 9, 15], [0, 14, 27, 15, 0, 0, -2, -4, 11, 0, 0, 0], [5, 19, 0, 0, 9, 19, 0, -5, 9, 13, 12, -9], [4, 0, 15, 1, -4, -15, -13, 0, 0, -15, -13, -1], [-21, 0, -8, 9, 4, -25, 9, 0, 0, 8, 9, -9], [19, 0, -3, 5, -8, -11, 6, 0, 0, 3, 6, -5], [0, -6, -6, 8, 0, 0, 4, -22, 2, 0, 0, 0], [2, -13, 0, 0, 13, -13, 0, -2, 13, 7, -7, 19], [0, 24, 8, 7, 0, 0, 8, 8, 11, 0, 0, 0], [0, 7, 0, -14, 0, 0, 2, -8, -1, 0, 0, 0], [-7, -20, 0, 0, 0, -20, 0, 7, 0, -2, -10, 12], [17, 0, 6, -13, 10, 13, 8, 0, 0, -6, 8, 13], [-14, -3, 0, 0, -17, -3, 0, 14, -17, -7, -3, -8], [32, 22, 0, 0, 6, 22, 0, -32, 6, 6, -26, -11], [0, -12, -16, 10, 0, 0, 3, 22, -6, 0, 0, 0], [32, 3, 0, 0, 14, 3, 0, -32, 14, 14, -2, 10], [-6, -18, 0, 0, 7, -18, 0, 6, 7, 7, 15, 26], [0, 30, 4, -7, 0, 0, 9, -24, 4, 0, 0, 0], [0, -16, -14, 13, 0, 0, -5, 32, -3, 0, 0, 0], [-34, 0, 6, 28, -8, 8, -5, 0, 0, -6, -5, -28], [-4, 0, -9, -20, 15, -7, -6, 0, 0, 9, -6, 20], [-21, -40, 0, 0, 2, -40, 0, 21, 2, -10, 23, 21], [10, 3, 0, 0, 8, 3, 0, -10, 8, -4, 5, 17], [0, -5, -10, -16, 0, 0, 2, -28, -7, 0, 0, 0], [6, 0, 1, -7, -7, 6, -3, 0, 0, -1, -3, 7], [0, 27, 0, -21, 0, 0, 9, 17, 3, 0, 0, 0], [0, 0, -2, -12, 17, -11, -3, 0, 0, 2, -3, 12], [0, 18, 15, 2, 0, 0, -14, 35, 14, 0, 0, 0], [0, -21, -2, -9, 0, 0, -9, -6, -15, 0, 0, 0], [5, -4, 0, 0, 5, -4, 0, -5, 5, 0, -14, -17], [-43, 0, -2, -2, 12, 6, 1, 0, 0, 2, 1, 2], [-31, 0, -6, 7, 2, -4, -1, 0, 0, 6, -1, -7], [0, 15, -4, 11, 0, 0, 8, 13, -6, 0, 0, 0], [24, 20, 0, 0, -2, 20, 0, -24, -2, 3, 7, -25], [-16, 0, -11, -26, -8, -25, 7, 0, 0, 11, 7, 26], [32, -1, 0, 0, 12, -1, 0, -32, 12, 18, -8, 0], [-20, 0, -7, -4, -6, 11, 7, 0, 0, 7, 7, 4], [6, 11, 0, 0, -4, 11, 0, -6, -4, 13, 12, 22], [-27, 19, 0, 0, -1, 19, 0, 27, -1, 3, -6, -6], [0, 4, 10, -3, 0, 0, 2, -14, 8, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1183_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1183_2_e_g();".
// 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_1183_2_e_g(:prec:=12)
    chi := MakeCharacter_1183_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1183_2_e_g();".
// 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_1183_2_e_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1183_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 42, 172, 178, 236, 86, 147, 48, 50, 10, 10, 2, 1]>,<3,R![1, 1, 7, 2, 33, 11, 55, -17, 47, -1, 8, -1, 1]>],Snew);
    return Vf;
end function;