
// Make newform 1183.2.e.i in Magma, downloaded from the LMFDB on 11 October 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_i();".
// 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_i();".
// 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 := [9, 0, 93, 0, 853, 0, 1050, 0, 952, 0, 334, 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], [-40872159, 0, 690027, 0, 6321845, 0, 17193085, 0, 6411671, 0, 1852321, 0, 246060, 0, 24498, 0], [0, 55035781, 0, -690027, 0, -6321845, 0, -17193085, 0, -6411671, 0, -1852321, 0, -246060, 0, -24498], [23829231, 0, 218482087, 0, 377649654, 0, 236538400, 0, 78484018, 0, 17340370, 0, 2170865, 0, 172099, 0], [-20454191, 0, 2790528, 0, 25566080, 0, 66564370, 0, 25929344, 0, 7490944, 0, 1000291, 0, 99072, 0], [5618970, 0, 441286822, 0, 514968195, 0, 494434675, 0, 174659083, 0, 45125185, 0, 5861765, 0, 539569, 0], [23341327, 0, -2881788, 0, -26402180, 0, -67022271, 0, -26777324, 0, -7735924, 0, -1048444, 0, -102312, 0], [0, -61326350, 0, 3480555, 0, 31887925, 0, 83757455, 0, 32341015, 0, 9343265, 0, 1246351, 0, 123570], [0, -48109836, 0, -441286822, 0, -514968195, 0, -494434675, 0, -174659083, 0, -45125185, 0, -5861765, 0, -539569], [-46491129, 0, -426433173, 0, -508646350, 0, -477241590, 0, -168247412, 0, -43272864, 0, -5615705, 0, -515071, 0], [366555, 0, 3361729, 0, 4460568, 0, 3722089, 0, 1287475, 0, 317920, 0, 40712, 0, 3598, 0], [0, 11237940, 0, 861328211, 0, 1029936390, 0, 988869350, 0, 349318166, 0, 90250370, 0, 11723530, 0, 1079138], [0, -19219314, 0, -176278948, 0, -219868809, 0, -196608895, 0, -68898667, 0, -17480377, 0, -2261795, 0, -205171], [0, 167644965, 0, 1094093084, 0, 1313214930, 0, 1125399698, 0, 390741062, 0, 98087264, 0, 12639109, 0, 1137374], [0, -359333319, 0, 12180871093, 0, 15781577445, 0, 15847248841, 0, 5653350919, 0, 1470474691, 0, 191553668, 0, 17689120]];
    Rf_basisdens := [1, 1, 14163622, 14163622, 42490866, 14163622, 42490866, 7081811, 14163622, 42490866, 14163622, 95271, 21245433, 3862806, 11588418, 127472598];
    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 := [[0, 0, 0, 0, 0, 0, 1, 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 := 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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 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, 1, 0, 0, -1, 1, 1, 0], [0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, -1, 0, 0, 0], [1, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, -2, 0, 0, -3, 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, 1], [0, -1, 0, -1, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -2, -1], [0, 0, 0, 1, 0, 0, 0, 0, -1, -1, 0, 0, 2, 1, -1, -2], [-2, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0, 3, 1, -2, -1], [-3, 0, 0, 0, 0, 0, -3, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, -3, 0, -1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, -1], [0, 0, 2, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, -1, -4, 0, 0, 3, 1, 0, 0], [0, 2, 0, -2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -2, 2], [-1, 0, 0, 0, -1, -1, -3, -1, 0, 0, 0, 1, 0, 0, 0, 0], [0, 6, 0, -1, 0, 0, 0, 0, -1, 7, 0, 0, -2, 1, 1, 2], [0, -1, 0, -3, 0, 0, 0, 0, 0, 3, 0, 0, -2, -2, -2, -1], [0, 2, 0, -1, 0, 0, 0, 0, -1, 3, 0, 0, 0, 1, 1, 2], [-1, 0, 0, 0, -3, 0, -1, 0, 0, 0, -2, -1, 0, 0, 0, 0], [1, 0, -2, 0, 1, 1, 5, -2, 0, 0, 2, 2, 0, 0, 0, 0], [5, 0, 2, 0, 5, 5, 3, 1, 0, 0, -2, -1, 0, 0, 0, 0], [0, -1, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, -1], [3, 0, 4, 0, 0, 5, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -6, 0, -1, -1, 5, -1, 0, 0, 6, 1, 0, 0, 0, 0], [0, 4, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, -2], [4, 0, -4, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 0, 0, -4, 0, 0, -2, -3, 0, 0], [0, -4, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 0, 0, -2, 0, 7, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 2, 0, -7, 0, 0, 0, 0, 0, 4, 0, 0, -9, 4, 4, 2], [0, 2, 0, -2, 0, 0, 0, 0, -1, 4, 0, 0, -1, 1, 2, 4], [0, 0, -2, 0, 0, 0, 7, 2, 0, 0, 2, -2, 0, 0, 0, 0], [-1, 0, 0, 0, 5, 0, -1, 0, 0, 0, 0, -2, 0, 0, 0, 0], [9, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, 2, 0, -7, -7, 1, -3, 0, 0, -2, 3, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 3, -3], [0, -6, 0, 1, 0, 0, 0, 0, 1, -7, 0, 0, 2, -1, -1, -2], [-13, 0, 0, 0, -5, 0, -13, 0, 0, 0, 4, 2, 0, 0, 0, 0], [6, 0, 2, 0, 0, -2, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, -1, 0, 0, 0, 0, -4, -3, 0, 0, -5, 4, 1, 2], [0, 7, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, -3], [0, 3, 0, -3, 0, 0, 0, 0, 0, 1, 0, 0, -6, 0, 6, 3], [4, 0, 4, 0, 4, 4, 15, 2, 0, 0, -4, -2, 0, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0, 2, 3, 0, 0, 7, -2, 1, 2], [0, 1, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, -1, 1], [8, 0, -4, 0, 0, -2, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, -2, 0, -3, -3, 3, -3, 0, 0, 2, 3, 0, 0, 0, 0], [3, 0, 0, 0, -5, 0, 3, 0, 0, 0, 4, -2, 0, 0, 0, 0], [-19, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 4, 0, 3, 0, 0, 0, 0, 0, 8, 0, 0, -1, 4, 8, 4], [-7, 0, 0, 0, -4, 0, -7, 0, 0, 0, -4, 2, 0, 0, 0, 0], [0, -6, 0, 3, 0, 0, 0, 0, -1, -9, 0, 0, 6, 1, -3, -6], [1, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, -8, 0, 3, 0, 0, 0, 0, 1, -11, 0, 0, 8, -1, -3, -6], [0, 4, 0, -1, 0, 0, 0, 0, 0, 5, 0, 0, 5, 0, 1, 2], [-13, 0, 0, 0, 3, 0, -13, 0, 0, 0, 2, -4, 0, 0, 0, 0], [5, 0, 8, 0, 5, 5, -11, 1, 0, 0, -8, -1, 0, 0, 0, 0], [0, 4, 0, -4, 0, 0, 0, 0, 0, 10, 0, 0, -8, -1, 8, 4], [0, 3, 0, -3, 0, 0, 0, 0, 0, 3, 0, 0, -6, 0, 6, 3], [-5, 0, 2, 0, 0, -7, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [13, 0, 0, 0, 5, 0, 13, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 7, -5, 0, 0, 2, -7, -5, -10], [0, 1, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, -1, 1], [0, -1, 0, 7, 0, 0, 0, 0, 0, -11, 0, 0, 8, -10, -2, -1], [-5, 0, 0, 0, -1, 0, -5, 0, 0, 0, 2, 5, 0, 0, 0, 0], [8, 0, 4, 0, 8, 8, 11, 3, 0, 0, -4, -3, 0, 0, 0, 0], [0, 8, 0, -3, 0, 0, 0, 0, 3, 11, 0, 0, -2, -3, 3, 6], [0, -3, 0, 7, 0, 0, 0, 0, 0, -5, 0, 0, 10, -6, -6, -3], [9, 0, 0, 0, -8, 0, 9, 0, 0, 0, 2, 1, 0, 0, 0, 0], [0, 2, 0, 6, 0, 0, 0, 0, 0, 8, 0, 0, 4, 1, 4, 2], [0, 1, 0, -1, 0, 0, 0, 0, 0, 11, 0, 0, -2, 4, 2, 1], [0, -14, 0, -6, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -2, 2], [-2, 0, -12, 0, 0, -6, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 2, 0, 0, 0, 0, -6, 0, 0, 0, 2, 6, -2, -4], [0, 3, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, -3, 3], [1, 0, 0, 0, 0, -3, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -2, 0, -1, -1, 13, 1, 0, 0, 2, -1, 0, 0, 0, 0], [3, 0, 4, 0, 3, 3, -5, 5, 0, 0, -4, -5, 0, 0, 0, 0], [0, -6, 0, 6, 0, 0, 0, 0, 10, -12, 0, 0, 14, -10, -6, -12], [0, -1, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 4, 2, -2, -1], [0, -2, 0, 1, 0, 0, 0, 0, 1, -3, 0, 0, 0, -1, -1, -2], [0, 8, 0, -1, 0, 0, 0, 0, -3, 9, 0, 0, -4, 3, 1, 2], [7, 0, 10, 0, 7, 7, 1, 1, 0, 0, -10, -1, 0, 0, 0, 0], [0, -3, 0, -3, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, -7, 7], [0, -1, 0, 5, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 1], [6, 0, -2, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, -3, 0, 0, 0, 0, 0, 11, 0, 0, 0, 4, -6, -3], [16, 0, 6, 0, 0, 8, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 7, 0, 0, 0, 0, 0, -3, 0, 0, 6, -8, 2, 1], [-3, 0, 0, 0, 1, 0, -3, 0, 0, 0, -10, 1, 0, 0, 0, 0], [3, 0, -2, 0, 3, 3, 5, -6, 0, 0, 2, 6, 0, 0, 0, 0], [0, -4, 0, -2, 0, 0, 0, 0, 0, 6, 0, 0, 2, -1, -8, -4], [-12, 0, -4, 0, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 2, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, -6, 6], [-17, 0, 0, 0, 5, 0, -17, 0, 0, 0, -10, 3, 0, 0, 0, 0], [-14, 0, -6, 0, -14, -14, -19, 0, 0, 0, 6, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -7, 0, -1, 0, 0, 0, 8, 6, 0, 0, 0, 0], [0, 11, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -3, 3], [0, 4, 0, -6, 0, 0, 0, 0, -9, 10, 0, 0, 1, 9, 6, 12], [0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 1, 0, 0], [-3, 0, 0, 0, -7, 0, -3, 0, 0, 0, 8, -1, 0, 0, 0, 0], [-1, 0, -2, 0, 0, -3, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 0, -2, 0, -7, -7, 5, -5, 0, 0, 2, 5, 0, 0, 0, 0], [0, 2, 0, -4, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 6, -6], [0, 8, 0, 5, 0, 0, 0, 0, -3, 3, 0, 0, 4, 3, -5, -10], [0, 5, 0, 2, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 1, -1], [0, 8, 0, -5, 0, 0, 0, 0, -11, 13, 0, 0, -4, 11, 5, 10], [-7, 0, 0, 0, 1, 0, -7, 0, 0, 0, 8, -3, 0, 0, 0, 0], [0, 2, 0, 4, 0, 0, 0, 0, 11, -2, 0, 0, -1, -11, -4, -8], [-13, 0, 0, 0, 5, 0, -13, 0, 0, 0, -8, 1, 0, 0, 0, 0], [-1, 0, -10, 0, -1, -1, -1, -1, 0, 0, 10, 1, 0, 0, 0, 0], [28, 0, 8, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -1, 1], [20, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, 0, -12, 0, 8, 8, 1, 0, 0, 0, 12, 0, 0, 0, 0, 0], [0, 7, 0, 6, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 3, -3], [0, -3, 0, 8, 0, 0, 0, 0, 0, -9, 0, 0, 11, 11, -6, -3], [-10, 0, -10, 0, 0, -2, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 4, 0, 0, 0, 0, 0, -22, 0, 0, 8, -13, -8, -4], [1, 0, -6, 0, 1, 1, 19, -2, 0, 0, 6, 2, 0, 0, 0, 0], [20, 0, 6, 0, 0, 16, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 5, 0, 0, 0, 0, 0, 3, 0, 0, 8, -6, -6, -3], [0, 3, 0, -2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 1, -1], [0, 8, 0, -3, 0, 0, 0, 0, 2, 11, 0, 0, -9, -2, 3, 6], [-3, 0, -12, 0, -3, -3, 11, 2, 0, 0, 12, -2, 0, 0, 0, 0], [31, 0, 6, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 12, 0, 0, 0, 0, 0, -8, 0, 0, 16, -9, -8, -4], [0, -14, 0, -2, 0, 0, 0, 0, -4, -12, 0, 0, 6, 4, 2, 4], [0, 1, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0, -7, 7], [0, 0, 0, 7, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, -2, 2], [12, 0, -4, 0, 0, -6, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [-25, 0, 0, 0, 2, 0, -25, 0, 0, 0, -8, 1, 0, 0, 0, 0], [0, -2, 0, 3, 0, 0, 0, 0, 3, -5, 0, 0, -2, -3, -3, -6], [0, -6, 0, 10, 0, 0, 0, 0, 0, -16, 0, 0, 16, -17, -12, -6], [15, 0, 0, 0, 15, 0, 15, 0, 0, 0, -12, -8, 0, 0, 0, 0], [0, -14, 0, 2, 0, 0, 0, 0, 9, -16, 0, 0, 7, -9, -2, -4], [-1, 0, 0, 0, 4, 0, -1, 0, 0, 0, 2, -10, 0, 0, 0, 0], [0, 4, 0, -5, 0, 0, 0, 0, -12, 9, 0, 0, -1, 12, 5, 10], [0, -4, 0, -2, 0, 0, 0, 0, 10, -2, 0, 0, 6, -10, 2, 4], [3, 0, 0, 0, 12, 0, 3, 0, 0, 0, -8, -6, 0, 0, 0, 0], [-1, 0, 2, 0, -1, -1, -7, -9, 0, 0, -2, 9, 0, 0, 0, 0], [0, -3, 0, 3, 0, 0, 0, 0, 0, -15, 0, 0, 6, -2, -6, -3], [0, -7, 0, -3, 0, 0, 0, 0, 0, -5, 0, 0, 4, 0, -14, -7], [8, 0, 2, 0, 0, 16, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 2, 0, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 6, 0, 5, 5, 31, 6, 0, 0, -6, -6, 0, 0, 0, 0], [3, 0, 0, 0, -1, 0, 3, 0, 0, 0, -2, 1, 0, 0, 0, 0], [-10, 0, 2, 0, 0, -4, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, -2, 0, -5, -5, -11, 8, 0, 0, 2, -8, 0, 0, 0, 0], [0, -3, 0, -1, 0, 0, 0, 0, 0, -23, 0, 0, 2, -4, -6, -3], [3, 0, 14, 0, 0, 7, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 6, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, -3], [0, -3, 0, -12, 0, 0, 0, 0, 0, -1, 0, 0, -9, -15, -6, -3], [3, 0, 8, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 7, -8, 0, 0, 13, -7, 0, 0], [9, 0, -6, 0, 9, 9, 25, -1, 0, 0, 6, 1, 0, 0, 0, 0], [0, 16, 0, -8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -4, 4], [0, -6, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, -6], [5, 0, 0, 0, -3, 0, 5, 0, 0, 0, -4, -2, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0, 0], [0, 16, 0, -4, 0, 0, 0, 0, -10, 20, 0, 0, -8, 10, 4, 8], [2, 0, -6, 0, 2, 2, -9, -2, 0, 0, 6, 2, 0, 0, 0, 0], [0, -14, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, -6], [0, 6, 0, 6, 0, 0, 0, 0, 4, 0, 0, 0, 4, -4, -6, -12], [-1, 0, 4, 0, -1, -1, -11, 0, 0, 0, -4, 0, 0, 0, 0, 0], [0, 13, 0, 2, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 5, -5], [-3, 0, 0, 0, -11, 0, -3, 0, 0, 0, -6, 8, 0, 0, 0, 0], [-17, 0, -2, 0, 0, -15, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -12, 0, -1, 0, 0, 0, 0, 6, -11, 0, 0, 1, -6, 1, 2], [0, 11, 0, 8, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 5, -5], [-6, 0, -18, 0, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-20, 0, -16, 0, -20, -20, 1, -1, 0, 0, 16, 1, 0, 0, 0, 0], [0, 9, 0, -8, 0, 0, 0, 0, 0, 35, 0, 0, -17, 7, 18, 9], [16, 0, 18, 0, 0, 16, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 2, 0, 0, 11, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, 4, 0, 0, 0, 0, 0, 12, 0, 0, 0, 1, 8, 4], [0, 8, 0, -5, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -2, 2], [11, 0, 0, 0, 13, 0, 11, 0, 0, 0, -4, -3, 0, 0, 0, 0], [-13, 0, 0, 0, -10, 0, -13, 0, 0, 0, 4, 5, 0, 0, 0, 0], [0, 21, 0, 4, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 3, -3], [0, 3, 0, 17, 0, 0, 0, 0, 0, -5, 0, 0, 14, 0, 6, 3], [0, -2, 0, 3, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, -4, 4], [16, 0, 16, 0, 16, 16, 19, 3, 0, 0, -16, -3, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, -4, -3, 0, 0, -15, 4, -3, -6], [-7, 0, -14, 0, -7, -7, -1, -2, 0, 0, 14, 2, 0, 0, 0, 0], [0, 6, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 10, 19, 12, 6], [-9, 0, 0, 0, 1, 0, -9, 0, 0, 0, 20, 1, 0, 0, 0, 0], [-9, 0, 0, 0, -10, 0, -9, 0, 0, 0, 16, 2, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 0, -10, 10], [11, 0, -2, 0, 11, 11, 3, -3, 0, 0, 2, 3, 0, 0, 0, 0], [0, 22, 0, -6, 0, 0, 0, 0, 0, 28, 0, 0, -12, 0, 6, 12], [-3, 0, -4, 0, -3, -3, 15, 9, 0, 0, 4, -9, 0, 0, 0, 0], [0, 15, 0, 4, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 1, -1], [8, 0, 8, 0, 0, 20, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, -3, 0, 0, 0, 0, 0, 17, 0, 0, -10, -6, 14, 7], [3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 8, -3, 0, 0, 0, 0], [0, 2, 0, 1, 0, 0, 0, 0, -5, 1, 0, 0, -8, 5, -1, -2], [-27, 0, 0, 0, -13, 0, -27, 0, 0, 0, 6, -1, 0, 0, 0, 0], [0, 0, 0, -13, 0, 0, 0, 0, 0, 8, 0, 0, -13, 8, 0, 0], [-11, 0, -4, 0, -11, -11, -45, -1, 0, 0, 4, 1, 0, 0, 0, 0], [0, 10, 0, -2, 0, 0, 0, 0, -22, 12, 0, 0, -10, 22, 2, 4], [31, 0, 0, 0, -13, 0, 31, 0, 0, 0, 10, -1, 0, 0, 0, 0], [-6, 0, 10, 0, -6, -6, 5, 0, 0, 0, -10, 0, 0, 0, 0, 0], [13, 0, 8, 0, 0, 7, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, -10, 0, 0, -18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, 0, 16, 0, 0, -3, 14, 0, 0], [0, 3, 0, 4, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, -5, 5], [0, -5, 0, -5, 0, 0, 0, 0, 0, -13, 0, 0, 0, -12, -10, -5], [-11, 0, 0, 0, 11, 0, -11, 0, 0, 0, 0, -5, 0, 0, 0, 0], [25, 0, 0, 0, 22, 0, 25, 0, 0, 0, -10, 2, 0, 0, 0, 0], [0, 8, 0, 3, 0, 0, 0, 0, 0, 10, 0, 0, -5, 16, 16, 8], [0, -8, 0, -5, 0, 0, 0, 0, -5, -3, 0, 0, 10, 5, 5, 10], [18, 0, 8, 0, 18, 18, 1, -2, 0, 0, -8, 2, 0, 0, 0, 0], [0, -12, 0, 1, 0, 0, 0, 0, 5, -13, 0, 0, -8, -5, -1, -2], [0, 1, 0, 13, 0, 0, 0, 0, 0, -3, 0, 0, 12, -4, 2, 1], [0, -1, 0, 21, 0, 0, 0, 0, 0, -9, 0, 0, 22, -12, -2, -1], [54, 0, 2, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [31, 0, 12, 0, 0, 5, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 6, 0, 1, 1, 17, -3, 0, 0, -6, 3, 0, 0, 0, 0], [15, 0, 0, 0, 11, 0, 15, 0, 0, 0, -2, 8, 0, 0, 0, 0], [9, 0, -4, 0, 9, 9, -33, 5, 0, 0, 4, -5, 0, 0, 0, 0], [0, -8, 0, 3, 0, 0, 0, 0, 0, -28, 0, 0, 11, 4, -16, -8], [17, 0, 0, 0, 6, 0, 17, 0, 0, 0, -18, -9, 0, 0, 0, 0], [5, 0, 0, 0, -11, 0, 5, 0, 0, 0, 8, -10, 0, 0, 0, 0], [0, -4, 0, 2, 0, 0, 0, 0, 1, -6, 0, 0, -1, -1, -2, -4], [25, 0, 0, 0, 8, 0, 25, 0, 0, 0, 2, -3, 0, 0, 0, 0], [0, 0, -12, 0, 0, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -13, 0, -20, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, -5, 5], [0, 1, 0, -11, 0, 0, 0, 0, 0, -5, 0, 0, -12, 6, 2, 1], [0, -4, 0, -6, 0, 0, 0, 0, 0, 4, 0, 0, -2, 3, -8, -4], [-2, 0, -10, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0], [18, 0, -12, 0, 0, 8, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -7, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 1], [-3, 0, -4, 0, -3, -3, 33, -4, 0, 0, 4, 4, 0, 0, 0, 0], [-9, 0, 0, 0, 9, 0, -9, 0, 0, 0, -12, -2, 0, 0, 0, 0], [0, 12, 0, -4, 0, 0, 0, 0, 0, 34, 0, 0, -16, 7, 24, 12], [0, 4, 0, -8, 0, 0, 0, 0, -18, 12, 0, 0, -14, 18, 8, 16], [7, 0, 4, 0, 7, 7, -1, 9, 0, 0, -4, -9, 0, 0, 0, 0], [0, -16, 0, -2, 0, 0, 0, 0, -3, -14, 0, 0, 13, 3, 2, 4], [0, 19, 0, 7, 0, 0, 0, 0, -21, 0, 0, 0, 0, 0, -7, 7], [0, 6, 0, 3, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 10, 0, 0, -2, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, -4, 0, 0, 0, 0, -20, 12, 0, 0, -14, 20, 4, 8], [0, -12, 0, 8, 0, 0, 0, 0, 0, -26, 0, 0, 20, -7, -24, -12], [0, -7, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, 3], [-19, 0, 2, 0, 0, -9, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [5, 0, 0, 0, -8, 0, 5, 0, 0, 0, 22, 3, 0, 0, 0, 0], [0, 15, 0, -7, 0, 0, 0, 0, 0, 31, 0, 0, -22, 22, 30, 15], [-21, 0, 0, 0, -12, 0, -21, 0, 0, 0, -8, -5, 0, 0, 0, 0], [0, 29, 0, 9, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, -1], [-55, 0, -6, 0, 0, -9, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 8, 0, 0, 0, 0, 0, -22, 0, 0, 14, -15, -12, -6], [-1, 0, 0, 0, 23, 0, -1, 0, 0, 0, -2, -6, 0, 0, 0, 0], [12, 0, 12, 0, 0, 8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 9, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 1], [-26, 0, -4, 0, 0, -6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, -14, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, -3], [1, 0, 0, 0, 1, 1, 33, 9, 0, 0, 0, -9, 0, 0, 0, 0], [0, 10, 0, -14, 0, 0, 0, 0, -20, 24, 0, 0, -22, 20, 14, 28], [0, 30, 0, 2, 0, 0, 0, 0, -10, 28, 0, 0, -6, 10, -2, -4], [0, 6, 0, 9, 0, 0, 0, 0, 19, -3, 0, 0, -2, -19, -9, -18], [-25, 0, 0, 0, 2, 0, -25, 0, 0, 0, 8, -2, 0, 0, 0, 0], [-5, 0, 4, 0, -5, -5, -27, 1, 0, 0, -4, -1, 0, 0, 0, 0], [0, -11, 0, -13, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, -5, 5], [0, 16, 0, 12, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, -2, 2], [-5, 0, -8, 0, -5, -5, 1, 2, 0, 0, 8, -2, 0, 0, 0, 0], [10, 0, 16, 0, 0, -10, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 4, 0, -15, 0, 0, 0, 0, 0, 34, 0, 0, -19, 18, 8, 4], [0, -4, 0, -5, 0, 0, 0, 0, 0, -12, 0, 0, -1, 0, -8, -4], [0, -14, 0, -18, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 1, 0, 0, 0, 0, 0, 43, 0, 0, -6, 2, 14, 7], [3, 0, 8, 0, 3, 3, 5, 10, 0, 0, -8, -10, 0, 0, 0, 0], [-7, 0, 0, 0, 9, 0, -7, 0, 0, 0, -6, -1, 0, 0, 0, 0], [0, -17, 0, -16, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -5, 5], [34, 0, -8, 0, 0, -12, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, 0, 15, 0, 0, 0, 0, 17, -21, 0, 0, 36, -17, -15, -30], [0, 0, 0, 5, 0, 0, 0, 0, 0, -12, 0, 0, 5, 2, 0, 0], [-27, 0, -18, 0, -27, -27, -5, -1, 0, 0, 18, 1, 0, 0, 0, 0], [27, 0, 0, 0, 10, 0, 27, 0, 0, 0, 6, 8, 0, 0, 0, 0], [0, -2, 0, 1, 0, 0, 0, 0, 21, -3, 0, 0, 8, -21, -1, -2], [0, 6, 0, 9, 0, 0, 0, 0, -22, 0, 0, 0, 0, 0, -2, 2], [27, 0, -12, 0, 0, -13, 0, -13, 0, 0, 0, 0, 0, 0, 0, 0], [-21, 0, 0, 0, 6, 0, -21, 0, 0, 0, 10, -6, 0, 0, 0, 0], [33, 0, 0, 0, 7, 0, 33, 0, 0, 0, -10, -5, 0, 0, 0, 0], [0, -4, 0, -12, 0, 0, 0, 0, 0, -20, 0, 0, -8, 1, -8, -4], [0, 26, 0, -3, 0, 0, 0, 0, 10, 29, 0, 0, -15, -10, 3, 6], [2, 0, -16, 0, 0, -24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 0, 0, 0, -6, 0, 7, 0, 0, 0, -10, 1, 0, 0, 0, 0], [7, 0, 6, 0, 7, 7, 19, 3, 0, 0, -6, -3, 0, 0, 0, 0], [-13, 0, -22, 0, 0, -3, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0], [-19, 0, -2, 0, -19, -19, -7, -4, 0, 0, 2, 4, 0, 0, 0, 0], [0, -16, 0, 1, 0, 0, 0, 0, -3, -17, 0, 0, 4, 3, -1, -2], [-20, 0, -10, 0, -20, -20, 3, 0, 0, 0, 10, 0, 0, 0, 0, 0], [0, 10, 0, -12, 0, 0, 0, 0, 0, 20, 0, 0, -22, -9, 20, 10], [-11, 0, 0, 0, 11, 0, -11, 0, 0, 0, -20, 4, 0, 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_i();".
// 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_i(:prec:=16)
    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_i();".
// 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_i( : 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, 0, 93, 0, 853, 0, 1050, 0, 952, 0, 334, 0, 85, 0, 11, 0, 1]>,<3,R![4, -20, 114, 62, 67, 6, 11, 2, 1]>],Snew);
    return Vf;
end function;