
// Make newform 1183.2.a.q in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1183_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_1183_2_a_q();".
// 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_a_q();".
// 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 := [29, 183, -157, -542, 262, 562, -199, -246, 80, 46, -15, -3, 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], [25085, 16187, 193411, 77279, -346540, -110863, 161452, 39197, -29197, -5010, 1816, 202], [10456, -56958, 193411, 91908, -346540, -110863, 161452, 39197, -29197, -5010, 1816, 202], [-14765, -263443, 165757, 573947, -239507, -384542, 125845, 98110, -26136, -9639, 1812, 266], [-73281, -263443, 180386, 573947, -239507, -384542, 125845, 98110, -26136, -9639, 1812, 266], [63537, 204863, -374170, -310269, 447600, 95671, -212101, 13387, 41570, -8033, -2788, 721], [-16033, -384615, -264148, 761552, 506165, -469197, -245916, 135962, 45138, -19049, -2807, 1025], [60552, 361943, -446758, -620810, 705799, 301924, -408596, -23075, 90086, -9303, -6474, 1181], [-68266, -425386, 225077, 930739, -201544, -690923, 76988, 214356, -13256, -29069, 825, 1429], [5396, -144501, -345129, 415204, 632025, -391458, -364728, 175064, 79215, -34485, -5633, 2354], [5226, 34553, 648376, -443637, -1236319, 627337, 725177, -324031, -158261, 66841, 11261, -4628]];
    Rf_basisdens := [1, 1, 14629, 14629, 14629, 14629, 14629, 14629, 14629, 14629, 14629, 14629];
    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_a();"
function MakeCharacter_1183_a()
    N := 1183;
    order := 1;
    char_gens := [339, 1016];
    v := [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_1183_a_Hecke(Kf)
    return MakeCharacter_1183_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], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, 0, -1, 1, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, 0], [2, -1, 0, 0, -1, -1, 1, -1, 0, -1, 3, 0], [2, -1, 0, 1, 0, 0, 0, 1, 0, -2, 1, -1], [2, 1, 0, 0, -2, -1, 0, -1, 2, 1, 2, 0], [2, -1, -1, 1, 1, 1, 0, 0, 3, 2, 0, 1], [2, 1, 0, 0, -2, 1, 2, -2, -2, -1, 0, -1], [1, 1, 1, 1, 2, 1, 0, 0, 0, -1, -1, 2], [-4, 1, 1, -2, 2, 0, 1, -1, -1, -1, -4, -1], [1, 1, -2, 2, 1, 1, 0, 0, 3, 2, 2, 2], [5, 0, -1, -1, -2, 0, -2, 0, 0, 2, 3, 1], [3, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, -1], [-1, 2, 0, -2, 2, -2, -1, 0, -2, 0, 0, 0], [-2, 0, 0, -1, 2, -1, 2, 0, -3, -4, -1, -2], [-3, 1, 1, -1, 0, 1, -2, 1, -1, 0, -1, 2], [1, -2, 2, -2, 0, 2, -1, 1, -1, -3, -1, 0], [0, 0, 2, 1, 1, -1, 0, 0, -1, -4, 0, -2], [2, 0, 1, 1, 1, 0, 2, -1, 0, 0, 0, 2], [1, 0, 3, -2, 1, 1, 1, 2, -2, -3, -1, -2], [-3, 1, 0, 0, -1, -3, -2, 1, 2, -1, 1, -2], [-4, 4, -1, -1, 2, 0, -1, 1, 1, 3, -3, 1], [3, -1, 1, 3, -3, 1, 1, 2, -1, -4, 4, 1], [1, 0, 2, -2, -2, 0, -2, 2, -1, -1, -3, 1], [-2, -1, 0, -2, -4, -1, -1, 0, -1, 2, -2, -2], [8, -1, 1, 0, 1, 0, 1, 1, 0, -2, 0, 2], [-2, 2, -2, -2, 0, -2, -2, -1, 2, 0, 3, -1], [0, 1, 0, -2, -3, 1, 1, -2, 1, 0, -4, 0], [1, 2, -1, 2, -4, 1, -1, 1, 1, 2, 4, 0], [1, 0, -1, 3, -2, -2, 2, 0, 2, 0, 0, 1], [0, 2, 2, 2, 5, 2, 3, 0, -6, -2, -3, 0], [-6, 2, 2, 4, 2, 0, 0, 1, 0, -4, -5, 1], [-3, -1, -2, -3, -2, -2, -3, 0, 5, 3, 5, -1], [2, 0, -1, 1, 1, 0, 4, 1, -1, -5, 0, -1], [7, -2, 0, 3, -3, -1, -2, 4, 2, -1, 5, 1], [4, 0, 1, 1, 4, 2, -1, 3, -3, -3, 1, -1], [2, 0, 2, 3, 2, 3, 3, -3, 1, 4, -6, 1], [0, -4, -2, 0, -7, 2, -1, 0, -1, -1, 1, 1], [-1, 6, 3, 1, 0, 0, 2, 2, -3, -3, 0, 0], [-1, -1, 0, 0, 7, 3, 4, -2, -2, 1, -6, 1], [-2, 0, 0, -2, -5, -4, -1, 2, -1, -1, 5, -1], [-7, -4, 1, -3, 11, -2, 3, 3, -4, -8, -6, -2], [3, -5, -3, 2, -2, 4, 0, -1, 2, 2, -1, 2], [-1, 1, 1, -2, 0, 2, -2, 4, -3, -1, -3, 0], [-4, -1, -2, -1, 3, -4, -3, 2, -2, -2, 4, 0], [5, -4, 1, -1, -4, -2, 0, -1, -5, -5, -1, -5], [11, 2, 0, 4, 2, 0, -1, 1, 4, 2, 5, 1], [-8, -6, 1, -2, 1, 3, -2, 3, -2, -3, -4, -3], [2, -1, -2, 2, -4, 5, -2, 0, 0, 7, -5, 1], [0, -2, 4, 0, 3, 4, 1, 0, -3, -1, -9, -3], [9, 0, 0, -1, -4, 1, -5, -1, 2, 5, -3, -2], [-2, 0, 4, -1, 0, 1, -1, 0, 2, 1, -1, -1], [-3, -2, -1, 0, 3, -1, -1, -1, -1, 0, -6, -4], [2, 1, -2, 0, 5, -1, -1, 3, -6, -1, 5, 0], [-2, 6, -3, 3, -4, -2, -4, -2, 5, 7, 5, 0], [-6, -1, -5, -4, -8, -2, -3, 1, 4, 4, 2, 2], [-2, 2, 4, 2, -1, 0, 1, 3, -2, -2, 4, 7], [3, 0, 4, -3, 0, 3, 1, 2, -9, -6, -4, -2], [1, 1, 1, 5, 5, -3, 5, 1, -2, -9, -5, -1], [-1, 2, -1, -2, 3, -1, -1, 4, -6, -5, 7, 0], [8, -6, 0, 1, -5, 5, -3, -3, 1, 6, -8, -1], [1, -3, 3, 5, 3, -1, 6, 0, -6, -9, -7, -4], [-3, -1, -2, 4, 8, 3, 5, 0, -2, -1, 6, 1], [-4, -3, -3, -2, 7, 4, -4, 0, 1, 3, -10, 2], [4, 1, -5, 3, 1, 1, -2, -4, 7, 8, 5, 5], [6, 2, 1, 2, -5, -3, -2, -1, 1, 4, 2, 0], [-5, 2, -6, -1, 5, -3, 0, -1, 7, 6, 2, 1], [-10, 1, 4, 1, 1, 0, 3, -2, 2, 2, -3, -2], [-4, 7, -2, 2, -3, 3, 1, 0, -3, 0, 4, 0], [-7, 6, -8, 0, -5, -4, -6, -1, 12, 16, 7, 5], [-1, 3, 1, 0, 2, 0, 2, 0, -4, 4, -7, 1], [6, -3, 4, -5, 2, -2, 0, 1, -2, 0, -4, -1], [-2, -1, 2, -4, -1, -3, -2, -1, -3, -4, 2, -1], [-3, 6, -2, 4, -1, -4, -6, -1, 10, 8, -2, 3], [-7, 1, -4, 3, 3, -2, -2, -1, 8, 12, 7, 5], [1, 3, 5, 1, -5, 3, 1, 2, -5, -2, 0, -1], [5, -2, -3, -5, -2, -2, -2, 1, -7, -3, 5, -5], [7, 4, 1, -1, -17, 0, -1, -5, 6, 6, 4, 0], [-9, 4, 2, -2, 2, 4, -4, -4, -3, 1, 3, -3], [1, 1, 2, -6, -3, -5, 2, -3, -4, -3, 3, -2], [11, -5, -2, 1, -4, 0, 3, -2, 3, 1, 21, 1], [3, 5, -6, 6, -3, -1, 2, -1, 3, 0, 3, -1], [-12, -5, 1, -3, 7, 1, 0, 5, -2, -3, -10, -5], [-4, -4, 0, -4, -2, 6, 1, -3, -4, 0, -2, 3], [-7, 4, -7, -1, 3, 0, 3, -2, 1, 5, -9, 0], [3, 7, 0, 0, 0, 3, -2, 0, -3, 4, 0, 4], [9, -5, -1, -2, 1, 0, -3, -1, 5, 5, 7, 3], [12, -5, 3, 2, 0, 4, 1, -1, -4, -6, -8, -4], [2, -5, -1, -8, -6, 2, -4, 0, -2, 2, 2, -3], [-10, -2, -1, 2, 9, -5, 4, 2, 2, -3, 0, -2], [-4, -2, -2, -3, 5, 5, -3, 0, 5, 8, -13, 2], [-2, -1, 3, -2, 10, -2, 3, -6, -3, -1, -1, -4], [-8, 1, 1, 7, 5, 1, -2, 3, 0, -5, -13, 3], [10, 0, 1, 1, -7, -4, -2, 2, 7, -3, 9, -2], [2, -5, 0, -5, -8, 0, 0, 0, -7, -7, 3, -1], [2, -4, -1, -2, -2, 1, -1, 5, 2, 1, 10, -3], [-2, 1, 1, 6, -7, 2, -3, -5, 14, 16, -2, 6], [-1, 7, -2, -1, -8, 2, -3, 0, 5, 13, 0, 1], [11, 1, -4, 2, -4, -3, 3, -6, -2, -9, 1, -5], [5, 1, -5, 0, 3, 0, -6, -2, -3, 7, -3, -2], [-1, 5, -2, -2, -12, -3, -5, 0, 5, 6, 5, -4], [0, -4, 3, -1, 8, 6, -1, 4, 2, 2, 0, 1], [1, -2, -5, 1, -1, 2, 3, -5, 6, 8, -2, -2], [-8, 5, 4, 0, -6, 1, -8, 2, 6, 11, -1, 1], [-1, 0, -1, 1, 7, -4, 5, -2, -6, -6, -3, -1], [-9, -4, 5, -1, 13, 2, 7, 3, -8, -12, -8, -4], [-20, 3, 3, 0, 11, -4, -4, 6, -2, 2, -8, 1], [11, 3, -2, -1, 6, 0, -1, 0, -6, 2, -5, 0], [-5, -3, 5, -3, 6, -3, 6, 2, -4, -5, -11, 0], [-24, 1, 0, -2, 16, -1, -4, 3, 3, 4, -14, -1], [4, -3, 2, -2, -7, -1, -10, -2, 2, -1, 1, -3], [18, -5, -3, 3, -2, 1, 7, -4, 6, 3, 4, 6], [17, 5, -2, 4, -12, -5, -1, 2, 2, 1, 13, 1], [-5, -1, 2, -2, 9, 1, 2, 1, -12, -7, -3, 0], [-5, 0, 0, 0, 14, -2, -3, 10, 1, 3, -10, 3], [-3, -6, 6, -4, 2, -8, 5, 2, -1, -13, 2, -3], [-2, 5, -4, 9, 0, -2, 0, 1, 6, 2, 18, -1], [10, -4, 0, 4, -10, 6, 5, 1, -2, 2, 0, -3], [6, 1, 0, -1, 14, 0, 8, -5, -2, -2, -6, 1], [-6, 3, 1, -3, 3, 1, 0, -4, 5, 6, -2, -1], [4, 5, 6, -2, -6, 3, 6, -3, -3, 0, -4, -1], [3, -9, 1, -4, -17, -2, -5, -4, 2, 6, 1, -5], [-1, -2, 6, -5, -1, 5, 2, 0, -11, -10, -5, -4], [-1, 12, -5, 6, -6, -7, -2, 0, 9, 6, 16, 3], [3, 4, 3, -4, -5, 5, -3, 0, -8, 7, -3, -8], [-8, 6, -1, -4, 1, -1, 0, -5, 1, -2, -2, 2], [-7, -3, 4, 1, -5, 2, -10, 3, 7, 1, -11, 0], [-12, 4, -2, 5, 0, 1, 4, -4, -2, 1, 8, 1], [-8, -3, 4, -4, -6, 1, 5, -3, -8, -7, -3, -6], [6, -7, 1, 3, 1, -5, 2, 0, 8, -3, 0, 0], [-8, -6, -1, 3, 10, 0, 1, 2, -4, -2, 2, 5], [1, 6, 1, 8, 7, -3, -3, 3, 3, 4, -6, 0], [-2, 1, -3, -4, -13, 0, -8, -1, 5, 3, 3, 5], [-10, 3, -6, -9, -2, -4, -2, -1, 0, 2, 8, -1], [7, -1, -4, -3, -3, 0, 0, 9, -3, -3, 1, 2], [11, -4, -3, 0, 5, -1, 1, 1, 7, -6, 17, 2], [-3, 1, -1, -5, -3, 1, -5, 1, 14, 11, -1, 9], [3, 0, 4, 1, 12, -1, 9, 0, -1, -4, -11, 0], [-8, 4, 6, -2, -2, 2, 0, -2, -4, 2, -3, 4], [13, -3, 0, 7, -6, 0, 3, -6, 3, -7, 14, -5], [6, 1, -3, 8, 1, -2, 4, 2, 4, 0, -2, -3], [13, -11, -5, -2, -8, 8, -2, 2, -5, -5, -5, -4], [3, 7, 0, 3, -4, -4, -1, 3, -6, -2, 2, 1], [18, 3, -3, -5, -4, -1, -3, -1, 7, 6, 4, 2], [-4, -2, -7, 3, 9, 0, 0, 4, 5, 11, 1, 6], [3, 6, 3, 3, 7, 4, 10, -1, 3, -7, -3, 7], [2, 5, 3, -1, 8, -1, 2, 0, 1, 12, -8, 3], [2, -1, 1, -6, -2, -8, -1, 0, 3, -3, -11, -2], [11, -8, -1, 7, -7, -2, 2, 0, -6, -12, 4, -9], [11, -5, 1, 3, -2, 3, 0, 5, -5, 2, 2, 0], [0, -3, 1, 0, 9, 10, 4, -5, 5, 1, -4, 3], [-9, 1, -2, 5, -16, 4, -7, 6, 5, 7, 2, 7], [9, -9, 5, 2, -11, -2, 7, -6, 0, -12, 7, -3], [1, -2, -3, 0, -9, -3, -13, 5, 1, 4, 6, 2], [8, 11, 3, -2, 3, -2, 6, 0, -2, 0, 4, 1], [7, -3, -9, -4, -3, 0, 1, -5, 5, 11, -3, 7], [2, 9, 3, 1, 14, -3, 9, -4, -6, -1, -8, 6], [15, 0, 1, 1, 1, 2, -1, 1, -4, -2, 5, -10], [-6, -4, 1, 0, 5, 9, -2, -2, -2, -1, -4, 2], [4, -3, -4, -1, 0, 8, 2, 4, -1, -1, 5, 9], [-4, -15, -2, -1, 8, 6, -5, 3, 5, 5, -7, 2], [10, -1, -5, 1, -4, 1, -5, -1, 11, 8, 10, 10], [-21, 1, 4, 0, 16, -3, -1, 5, -6, -3, -8, 2], [7, 6, -2, 2, -3, -4, 12, -8, -3, -3, 5, -5], [1, 11, 0, -6, 3, -3, -4, -3, 11, 18, 3, 9], [18, 9, 7, 2, -5, 6, 8, 0, -6, -10, 4, 1], [-4, 5, 7, -5, -4, 1, -1, 3, -7, 0, 14, -4], [-2, 3, 4, 1, -5, -2, 3, -7, 7, 9, 1, -2], [-1, -4, -1, 2, -10, 5, 0, -5, 5, 4, 15, 4], [14, 14, -9, 4, -3, -9, -2, 1, 11, 8, 13, 6], [-19, 8, -3, 0, 7, -1, -5, 5, -3, -2, -6, -10], [5, 4, -7, 9, -7, 8, -1, -3, 6, 8, 12, 8], [-6, 1, -4, -3, -2, 0, -8, 6, 0, 2, 1, 0], [12, 0, 4, 8, -11, 4, 4, -2, 2, -6, 10, 0], [-10, 3, 2, -3, -1, -4, -3, -4, 2, -4, -14, -12], [1, -9, -2, 11, 9, 2, 6, -1, 9, -11, 11, 4], [3, 4, -1, 1, -7, -12, 2, -1, 9, -5, 7, 5], [-17, -9, 1, -5, -4, -1, -4, -1, 1, -2, 12, -4], [-1, -14, 8, -4, -3, 2, 0, 2, -16, -14, -1, -12], [-25, -6, 0, -8, 15, 2, -4, 2, -7, -5, -15, -11], [11, 6, 1, 1, 6, -4, 6, 4, 0, -6, 3, -3], [-9, -6, 0, -4, -20, 0, -1, -2, 8, 2, 0, -2], [2, 4, 2, -1, -3, -1, 4, -2, 9, 8, 8, 12], [-14, 10, -1, -4, 5, -9, -8, -1, 9, -2, 4, -2], [-28, 1, -3, -1, 13, -1, -15, 1, 13, 10, -8, 6], [-10, 0, 1, -5, 8, -6, -9, 8, -8, -4, -10, -1], [2, -4, -9, 11, 11, -2, 6, 0, 7, -1, 7, -2], [3, 8, 9, 9, 7, 4, 7, -6, -1, -3, 4, 0], [-17, 7, 0, 2, 2, -9, 5, -7, 9, 4, 3, 1], [7, -5, -1, -5, 1, -5, 7, -3, -2, -9, 17, -1], [-19, 2, 0, -1, 21, -3, -2, 7, 5, 2, 4, 3], [16, 1, 10, 0, 5, -5, -1, 2, -3, -10, 8, -2], [-13, -11, -3, -14, 5, -10, -7, 0, -2, -6, -11, -9], [-1, -13, -3, -5, -4, 7, -3, 4, -7, -6, -2, -13], [3, 12, -4, -3, 5, 5, -4, -4, -6, 5, -23, 3], [-16, 0, 1, -6, -5, -1, -2, -2, -8, -11, -3, -8], [4, 9, -2, 6, -3, -11, 1, -4, 8, 3, 8, 7], [-3, -3, 7, -3, -10, 7, -4, 0, 6, 1, 1, 0], [10, 2, -10, 2, -3, 0, 1, -2, 9, -1, 25, -1], [6, -4, 9, 5, 2, 4, 9, -10, 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1, 6, 8, 2, 10, -2, 6, 14], [22, -9, 5, 10, 3, 10, 5, 4, 1, -5, -23, 2], [37, 6, 9, 4, 2, 15, 12, -2, -21, -4, -2, -1], [8, -15, 14, 2, 22, 21, -4, -4, -2, 3, -35, 1], [19, 9, -13, 7, -6, -1, -4, -9, 13, 4, 21, 6], [1, 21, 11, -1, -11, 9, 10, -6, -6, 5, 3, 8], [10, -25, 12, 1, 3, 4, 15, 5, -15, -23, 5, -10], [26, -17, 13, -11, 6, -3, 15, 2, -21, -34, -3, -7], [-14, -2, -1, 5, 3, 2, -12, 2, -13, -7, -1, 4], [18, 0, 1, 0, -7, -1, -12, -12, 6, 9, -7, -12], [7, -2, -13, 5, -24, 4, -4, -3, 21, 11, 9, 1], [20, 5, -3, -10, -17, -2, 2, -16, -8, 10, 6, -13], [-4, -7, 5, -4, -5, -6, 1, -6, 11, 5, 19, -6], [1, 10, -2, -5, 17, -3, 3, 2, -9, -18, 6, -2], [18, 3, -10, 18, -2, -11, 0, 1, 15, -4, 16, 13], [10, 4, 22, -7, 8, 5, -2, 11, -10, -13, 1, 0], [-3, 2, -1, -14, -1, 3, -3, 7, 3, 4, 0, 4], [26, 2, -3, 12, 5, -1, 16, -5, -10, -19, 10, 5], [6, 10, 1, 0, -10, -7, 1, -9, 4, -9, 14, 5], [14, 0, 13, -2, -7, 17, -1, 5, -6, -5, -5, 9], [-6, 5, -10, -5, -23, 2, -3, -6, -2, 8, 19, 0], [12, 6, -3, 10, 9, 9, 4, -2, 10, 17, 19, 16], [-34, 14, -7, -5, -9, -6, 4, -5, 4, 4, 2, 2], [1, 6, 11, 4, 23, -9, 19, -7, -29, -30, -10, -10], [-7, -10, 8, 8, -15, -2, 4, 8, -9, -23, 11, -11], [0, 4, 8, 3, 4, -13, 2, 3, -16, -17, 17, -8], [-16, -13, -15, -6, -14, -8, -3, 1, -3, -15, 14, -17], [-20, 5, -11, 7, -14, -9, -9, 5, -3, 10, 4, -2], [-23, -5, 0, 3, 36, 0, 7, -1, 3, -7, 0, 0], [-20, -9, -14, -1, 25, 14, 1, 7, 11, 21, -25, 12], [36, -15, -8, 17, 8, 2, 12, 2, -3, 3, 13, 1], [-12, -1, 2, -4, 2, -7, 12, 5, -9, 0, 18, -3], [4, 10, -6, 12, 3, 6, -8, 8, 4, 8, 16, 6], [17, -1, 10, -10, 7, 9, 6, 8, -13, -14, 2, -4], [26, 6, 4, 0, -17, 12, 1, -6, 9, 13, 21, 17], [1, 2, 7, 3, -36, 12, 0, 9, -5, -5, 1, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1183_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1183_2_a_q();".
// 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_a_q(:prec:=12)
    chi := MakeCharacter_1183_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_1183_2_a_q();".
// 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_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1183_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![29, -183, -157, 542, 262, -562, -199, 246, 80, -46, -15, 3, 1]>,<11,R![-2701133, -900676, 1617506, 505685, -339321, -112394, 30684, 12005, -994, -613, -13, 12, 1]>],Snew);
    return Vf;
end function;