
// Make newform 4598.2.a.bx in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4598_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_4598_2_a_bx();".
// 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_4598_2_a_bx();".
// 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 := [-4, 20, -16, -32, 26, 16, -10, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-10, 12, 12, -8, -2, 1, 0, 0], [4, -28, 6, 22, -7, -3, 1, 0], [10, -28, 4, 22, -7, -3, 1, 0], [4, 2, -30, 6, 22, -7, -3, 1], [12, -34, -14, 40, 7, -12, -1, 1], [-8, 38, 24, -58, -20, 21, 3, -2]];
    Rf_basisdens := [1, 1, 2, 2, 2, 2, 2, 2];
    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_4598_a();"
function MakeCharacter_4598_a()
    N := 4598;
    order := 1;
    char_gens := [3269, 3631];
    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_4598_a_Hecke(Kf)
    return MakeCharacter_4598_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 := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, -1, 0], [0, 0, 0, -1, 0, 0, 0, 0], [-2, 1, 0, 1, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 1, -1, 0, -2, -1, 0], [1, 0, 0, 2, 0, 2, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, -2, -1, 0, -1, 0, -1], [-2, 2, -1, -1, -1, 0, 2, 0], [1, 0, -1, 0, 2, 0, 2, 0], [-4, 0, -1, -1, 1, 0, 0, 2], [-3, 0, 1, 1, 0, -2, -1, 3], [-2, 0, -3, -1, -2, 2, 4, -1], [-1, 1, -3, 0, -1, 0, -2, -1], [-4, 1, 1, 1, 3, 0, -4, 0], [5, 1, -1, -1, -2, 0, 1, 0], [6, -2, -3, -1, -4, 2, 6, -2], [-3, -1, 3, 3, 0, 0, -1, 3], [1, -1, 2, 0, -3, 0, -1, -1], [-2, -1, 1, 1, -3, 2, 0, 1], [3, -1, -3, -3, -2, -1, 9, -2], [0, 1, 0, 0, 2, 1, 1, -1], [-2, -3, 1, 0, 1, -3, 1, -1], [-2, 1, 2, 0, -2, -2, 2, -1], [6, -2, 2, 3, -2, -1, 4, -1], [1, -1, -2, 0, -3, 3, 3, -1], [1, -5, -1, 1, 0, 2, 3, -1], [4, -2, 3, 0, 3, 1, -1, -1], [-1, 0, 0, 2, -3, 1, 1, -1], [-2, 1, 4, -1, 2, -2, -1, 4], [3, -6, 1, 2, -6, 2, 3, -1], [0, -1, -3, -1, -5, 2, 5, -1], [-5, 0, 1, 1, -1, 1, -3, 3], [-4, 4, 2, -2, 0, -1, 2, 3], [-5, 3, -1, -1, 2, -2, -3, -2], [-4, 4, 0, -1, -2, 0, -4, -2], [5, -3, -2, 4, -3, 4, -2, -1], [1, -3, -6, 0, -1, 2, 9, -3], [-1, 5, -5, 1, 0, 2, 1, 2], [3, -6, 2, 5, -1, -1, 0, 1], [-3, -4, -3, -3, -2, 2, 7, 0], [3, 2, -7, -2, 2, -1, -5, -1], [0, -4, -1, -2, -1, -1, 7, -4], [4, 0, -4, 2, 2, 2, -4, 0], [2, -3, -4, 1, -6, 2, 8, 0], [-9, 4, 0, -1, 1, -1, -4, -4], [5, 1, 1, 1, -2, 1, 5, -4], [-5, 4, 7, -3, 0, -1, -7, 3], [-2, -2, -3, -2, 0, 5, -1, -2], [-5, 5, -1, 0, 0, 2, 5, 1], [-2, 2, 6, 1, -1, -3, -5, 4], [3, 0, 6, -1, 7, -1, -10, 1], [-1, 1, -4, -5, 0, 1, -2, 1], [0, 2, 7, 4, 7, -1, -5, -2], [-7, -4, 6, -2, 3, -6, -9, -1], [-16, -3, 3, -1, -1, -4, -10, 6], [-7, 3, 1, -2, 1, -2, -2, 5], [-26, 6, 2, -5, 2, -2, -11, 5], [1, -2, -5, 2, -4, 4, 2, -3], [-7, 4, 7, 2, 6, -4, -5, 5], [-7, 6, -7, -3, 0, 2, 5, -4], [7, -1, 1, -3, 0, -4, 9, 4], [-7, 0, 2, -3, -2, -7, 2, 3], [-3, 4, -2, 8, 8, 2, -5, 0], [-2, -11, 6, -1, -2, -1, -1, -1], [0, 3, 2, 0, 4, 3, -4, 0], [-14, -2, 1, 0, 3, -3, -5, -4], [5, 1, 0, -5, 2, 3, 6, -1], [0, -10, -4, 4, 0, 0, 7, 1], [14, -4, -2, 2, 0, -2, 17, -2], [16, -2, -6, 7, 3, 11, 9, -4], [2, -2, -1, -10, -3, -3, -2, 0], [-18, 12, 4, -1, 6, -5, -13, 3], [8, -10, 9, 2, 1, 0, 9, -1], [11, -1, -6, -2, -7, 6, 17, -5], [-14, 0, 5, 3, -4, -3, 0, 5], [4, 0, 1, 2, 2, -1, -9, 4], [19, -2, 3, 2, -4, 2, 8, 1], [-5, -3, 1, 2, 2, 8, 4, 0], [2, 4, -3, -3, -2, 4, 2, -3], [-1, 4, -6, 0, 5, -2, -1, -4], [13, 1, 5, -1, 8, 0, 1, 0], [-12, -6, -2, 1, -4, 0, -5, -2], [-9, -7, 4, 4, 1, -5, -5, 5], [-8, 2, 7, -2, 7, -1, -6, 7], [-2, 10, 4, 5, 10, 0, -9, 2], [-1, -8, -4, 5, 3, 6, 7, 0], [-6, -6, -2, 3, 2, 2, -6, 2], [9, -5, -4, -1, -11, 6, 21, -3], [-9, 4, 3, 0, 12, -8, -21, 1], [-3, -6, 8, -3, 0, -13, -6, 5], [-17, 3, -3, -5, 2, -6, -13, 4], [3, 0, 1, 2, 0, -6, 5, 9], [4, 7, -8, -2, -4, 11, 11, -7], [4, -3, 5, 1, -5, -3, -8, -2], [4, -2, 12, 7, 2, 1, -7, 7], [-9, 4, 5, 2, -4, -2, 6, -3], [5, 1, -10, -3, -9, 6, 20, -4], [6, -4, 1, 0, 0, -4, -5, -1], [-11, 3, 4, 5, 5, -4, -6, 4], [6, 4, 0, 1, 12, 1, -2, -1], [7, 8, 1, -5, -2, 1, 13, -2], [8, 0, 4, -6, 4, -2, 4, 4], [-9, -6, -1, 4, -8, 2, 1, 1], [2, -1, -9, -3, 1, 0, 18, -3], [-5, 5, 1, -2, -10, 1, 4, 4], [11, 0, -4, -8, 4, 0, 19, -4], [-9, 12, 13, 6, 6, -1, -28, 8], [-6, 3, -4, -1, 4, -2, -11, -3], [-5, -2, 1, 6, 4, 3, -16, -4], [-18, 0, 5, -5, -2, -12, -4, 0], [2, -7, -3, 1, 3, 0, 6, -7], [-6, 1, 10, -5, -1, -4, -5, -1], [-19, 11, -3, -1, 0, -1, -13, 3], [10, 3, -5, 0, -1, 11, 9, -9], [-2, -1, -10, -1, -9, 0, 19, -3], [12, -3, 9, -1, -1, -3, 6, 2], [0, 4, -11, 1, 2, 0, 6, -6], [10, 5, 8, -6, 2, 3, 4, 2], [-1, -2, 3, 0, 8, -5, -6, 9], [-17, -3, 5, 1, -2, -6, -17, 6], [-6, 0, 3, -2, 7, -1, -19, 3], [12, -5, -3, 4, 3, -5, 9, 0], [-11, 9, -2, -2, -1, 0, -12, -3], [12, -10, 9, -1, 4, -3, -10, -1], [-4, 0, 4, -5, 8, 2, 0, 0], [15, -4, 7, -4, 4, -3, 11, 3], [-14, 7, 7, -6, 2, -10, -7, 2], [6, 8, -10, -2, -6, 4, 15, -7], [-7, 1, -10, -8, -1, 2, 0, -5], [-18, 8, 13, -1, 13, -6, -22, 7], [17, 3, 9, 5, 4, -3, -11, 2], [6, 0, 4, 4, 2, 4, -4, -8], [-2, 5, -5, -9, 3, 0, 4, -1], [10, -2, 4, 2, 2, -2, -3, 4], [-10, 10, 3, -3, 3, -2, -18, 2], [7, 7, -3, 1, -2, 0, 15, 2], [-8, -9, -3, 2, -1, -1, -11, 1], [-5, 7, 5, 8, 8, 0, -19, 11], [6, 10, -5, -8, 5, 4, 11, -2], [-8, 0, 13, -2, -2, -1, -3, 8], [-15, 6, 3, -4, 2, -5, -13, -5], [-10, 8, 1, -10, 1, -12, -5, 3], [7, 8, -6, 0, -7, 4, 3, -2], [-4, -7, -3, 8, 5, -5, -5, -5], [4, 4, -3, -10, 0, -7, -9, -6], [15, -3, 6, 7, -1, 1, -18, -4], [23, 3, 0, 7, 2, 9, 4, -3], [-12, -16, -2, 0, -8, 1, 0, -4], [-4, 9, 2, -5, 6, -1, 1, 1], [3, 3, -13, 0, -4, -2, 6, -3], [4, 7, -2, 4, -4, 7, -3, -3], [6, -4, -4, 0, 0, -4, -8, -10], [0, 6, 9, 10, 5, -2, -5, 11], [11, 5, -4, 0, 11, -6, -11, 3], [-5, 12, -1, 2, 6, 3, -13, -1], [-1, -7, 13, -3, 7, -14, -17, 1], [-1, 0, -2, 2, -4, 2, 9, -4], [-9, -3, 4, -2, -9, 0, -7, 2], [-6, 7, 6, 0, 12, 3, -25, 5], [-12, 0, -11, -7, -1, 1, 2, -2], [-11, 4, -12, -5, -2, 1, 8, 1], [10, -2, 0, -10, 6, -4, -22, -5], [-6, -7, 1, 1, 5, -3, 6, -1], [-13, 6, 10, 3, 9, 0, -24, 7], [26, -1, -7, 6, -5, 4, 27, -7], [0, 10, 2, 3, 10, 9, -16, 3], [4, -5, -8, -6, -8, 2, 6, -5], [14, -10, 7, 2, 0, 2, 3, 1], [3, 11, 6, 7, 7, 8, -18, 5], [-8, 2, -1, 9, 3, 16, -10, -6], [11, -18, 1, 4, -10, -1, 23, -1], [13, 2, 6, -6, -10, 4, 21, 2], [-2, -4, 2, 12, 6, 6, -12, -2], [9, 3, -1, -6, 2, -4, -1, 1], [-36, 15, 4, -1, 9, -2, -25, -1], [-32, 0, 4, 0, -6, 2, -3, 7], [8, -7, 3, 16, -9, 13, 11, 5], [8, -4, -8, -3, 2, -3, 16, -1], [26, -11, -16, 3, 6, 4, 8, -8], [8, 4, 0, 2, 6, -2, -2, 9], [-18, -5, 6, 5, 14, -7, -23, 3], [-16, 12, -4, 0, 4, 16, -16, 0], [7, -8, -1, -6, 6, -11, 5, -7], [-14, 10, -2, -8, 10, -7, 1, 6], [-8, -13, 3, 3, -1, 8, -4, 4], [2, 10, -9, -4, 5, 0, 9, -5], [27, 2, -10, 3, 3, 1, 16, -5], [-10, 16, -9, -4, -1, 3, 6, -6], [-26, 1, -10, -2, 8, 2, 10, 3], [13, -8, 3, -7, 3, -9, 1, 7], [13, -11, 7, -1, 0, -6, 5, 0], [-9, -12, 2, 7, -13, 7, -6, -5], [6, -5, 0, -5, -5, 2, 15, -3], [-1, -9, -3, 5, 10, -4, -11, -4], [-4, 7, 1, -3, 3, -2, -3, -9], [-10, 18, 13, -1, 0, -2, -24, 4], [14, -8, 0, 5, 1, -6, 11, 8], [0, -9, 11, 12, 0, 6, -7, 0], [-1, 0, 3, 11, -10, 0, -9, 2], [4, -4, 10, 6, -4, -5, 10, 2], [-13, 5, 6, -2, -3, -2, 9, 2], [10, 5, 3, -2, 5, -11, 3, 1], [-10, 0, 11, -8, 1, -19, -26, 3], [8, 8, -8, -13, -6, -2, 1, 1], [0, 8, 11, 2, 5, 5, -10, 6], [-32, -3, 5, -8, -3, -7, -7, -8], [6, 4, 10, 2, -2, 0, 1, -6], [-5, 7, -14, -5, -1, 6, -14, 0], [-7, 1, -15, -10, -8, 4, 33, -9], [-10, -2, -4, -2, 4, -9, 3, 0], [-24, 3, 3, 0, 8, 2, -19, 6], [12, 1, -14, 12, -8, 9, 10, -7], [0, -1, 0, -3, -6, -1, -5, 0], [18, -4, -7, 1, 1, 3, 38, -4], [4, 2, 2, 3, 7, 1, -1, -4], [-18, 24, -6, -12, 8, -4, -14, -2], [-12, 2, -7, -13, 1, 9, 18, 3], [26, -14, 1, -1, -10, -1, 28, -9], [-18, -10, 4, 9, -4, 5, -13, 5], [-6, -1, 3, -2, 1, -14, -23, -7], [-25, 13, -10, -8, 7, 3, -13, -4], [-15, 16, 4, 7, -2, -7, -20, 5], [-12, 2, 11, -2, 3, 8, -9, -1], [-24, 6, 8, 8, 0, 6, -5, 5], [-12, 10, 0, 3, -4, 2, -9, 12], [0, 0, 10, 2, 4, -15, 4, 14], [-4, -1, -4, -7, -4, 10, 31, -4], [-12, 1, -6, 2, 0, 13, -3, -3], [-10, -10, 13, 11, -6, 1, -16, 9], [-4, -14, 7, 16, -7, 0, -1, 0], [-1, 3, 13, -1, 2, -2, -15, 6], [6, 9, -8, -5, 0, 14, 7, -11], [-10, 2, 15, 5, 8, -4, -40, 5], [-32, 10, 10, 12, 14, -4, -18, 10], [4, -4, 10, 9, 7, 6, 3, 4], [-6, -4, -1, 7, 2, 10, 4, 8], [7, 6, 3, -9, 7, -1, 23, -3], [15, -8, 12, -13, 11, 0, 10, -1], [-17, 14, 10, -7, 9, -19, -22, 0], [-1, -8, 1, 2, 6, 6, -23, 5], [33, -11, -12, 5, -7, 16, 27, -7], [-24, 11, 11, 5, 3, 0, -14, 8], [16, 2, -16, 4, 6, 12, 12, -10], [-9, 18, -12, -4, 15, -2, -13, 3], [21, -4, -10, 4, -21, 16, 25, -1], [-10, 4, -14, 6, -8, 16, -4, -15], [-20, -2, 2, 12, -4, 4, -14, -4], [-5, -23, 16, -9, -5, -10, -3, -5], [-12, 2, 10, 10, 8, -8, -29, 13], [-13, -14, 0, -5, -5, -2, -6, -6], [2, -4, 3, 2, 7, 3, -16, 0], [-25, -6, 9, -1, 8, -12, -17, 5], [-26, 20, 9, -4, 10, -8, -29, -1], [-26, 7, 8, 5, -1, -2, -35, 11], [-26, 16, 11, 9, 18, -11, -46, 11], [-17, 12, -1, -3, 8, 3, -17, -12], [1, -8, -2, -6, -9, -8, 13, -4], [-11, -14, 8, 2, 0, -4, -21, 4], [-2, -6, 1, 11, -7, -8, -8, 1], [3, 9, -12, -5, -7, 8, 18, -14], [-28, 6, 7, 6, -12, 6, -3, -5], [-5, 7, -5, -5, -6, 8, 5, -8], [25, 7, -9, 2, 8, 12, 14, -13], [10, 5, 21, 2, 1, -7, 1, 8], [3, -2, -5, -10, 2, 11, 10, -7], [-2, -6, -10, 1, -21, 6, 1, 0], [21, -17, -10, -5, -5, -6, 48, -3], [14, -2, -12, -8, 10, -3, 13, -4], [-27, 15, 6, -3, -3, -5, -8, 7], [-11, 14, 16, -17, 15, -5, -22, 2], [18, 5, -11, 9, -7, 20, 11, -1], [-26, -3, 24, 5, 14, -8, -16, 12], [-8, -5, -12, 12, 6, 4, -8, -5], [-8, 3, 13, 2, 6, -18, -25, 0], [20, -4, 9, 1, 0, 0, 0, -3], [1, -2, 2, -8, -3, 2, -7, 2], [-13, -24, -16, -2, -15, -1, 29, -7], [-4, -11, -12, -12, 0, 3, 5, 1], [2, 10, -5, -1, -5, -10, -16, 5], [11, 10, 8, -7, 4, -5, 0, 9], [0, -13, -3, 12, -6, 12, -7, -8], [-3, 10, 14, 11, -1, -3, -44, 5], [-18, 6, -20, -6, -6, 0, 8, 2], [11, 2, -21, -2, 6, 3, 13, -3], [18, 16, -14, 0, -4, 6, 29, -2], [25, -3, 4, 4, -5, -2, 17, 0], [-5, 5, -5, 0, -11, 6, 44, -1], [-28, 25, -5, -11, 3, -5, 2, 7], [-6, 10, 16, -4, 22, -14, -23, 13], [-9, -15, -9, -8, -2, -3, 20, -6], [21, 18, -16, -14, -9, -1, 23, -5], [33, 0, -3, -9, 4, 5, 1, -6], [19, -11, -21, -4, -6, 17, 26, -17], [8, 8, -21, -4, 8, 2, 1, -5], [-35, -10, 8, 3, 13, -14, -18, -1], [-12, -6, -16, -4, -10, 11, 19, -8], [12, -3, -15, -10, -17, 17, 33, -14], [-32, -7, -7, 2, 5, -6, 1, 2], [-3, 1, 3, -5, -9, -12, 15, 15], [-29, -3, 11, 7, 6, -12, -45, 12], [28, -1, -7, -2, -4, 10, 9, -20], [4, -4, -5, 8, -11, 12, 7, -2], [-32, 6, -3, -7, -2, 6, -14, 1], [21, 7, 1, 0, -4, 0, 4, 8], [25, -11, 20, 21, 1, 6, -8, 1], [18, 7, 9, -4, 9, 5, 13, 7], [37, -7, 7, -12, -5, -4, 18, 5], [8, 0, -6, 8, 10, 9, 7, 8], [4, -8, -7, -8, -9, 9, 24, -6], [-54, 16, -1, -3, 6, -7, -14, 15], [-23, -13, 0, 9, -3, -11, -16, 4], [-4, -2, 5, 0, 11, -4, 1, 6], [-18, 3, 22, -3, 4, -10, -22, 18], [-27, 20, -1, -7, 0, -10, 5, 13], [7, -16, 25, -5, 5, -17, -31, 1], [-20, 6, -5, -9, 9, 0, 8, 15], [-18, -6, 12, -12, 12, -16, -5, 6], [-25, 3, 5, -6, 4, -2, -8, -11], [16, -21, -7, 6, -3, 13, 3, 5], [18, -4, -6, 9, -11, 14, 17, 8], [-12, 11, -3, -10, -5, -3, -5, -9], [-15, -2, -8, -23, -5, -13, 24, -7], [38, 1, -7, 5, -7, -10, 13, -13], [29, 0, -4, 6, 5, -1, -9, -3], [-10, -3, 21, 14, 9, 7, -27, 16], [10, -6, 14, -2, -14, -4, -4, -9], [2, 13, -2, 3, -2, -2, -23, -2], [-14, 12, 0, -3, 8, -13, -10, 11], [44, -10, 11, 3, -4, -5, 18, 5], [-1, -16, 13, 15, -8, -4, -31, 6], [-20, -18, 8, 13, -9, 3, -5, 0], [5, -11, 4, 10, -1, 4, 12, 5], [3, -2, 5, 22, 2, 6, -25, 1], [8, -6, 11, 26, 13, 17, -32, 1], [-11, -18, 13, -15, -8, -16, -3, -2], [2, 7, -9, 4, 11, 5, -3, -5], [11, 28, -2, -13, -1, -4, -13, -4], [2, -16, -9, -2, -5, 1, 12, 0], [12, -6, -11, -7, -3, 12, 38, -12], [-15, 16, -4, 5, -1, 0, -20, 10], [-26, 2, -6, 19, 10, 11, -23, -9], [17, -19, -14, -6, -5, 0, 25, -17], [-14, -11, 13, 9, 5, 10, -2, 0], [7, 11, -10, 1, 7, 14, 8, -12], [50, 2, -8, 1, -6, 3, 6, 9], [0, -9, 19, 0, 9, -3, -23, 3], [-2, -1, -18, 3, 0, 29, 35, -6], [20, -32, 11, 10, -4, -7, 23, 0], [-49, 20, -4, 9, -5, -5, -22, 10], [40, 3, -13, -9, -3, 15, 40, -17], [-40, 8, 12, 8, 4, -4, -12, 8], [-14, -6, 3, 7, 3, -13, 6, 15], [-9, 8, -12, 16, -5, 2, 1, 2], [9, -11, -6, 11, -5, 11, -8, -6], [8, -12, 4, 24, 6, 8, -18, 0], [24, 29, -9, 8, 15, 2, 5, 8], [8, -11, 1, 3, -15, 0, 13, -1], [24, 3, 1, -6, 5, -4, 17, 15], [12, -8, -7, -11, -10, -10, 28, -10], [-29, 19, 9, -12, 8, -10, -27, -9], [1, 22, -7, 2, -4, 4, 14, 6], [-14, -1, -3, -4, -7, 1, -11, 7], [-10, 4, -12, 0, -4, 16, -6, -12], [-26, -10, -17, -1, -5, -3, -16, -2], [10, -2, 4, -4, -10, 3, 41, -2], [1, -27, -11, 1, 0, -1, 21, 8], [-4, -14, -3, -5, -7, -1, -6, -12], [-18, 1, 2, 6, -18, 3, 13, 1], [30, -25, 13, 6, -7, -7, 17, 5], [6, -24, -22, -8, -18, 12, 30, -2], [-24, 16, 15, 8, 15, -15, -41, 7], [-4, -24, -8, 10, 14, -8, 4, -6], [-22, 4, -14, -1, -6, -13, -16, -5], [-12, -26, 18, 8, -10, 2, -6, 0], [40, -4, 8, -7, 1, -5, 3, 6], [-23, 10, -8, 1, -1, 6, 19, 2], [3, -7, 7, -5, 1, -2, 5, 7], [11, -21, 2, -10, 9, -5, 23, 2], [-8, 6, -17, -2, -7, -5, 13, 9], [-33, 10, 4, -5, -11, -8, -19, 2], [8, -18, -13, 0, -17, 7, 16, -6], [16, 1, 5, -4, 6, -12, -21, -11], [-19, -7, 13, 17, -4, -18, -9, 15], [47, -21, 1, 9, -10, -1, 49, 0], [20, -1, -15, 6, 16, 14, -13, 4], [-20, 16, -7, 0, -4, 1, -15, 16], [1, -10, 0, 10, -13, 1, 25, -4], [10, 12, -13, -7, 3, 0, 30, -1], [-3, 12, 12, 8, 25, -11, -37, 1], [-5, 5, -21, -5, -10, 18, 35, -2], [-4, 11, 3, -2, 17, 6, -51, 4], [-10, 20, 21, -2, 1, -13, 2, 11], [-21, 16, -16, -3, 5, 8, 6, -9], [-5, -3, -2, 8, 11, -1, 9, -2], [30, 5, 12, 1, 10, -5, 3, 10], [-28, 5, 6, 13, -14, -4, -19, 17], [25, 8, -5, -15, -16, -15, 13, 4], [18, 6, 3, 13, -2, 0, -12, 0], [-9, 8, -17, -3, 0, 0, -13, 0], [19, 2, -10, -18, 8, 16, 29, -12], [-16, -19, 11, 6, 3, -10, -19, 9], [39, 6, -17, -4, -16, 12, 28, -29], [-11, 4, -2, 13, -5, -5, -6, 7], [-14, -7, -17, -10, -10, 18, 3, -14], [-2, -14, 25, -2, -14, -10, -1, 1], [18, 12, 9, -9, 13, 3, -22, 2], [30, -1, -10, 2, 0, -9, 16, 0], [-9, -6, 7, 11, -15, -1, -19, 5], [7, -24, -1, 1, -6, 2, 23, -7], [-64, 16, 7, 7, -16, -5, 10, 11], [2, 3, -33, -6, -17, 7, 25, -14], [-42, 11, 1, -12, 9, 0, -35, 6], [-3, -18, 18, -9, -1, 2, -6, -6], [-20, 12, -9, -19, 4, -9, 14, -13], [12, 11, -19, 8, -3, 3, 13, 7], [13, 19, -10, -2, 1, -11, -3, 9], [14, 6, 8, 0, -6, 0, 33, 8], [28, -18, 11, 9, -5, -8, 4, -5], [24, -15, -3, -8, -1, 3, 5, 7], [23, 1, 2, 0, -17, 10, 41, 2], [14, 7, 13, -5, -1, 9, 10, -4], [-46, 5, -16, -6, 10, -1, -16, 5], [-10, -13, -5, -12, 3, -9, 9, 5], [-10, -3, 3, 10, -23, 19, 33, -1], [-13, -6, 12, -5, -3, -9, -12, 16], [26, -6, -18, -3, 7, 4, 13, -12], [-2, -3, -6, 0, 8, 21, -15, -5], [15, -18, -16, -2, -5, 14, 39, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4598_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4598_2_a_bx();".
// 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_4598_2_a_bx(:prec:=8)
    chi := MakeCharacter_4598_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_4598_2_a_bx();".
// 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_4598_2_a_bx( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4598_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-44, 332, -332, -108, 154, 8, -22, 0, 1]>,<5,R![209, 224, -256, -152, 107, 32, -18, -2, 1]>,<7,R![1189, 4152, 4018, 902, -501, -246, -12, 8, 1]>,<13,R![-33344, 46080, 34048, -608, -3496, -472, 68, 18, 1]>],Snew);
    return Vf;
end function;