
// Make newform 4598.2.a.cb 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_cb();".
// 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_cb();".
// 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 := [-2, -28, -90, 32, 75, -4, -16, 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, 1, 0, 0, 0, 0, 0, 0], [-2, -66, -24, 64, 11, -15, -1, 1], [-18, -40, 44, 20, -13, -2, 1, 0], [-18, -166, 56, 108, -15, -19, 1, 1], [-32, -218, 8, 196, 14, -45, -2, 3], [-98, -610, 232, 448, -53, -87, 3, 5], [-78, -582, 72, 496, 13, -109, -3, 7]];
    Rf_basisdens := [1, 1, 8, 4, 4, 4, 8, 8];
    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], [1, 0, 0, 0, 0, 0, 1, 0], [0, 0, -1, -1, -1, 1, 1, 0], [1, -1, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 1, -1], [-1, -2, 0, 0, 0, 1, 1, -1], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -2, 1, 0, -1], [-1, -2, 1, 0, 1, -1, -1, -2], [-1, -1, 0, 0, 1, 0, 1, -3], [3, -1, 1, 1, 2, 2, 0, -1], [1, -2, 2, 0, 1, 0, -2, 0], [1, 3, 0, 0, -1, 0, -1, 1], [-1, -3, -1, 1, 3, 1, 2, -1], [4, -2, -1, -1, -2, 3, 2, 0], [-3, -2, -1, -1, 2, 2, 3, -3], [0, 0, -1, 1, -1, 1, -1, -2], [2, -2, 1, 1, 2, 4, 0, -1], [-2, -3, 2, 2, 0, 2, -1, -3], [-1, 0, 2, -2, 0, -5, -1, 3], [-2, -1, 0, 0, -2, 2, -1, 1], [4, 4, -1, 1, -1, -5, -3, 4], [1, 2, 2, 0, -1, 0, 0, 2], [1, 1, 0, -2, 1, -6, 1, -1], [-9, 1, -1, -1, -2, 1, 2, -1], [-2, -1, 0, -2, -4, 0, 1, -1], [-2, -3, 3, -2, 0, 1, 3, -2], [3, 3, -2, -3, 0, -4, 2, 0], [-1, 0, 0, -2, -3, -2, -4, 0], [0, 0, -2, 0, 2, -2, 2, -2], [1, -5, -2, -4, -1, 0, -1, 3], [3, 2, 1, 1, 2, -3, -2, 1], [-4, 4, 0, 0, -4, -4, -2, 4], [2, 4, -4, 2, -2, 4, 2, 0], [5, -4, 2, -4, -1, -2, 0, 2], [7, 5, -3, 1, 2, -5, 0, 3], [5, 3, -2, 0, 1, -6, -1, 1], [-5, -1, 2, 4, 1, 2, -7, -1], [-5, -5, 0, -2, 2, 6, 3, -3], [-2, -1, 2, 0, -6, 5, 1, -1], [4, 0, -2, 2, 0, -2, 4, 2], [5, 2, -2, -5, -6, -4, -3, 7], [7, -1, 2, 2, 1, -6, -5, 1], [3, 3, 0, -4, 1, -2, -1, 1], [3, -2, 1, -1, 0, 3, 2, -1], [8, 2, -2, -7, -1, -6, 2, 4], [-5, -4, 4, 0, -1, -2, -4, -4], [8, 0, -4, -2, 4, -2, 1, 1], [3, 3, 0, 0, 5, -2, -5, 1], [-1, 10, -1, -1, -2, -8, -3, 2], [8, 1, 3, 1, 2, -3, -1, -3], [4, -9, -2, 0, 2, 6, 5, -1], [-4, 6, -2, 2, -4, 2, 0, 8], [0, 4, -6, 4, 0, 0, -2, 2], [-7, 0, 5, -3, -4, 1, 3, -2], [9, 4, 0, 2, -1, -1, 2, 4], [9, 4, -2, 0, 4, 1, -3, -1], [4, -4, 1, -1, -7, 7, 3, 4], [1, 0, -4, -6, -1, -8, 2, 4], [8, 0, -2, -2, -4, -4, 0, 4], [-15, -1, 0, -3, 0, -2, 3, 1], [-10, -4, -2, -2, 3, 4, 2, 2], [5, 0, -1, 1, -4, 5, 0, 3], [2, -1, -8, -3, -3, 2, 0, 4], [-2, -2, 3, -3, 2, 2, 0, -1], [11, 1, 0, 2, -3, -1, 1, 4], [-10, 3, 2, -4, -2, -6, -3, -1], [7, 5, -3, 1, 4, 1, -4, 5], [-5, 1, 2, 6, 7, -10, -3, -1], [2, -11, -2, 1, 7, 6, 1, -5], [15, -2, -2, -4, -2, -3, 3, 5], [3, -9, 1, -1, 3, -1, -4, -5], [-13, 2, -1, 2, -3, 1, 3, 0], [5, -1, 4, -2, -1, 3, -5, 2], [-6, 0, -2, -6, -8, 6, 4, 2], [-7, -5, 1, -3, 2, -3, -4, -5], [2, 0, 2, 4, 2, 2, -2, 0], [-6, 1, 2, 2, -2, -10, -3, 1], [-8, -6, 2, 0, 6, 2, 6, -6], [-6, -2, -1, 3, 1, 3, 5, -6], [-2, 2, 0, -6, 0, -1, 4, -3], [10, 9, -4, -4, -2, -6, -1, 7], [15, -1, -2, 2, 7, 0, -7, 1], [0, 10, -2, 4, -2, -2, 4, 4], [-9, 11, 0, 0, -3, -8, -3, 5], [-6, 0, -4, -2, -2, 10, 2, 6], [3, 0, 0, -4, 1, 1, 6, 1], [-5, -3, 5, -5, -4, -1, 4, 1], [11, -4, -1, -1, -4, -3, -3, 2], [-9, 15, 0, 0, -3, -12, -3, 7], [-3, -3, 2, -6, -13, 5, 5, 1], [3, 1, -2, 8, 5, -2, -5, -3], [-5, 1, 4, 0, -5, 0, 5, -3], [-3, -3, 3, -9, 0, -3, 4, -5], [-7, 1, 1, -7, -2, 0, 0, -4], [-5, 11, -1, 7, 4, -4, -6, -1], [-1, -7, -4, -2, -7, 2, 5, 5], [-4, 3, 1, 6, 4, -1, 3, -12], [-2, -6, 0, -2, 2, -2, -4, -6], [-10, -3, -1, 5, 4, 9, 4, -11], [-6, -8, -2, 0, 2, 6, -2, -2], [-16, 11, 5, 7, 0, -1, -6, 1], [-2, 0, 6, 0, -2, 4, 4, -2], [-4, 2, -5, -11, -1, -7, 3, 4], [1, 2, 5, 9, 10, -3, -6, -7], [-11, 5, -3, -3, -2, 9, 6, -1], [-1, -8, -1, -1, 4, 6, 1, 2], [8, 2, 4, 2, 0, 0, 2, 0], [-10, 0, 10, 12, 4, 4, -4, 0], [-9, 8, 0, 0, -7, -6, -4, 8], [5, -1, -1, -1, -4, -1, 4, 11], [-18, 0, -1, -1, 2, 5, -5, 4], [8, 6, 0, -4, -2, -4, 0, 6], [4, -9, -6, 0, 5, -2, 1, -1], [4, -1, 6, -4, 0, -2, -5, -5], [10, 6, -8, 2, 0, 4, 6, -2], [-4, 2, 1, -2, -2, 5, 3, 4], [9, 9, 0, 6, 1, -2, -5, 1], [8, 5, 1, 1, -3, 7, 9, -2], [8, -3, -6, -8, -1, 2, 8, -9], [-12, 11, 0, 4, -10, -2, -3, 11], [-7, -6, 9, 6, -3, 11, 2, -9], [-14, 0, 8, 4, -4, -2, -2, -2], [5, -11, -1, -1, 0, -5, -2, -5], [-12, 0, -2, -2, 2, -2, 4, 4], [6, -4, 0, 6, 6, 0, -2, -2], [-6, -4, 2, 16, 10, 10, -3, -7], [9, 2, 2, 5, 8, 0, -2, -2], [17, 7, 0, -2, -5, 6, -1, 11], [5, -5, -1, -5, 4, -3, 0, 5], [7, -9, -2, -2, -9, 12, 9, -5], [21, 3, -2, -2, -3, 6, -5, 5], [5, 9, -3, -7, -4, -13, 0, 15], [21, 0, -4, 0, 7, -11, -2, 1], [11, 2, 7, -1, 5, -11, -6, 6], [-2, -4, -2, -8, 6, -7, 0, 3], [-4, -5, 7, -8, -2, 3, 0, -1], [-12, 0, 5, -5, -4, -4, -6, -1], [10, -4, -10, -6, -1, 6, 7, 0], [-12, 3, 5, -2, 2, -15, -4, 1], [-12, 0, -3, -7, -1, 5, 7, -6], [20, -4, -3, -4, 6, 5, -1, 0], [0, -7, -3, 1, 1, -5, 9, -2], [10, 6, 3, -3, -6, 2, 0, 3], [-3, 4, -6, 6, 7, 4, 2, -8], [-5, 1, 8, 10, -1, 2, -13, 9], [13, -3, -8, -10, -9, -2, 5, 1], [17, -11, 1, 3, 18, -3, -4, -1], [5, 1, 6, 6, 9, -6, -11, 5], [-6, -3, 9, 15, 7, 5, -1, -14], [-19, 4, 0, 0, -10, 8, -2, 2], [22, -4, 5, -3, -7, 1, -5, 6], [-8, 1, 2, -2, -2, -2, -13, 3], [-9, 4, -5, -1, -12, 6, -3, 11], [-17, -2, 2, 8, 5, 6, 0, -8], [-10, 0, -3, 7, 2, 4, -2, 5], [-19, -3, -1, 2, 9, -3, 4, -3], [24, 2, 0, 8, 1, -4, -3, 4], [10, -4, -8, 0, 6, 4, 9, -5], [1, -3, 2, -12, 3, -6, -1, 3], [5, 0, 4, 0, -3, -6, 2, 10], [0, -1, 7, 15, 9, -12, -10, -3], [-14, -9, 12, -2, 4, -9, -5, -5], [-16, -4, 0, 6, 4, -2, -8, -4], [9, 8, -10, 2, 5, -18, 0, 4], [2, 5, -4, 12, 6, 4, -1, -5], [4, 18, 7, 1, -5, -11, -3, 6], [-8, -7, 2, -8, -6, 14, 11, -1], [-8, -2, -6, -6, -2, 8, 2, -8], [-12, -2, -2, 8, -6, 8, 5, -9], [0, -2, 1, -15, -4, -3, 8, 0], [-1, -8, -3, -3, 8, 1, 5, 0], [0, 8, 6, 2, 7, -10, -15, 4], [-7, 1, -4, 2, -7, -8, 5, 3], [-18, 2, 3, -5, 0, -15, -9, 4], [10, 4, -7, -3, -1, 7, 9, -4], [-13, -5, -6, 0, -1, -2, 13, -3], [0, -5, 10, 12, 4, 18, -5, -13], [0, -2, -5, 1, 11, -3, -1, -6], [6, -4, -13, -14, -4, -9, 4, 9], [6, -1, -3, 2, 4, 7, 1, -2], [7, -8, 4, -2, -9, 3, 4, -10], [1, -4, -10, -10, -1, -6, 8, 10], [-16, -2, 3, 4, 0, -13, 5, 4], [-10, 4, -5, -5, -5, -9, 1, 4], [-15, 7, 6, -1, -6, 0, 2, -4], [2, -1, 6, -4, -4, 7, 7, -3], [-3, -8, 2, -4, -5, 12, 0, 0], [10, 9, 3, -3, 12, -7, -5, -3], [11, 14, 4, 2, 3, -8, -8, 6], [8, 12, -5, -5, 5, -11, 5, -8], [-14, 6, 6, 18, 12, -4, -4, 2], [0, -16, 3, 5, 4, 14, -2, 1], [16, 2, 3, 1, 7, -3, -3, 2], [9, -9, 6, 4, 9, -2, 1, 1], [5, 2, -2, -6, 8, -15, -1, 3], [7, 1, -9, 7, 10, -1, -8, 1], [17, 0, -3, 5, 1, -1, -2, -2], [-20, -6, 3, 5, 4, 0, 0, 1], [-21, -12, 3, -1, 1, 5, -1, 0], [16, 4, 0, -10, -18, -2, -4, 8], [-12, -10, 5, 3, 0, -8, -16, 1], [0, 4, 4, 14, 8, 8, 4, -12], [-1, -5, 8, -2, -1, 0, 5, -9], [-8, -2, 6, -4, -10, 4, -2, 0], [-7, -6, 8, 7, 0, -2, -9, 7], [8, -6, 6, 3, -3, 10, -2, -4], [-11, 4, 1, 1, 3, 11, 0, -10], [3, -4, -13, -13, -10, 6, 3, 13], [1, -4, -5, 7, 7, 1, -8, -6], [5, 3, -4, 4, 11, -8, -5, 3], [5, 1, -3, 7, 11, 15, -4, -5], [-1, 21, -7, 7, 5, -11, -1, -5], [-2, -10, -6, 2, 8, -6, 2, -4], [6, 8, -4, -4, 4, 2, 0, -6], [-2, -3, 5, 7, 6, 7, -3, 11], [-11, 7, -2, 0, -19, 10, 5, 11], [2, -5, 2, -2, -2, -12, -9, -9], [-6, -8, 3, 11, 1, 5, -3, -6], [-35, -7, 8, -2, -6, 6, -1, -11], [4, -12, 10, 4, 10, 6, -4, -12], [7, 0, -6, -14, -5, -6, 2, 4], [-5, 0, 3, -11, 2, 1, 2, -3], [-16, 20, 6, 3, -5, -14, -2, 0], [-14, 4, -1, 3, -1, 3, 5, 10], [11, -2, 4, 6, 19, -8, -14, 4], [-3, 3, 2, -14, -7, -2, -3, 5], [-11, 15, 0, 2, -5, 0, 1, 7], [-19, 1, 7, 9, -2, 23, -2, -5], [7, -1, 0, -12, -15, 6, 3, 7], [18, 0, 14, -9, -3, -8, 1, 3], [-1, 4, 4, 10, 11, -12, -6, -4], [0, 12, 4, 12, 0, -12, -6, 6], [9, -3, 8, -6, -9, 0, -5, -3], [16, -11, 0, -12, -4, -9, -1, 14], [-11, -6, 1, -3, 2, 19, 1, -8], [-9, -1, 5, 15, 4, 5, -4, -9], [15, 19, -9, -3, 2, -17, -2, 1], [11, 12, -12, 2, 7, -18, 6, 8], [-30, -1, 8, 6, -10, 15, 1, -3], [13, 17, -3, 3, -8, -7, -6, -3], [-4, -22, 3, -2, 2, 15, 18, -5], [6, -8, -3, 5, 10, 9, 10, -12], [-1, -4, 2, -8, 3, 4, -8, -2], [6, -1, -11, -7, 1, 10, 8, -11], [-11, 15, 1, 11, 3, 7, 4, -13], [-4, 0, 4, -4, 6, -14, -10, 6], [-12, 11, -1, 5, -2, -11, -2, 7], [21, -15, 2, 0, 12, -10, 0, 1], [23, 5, -8, 0, -7, 22, 13, 3], [-1, -22, 7, 3, 7, 5, -8, -12], [1, 4, 3, 5, 12, -15, -10, 9], [-15, -2, -2, -8, -1, 6, 12, -6], [-22, -12, 4, -12, 4, -4, -6, -2], [13, 9, 9, -1, -10, -9, 0, 9], [4, -14, -1, -1, 9, 9, 1, -4], [8, 0, -20, -16, -5, -10, 6, 8], [4, -3, -7, 11, 13, 9, -16, 5], [11, -8, -7, -3, 20, 6, 5, -16], [-11, -17, -6, -12, 7, 10, 3, -9], [4, -7, 1, 3, 18, 15, 8, -13], [-21, -11, 4, -8, 3, 0, 3, 3], [-4, -7, -1, -3, -1, -1, -5, -16], [5, 4, -13, -9, -18, 13, 11, 4], [-4, 2, -9, 15, -2, 4, -2, 3], [0, 5, 2, 8, 15, -18, -16, -7], [-8, 10, -10, 2, -2, -4, 3, 9], [-8, 14, -18, -1, -7, -2, 6, 12], [-20, -14, -9, -5, -5, -1, -11, -2], [-18, 2, -3, -7, -8, 1, 10, 0], [7, -16, -3, -7, -12, 10, 17, 3], [16, 3, -6, -9, -5, 0, 16, 2], [-11, 18, 0, 6, -6, -5, 9, 9], [11, 10, -8, -2, 7, -8, 12, 10], [3, 18, -8, 3, 8, -2, 4, -6], [27, 1, 6, 6, -5, 18, 3, 1], [-11, 5, -14, 7, -8, 4, 9, -7], [-23, -8, 2, -2, -21, 0, -2, 8], [12, -4, 3, 1, 13, -7, -1, -12], [11, -5, -6, 0, 5, -12, -5, -7], [3, 2, -10, -25, -4, -12, 2, 6], [-13, 10, 12, 14, 18, -3, -9, -13], [-27, -11, -2, -7, -4, 12, -1, -5], [6, 0, 14, 2, -6, 8, 4, 0], [4, 1, 10, 16, 17, -10, -9, -19], [-18, 0, 10, 18, 8, 0, 11, -23], [-23, -14, 9, 13, 13, 7, 3, -18], [8, 4, -1, -4, 2, 3, 17, -4], [-29, -6, -6, -2, 1, 12, 2, 2], [-10, 10, -1, 3, -7, -3, -15, 2], [19, 13, 9, -7, 0, -15, -2, 9], [10, 17, 10, 2, -10, -14, 1, -3], [-6, -7, 1, 3, 10, -9, -4, -5], [19, -13, 2, -4, 2, 4, 1, -11], [-38, 4, 1, 5, 3, 21, 1, -14], [-27, -2, 8, 4, -3, 6, 0, 2], [-24, 14, 6, 14, -6, 4, 0, 2], [24, 11, -1, -6, 0, -19, -1, 14], [20, 14, -2, 2, 20, -14, -2, 0], [-22, 2, 18, 18, 8, 11, -12, -14], [-2, 3, -5, -15, -11, -5, 9, 14], [21, -13, 2, 4, 16, 18, -2, -5], [6, 14, 9, 5, 4, -10, -18, 5], [16, -10, -19, -5, 5, 15, 7, -4], [16, 5, 8, 14, 4, -10, -5, -5], [31, 4, 6, 2, 0, -14, 3, 2], [-14, 18, -8, -2, -2, -5, 10, -17], [-7, 0, 11, 11, 8, -3, -1, -4], [23, -15, -1, -11, -8, 1, -2, -3], [18, -9, 5, 5, 11, -6, 6, -13], [-23, -11, -1, 5, 12, -15, 8, -5], [9, 10, 6, 12, 1, -14, 0, 4], [16, 9, -14, -8, -4, 7, 3, 5], [-1, 7, -4, 4, 19, -21, -7, -1], [-3, -9, 4, -4, 3, -22, -7, -3], [22, 8, -7, -11, -1, -9, -9, 12], [7, -19, -14, -12, 5, 32, 17, -17], [30, -9, -6, -3, 1, -2, 5, -1], [-3, -20, 3, 7, 14, 13, 13, -18], [20, -1, -3, -12, 0, 7, 3, -8], [3, 7, -7, -8, 5, -5, 12, -1], [-36, -3, 7, 9, -13, 11, -9, 8], [44, -14, 10, -8, -6, 8, -2, -2], [-30, 4, 8, 0, -1, 4, -1, -6], [29, 2, -4, 8, 1, -4, -4, -6], [19, 17, -3, 15, 5, -3, 3, 1], [17, 13, -4, -14, -11, -9, 3, 6], [-7, -1, -6, -10, -3, -2, 3, -5], [3, -7, -15, 5, 10, 1, -10, -3], [-7, 9, 3, 1, 4, -25, -8, 11], [10, -1, -15, -3, 7, 16, 14, -12], [-3, 11, -14, 12, 10, 10, 8, -7], [27, 19, 0, -4, -3, -24, -3, 25], [20, -12, -1, -1, -7, 9, 13, -2], [-11, 5, -2, 10, 1, 6, -5, 1], [-14, 4, 7, -5, -16, -9, 12, 10], [-41, -9, 12, 4, -3, -8, 1, 3], [15, -8, -11, -15, -2, -3, 0, 15], [35, -5, -3, -11, -20, 6, 6, 8], [12, 5, 0, 2, 1, -12, 2, -9], [5, -17, 3, -14, 5, -5, 5, 6], [3, -13, 6, 7, 2, 28, 0, -6], [2, 4, 0, 10, 6, 6, -22, 14], [24, 14, -1, 3, 0, -1, 12, 4], [18, 7, -5, 13, 18, -15, -4, -3], [32, 4, -6, 2, -6, -14, -4, -2], [36, -6, -5, -1, 6, -6, 14, 3], [16, 19, 8, 10, -12, -4, -1, 13], [-2, 10, -2, -8, -6, -16, 12, 6], [4, -2, -10, -18, -6, -18, 4, 10], [-8, 13, -14, -8, -18, 2, -1, 21], [17, -1, 3, -9, -8, 7, 6, 3], [24, -15, -2, -2, 18, 0, -11, -5], [5, -2, -5, -9, -11, 13, 7, 8], [-8, 2, -2, -10, 0, 4, -4, -4], [14, -6, -5, 4, -2, -1, 0, -3], [3, 15, -11, -3, -8, 7, 0, 11], [25, 12, -10, 10, 9, -5, -8, 9], [0, 17, -12, 10, 10, -8, 9, -5], [-11, -21, 3, -3, 10, -1, 0, -15], [-33, 0, 13, 1, 2, 12, -5, -10], [10, -18, -10, -12, 24, -8, -4, 0], [-2, 3, -3, -6, 4, -11, 4, 1], [25, 19, -5, -5, -2, -13, -14, 17], [7, -5, 4, 14, 3, 20, 15, -15], [-9, 3, 14, 0, -11, -20, -3, -1], [12, 2, -14, -28, -18, 10, 18, 8], [8, 21, -6, 4, -4, -8, -5, 5], [-4, -4, -4, -2, 12, -28, -4, 4], [26, -14, 11, 7, 14, 10, -8, -15], [-31, -12, 0, -2, -2, 31, -5, -13], [-3, -1, -2, -27, -12, -4, 8, 10], [14, 19, -7, 5, -7, -2, 0, 9], [35, -9, 9, -5, 0, 9, -10, -1], [18, -30, 3, -1, 4, 16, 6, -17], [-13, 6, -7, -7, 4, -15, 7, -10], [10, 16, -8, 20, 16, 4, 10, -10], [9, 22, -1, 11, -5, -7, -6, 8], [0, 13, -6, -14, 0, -12, -3, -9], [11, 8, -2, 8, -13, 7, 0, 4], [-28, 8, -19, 9, -2, -13, 11, -2], [31, 10, 12, 6, 0, -11, -3, -7], [-19, 16, -5, -1, -14, -6, -3, 6], [-4, -4, -16, -3, -7, -4, 4, -8], [-4, -2, -6, 10, 16, 10, 12, -14], [-13, 2, 7, 9, -4, 29, -2, -3], [19, 13, 1, -1, 0, -1, -14, -3], [15, -1, -1, -1, -12, 9, -4, 15], [17, -8, 4, 22, 13, -6, -16, 0], [-28, -2, 2, -12, 8, -4, 2, -14], [-2, 5, -18, -4, 5, -2, 24, -15], [4, 2, -2, 6, 12, -18, -20, 2], [-11, -19, 1, -13, -10, 41, 10, -3], [21, 3, -6, -14, 7, -18, 13, -9], [-19, 6, -2, -8, -5, -12, 6, 18], [43, 22, -6, -8, -7, -8, 0, 10], [27, 8, 12, 8, -11, -4, -16, -4], [2, 27, -3, 7, -1, -9, -6, 15], [-9, 5, 6, -4, 9, -6, -9, -3], [24, -4, 0, 8, 26, -4, -20, -12], [27, -8, 2, -2, 0, -2, -8, 14], [4, -12, -6, -14, 2, 22, 2, -8], [8, 7, -16, -14, 0, -6, 5, 3], [-23, -1, -2, 4, 18, -4, -13, -1], [13, -30, 0, -6, 0, 23, 7, -13], [6, -16, 20, 16, 8, 12, -18, -28], [29, 12, -4, -12, -15, 6, -2, 10], [33, 3, 0, -10, 5, -3, 13, -2], [16, 12, -13, -11, -5, -25, -3, 10], [16, -10, 4, 8, 24, -8, -18, -2], [-1, -7, 16, 6, -11, 6, -15, 7], [-25, -19, 10, 0, -5, 5, 11, -8], [2, -8, 1, 11, 23, 21, 3, -12], [18, 8, -6, 2, 17, 6, -3, -2], [7, 9, -1, -11, 6, -11, 0, 11], [-10, 2, 0, 24, 22, 12, -12, -26], [0, -19, -6, -14, -14, 36, 23, -15], [-19, 16, -11, -3, -18, 1, 2, 1], [-13, 10, 2, 16, -3, -12, -8, 8], [-23, 7, -1, -1, -8, 18, 4, 9], [-9, -4, -3, 2, -7, -1, -9, 14], [-2, -32, 2, -4, -2, -2, -1, -9], [-2, -16, -2, -12, -10, 4, 19, 9], [-14, -13, -6, -16, -14, 8, -5, 1], [-40, -13, 3, 15, 19, 4, 8, -18], [5, -20, -1, 7, -6, 24, 13, -11], [-4, 28, -9, -5, -3, -20, 5, 11], [8, 6, 11, -1, -19, -11, -11, 14], [-36, -21, -6, 2, 0, 29, -7, -2]];
    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_cb();".
// 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_cb(: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_cb();".
// 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_cb( : 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, -368, 557, -80, -183, 72, 9, -8, 1]>,<5,R![1408, 1536, -1984, -288, 470, 12, -38, 0, 1]>,<7,R![-1388, 5408, -867, -1716, 387, 152, -37, -4, 1]>,<13,R![-368, 96, 1088, -72, -451, -84, 34, 12, 1]>],Snew);
    return Vf;
end function;