
// Make newform 4864.2.a.bm in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4864_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_4864_2_a_bm();".
// 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_4864_2_a_bm();".
// 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, 12, -22, -36, 31, 28, -8, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-162, -126, 412, 256, -205, -39, 37, -5], [-58, -438, 308, 308, -205, -39, 37, -5], [10, 522, -28, -600, -161, 117, 25, -9], [98, -22, -4, -30, -114, 13, 24, -5], [-138, -142, 220, 428, 251, -91, -59, 15], [38, -268, -304, 424, 267, -104, -35, 10], [38, -294, -460, 736, 579, -195, -87, 23]];
    Rf_basisdens := [1, 52, 52, 52, 26, 52, 26, 52];
    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_4864_a();"
function MakeCharacter_4864_a()
    N := 4864;
    order := 1;
    char_gens := [3839, 2053, 4353];
    v := [1, 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_4864_a_Hecke(Kf)
    return MakeCharacter_4864_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, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 0, 0], [-2, 1, 0, 0, 0, 0, 0, -1], [0, 0, -1, 0, 1, 1, 1, 0], [-2, -1, 0, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -1, -1, -1, 0], [0, 0, 0, 0, -2, -1, -1, 0], [0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 1, 0, -1, 2, 0, 0], [2, 1, 0, 1, 0, 0, 0, 1], [-2, 1, 0, 2, 0, 0, 0, -3], [0, 0, 1, 0, -2, -1, -3, 0], [0, 0, -1, 0, 0, -2, -1, 0], [-3, 3, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, -2, -2, -1, 0], [-5, -2, 0, -1, 0, 0, 0, 2], [0, 0, -2, 0, 2, -2, 0, 0], [1, 2, 0, 2, 0, 0, 0, -2], [0, 0, 2, 0, 1, 3, 0, 0], [-6, 0, 0, -2, 0, 0, 0, 2], [0, -1, 0, 3, 0, 0, 0, -1], [4, -2, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, -2, 0, -2, 0], [0, 0, -2, 0, 0, 2, -2, 0], [-7, -3, 0, 0, 0, 0, 0, 3], [0, 0, 0, 0, 4, -1, -1, 0], [-8, -1, 0, -3, 0, 0, 0, -1], [0, 0, 0, 0, -1, -1, 2, 0], [-10, -7, 0, 0, 0, 0, 0, 1], [-1, 0, 0, 0, 0, 0, 0, 4], [-10, -5, 0, -2, 0, 0, 0, 3], [0, 0, 0, 0, 2, 0, -1, 0], [0, 0, 0, 0, 0, 0, 2, 0], [0, 0, -4, 0, 2, 0, 0, 0], [4, -4, 0, 4, 0, 0, 0, 2], [0, 0, 4, 0, -3, -1, 0, 0], [0, 0, 1, 0, 0, -3, -2, 0], [-8, 3, 0, -1, 0, 0, 0, -3], [0, 0, 3, 0, 2, 1, 2, 0], [0, 0, -2, 0, -1, 0, 6, 0], [4, 3, 0, 1, 0, 0, 0, 1], [0, 0, 0, 0, -4, 0, -4, 0], [0, 0, -1, 0, -1, -5, -4, 0], [-8, -2, 0, -1, 0, 0, 0, -1], [0, 0, 6, 0, -3, 1, -4, 0], [-9, -1, 0, 2, 0, 0, 0, 1], [0, 0, 4, 0, -4, -4, -5, 0], [-2, 11, 0, -2, 0, 0, 0, -1], [0, 0, 3, 0, 3, 3, 4, 0], [8, -3, 0, 1, 0, 0, 0, 7], [-6, 1, 0, -2, 0, 0, 0, -3], [2, -4, 0, -2, 0, 0, 0, 4], [0, 0, 3, 0, 0, -1, 1, 0], [0, 0, -3, 0, -1, 2, -2, 0], [0, 0, -1, 0, 1, -1, 1, 0], [0, 0, -2, 0, -6, -4, -3, 0], [-2, 0, 0, -2, 0, 0, 0, 6], [8, 7, 0, 4, 0, 0, 0, -5], [0, 0, -4, 0, -1, 0, 5, 0], [6, -1, 0, 1, 0, 0, 0, 3], [0, 0, -5, 0, -1, 3, 0, 0], [-11, -7, 0, -8, 0, 0, 0, 1], [0, 0, -4, 0, 1, -2, 7, 0], [-15, -6, 0, -3, 0, 0, 0, 6], [0, 0, 0, -2, 0, 0, 0, 10], [-12, -5, 0, -4, 0, 0, 0, 5], [0, 0, 4, 0, 2, 2, -1, 0], [-4, 6, 0, 1, 0, 0, 0, -1], [0, 0, -4, 0, 3, 0, 2, 0], [0, 0, -3, 0, -4, -1, -6, 0], [0, 0, 2, 0, -1, 2, 5, 0], [4, 0, 0, 1, 0, 0, 0, -11], [0, 0, 2, 0, 5, 1, 6, 0], [0, 0, -2, 0, 4, 4, -3, 0], [0, 0, -2, 0, 2, -4, 5, 0], [10, 15, 0, 1, 0, 0, 0, -1], [-8, -1, 0, -1, 0, 0, 0, -5], [8, 8, 0, 8, 0, 0, 0, 2], [0, 0, -1, 0, -2, 0, -1, 0], [0, 0, 2, 0, 0, 4, 2, 0], [0, 7, 0, 7, 0, 0, 0, 1], [0, 0, 6, 0, -5, -5, -4, 0], [8, -1, 0, 2, 0, 0, 0, -5], [4, 4, 0, 4, 0, 0, 0, -8], [3, -4, 0, -2, 0, 0, 0, -6], [0, 0, 0, 0, -4, 2, 5, 0], [0, 0, 3, 0, -6, 5, 1, 0], [4, -7, 0, 0, 0, 0, 0, 11], [0, 0, -5, 0, 4, 1, -2, 0], [0, 0, 2, 0, 6, 4, -6, 0], [-12, 0, 0, 2, 0, 0, 0, 4], [-2, -5, 0, -4, 0, 0, 0, 3], [0, 0, 0, 0, 1, 2, -7, 0], [0, 0, -1, 0, 8, 3, 0, 0], [6, 7, 0, -3, 0, 0, 0, -1], [-4, -6, 0, 1, 0, 0, 0, -7], [0, 0, 0, 0, 0, 2, -9, 0], [6, -14, 0, -2, 0, 0, 0, 0], [0, 0, 2, 0, 2, 4, -5, 0], [16, 5, 0, -3, 0, 0, 0, 3], [10, -10, 0, 2, 0, 0, 0, 4], [16, -2, 0, -8, 0, 0, 0, 2], [-14, -3, 0, 7, 0, 0, 0, -6], [-18, -19, 0, -2, 0, 0, 0, 7], [-2, 4, 0, 6, 0, 0, 0, 0], [0, 0, -4, 0, -9, -3, -2, 0], [0, 6, 0, -6, 0, 0, 0, -8], [0, 0, 2, 0, 0, -4, -10, 0], [0, 0, 2, 0, 2, -6, -3, 0], [22, 9, 0, 8, 0, 0, 0, -3], [-6, -10, 0, -10, 0, 0, 0, 6], [0, 0, 3, 0, 4, 3, 1, 0], [10, -2, 0, -2, 0, 0, 0, 8], [-4, -9, 0, -10, 0, 0, 0, 5], [0, 0, -1, 0, -5, -5, 6, 0], [0, 0, -4, 0, 0, -4, 3, 0], [-12, 0, 0, -5, 0, 0, 0, -9], [0, 0, 4, 0, -3, 6, 1, 0], [18, 16, 0, 10, 0, 0, 0, -6], [0, 0, 6, 0, -8, 1, 1, 0], [-16, -2, 0, -6, 0, 0, 0, -2], [18, 15, 0, 0, 0, 0, 0, -5], [0, 0, -2, 0, 2, 0, -6, 0], [0, 0, 0, 0, 10, 8, 4, 0], [0, 0, 0, 0, 1, -4, 8, 0], [0, 0, -1, 0, 5, 1, 6, 0], [0, 0, -4, 0, 0, -2, 4, 0], [0, 3, 0, -2, 0, 0, 0, -13], [0, 0, -8, 0, 1, 5, 8, 0], [0, 0, -2, 0, -2, 8, 8, 0], [0, 0, 6, 0, 0, 2, -7, 0], [3, 6, 0, -4, 0, 0, 0, 6], [7, 4, 0, 2, 0, 0, 0, -2], [0, 0, 6, 0, -7, -4, 3, 0], [7, -5, 0, 2, 0, 0, 0, -7], [0, 0, 4, 0, -2, -1, -7, 0], [-8, -3, 0, 7, 0, 0, 0, -2], [2, 6, 0, 7, 0, 0, 0, 7], [0, 0, 0, 0, 2, -2, 1, 0], [0, 0, -2, 0, -5, -6, 4, 0], [8, -7, 0, -6, 0, 0, 0, 0], [0, 0, -6, 0, 9, 4, 5, 0], [0, 0, 0, 0, -2, 4, -8, 0], [0, 0, -6, 0, -2, 0, 6, 0], [18, 2, 0, 4, 0, 0, 0, 0], [10, 9, 0, -6, 0, 0, 0, -5], [0, 0, -2, 0, 9, 7, 6, 0], [0, 0, 3, 0, 7, 1, 9, 0], [-30, -9, 0, 0, 0, 0, 0, -1], [18, 11, 0, -2, 0, 0, 0, -5], [0, 0, -2, 0, 4, -8, 0, 0], [4, -13, 0, -6, 0, 0, 0, 8], [0, 0, 2, 0, -10, -2, -4, 0], [0, 0, -2, 0, -5, -4, 7, 0], [-1, -1, 0, -2, 0, 0, 0, -11], [-4, 5, 0, -5, 0, 0, 0, -6], [0, 0, -5, 0, 11, 3, 3, 0], [0, 4, 0, -10, 0, 0, 0, -2], [22, 13, 0, 7, 0, 0, 0, -11], [0, 0, 3, 0, 0, 9, 8, 0], [4, 8, 0, 8, 0, 0, 0, 8], [34, -1, 0, -1, 0, 0, 0, -3], [0, 0, 4, 0, 4, 10, 6, 0], [0, 0, 0, 0, -3, -3, -10, 0], [0, 0, -2, 0, 6, 0, -7, 0], [2, -9, 0, -9, 0, 0, 0, -3], [0, 0, 8, 0, -8, -4, -11, 0], [8, 21, 0, -2, 0, 0, 0, 2], [0, 0, 0, 0, 4, 5, 9, 0], [0, 0, -1, 0, 0, 1, 10, 0], [-10, -2, 0, -1, 0, 0, 0, -1], [0, 0, -4, 0, 2, -6, 4, 0], [-6, 13, 0, 8, 0, 0, 0, 1], [-8, -8, 0, -12, 0, 0, 0, 2], [0, 0, -10, 0, 6, 6, 3, 0], [0, 0, 4, 0, 0, -10, -10, 0], [0, 0, 10, 0, -10, -2, -7, 0], [0, 0, 5, 0, -9, -5, 6, 0], [-26, -18, 0, -5, 0, 0, 0, 7], [0, 0, -9, 0, -6, 1, -2, 0], [-18, -26, 0, -2, 0, 0, 0, 6], [0, 0, -3, 0, 1, -1, 4, 0], [0, 0, 4, 0, 8, 14, 9, 0], [0, 0, -4, 0, -5, -4, -5, 0], [34, 16, 0, -2, 0, 0, 0, -6], [24, -7, 0, 3, 0, 0, 0, 1], [0, 0, 8, 0, -7, -6, -17, 0], [-20, 5, 0, 1, 0, 0, 0, 7], [0, 1, 0, -4, 0, 0, 0, -15], [-22, 1, 0, -5, 0, 0, 0, -3], [0, 0, 3, 0, 4, 4, -1, 0], [-14, 4, 0, -4, 0, 0, 0, 0], [6, -4, 0, 6, 0, 0, 0, 0], [-3, 8, 0, 4, 0, 0, 0, -4], [0, 0, 0, 0, -10, -2, -6, 0], [13, 13, 0, -6, 0, 0, 0, 3], [0, 0, 3, 0, 5, -1, -2, 0], [0, 0, 11, 0, -10, -11, -16, 0], [0, 0, 4, 0, -5, -1, -2, 0], [0, 0, 3, 0, -4, 10, 3, 0], [-6, -9, 0, -9, 0, 0, 0, 3], [-20, -4, 0, -2, 0, 0, 0, 4], [0, 0, 0, 0, 6, 0, 3, 0], [0, 0, 10, 0, -9, 4, -6, 0], [-21, -7, 0, 8, 0, 0, 0, -1], [10, 5, 0, 15, 0, 0, 0, 0], [-17, -9, 0, -12, 0, 0, 0, -3], [-10, 6, 0, 1, 0, 0, 0, 1], [0, 0, 6, 0, 12, -8, -4, 0], [0, 0, 3, 0, -6, 5, -5, 0], [-9, 3, 0, 12, 0, 0, 0, -13], [0, 0, 8, 0, 1, 1, 2, 0], [-2, 3, 0, -1, 0, 0, 0, -5], [0, 0, 3, 0, 13, 15, 13, 0], [20, 2, 0, -2, 0, 0, 0, 2], [0, 0, -8, 0, -5, -13, -4, 0], [0, 0, 0, 0, 10, 4, 3, 0], [0, 0, -1, 0, -14, -3, 2, 0], [0, 0, -6, 0, -9, 7, 4, 0], [12, -15, 0, -9, 0, 0, 0, -7], [0, 0, -10, 0, 11, 12, 20, 0], [2, 10, 0, -3, 0, 0, 0, -9], [0, 0, 2, 0, 6, -2, 13, 0], [36, -2, 0, -2, 0, 0, 0, 4], [0, 0, -12, 0, 11, 9, 10, 0], [0, 0, 0, 0, -12, -6, -12, 0], [-26, 1, 0, -2, 0, 0, 0, 17], [0, 0, 4, 0, 4, 12, 7, 0], [-1, 0, 0, -3, 0, 0, 0, -16], [0, 0, -4, 0, -5, -7, 6, 0], [4, 18, 0, 2, 0, 0, 0, 4], [-10, 7, 0, -14, 0, 0, 0, -5], [0, 0, -2, 0, 7, -2, -6, 0], [-30, -5, 0, -4, 0, 0, 0, -1], [0, 0, 2, 0, 16, 12, 11, 0], [-28, -2, 0, 4, 0, 0, 0, -2], [0, 0, 6, 0, 1, -3, -14, 0], [14, -10, 0, 4, 0, 0, 0, 0], [-14, -21, 0, -16, 0, 0, 0, -5], [0, 0, -10, 0, -7, 2, -5, 0], [0, 0, 0, 0, 19, 10, 1, 0], [-4, -9, 0, 7, 0, 0, 0, 1], [0, 0, 9, 0, 8, 9, 1, 0], [0, 0, 3, 0, 4, 11, 1, 0], [-30, 7, 0, -11, 0, 0, 0, -9], [10, 14, 0, 6, 0, 0, 0, 4], [0, 0, -4, 0, -9, -6, -10, 0], [0, 0, 4, 0, -12, 0, 2, 0], [-30, -1, 0, -5, 0, 0, 0, 16], [0, 0, -4, 0, -3, -10, -9, 0], [-26, -13, 0, 1, 0, 0, 0, -1], [0, 0, 4, 0, 10, 4, 17, 0], [-12, 12, 0, 14, 0, 0, 0, 4], [0, 0, 10, 0, 0, -2, -3, 0], [2, 17, 0, 9, 0, 0, 0, -7], [0, 0, -5, 0, 7, 4, -4, 0], [-10, 12, 0, -2, 0, 0, 0, -8], [0, 0, -2, 0, 9, -9, 2, 0], [-20, 2, 0, 16, 0, 0, 0, -14], [0, 0, 6, 0, -4, -8, -1, 0], [0, 0, -9, 0, -3, -3, -5, 0], [-18, -29, 0, -10, 0, 0, 0, 7], [42, -9, 0, 0, 0, 0, 0, 0], [0, 0, 9, 0, -8, -10, -7, 0], [-5, 2, 0, 6, 0, 0, 0, -10], [31, -9, 0, 6, 0, 0, 0, 15], [0, 0, -10, 0, -8, 4, -7, 0], [0, 0, -5, 0, 4, -2, -1, 0], [-24, 13, 0, 10, 0, 0, 0, -2], [-24, -13, 0, -9, 0, 0, 0, 12], [0, 0, -13, 0, 1, -1, 4, 0], [4, -13, 0, 11, 0, 0, 0, -9], [0, 0, -11, 0, 5, -5, 13, 0], [-40, -5, 0, -12, 0, 0, 0, -9], [0, 0, 17, 0, -16, 0, -5, 0], [-30, 16, 0, -12, 0, 0, 0, -8], [-4, -13, 0, 2, 0, 0, 0, 13], [0, 0, 6, 0, -16, -12, -12, 0], [0, 0, 7, 0, 5, 7, -1, 0], [0, 0, -6, 0, 17, 6, 2, 0], [0, 0, -8, 0, -1, 6, 1, 0], [14, -13, 0, 0, 0, 0, 0, 15], [0, 0, 1, 0, -6, -13, -1, 0], [0, -7, 0, 1, 0, 0, 0, 18], [0, 0, -7, 0, -6, -5, -8, 0], [0, 0, 7, 0, -5, -3, 4, 0], [6, 12, 0, 0, 0, 0, 0, -2], [0, 0, 8, 0, -12, -10, -4, 0], [10, 9, 0, 6, 0, 0, 0, -9], [0, -8, 0, -4, 0, 0, 0, 0], [-6, 6, 0, 8, 0, 0, 0, 8], [0, 0, 5, 0, 14, 7, 10, 0], [0, 0, 0, 0, -14, -12, -6, 0], [0, 0, -10, 0, 4, 0, 10, 0], [0, 0, -18, 0, 10, -8, 9, 0], [6, -17, 0, -2, 0, 0, 0, 6], [-12, -3, 0, 6, 0, 0, 0, 15], [30, -9, 0, -6, 0, 0, 0, -13], [0, 0, 8, 0, -7, 2, -1, 0], [0, 0, 7, 0, -15, 5, -4, 0], [-33, -4, 0, -11, 0, 0, 0, 4], [-44, 9, 0, 4, 0, 0, 0, 5], [42, 17, 0, -7, 0, 0, 0, -1], [9, 11, 0, 10, 0, 0, 0, 15], [0, 0, -9, 0, 2, -2, 19, 0], [0, 0, -3, 0, -6, -5, 1, 0], [0, 0, 2, 0, 6, 6, 6, 0], [0, 0, 9, 0, 8, 17, 9, 0], [0, 0, -4, 0, 10, -4, 6, 0], [-10, 4, 0, 14, 0, 0, 0, 0], [18, 23, 0, -5, 0, 0, 0, -13], [0, 0, -21, 0, 14, 7, 10, 0], [-4, -17, 0, 9, 0, 0, 0, -3], [-18, 13, 0, 0, 0, 0, 0, -3], [0, 0, 12, 0, 1, -5, 10, 0], [-18, -5, 0, 7, 0, 0, 0, 8], [2, -3, 0, -4, 0, 0, 0, -19], [5, 6, 0, 19, 0, 0, 0, -6], [-32, -33, 0, -13, 0, 0, 0, 6], [0, 0, 5, 0, -1, -3, 3, 0], [0, 0, 8, 0, -7, -1, 10, 0], [23, 21, 0, -6, 0, 0, 0, 5], [-24, -31, 0, 5, 0, 0, 0, 9], [-12, -10, 0, -3, 0, 0, 0, -5], [-13, -12, 0, -15, 0, 0, 0, -6], [0, 0, -8, 0, -3, -9, 16, 0], [0, 0, -2, 0, -12, -12, -9, 0], [0, 0, -6, 0, -2, -4, 21, 0], [0, 0, 0, 0, -5, 0, 15, 0], [0, 0, -6, 0, -3, 12, 7, 0], [-14, 27, 0, 2, 0, 0, 0, 11], [-24, -21, 0, 4, 0, 0, 0, 7], [-50, -15, 0, 4, 0, 0, 0, -3], [0, 0, -3, 0, 18, 0, 1, 0], [24, 9, 0, 15, 0, 0, 0, 9], [23, 10, 0, -6, 0, 0, 0, -6], [0, 0, -5, 0, -9, -9, -11, 0], [0, 0, 3, 0, 4, 8, 5, 0], [10, 17, 0, 7, 0, 0, 0, -18], [0, 0, -1, 0, 2, 4, 13, 0], [0, 0, -10, 0, 17, 7, 12, 0], [0, 0, 5, 0, -5, -11, -13, 0], [-24, -12, 0, -12, 0, 0, 0, 12], [0, 0, 14, 0, 4, 0, 8, 0], [-45, -6, 0, -5, 0, 0, 0, -8], [0, 0, -14, 0, 0, 6, -4, 0], [0, 0, 6, 0, -18, 2, -1, 0], [-4, 3, 0, -5, 0, 0, 0, 1], [40, 6, 0, 6, 0, 0, 0, -14], [0, 0, 6, 0, 14, 10, 9, 0], [0, 0, -4, 0, 0, -2, 12, 0], [0, 0, -11, 0, -3, -8, -2, 0], [-20, 10, 0, -10, 0, 0, 0, -22], [0, 0, -5, 0, -1, 1, 7, 0], [-4, 6, 0, 6, 0, 0, 0, -16], [5, -16, 0, 10, 0, 0, 0, -10], [0, 0, 0, 0, 11, -7, 4, 0], [0, 0, 4, 0, 6, 4, -19, 0], [-2, 6, 0, -12, 0, 0, 0, -6], [0, 0, 0, 0, 19, 17, 4, 0], [-63, -12, 0, 1, 0, 0, 0, 10], [8, -13, 0, -6, 0, 0, 0, 5], [4, -13, 0, -17, 0, 0, 0, 1], [0, 0, 4, 0, -2, 16, 8, 0], [0, 0, 8, 0, -18, -4, -12, 0], [-14, -21, 0, -7, 0, 0, 0, 19], [-46, 10, 0, 2, 0, 0, 0, 4], [-19, 8, 0, 1, 0, 0, 0, -4], [0, 0, 10, 0, -23, -10, 0, 0], [0, 0, 1, 0, -2, 14, 13, 0], [0, 0, 5, 0, 10, 7, 7, 0], [0, 0, 2, 0, 22, -2, -5, 0], [-11, 19, 0, -4, 0, 0, 0, 9], [0, 0, 1, 0, 2, 9, 2, 0], [42, -13, 0, 4, 0, 0, 0, -1], [-15, 12, 0, 16, 0, 0, 0, -8], [-22, -2, 0, -14, 0, 0, 0, 12], [0, 0, 11, 0, 0, 3, -2, 0], [-34, 15, 0, 4, 0, 0, 0, 11], [0, 0, -1, 0, 11, 13, 14, 0], [-18, -7, 0, -4, 0, 0, 0, -11], [27, 22, 0, 17, 0, 0, 0, -6], [0, 0, 8, 0, -20, -4, 0, 0], [0, 0, -2, 0, -13, -8, -17, 0], [0, 0, -4, 0, 6, -4, 20, 0], [0, 7, 0, -6, 0, 0, 0, 11], [0, 0, 4, 0, 5, -5, -20, 0], [-26, -14, 0, -8, 0, 0, 0, -12], [0, 0, -7, 0, 11, -17, 1, 0], [-8, 18, 0, -10, 0, 0, 0, -8], [-18, -5, 0, -12, 0, 0, 0, -5], [0, 0, 0, 0, 6, -2, -18, 0], [-28, -8, 0, 2, 0, 0, 0, -16], [0, 0, 8, 0, -13, -3, -10, 0], [-6, -3, 0, 7, 0, 0, 0, -1], [-8, -19, 0, 1, 0, 0, 0, 11], [0, 0, -10, 0, 10, 4, 10, 0], [0, 0, -9, 0, 1, 2, 18, 0], [30, 1, 0, 12, 0, 0, 0, 5], [0, 0, -4, 0, -16, -6, 14, 0], [12, 12, 0, 10, 0, 0, 0, -32], [0, 0, -7, 0, 1, 12, -10, 0], [0, 0, 9, 0, -5, -7, -12, 0], [0, 0, -8, 0, 0, -10, 1, 0], [52, -3, 0, -11, 0, 0, 0, -1], [11, 10, 0, 15, 0, 0, 0, -20], [-10, -5, 0, 4, 0, 0, 0, 3], [12, 3, 0, -5, 0, 0, 0, -9], [0, 0, -14, 0, 10, -8, 7, 0], [-40, -10, 0, -5, 0, 0, 0, 23], [20, 11, 0, 2, 0, 0, 0, 13], [-8, 2, 0, 15, 0, 0, 0, 7], [0, 0, 8, 0, 8, -6, 8, 0], [0, 0, 8, 0, 6, -6, -16, 0], [0, 0, 4, 0, 3, -9, -8, 0], [12, -6, 0, 13, 0, 0, 0, 9], [0, 0, 12, 0, -22, -6, -2, 0], [0, 0, 21, 0, -13, -2, -2, 0], [0, 0, 10, 0, -2, 7, -11, 0], [0, 0, 2, 0, 13, 16, -2, 0], [-7, 6, 0, 15, 0, 0, 0, -2], [-10, 12, 0, -14, 0, 0, 0, 0], [0, 0, 15, 0, -13, -7, -17, 0], [6, -9, 0, 14, 0, 0, 0, -4], [0, 3, 0, 15, 0, 0, 0, -2], [-38, 5, 0, 10, 0, 0, 0, -13], [0, 0, -7, 0, -19, -1, -12, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4864_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4864_2_a_bm();".
// 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_4864_2_a_bm(:prec:=8)
    chi := MakeCharacter_4864_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_4864_2_a_bm();".
// 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_4864_2_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -8, -5, 2, 1]>,<5,R![656, 0, -712, 0, 249, 0, -30, 0, 1]>,<7,R![41, 0, -1072, 0, 326, 0, -32, 0, 1]>,<11,R![-32, 0, 25, 10, 1]>],Snew);
    return Vf;
end function;