
// Make newform 5054.2.a.bf in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5054_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_5054_2_a_bf();".
// 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_5054_2_a_bf();".
// 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, -32, -72, -24, 50, 16, -12, -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], [6, 8, -6, -2, 1, 0, 0, 0], [-4, -18, 4, 4, -1, 0, 0, 0], [-2, 16, 10, -8, -2, 1, 0, 0], [4, 18, 8, -8, -2, 1, 0, 0], [-16, -64, -32, 48, 17, -12, -2, 1], [24, 116, 86, -84, -43, 22, 5, -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_5054_a();"
function MakeCharacter_5054_a()
    N := 5054;
    order := 1;
    char_gens := [1445, 1807];
    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_5054_a_Hecke(Kf)
    return MakeCharacter_5054_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, -1, 0, 1], [1, 1, 0, 0, 0, 0, 1, -1], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, -1, 0, 0], [-1, -2, -1, -1, 0, -1, -1, 2], [0, -1, -1, -1, -1, 0, -1, 2], [0, 0, 0, 0, 0, 0, 0, 0], [3, -1, 2, 1, 0, 2, -1, -2], [-2, 0, -1, 0, -2, 0, 1, 0], [-2, 0, 0, 0, -1, 1, 1, 1], [3, 1, 3, 2, 1, 3, 1, -2], [-5, -2, 0, 0, -2, 1, -2, 1], [0, 1, -2, -1, -2, 2, -1, -3], [3, 1, 0, 0, 0, 2, 3, -1], [-4, 1, -1, 1, 0, 0, -3, -3], [-2, -3, -1, -2, 0, 2, -5, 2], [0, 0, 2, -2, -1, -1, -2, 0], [4, 1, 1, 3, 2, 0, 4, -5], [1, -2, 4, -1, -1, 3, -2, -1], [-4, -3, 0, 0, 0, -1, -4, 1], [-2, 3, -1, 2, -1, -1, 1, 1], [6, -4, 1, -1, 1, -1, 1, 7], [0, 2, -2, 1, 0, -1, 7, -1], [-6, 3, -1, -2, 0, -2, -3, 2], [-3, -3, 1, 2, 5, -4, 0, 0], [1, 2, 4, 2, 2, 5, 5, -8], [1, 0, 3, 1, 3, 1, 4, -2], [0, 0, 4, -1, 5, -1, 4, 2], [-1, 1, -3, 2, 3, -1, -2, 2], [5, -1, -3, -1, 2, 0, 7, 0], [2, -1, -3, -1, -1, -2, -5, 0], [5, 2, 6, -1, 1, 1, 6, 3], [2, -3, 2, 1, -3, 3, -3, -2], [-6, -3, 1, -1, 0, 4, -7, 2], [-6, 0, -5, -4, -2, -2, -7, 4], [-1, 3, -2, 1, 2, -1, -5, 1], [0, 5, -2, 3, 0, 0, 6, 1], [-6, -2, -5, -1, 1, -5, -3, -3], [-3, 9, 0, 2, -2, 0, 2, -8], [-5, 0, 2, -1, -1, 5, -4, 2], [-2, -2, 3, -1, 4, -2, 8, -2], [3, 2, 0, 0, 6, -2, 3, -1], [5, 3, 2, -2, 1, 3, -6, -2], [-2, 0, -1, 0, -6, -2, -7, 1], [-5, 1, -6, -1, 0, -3, -3, 5], [-9, -4, -3, -1, -1, 5, -14, 0], [-1, 3, -4, 2, -2, 0, 0, -4], [10, -3, -6, 1, -1, 1, 2, -1], [-3, -5, -6, 1, 1, 0, 3, 5], [-14, -1, 0, -1, 2, -4, 1, 3], [-4, 3, 3, 4, 3, -1, 8, 0], [5, -3, 3, -2, -2, 9, -5, -5], [-3, -3, 3, -1, -2, -4, -8, 2], [-1, 0, 1, 2, 1, 7, 8, -5], [1, 2, 2, -4, 0, 2, -5, 3], [-5, 3, 2, -2, -4, 4, -6, -2], [-7, -1, -7, -1, -1, -9, 4, 2], [2, 8, 8, 2, 0, 8, 6, -6], [-5, -4, -3, 3, 3, -3, 0, 4], [-1, -1, 9, 0, 5, 0, 3, 1], [4, -4, 5, 1, 0, 2, -6, 6], [-6, -4, -1, -1, 1, -7, 1, 5], [2, -8, 5, -1, 0, -4, 3, 7], [-14, -4, 0, -1, 1, -6, -7, 7], [0, -2, 1, 2, 2, -6, -11, 2], [-7, 5, 4, 2, 3, -3, 4, -6], [6, 0, 11, 0, 0, 2, 6, 6], [5, -1, -2, 4, 3, -7, 10, 0], [-5, -7, -5, 3, 3, -5, 8, 4], [-9, 3, -6, 1, 0, -1, 2, 2], [-4, 7, -6, 0, 1, -5, -3, 1], [-8, 2, -1, 0, 2, -7, -5, 3], [-3, 5, -5, 1, 0, -2, 7, -2], [-11, 5, -3, 2, -5, 3, -8, -10], [-7, 1, -1, 1, 5, -1, -11, 5], [-3, 3, -11, 0, -1, -3, 7, 3], [-7, -5, 9, -2, -8, 9, -4, 0], [3, -4, -2, -1, -3, -5, 2, -3], [-17, 5, -2, -1, 1, -10, 2, 0], [9, -3, -3, -4, -5, 0, 5, 7], [3, 8, 3, 1, 3, 3, 17, -1], [6, -5, 0, 5, -6, 8, -10, -1], [6, 4, -3, 2, 1, 0, 12, -4], [-24, 0, 4, 0, -3, 1, -5, 1], [9, -3, 4, 0, 3, -1, -5, -2], [1, -3, 6, 0, -5, 7, -2, -12], [-18, -1, -2, -1, -6, 2, -8, -7], [9, -3, 6, -2, 1, -3, 5, 5], [0, 2, -5, 2, -4, -4, -1, -3], [9, 2, 15, 6, 7, 11, -1, -10], [-1, 7, -1, 5, -6, 2, 0, -8], [-4, -4, -9, -4, -2, -4, -3, 0], [-8, -4, -2, 0, 0, -4, 10, 6], [-3, -3, -2, 3, 0, 2, 6, 1], [13, 1, 3, 3, -2, -2, 14, -4], [13, 9, 0, 2, 1, -1, 23, -1], [4, 1, 2, 5, 5, -3, 10, 1], [-16, -1, -1, 3, 4, -3, -2, -1], [-1, -1, 2, 7, 2, 8, 3, -4], [-3, -8, -1, -3, -3, 1, -8, 4], [6, 4, 6, 1, 1, 13, 7, -16], [-7, -3, -5, -3, 0, -4, 4, -2], [1, -16, -4, -5, 1, -7, -10, 16], [4, -3, 6, -3, -4, 4, 2, -1], [-8, -1, -11, 2, 2, -12, 8, 1], [-8, 9, 4, 1, -1, 3, -15, -4], [0, 3, -1, -3, 5, -4, 3, -8], [13, -6, 1, -1, 3, -11, -5, 13], [-6, -5, -3, 6, 12, -12, 5, 6], [-23, 5, 9, 7, 4, 2, 2, -6], [4, -2, -10, -2, 0, -4, -6, 12], [-6, -12, 2, -9, -1, -3, 1, 13], [-12, 4, -9, -4, 2, -13, 6, 4], [-16, -5, -8, -1, -8, -6, -10, 5], [0, -1, 3, -5, -4, 10, 0, -1], [-3, -9, -4, -2, -5, -1, -17, -5], [0, -2, -1, -8, -1, 0, -11, 1], [3, -3, -1, -1, -2, 6, -6, 7], [-1, 4, 7, -1, -3, 15, 6, -6], [-5, -1, -10, 2, 4, -12, -2, 12], [-17, 4, -3, -7, -7, -3, -10, 3], [-14, 4, -9, -7, -7, -5, -15, 9], [12, 2, -10, 0, 0, 0, 18, -3], [0, 3, -4, -3, 5, -1, 7, -4], [5, -8, -7, 0, 2, -4, 4, 8], [-10, -15, 8, -6, 1, 5, -15, 9], [7, 1, -12, -4, 6, 0, 3, -9], [11, 7, -2, 0, -6, 0, 11, -1], [9, 7, 20, 4, -4, 10, 7, -3], [-9, -12, -6, 1, -7, 5, -16, -3], [-13, 4, -6, -6, 2, -12, -4, 1], [6, 10, 4, 6, -6, 6, -4, -10], [-9, 5, -10, -4, -3, -13, -2, 0], [-6, 3, 3, -6, -2, 4, -8, 15], [11, -11, 11, -3, 0, 6, -6, -6], [-12, 5, 5, -3, -6, 3, -13, -10], [-1, -13, 2, -9, -3, 0, -12, 2], [0, 1, -12, 8, -7, 6, 8, -9], [-5, 10, 0, 0, -4, 6, 13, -20], [-6, 14, 13, 3, 3, 5, 11, -5], [-11, -3, -4, 1, 4, 0, 1, 8], [12, -6, 10, -7, 3, -5, -5, 16], [-10, 9, 1, 9, 6, -2, 8, 1], [-5, 3, -5, 1, 5, 1, 3, 5], [6, -14, -5, 1, -6, 0, 0, -2], [-3, 1, 9, 5, -1, -3, -4, 2], [24, 7, 5, 5, 6, -1, 20, -13], [-6, 3, -1, -2, -4, 10, 0, -9], [18, -13, -3, -11, -2, 4, -13, 11], [27, 0, 3, 4, 2, 0, 29, -1], [-11, 19, -1, 4, 1, -4, 5, -15], [-5, -6, 14, -4, -6, 10, 1, -12], [-2, -4, -9, 1, 7, -17, 15, 7], [10, 7, -8, -3, 4, -12, 22, 12], [-5, 6, -5, -4, 6, -6, -1, 5], [1, -9, -3, -4, 5, -5, -7, -1], [24, -3, 8, 7, 5, -1, 19, -8], [-3, 9, 5, 4, 1, 2, 17, 13], [18, 3, 2, 2, -2, -5, 3, -6], [13, 6, -8, 7, 1, 5, 14, -5], [0, 12, -5, -4, -6, 2, 0, -7], [17, -6, 13, -7, 1, -3, 3, 18], [-11, 8, -2, -1, 4, -8, -1, -6], [12, 4, -9, -2, 8, -12, 8, 17], [9, 0, -3, -3, 9, -14, -2, 9], [7, 3, 2, -4, 7, -13, 10, 14], [9, -3, -5, -4, -2, 7, -6, 6], [-19, 4, -3, -5, -9, -1, 0, -14], [-8, -4, -14, 3, 4, 3, 12, 12], [-3, -3, 13, 2, 5, -5, -10, -10], [-4, -4, 2, 2, -4, 0, 0, 6], [-5, -1, 11, 3, -8, 12, -4, -1], [-1, -3, 0, 3, 4, 0, -11, -5], [-11, 13, -9, 7, 4, -2, -2, -14], [5, 7, -5, -7, -6, 2, 14, 0], [1, -5, -12, 1, -14, 6, -19, 9], [-9, 1, -6, 3, 0, -2, -19, 14], [10, 5, -6, 6, 10, -1, 13, -2], [-17, -9, -11, -5, -6, 8, -22, 6], [-1, 5, 3, -3, -2, -4, -6, 23], [12, 5, 4, 3, 1, 1, 14, 7], [-5, -13, -6, -3, -4, -6, -28, 8], [-1, -3, 25, 7, 2, 10, -22, -14], [15, -4, 2, -7, -11, 7, 0, -3], [-4, 2, 9, 9, 5, 1, 13, -11], [-13, 6, -10, -3, 3, -11, -10, 1], [1, 3, -15, -7, -12, 4, 6, 12], [-12, 6, -13, -4, 0, 2, -8, -7], [2, -2, -4, -7, 0, -11, 10, 8], [-8, -15, -6, 4, 7, 0, 3, 6], [-2, 5, -8, -1, -6, -8, 10, -9], [-16, -1, -22, -1, -2, -20, -6, 10], [3, -4, -4, -2, 11, -16, 16, 15], [-19, -7, 2, -5, 2, 0, 0, 15], [18, 3, 2, -6, -11, 0, -7, 4], [17, 3, -12, 7, 0, 4, -1, 5], [-4, 8, 9, 12, 0, 4, -9, -15], [-1, -12, -15, 0, -5, -9, -11, 0], [25, 1, -10, 3, -1, -4, 1, -3], [-3, 3, 1, -2, -3, -5, -21, 2], [-8, 0, 11, 3, 0, 2, 10, -6], [-13, -7, -1, -1, -5, 11, -5, 11], [-15, 4, -5, 1, -1, -8, 19, 6], [-2, 1, -2, -4, 0, -13, -16, 13], [3, -7, 5, 0, -3, 10, 8, -8], [22, 4, 2, 12, 4, 12, 22, 2], [11, -3, -15, -10, -5, -5, -10, 18], [6, 12, -8, 2, 6, 4, 7, -20], [-1, -14, 7, -7, -3, 10, -25, -4], [-23, 1, -2, -1, 0, -6, -26, 13], [13, 3, 4, -4, 14, -16, 23, 17], [10, 5, 11, 4, -3, 7, 6, 10], [12, -8, 18, -4, 0, -4, 12, 12], [6, 19, -7, 3, -8, -3, 3, 0], [-28, 8, 18, 9, 8, 7, 5, -11], [12, -1, 10, -7, 0, 8, 4, 1], [2, 8, -12, 2, -10, -2, -10, 10], [0, 6, 4, 12, 0, -2, 20, -8], [-5, 1, -9, -4, -7, -9, 7, 5], [-23, 1, -10, -12, -13, -7, -22, 6], [-23, 4, 21, 1, -2, 7, -7, -2], [13, 3, -5, 4, -1, 5, -18, -6], [0, 3, 3, -3, -14, 10, -3, -10], [-3, -19, 7, -5, -8, 0, -8, 12], [18, -12, -14, -1, 1, -5, 23, 2], [-8, -3, -4, 3, 7, -5, -12, -13], [-4, -6, 0, 5, 7, -15, 14, 8], [-19, -8, -1, -3, -4, -11, 9, 18], [-21, -2, -10, -15, -9, -7, -23, 19], [24, 7, -2, 12, 11, -17, 23, -15], [-23, 3, 14, 4, 9, -1, 5, -9], [-6, -7, -1, -3, -12, 2, -5, -7], [18, 9, 1, -1, 0, 10, -2, -11], [-11, -11, -15, -7, -18, -6, -22, 8], [-16, -14, 3, -5, 2, -16, -23, 13], [-28, 4, -19, 4, -6, -1, -12, -16], [-5, 3, -4, -5, -6, -8, -7, -10], [19, -4, 8, -1, -7, 19, -12, 1], [-12, 5, 4, 6, 10, 7, -13, 0], [2, 13, 5, -9, -12, 8, -4, -8], [17, -3, 17, 8, 5, -2, 13, -1], [-2, 2, -14, -2, -9, -1, -28, 18], [23, 6, 19, -2, 6, 8, 17, -11], [2, -12, 10, -4, -2, 0, -26, -2], [-5, -11, -10, -5, -4, 5, -25, 21], [-9, -11, -3, 0, -3, 4, -24, -2], [13, -2, -11, -6, -5, -3, 0, -7], [9, 10, 23, 3, 13, 3, 10, -6], [16, -12, -22, -7, -7, -7, -5, 14], [10, -4, -17, -11, -9, -7, 0, 15], [0, 14, 8, 0, 8, -18, 16, -8], [-20, -7, -24, 0, -9, -11, -13, -1], [-41, -6, -8, -7, -2, -12, -23, 10], [-18, -4, -18, 6, -5, 1, -24, -6], [5, 8, 16, 4, 4, 20, 11, -6], [-4, 2, 15, 1, 5, -19, 10, 7], [-2, -5, -10, 0, -9, 12, -13, -2], [-17, -4, 12, 2, 11, -12, 12, -5], [16, 13, 16, -2, 14, -1, 18, -3], [17, -14, 1, -9, 5, -9, -13, 31], [-32, 13, -8, -5, -16, 6, -10, -26], [-5, 7, -4, -10, -1, -1, -15, 13], [-37, 5, 4, -2, -1, 3, -13, -3], [0, -1, -2, -8, -6, 3, 14, -3], [-3, 2, -10, 6, 10, 3, -10, 5], [0, -3, -2, 4, 9, 7, 14, -7], [0, 14, 0, 8, 2, 2, -18, -12], [15, 25, 2, 12, 7, 7, 34, -13], [-10, 6, -10, 1, 7, -19, 15, 0], [-4, 8, 1, 5, -2, 4, 2, -6], [-6, -6, -11, -6, -8, -20, -24, 20], [-24, -3, -9, -2, -8, -6, 4, 5], [18, -8, -9, 4, 0, 2, 9, -17], [-15, 12, -3, -4, 13, -21, 12, 9], [24, 2, 20, 4, 16, 6, 16, -22], [11, -12, 6, 10, 12, -18, 8, 2], [-12, -7, -3, -5, -14, 8, -23, -19], [-23, -3, -2, -8, -15, 5, -28, 23], [16, 15, -27, -4, 0, -20, 30, 13], [26, 6, 7, -9, -1, -9, 19, 9], [33, 8, -4, 4, 0, -6, -5, -1], [-13, -5, 16, -2, 12, -2, -3, -3], [-42, 5, 4, -2, 2, -11, -23, -2], [9, 10, -5, 3, -2, 11, 18, -13], [11, 10, -11, 2, -8, 8, -10, -8], [-17, -12, -16, -1, -7, -5, -31, 13], [14, 10, -1, 7, -1, 13, 25, -18], [-25, 5, -12, -8, -6, -12, 9, -5], [21, -4, -7, 0, 3, 7, 20, 1], [4, 12, 16, -2, -2, 0, 8, -10], [23, 6, 4, 10, 17, 8, 29, -6], [4, -4, 6, -4, 10, -8, 14, 10], [1, -3, 2, -6, -9, 1, -17, 5], [10, 4, -11, 2, 4, 8, 19, 4], [8, 0, 15, 3, 12, 4, 7, 3], [-12, -4, -7, -5, 11, -19, 11, 9], [2, 6, 1, -6, 8, 1, 7, 1], [2, -6, 7, -3, -9, 21, 5, -11], [5, 6, 11, 0, 5, 13, 3, -4], [15, 4, 21, 10, 17, -1, 21, -2], [-16, 14, 2, -9, -7, 7, -10, 4], [8, -5, -15, -3, 12, -14, 13, 22], [10, -9, 6, 1, -6, 18, 8, -5], [30, -10, -6, -6, -18, 10, 6, 14], [-16, 9, -22, -7, 4, -16, 12, 8], [-4, 16, -6, -10, -6, -2, -10, 5], [-21, 4, -5, -5, -5, 2, 8, -11], [6, -3, -1, -11, -22, 6, -11, -16], [-29, -15, -8, -12, -3, -1, -40, 4], [13, -9, 9, 6, 0, 17, -18, 18], [-9, 18, 5, -4, -12, 10, 8, -18], [2, 18, -11, 8, -6, -8, 30, -7], [15, -8, -14, 5, 3, -3, 32, -13], [8, -4, -4, 3, -15, 21, -3, 0], [-8, -6, 19, 4, -5, 20, -28, -12], [7, 5, 7, 8, 5, 14, -1, -3], [5, 10, 5, -19, -3, -5, -13, 9], [11, 9, -5, -4, -5, 0, 27, -7], [9, 21, 8, 5, -4, 14, 14, -17], [-1, -5, 6, -7, 15, -2, 20, 12], [2, 15, -28, 3, 0, -16, 18, 11], [20, 2, 15, 3, -6, 2, 6, 8], [-23, 2, 5, 10, 4, -2, -7, -7], [12, -3, 16, 6, 1, 7, 16, -12], [-5, 10, -10, 7, 8, -10, 17, 12], [15, -6, -14, -1, 11, -1, 2, 10], [-5, 11, -11, 15, 4, -2, 10, -24], [-13, -8, 7, 0, -2, -10, -9, 7], [27, 15, 2, -6, 7, -11, 14, -2], [2, 11, 13, 11, 10, -4, 13, -22], [23, 4, -20, 7, 5, -3, 12, -19], [11, 15, 0, -6, -6, 2, 25, -19], [-31, 21, -15, 7, -10, -4, 8, -12], [-14, 7, -26, -18, -15, -1, -10, 18], [32, 11, -9, -7, -1, -2, 25, -12], [-17, -7, 0, -6, -3, -1, -6, 32], [4, 7, -14, 11, 9, 11, 14, -23], [10, 6, 16, 5, -1, 22, 3, -15], [-15, -15, -24, -3, -2, -17, -19, 1], [22, -6, 16, -4, -4, 28, -2, -6], [-20, 3, -10, 0, 17, -37, 2, 2], [-26, 0, 4, 2, 4, -14, 8, 24], [7, 14, 26, 11, 11, 5, 18, -31], [7, 11, -11, -2, -9, 11, -3, 0], [-20, -12, 4, 4, -6, 10, -26, 2], [1, 3, 23, 7, -4, 8, 12, 18], [25, -23, 12, 0, -7, 11, -6, 1], [-36, 9, 2, 4, 17, -4, 0, -5], [11, 1, -5, 10, 0, 5, -12, 0], [-16, 25, 24, 10, 15, -6, 29, -18], [6, 19, 18, 5, 16, -6, 18, -32], [-1, -13, -18, 1, 2, -10, 1, -4], [-25, -13, 8, -6, -5, 3, -15, -9], [-4, 9, -19, -2, -3, -1, 29, 5], [-25, -13, 1, -10, -8, -13, 1, 25], [14, -10, -2, -8, 12, 0, 8, -8], [-13, -17, -13, 5, 4, -2, -4, -10], [-19, -11, -2, 4, 13, -15, 11, 6], [5, -20, -19, 4, 10, -6, 21, 17], [2, 1, -23, -1, -4, -10, -3, 0], [14, -6, 8, 1, -9, 14, -3, 9], [-28, 11, -4, -1, 2, -6, -10, -9], [7, 5, 9, 2, 1, 7, 10, 4], [16, -6, -2, 20, 4, 10, 28, -14], [-10, -13, -11, -5, -8, -26, -12, 26], [-6, -10, 26, -2, -7, 23, -16, -6], [14, -14, -8, 2, -8, -12, -4, -2], [-7, -24, -4, 3, -7, -5, 8, 17], [-30, -25, 6, -11, -4, 4, -26, 23], [-4, -9, 12, -12, -3, 6, -23, 2], [21, -5, 28, 2, 19, 1, 0, -10], [-2, 4, 20, -2, 4, -8, -8, -12], [9, 3, 6, -11, 1, 4, -3, 9], [-1, 10, -2, -9, -7, 3, -2, -13], [41, -5, -19, 1, -14, 30, 4, -10], [25, 5, 11, 2, 3, 11, 13, -1], [5, 15, 3, 9, 6, 32, 16, -36], [-1, 7, 9, 12, 17, 14, 19, -19], [-1, -7, 17, 6, 16, -9, 5, 15], [14, 5, 10, -6, 4, -3, -9, 0], [-15, 6, -4, -6, -16, -2, -30, 18], [-6, -8, 4, -13, 7, -7, 15, 32], [-19, 7, -12, 17, -2, -2, 17, -11], [1, 0, 8, -10, 4, -3, 8, 11], [-28, -16, 1, -3, -2, 6, -40, 8], [-18, 16, -17, 13, -1, -27, -2, 3], [-20, -6, -10, 1, 7, 4, -10, -28], [37, 9, 4, 1, 2, 4, 6, -3], [-13, -9, -4, 2, 3, -11, -17, 15], [34, -11, 18, 5, 0, 4, -4, -1], [14, -12, -19, -7, -13, 1, -19, -19], [-2, -8, 6, -12, -6, -14, 0, 24], [10, 3, 6, 5, 3, 29, -1, -14], [-30, 0, 19, -6, 2, -16, -11, 2], [10, 13, 21, 5, 4, 24, 18, -3], [-9, -33, -5, 3, 3, 3, -33, 37], [4, -6, 8, 5, 4, 9, -19, -5], [9, 8, 12, 5, -1, 9, 42, -6], [-11, 17, 7, -4, 5, -3, 5, -17], [6, 6, 6, 4, 21, -3, 7, -1], [-13, -1, -2, -7, 0, 26, -15, 3], [-1, 7, -22, -18, -5, -9, -20, 26], [-12, 12, 15, 13, 1, 3, 7, -23], [3, 5, 16, 14, 28, -6, 14, -22], [-25, -1, -1, -9, -25, 9, -26, -10], [0, -15, -7, 8, 1, 15, -14, -2], [-24, 12, 15, 10, -2, 12, 18, -25], [3, 19, -22, 2, 3, -17, 5, 1], [32, -9, 10, 9, 3, 15, 29, -16], [-43, -11, -18, -18, -13, 3, -59, 7], [-20, -9, -9, -20, -9, -5, -8, 24], [5, -9, 2, -4, -11, 7, 2, 29], [25, -16, -4, -6, -8, 24, -7, -10], [5, -2, 4, -10, -14, 14, -22, -16], [14, 0, -2, 2, -9, 9, -30, -12], [-19, -5, -9, 2, 11, -27, 17, 0], [-50, -12, 3, -15, -6, -14, -44, 18], [-40, -2, 3, -1, 1, 13, 9, 3], [16, -4, 26, -2, 15, -15, 21, 25], [19, -6, -18, -14, 2, -10, -10, 27], [-33, -14, 18, -12, -6, -9, -9, -2], [21, -13, 6, -9, 16, -14, -3, 32], [8, 0, -1, 4, 7, 4, 28, -22], [-16, 11, -5, -10, -14, -4, -29, -4], [-16, -7, -24, 2, 5, -20, 4, 31], [2, -13, 18, 1, -6, 4, -8, -1], [-25, 4, 4, 5, -13, 13, -2, -21], [22, 1, -11, -6, 14, -2, 0, -9], [13, -5, -3, 5, -10, 8, -4, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5054_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5054_2_a_bf();".
// 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_5054_2_a_bf(:prec:=8)
    chi := MakeCharacter_5054_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_5054_2_a_bf();".
// 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_5054_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5054_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -34, 2, 98, 15, -38, -8, 4, 1]>,<5,R![-79, 46, 122, -82, -50, 42, 2, -6, 1]>,<13,R![181, 144, -888, 422, 355, -182, -38, 6, 1]>],Snew);
    return Vf;
end function;