
// Make newform 4730.2.a.ba in Magma, downloaded from the LMFDB on 13 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4730_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_4730_2_a_ba();".
// 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_4730_2_a_ba();".
// 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 := [16, -124, 97, 327, -291, -154, 138, 22, -21, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0], [864, -4673, -1627, 7253, 338, -3014, 38, 445, -5, -21], [13136, -23529, -33663, 66987, 2876, -37474, 2048, 6621, -185, -349], [-181, 6765, -20073, -41116, 29990, 29324, -8002, -5855, 501, 328], [-15852, 53537, -6483, -151573, 57104, 96122, -18052, -18331, 1187, 1005], [1294, -3728, -3077, 10270, -380, -5747, 422, 1011, -33, -53], [-1828, 45342, -19797, -116564, 40574, 68527, -11893, -12605, 768, 679], [-9182, -50149, 126633, 180291, -157408, -130692, 40648, 26525, -2527, -1497]];
    Rf_basisdens := [1, 1, 1, 214, 2354, 1177, 2354, 107, 1177, 2354];
    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_4730_a();"
function MakeCharacter_4730_a()
    N := 4730;
    order := 1;
    char_gens := [947, 431, 1981];
    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_4730_a_Hecke(Kf)
    return MakeCharacter_4730_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], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 1, 0, -1, 0, 0], [0, -1, 0, -1, -1, 0, 1, 0, 0, 0], [1, 0, 0, 0, 1, 1, -1, -1, 0, 0], [-1, 0, 0, 0, 1, 1, 0, 1, 0, 0], [1, -1, -1, 1, -1, -1, 1, 1, 0, 0], [0, -1, 0, 1, 0, -1, 0, 1, 0, 1], [1, 0, 1, 0, 0, -2, -1, 0, 0, 1], [2, 1, -1, 1, 0, 0, 1, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -1, -1, 0, 0, -2, -1], [0, -1, -2, -1, -1, 1, 1, -1, 0, -1], [0, -1, -1, -1, 0, 2, 1, -2, 1, 0], [2, 1, 1, 1, 1, 0, 1, -1, 0, 0], [1, 1, 0, 1, 1, 0, 0, 1, 0, 1], [2, 1, 0, 1, 1, 1, 0, 1, 1, 1], [3, -2, -1, 0, 0, 0, 1, 1, 0, -1], [0, 0, -1, -1, 0, -2, 1, 2, 0, 0], [0, -1, 0, 1, 0, -1, -2, -1, 1, 1], [4, 0, -1, 0, 0, 1, -1, -2, 0, -3], [1, 0, 1, -2, 0, 0, 1, 1, 0, 1], [1, 1, 2, 0, 0, -1, -2, 1, 2, 2], [4, -1, 0, -1, 0, -1, -1, -1, 0, -1], [0, 1, -3, -1, 0, 1, 2, 1, -2, -1], [3, 0, -1, -2, 1, 4, 0, -3, 2, 1], [0, 0, 1, -2, 1, 1, 2, 1, 0, 1], [1, 2, 0, -2, 0, 1, 0, 0, -2, 0], [4, 2, 0, 0, 0, 0, 0, -1, 0, -2], [4, 3, 0, 1, 0, 1, 0, -2, -2, -1], [-1, 2, 0, 2, 2, -1, -2, 0, -2, 0], [3, 3, 2, -3, 1, 2, 0, -2, 2, 1], [5, 3, -1, 1, 0, 3, -1, -3, 0, -2], [8, -2, -2, -2, -2, 2, -2, -3, 0, -2], [2, -2, -3, 0, -3, 1, 0, 0, 0, -1], [-3, 1, 1, -1, -2, -1, 1, 2, 0, 0], [0, 1, 0, 3, 1, -3, 0, 1, 0, 1], [-1, 3, 1, 1, -1, -1, -1, 2, 0, 2], [-1, 4, 1, 0, 2, -2, 5, 1, 0, 1], [-2, 2, 1, 0, -1, -6, 0, 2, 0, 0], [-1, 0, 0, 2, -1, -5, 0, 3, -3, 0], [5, 1, 3, 1, 1, -3, -1, 2, 2, 0], [5, 2, 0, 0, 3, 2, 1, -2, 2, 3], [0, 3, -1, 0, -2, -1, 3, 1, 0, -1], [1, -1, 2, -3, 2, 2, 0, -2, -2, 1], [-6, -2, -1, 4, 0, -3, 1, 1, 0, 3], [7, 2, -1, 0, -2, 0, -1, -3, 0, -1], [3, 2, 2, 2, 4, -1, 0, 1, 0, 0], [6, 2, 2, 2, 1, -2, -3, 2, 0, 2], [1, 1, 2, 2, -1, -7, 2, 6, 0, 2], [-2, 2, 0, 2, 2, -1, -5, 0, 0, 3], [-8, 2, -2, 0, -3, 2, 1, -2, -2, -2], [-1, -1, 0, -1, -2, -1, -1, 3, 0, 0], [4, 1, 1, 1, 0, 0, -1, 1, 0, -2], [6, -3, -2, 1, 2, 3, 2, -2, 0, -3], [0, 0, -2, 0, -2, 1, 1, 2, 0, -3], [-9, 3, 1, 3, 0, -3, 3, 5, 0, 0], [0, -7, 0, -1, -2, 2, -3, -2, 2, 0], [-5, -2, 2, 0, -1, -1, -2, -1, 0, 0], [1, -2, 0, -2, 4, 5, 0, -3, 1, -2], [-4, -3, 1, -3, 0, 2, -1, -2, 0, 0], [4, 4, -2, 0, -1, 0, 1, 2, -4, 0], [-2, 0, 0, 1, 4, -3, 2, 4, 0, 1], [-5, 3, -2, -1, -1, 0, 4, 1, -2, -1], [0, 5, 1, 1, 3, 3, 0, -1, 0, -1], [-5, -2, 3, -2, -2, -2, -5, -3, 2, -1], [3, 3, 0, -1, 5, 0, 0, 1, 0, 1], [2, 3, 3, -3, 2, 2, -3, -3, -2, 2], [3, 2, 0, -2, 0, 1, 2, 0, 0, -2], [-13, 2, 1, 0, 4, 4, 0, 0, 0, 1], [13, -3, -2, -1, 0, 0, 2, -3, 3, -1], [5, 3, -2, -1, -2, 0, 0, 3, -3, -3], [-6, -1, -1, -1, -4, -2, -1, -1, -2, 0], [0, 9, 2, 1, 4, 3, -2, -5, 3, 3], [2, -1, -4, 2, -4, 0, 6, 3, 2, -2], [-2, 6, 4, -2, 0, -5, 1, 1, 0, 3], [4, 5, 1, -3, 0, 4, 0, -4, 2, 2], [3, -2, -1, -4, -3, 2, 1, -4, -2, -3], [0, 1, -2, -1, 4, 4, -1, -4, 2, 0], [-4, 6, -1, 5, 2, -6, 5, 6, -2, 0], [-1, -6, 0, 0, -2, -3, 2, 4, 2, 0], [17, -3, 0, 1, -3, -2, 1, -2, 0, -1], [4, -3, 3, 1, 3, 2, -3, -6, 0, 0], [5, 2, -2, 0, 6, 5, 2, -4, 4, 2], [2, 0, 3, 2, 4, 3, -1, 2, 2, 1], [-1, 1, -1, 3, -1, -5, 0, 4, 4, 2], [-1, -2, 1, -2, -2, 6, 3, 1, 0, -1], [-10, 10, -1, 2, 5, 5, 0, -4, -2, -1], [-5, -6, -2, 0, -2, -1, 4, 2, 2, 0], [5, -3, 0, -2, -5, -3, 6, 2, 2, 0], [0, -3, -1, 2, -4, -1, -1, 5, -2, -1], [-4, 2, 0, 0, 3, -2, 0, 3, 0, 0], [-14, -1, -1, -1, -2, 5, -4, -1, -2, 1], [7, 4, -2, 2, 6, 1, -4, -4, 4, 0], [-7, 0, -5, 0, 2, 2, 7, 5, 2, -1], [5, -3, -2, -3, 2, 4, -2, -2, 2, -1], [-5, 2, 0, -1, -2, -4, -4, -2, -2, 3], [-10, 0, 1, 0, -2, -1, -3, -4, -1, -1], [0, -3, 1, 1, 1, -4, -5, 2, 0, 0], [-3, -3, -4, -1, 0, -2, 2, 0, -2, -7], [12, 5, -2, 1, 4, 1, 2, -4, 0, -1], [1, -6, -3, -8, -1, 6, 6, -2, 4, 1], [4, 3, -2, 0, 4, 2, 2, 3, -2, -2], [-5, 1, 1, 6, 0, -2, -5, 1, 0, -1], [2, 2, 3, 0, 2, 3, -5, -5, 4, -1], [3, 3, -3, -1, -2, 1, 5, -2, 2, 0], [-10, 1, 1, -2, 1, 5, -1, -1, 0, 1], [-3, 3, -1, -3, 2, 0, 2, -6, 2, -1], [-1, -10, 1, -4, -7, -4, 3, 2, 2, 1], [-2, 1, 1, -1, 4, -2, -5, -1, 2, 2], [-10, -2, 1, -4, 0, 4, 4, -1, -2, 0], [1, -1, 3, -1, 1, -3, -5, 2, 1, 2], [-4, 0, -6, 2, -2, -6, 2, 3, 0, -2], [11, 2, 2, 2, -5, -9, -2, 4, -2, 0], [-3, 1, 6, 1, 4, 0, -2, 0, 4, 3], [-4, -6, 4, -4, -6, -2, 0, 3, -2, -2], [6, -3, 4, -1, 0, 1, -6, 1, 4, 5], [-4, -2, -2, 0, 2, -2, 8, 0, 2, -4], [13, 2, 0, -4, 0, 7, 0, -3, 0, -2], [-8, -1, 1, -3, -4, 0, -3, 2, 5, 0], [-4, -10, -2, -2, 1, 0, 4, 2, 2, -4], [-3, -2, 0, 6, 5, -3, -4, 5, 2, 6], [7, 3, 7, -3, 5, 5, -2, -6, 4, 4], [1, 3, -7, 3, -4, -1, 1, 3, -4, -4], [0, -1, -2, 3, 0, -1, -4, -3, 4, 1], [-5, 2, -4, -2, -2, 3, 6, 4, 2, -4], [2, -8, -2, -4, -2, 1, 7, 2, 4, -1], [2, 3, 0, -2, 0, 4, 4, -1, 2, 4], [-9, -4, 1, 0, 4, 0, 1, 5, 3, 5], [15, 3, -1, 1, -1, -1, -1, -4, 1, -6], [12, 2, 2, 2, 4, 0, -2, -2, 6, 2], [-7, 4, 0, 2, 3, -7, 0, 1, -2, 0], [14, -6, -2, -2, -6, -6, -4, 0, 0, -2], [-10, 8, 5, 8, 0, -5, -5, 2, 0, 5], [4, -3, 6, 1, -2, -2, -5, -2, 4, 4], [-4, 1, 6, 0, 0, -6, -8, -1, -4, 2], [7, 5, 1, 1, 8, -3, -1, 1, 2, 0], [-4, 3, 7, 2, 0, -7, -1, 7, -2, 1], [-2, 1, -1, 6, 0, -3, 1, 2, -6, -1], [11, -2, -4, 2, -7, -7, -4, 1, -4, -4], [0, 6, 4, 0, 6, 10, 0, -8, 6, 6], [-1, -9, -4, 1, 0, -2, -2, 4, -6, -3], [1, -13, 2, -1, -1, -2, 2, 5, 0, 3], [3, 2, 0, 4, -5, -1, -4, -1, -4, -4], [-4, -6, -5, 6, -1, -1, 3, 6, -1, -1], [4, -5, 1, 1, -3, -8, -3, 4, 6, 2], [-2, -4, -1, -2, 2, -1, 5, -1, 4, -3], [3, -3, 5, -3, -5, -3, -3, 0, -2, 2], [-11, -6, -6, -2, -6, -3, 2, -1, -1, 0], [4, -2, -1, 2, 3, 3, 1, -7, 2, -1], [-12, -11, 0, -9, -2, 6, 5, -2, 4, 0], [-12, 4, 3, -2, 5, 1, 3, -1, -2, 5], [3, 0, -1, -4, 1, 5, 8, 1, -4, -4], [10, 1, 4, 6, 4, 2, -2, -1, -2, -2], [12, 2, -3, -4, -2, -3, 3, 0, 0, -3], [-6, 0, 7, -4, 0, 5, -3, 1, 0, 1], [5, -7, 0, -3, -5, 8, -5, -7, -4, -7], [3, -2, -5, -8, -5, -2, 3, -3, 0, -5], [13, 4, 1, -2, 4, 0, 1, -2, -2, -5], [-2, -5, 3, 5, 2, 2, -9, -4, 0, -2], [-15, 7, 0, 3, 4, 1, -3, -2, 0, 2], [3, 2, -2, -4, -8, -3, 0, 4, -4, 0], [-21, -1, 5, -1, 4, 1, -1, -5, -4, -2], [5, 3, 4, 1, -3, -6, -8, 1, 2, 7], [-2, -10, 0, 0, 0, 2, -6, -1, 5, 4], [4, -3, -4, -5, -3, 7, 2, -5, 2, 3], [13, -6, 1, 5, 2, 3, -3, 3, -4, 0], [-5, 2, 0, -2, 7, 3, -3, 3, 0, 4], [10, -1, 2, -5, 2, 13, -2, -7, 5, -1], [16, -3, -4, 1, 2, -1, 0, -4, -4, -5], [15, 4, -4, -2, 3, -7, 0, 0, -4, 0], [2, 10, 6, 2, 5, 1, -3, -1, 2, -1], [-13, 5, 4, 5, -2, -8, 8, 6, -1, 3], [-15, 2, -8, -2, -6, -3, 6, 4, -4, -4], [-20, -2, -1, -8, 2, 7, 5, 2, -1, 3], [-6, -3, 1, -7, 4, 6, 1, 0, 2, 2], [8, 3, 3, 7, 1, -4, -7, 3, -4, -2], [13, 8, -3, 2, 2, 0, 3, -1, 0, -1], [-14, 1, 3, 5, -8, -8, 1, 4, -6, 0], [6, -4, -1, -6, -10, -5, -3, 0, -2, 1], [0, 0, 0, -8, -2, 4, 0, 1, -6, -4], [-11, -1, -6, -5, -10, 6, 4, 0, -4, -3], [-5, -2, 6, 2, 0, 5, -4, 3, -1, 6], [-12, -8, -4, 2, -8, -9, 3, 8, 2, -1], [3, -10, -1, 0, -9, -12, 2, 7, 2, -5], [12, 0, 8, 4, 4, 0, -6, -6, 6, 6], [7, -10, 0, 0, -8, -5, 2, 2, -4, 0], [14, 3, -2, 1, -2, 1, -4, -2, -2, -3], [-8, 1, 4, -1, 0, -3, -6, -4, -4, -3], [-1, 5, -2, 5, -2, 4, 0, -3, 3, -5], [7, 2, -6, -2, -6, 0, -3, 2, 0, -3], [-24, 5, 4, -1, 6, -3, 0, 4, -4, 5], [-2, 7, -2, 10, 2, -6, -2, 5, -4, 4], [6, 2, -3, 8, 8, 5, 3, -1, 0, -3], [8, 2, -4, -4, 1, 12, 3, -8, -4, 0], [11, -5, -4, 5, -7, -2, 4, 4, -3, -7], [9, -3, -3, 1, -8, -2, -10, 2, 4, 1], [9, -13, -8, -1, -2, 4, 6, -1, 6, -1], [1, -6, 4, 0, -2, -1, 0, 4, 10, 2], [26, -4, 3, -2, 6, 1, 1, -2, 8, 3], [0, 3, 2, 1, 1, -11, 0, 5, 2, 5], [-7, 2, 4, 4, -1, -9, -4, -1, 2, 8], [-3, 0, 2, -10, 4, 13, -2, -11, 6, -4], [-5, -1, 4, 7, 3, 0, 2, -4, -3, 1], [-6, 0, 4, 0, -4, -8, 8, 10, -4, 6], [24, -7, -1, -4, 7, 5, -3, -7, 6, -1], [-5, 9, -3, 1, 0, 8, 4, -5, 0, -1], [3, 8, -3, 6, -4, -6, -1, 2, -4, 3], [-4, 4, -5, -4, 4, 3, 3, -1, 4, -7], [11, -4, 3, -1, 9, 3, 1, -1, 0, 4], [-5, 4, -4, 0, -5, 7, 4, -10, -4, -8], [1, -12, 1, 0, -2, -2, -1, 7, -4, -3], [-16, -5, -2, -5, 2, 1, -6, 3, 2, 1], [-1, 1, -7, 5, -4, -4, 10, 4, 8, -3], [10, -6, 6, 5, 3, -9, -4, 0, 4, 7], [-12, 7, -2, 1, 2, 3, 2, 2, -2, -1], [10, 3, -1, 3, 3, 0, 4, 4, 6, 4], [-1, 1, 2, -1, -1, -12, -4, 7, 0, 5], [19, -6, -2, -2, -6, -3, -8, 0, 0, -2], [29, -12, -3, -2, 4, 6, -3, -4, 0, -3], [-2, 0, 2, -7, -1, 5, 2, 0, 8, 3], [-15, 3, 7, -3, 2, -5, -1, 5, -4, 2], [8, 2, -1, -3, 4, 10, -3, -9, -4, -4], [-18, 2, -1, 2, 2, 9, 1, -5, 4, -3], [7, 14, 2, 0, 1, -8, -1, 0, -6, -1], [-2, -3, -8, -1, -4, 0, 3, 2, 0, -8], [2, 4, 0, 0, 5, -2, 1, -6, -2, -2], [9, -6, 0, 2, -2, 1, 4, 0, 4, -2], [-7, -3, -1, -1, -6, -9, -5, -2, 0, -4], [22, -3, 0, -4, 6, 10, -10, -5, 2, -2], [14, 1, 2, -3, -8, -1, -1, 3, 0, 7], [-12, -3, 0, -5, 3, 3, -2, 5, 4, 1], [-16, 3, -2, -1, 0, -7, 0, 7, 0, 3], [-10, 4, 1, 10, 2, -7, 3, 10, 0, 3], [6, 0, -3, 2, -5, -1, 3, 0, -6, -11], [3, 0, 9, 4, -3, -12, 3, 4, -2, 3], [3, 3, 3, 5, 2, 5, -5, 2, 6, 0], [0, 6, -6, 6, -1, -6, 1, 7, 0, 2], [-4, -4, -7, 2, -1, -9, 4, 9, 6, 1], [9, -4, -1, 3, 1, -3, 1, 7, 0, 0], [11, -7, -1, -9, 2, 13, 1, -7, 4, 0], [17, -6, -5, -2, -8, 4, 9, -3, 6, -3], [20, 1, 2, -2, -4, 2, -12, -7, 0, -4], [-4, -2, 1, -8, -6, 7, 7, 3, 6, 3], [-10, -6, -3, -6, -4, -4, -2, -3, -8, 0], [-1, -7, -8, -11, -6, 8, 2, -1, 8, -1], [-9, 3, -6, 3, -4, -4, 4, 9, -6, -9], [-4, -7, 1, -1, -3, -4, 3, 4, 6, 0], [-10, -2, 7, -4, 4, 3, -1, -7, 2, -1], [-6, 7, 1, 3, 2, 2, -4, 2, -2, 4], [-2, -2, -2, -2, -2, 12, 6, -11, -4, -6], [-1, 8, -3, 0, 6, 6, 3, -4, -11, -1], [31, -1, -2, -4, -2, 9, -4, -2, 0, -6], [22, 1, 3, 3, 8, -5, -6, 3, 6, 5], [-4, 12, -1, 2, 10, 7, -5, -10, 1, 1], [16, 6, 4, 6, 10, -6, -2, 2, 4, 0], [18, 3, -5, -3, 3, 5, 2, -5, -10, -5], [20, 2, 5, 6, 2, -9, -9, 6, 3, 5], [-15, 10, 5, -6, 6, 6, 5, -5, 0, 1], [-1, 9, 4, 1, -2, 4, -6, -1, -6, 1], [4, -2, -1, -2, 3, 1, 3, 0, 6, 1], [-1, -1, -3, -8, -4, 8, -1, -6, 4, 1], [-12, 5, -1, 1, -1, 2, 3, -4, -2, 0], [12, 3, -2, -9, -4, 16, -3, -10, 4, 0], [-6, 16, 4, 4, 8, -3, 1, 9, 0, 1], [-1, -15, -5, -7, -2, 1, 11, -3, 10, 2], [2, -9, -1, -3, -1, -4, -3, 10, -2, 2], [-9, 1, 3, 9, 6, -5, -9, 3, 2, -2], [14, 2, -2, 2, 2, 6, -6, -7, 2, -6], [6, 4, 1, 5, 6, 6, -11, -6, -2, -4], [17, -1, -2, -3, 8, 8, 2, 2, 4, 7], [-4, 4, 9, -6, 9, 7, 1, -4, 3, 7], [35, 11, 5, 3, 2, -3, 1, -9, 0, -2], [8, -1, 9, -5, -3, -4, -19, -6, -4, 6], [12, 5, -4, -5, -1, -1, 4, -13, -2, -1], [42, -13, -7, -7, -2, 10, 1, -9, 2, -10], [21, -13, -4, 0, -6, -3, 2, -4, 4, 0], [1, 9, 3, 5, 2, -1, -15, -4, -2, 0], [-2, -4, 2, -10, 2, 8, -6, -2, 4, -6], [-16, -4, -5, -6, -8, 5, 3, -5, -4, -3], [2, -3, 0, -7, -5, 7, 4, -4, 4, -1], [-10, -1, -8, -5, -14, 3, 4, 0, -2, -3], [0, -1, 2, 1, 11, 5, 8, -4, -4, -3], [-7, -10, 3, -8, -6, 2, 3, -5, -6, -5], [-3, -11, -5, -3, -8, -6, 6, 9, 0, 9], [13, -12, -7, -4, -7, 7, 8, -9, -4, -8], [12, 4, -5, 0, -10, -7, -5, -1, -6, -5], [12, -14, -1, 2, 2, 1, -1, 1, 0, -1], [-27, 3, 1, 7, 1, -1, 5, 6, 4, 0], [-1, -6, -2, 2, -3, -11, 2, 6, -2, 12], [-11, 0, -5, 2, -1, 0, 0, 6, -8, -5], [-9, -1, 2, 9, -7, -18, 6, 17, -2, 3], [11, 7, -4, -2, 2, 9, 4, 5, -4, -2], [-5, -6, -4, -2, -8, -3, 10, 4, 6, 2], [1, -13, -4, -7, -4, 0, -2, -3, 0, 3], [2, -4, 6, -10, -6, 0, -6, -2, 0, 2], [6, -14, 5, 6, -2, -3, -11, 2, -1, 1], [23, -4, 2, 0, -3, 5, -10, -7, -1, -6], [8, 0, 2, 6, 2, 0, -2, -4, 6, -6], [-5, 5, 10, -3, 3, 2, -7, 0, 4, 9], [-6, -11, 0, -3, 1, 7, 6, -3, 0, 7], [21, 5, 1, 1, -5, -9, 1, 10, -4, 0], [-7, 11, 2, 1, 3, 0, -1, 0, 2, -1], [-13, 9, 0, -5, -5, 8, 4, 1, 2, -1], [-17, -4, -6, 0, -8, -6, 7, 10, 4, 3], [3, -2, 6, 6, 5, -8, -3, 2, -8, 3], [7, 5, 5, 7, 2, -8, -6, 5, -2, 5], [-24, 4, -7, 0, -12, -5, 3, 8, -5, -7], [20, -10, -5, 2, -2, -9, 5, 12, -1, 7], [15, -2, 0, 2, -6, 1, -6, -10, 4, 0], [9, -5, 7, 6, 2, -8, -11, 1, 2, 7], [8, 7, -4, 1, -4, 5, -2, 3, -4, 3], [-9, -18, -4, 4, -2, 3, 6, 12, 2, -2], [-3, 3, 3, 3, 6, 2, 2, -7, 0, -1], [12, -21, -2, -1, -10, 3, 6, 2, -4, -5], [-9, 3, 10, 9, 8, -4, -6, 0, 2, 3], [9, -8, 0, 6, 0, -3, -7, -9, 2, -2], [-14, -5, 9, -5, 4, -4, 3, 6, 6, 16], [15, 9, 5, 4, -2, -14, -1, 4, -6, -1], [9, -3, -5, -7, -7, -3, 1, -4, -12, -8], [10, 7, -5, 6, -5, -3, 3, -1, -8, -13], [-6, -6, 1, 2, -10, -7, -7, -4, -6, -9], [-6, 10, 5, 6, -4, -10, -6, 6, 4, 6], [13, -10, -2, 6, 11, 5, -4, 5, -2, 6], [-2, 1, 6, 3, -7, -9, 4, 11, -8, 3], [13, -5, 4, -13, 0, 20, -10, -17, 2, -7], [17, -1, -8, -3, 6, 6, 12, -1, 5, 1], [36, -3, 3, 7, -5, -14, -11, -1, 4, 0], [14, -2, -1, 0, -9, -5, -14, -11, 2, -7], [-12, -3, -2, 1, 8, 9, 0, 0, 8, 1], [0, 5, 5, 3, -6, -6, -1, 3, -2, 2], [15, -9, 1, 7, -6, -13, -3, 2, 10, 2], [0, -6, -1, 2, -9, -13, 3, -1, 0, -5], [3, -11, 1, 1, -6, -9, -7, 10, -7, -4], [-14, 4, 4, 4, 10, -4, 2, 1, -4, 6], [-5, -2, 4, 4, 4, -3, -4, 3, 0, 2], [-17, 4, 12, 8, 6, -13, -2, 5, -6, 2], [-5, -1, -3, -2, -4, 8, -3, -10, -2, -3], [1, -6, -6, -1, -6, -6, -8, -1, -6, 1], [2, -13, -6, -7, -7, -7, 6, 6, -6, -7], [-8, -11, -8, -5, -9, -9, 4, 7, -1, -3], [3, -14, 7, -3, 2, 3, -9, -5, -2, 6], [9, 5, -1, -3, 0, 7, 7, 2, 8, 0], [4, 4, -1, -4, 0, -3, 13, -2, 1, 1], [-30, -10, 5, 6, -9, -13, 5, 5, 0, 1], [-1, 17, -2, 11, 6, -2, -6, 4, -4, 3], [12, 0, 0, -1, -6, -3, -14, -14, 2, -9], [-12, 17, -5, -1, 2, 8, 11, -2, -14, 2], [14, -6, -1, 5, 2, -14, -1, 6, -6, 0], [-3, -7, -2, 5, 8, 4, -10, -12, 0, -7], [3, -8, -2, -4, 10, 13, 4, -5, 2, -8], [12, -14, 5, -1, 4, -2, -1, 13, 4, 4], [4, 6, 2, 6, -8, -12, 0, 1, 4, 8], [14, 3, 4, 1, 3, -3, -4, -5, -5, 5], [-6, -11, 5, -4, -8, 1, -7, 1, 8, 3], [7, 1, 2, 9, 1, -2, -10, 8, 3, 9], [4, -5, 0, -11, -4, -11, -1, -1, -2, -3], [-2, 4, 0, -4, 4, 10, 4, -6, -12, -2], [43, -4, 3, -10, 4, 12, -4, -18, 0, -1], [9, 12, -3, -2, 0, 0, 1, 5, 0, -5], [-31, 3, -8, 10, 2, -3, -6, 2, -4, 0], [0, 18, 3, 0, 1, -7, 1, 9, -2, 1], [1, -8, 4, -3, 0, 4, -10, -1, 10, 7], [-3, 11, 10, 3, 16, 6, -2, -13, 5, 9], [12, -5, -1, -11, 14, 18, -7, -14, 2, 0], [22, -11, 0, -5, 10, 11, 3, -1, 6, 3], [2, -4, -11, 8, -5, -7, 1, 10, -12, -3], [-1, -6, -1, 4, -3, -12, 1, 10, 6, -1], [6, 6, 6, -2, 2, 10, 2, -8, 4, 4], [14, -13, -1, -7, 1, 6, 1, -5, 8, 2], [13, -6, -6, 4, -10, -3, 0, 4, -10, -12], [5, -6, -15, 6, -8, 4, 7, 1, -10, -11], [-5, -6, 3, 10, -5, 2, 1, 0, 0, -3], [1, 2, -4, 7, 0, 6, 0, -3, -2, -1], [-6, -7, -7, -1, -5, 11, 10, 3, -2, -7], [35, -10, 0, -2, -13, -7, 4, 5, -4, -2], [-14, -8, 4, -10, 6, 18, 2, -7, 4, 4], [15, 19, 0, 5, 2, -2, -8, -8, -4, -3], [-9, -3, -2, 11, 6, -4, 12, 0, -4, -1], [2, -2, -5, 6, -4, 7, -3, -9, 4, -7], [4, -12, 0, 8, -2, -4, 2, 5, 2, -4], [-2, 10, -7, 4, -8, -3, 1, 3, 4, -9], [9, -6, -3, -1, -1, -5, 3, -3, -4, -2], [26, -1, 5, 3, -3, -2, -13, -1, -1, -4], [-5, -10, -5, -3, 0, 15, -5, -7, -8, -10], [-3, 5, -7, 7, 6, 15, -5, -4, -1, -4], [-12, 7, 3, 7, -2, 8, -8, -8, 2, -4], [3, 3, -10, -7, 8, 14, 2, -4, 5, -3], [-19, 12, 3, 4, 12, 2, 5, 4, -6, 1], [29, -7, 4, 5, -1, -2, -10, 3, 10, 3], [1, 7, 8, -5, 5, 6, -10, -7, 4, 11], [-3, 1, 0, -1, -7, 6, 6, 6, -7, 1], [0, 13, 3, 9, 5, -14, -1, 11, 5, 8], [5, 1, -5, -4, -14, 2, 11, 6, -4, -9], [41, -4, 4, -6, -6, -1, 8, 5, 2, 2], [-9, 4, -7, -2, 3, 8, -3, 2, -2, 3], [-3, -5, 1, -1, 1, -13, 7, 15, 0, 8], [-15, -7, -8, -3, -9, -16, 7, 17, 4, 5], [-31, 8, 9, -6, 2, 2, -3, -1, 8, 7], [5, 10, 4, -2, -10, -17, -2, 2, 2, 2], [-6, -3, 2, 9, 2, 1, -6, -4, -4, 1], [3, -1, 0, -7, -7, 2, 0, -1, 0, -1], [30, 2, 0, 6, 12, 8, -4, -14, 16, -2], [-7, -8, -7, 0, -2, -12, -1, 16, 2, 7], [16, -2, -6, -6, -4, 2, 6, -4, -2, -8], [4, 4, 12, 12, 11, -4, -7, 3, 0, 8], [-16, -8, -4, -4, 2, 19, 9, -2, 0, -3], [6, 10, 0, 0, -9, -1, -5, 1, -12, -1], [6, -9, -1, 1, -2, 4, -13, -2, -3, -6], [-15, 2, 8, -2, -2, -3, 2, 2, 8, 8], [38, -2, 1, 2, -10, -9, -5, 7, -2, 1], [-18, -4, 10, 0, 4, -16, 8, 8, 4, 12], [-12, -9, 14, 1, -8, -5, 2, 10, -6, 5], [5, 12, 6, -5, 12, 0, 6, 1, 12, 9], [-7, -14, 1, -4, -4, -8, -2, 8, -8, 3], [-13, -5, 7, 1, 6, -9, -5, 3, 4, 10], [-31, -7, 8, -1, 0, -19, 1, 5, 2, 8], [3, 10, 0, 8, -2, 0, 13, 6, -2, 1], [3, 3, 3, -11, 11, 3, 11, -6, 6, 8], [-19, -6, -5, -4, 6, 12, -3, -1, 4, 7], [29, 3, 3, 5, 16, 13, -11, -18, 4, -6], [4, -13, -8, 3, 0, -3, -8, -2, 0, -1], [22, -6, -5, -2, -6, 13, 5, -2, -2, -5], [-12, -8, -6, -18, 4, 22, 6, -15, 12, 4], [10, 7, 1, -3, 0, 12, 2, -8, 0, -8], [6, -18, 4, -10, -4, -4, 4, 5, 7, 4], [2, 11, -5, -3, 8, 4, 3, -13, -4, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4730_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4730_2_a_ba();".
// 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_4730_2_a_ba(:prec:=10)
    chi := MakeCharacter_4730_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_4730_2_a_ba();".
// 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_4730_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4730_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 124, 97, -327, -291, 154, 138, -22, -21, 1, 1]>,<7,R![-24880, -30944, 20827, 13673, -6149, -2106, 804, 134, -47, -3, 1]>,<13,R![-1664, -4928, 736, 7200, -192, -2856, 384, 284, -46, -6, 1]>],Snew);
    return Vf;
end function;