
// Make newform 4014.2.a.z in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4014_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_4014_2_a_z();".
// 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_4014_2_a_z();".
// 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 := [69, 126, -98, -148, 54, 46, -12, -4, 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], [108, -63, -97, 64, 4, -11, 2, 0], [84, 27, -165, -47, 62, 2, -7, 1], [-40, -29, -52, 27, 35, -9, -4, 1], [4, -7, -63, 27, 35, -9, -4, 1], [48, -351, -284, 228, 142, -47, -15, 4], [309, 222, -394, -63, 128, -7, -12, 2]];
    Rf_basisdens := [1, 1, 33, 33, 11, 11, 33, 33];
    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_4014_a();"
function MakeCharacter_4014_a()
    N := 4014;
    order := 1;
    char_gens := [893, 2233];
    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_4014_a_Hecke(Kf)
    return MakeCharacter_4014_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, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0], [-1, 1, 0, -1, 0, -1, 1, 0], [-2, 2, 0, -1, 1, -2, 1, 1], [1, 1, 1, 0, 0, 0, 0, 1], [0, -1, 1, 1, -1, 2, 0, -2], [0, 0, -2, -2, 0, -2, 0, 0], [0, 0, -2, 0, 0, -2, 0, 2], [0, -2, 0, 0, 0, 0, 0, 0], [2, 1, 1, 1, -1, 2, 0, 0], [0, 2, 0, 0, 0, -2, 0, 0], [-2, 0, 2, 0, 0, 0, 0, 0], [4, -2, 2, 2, -2, 4, -2, -2], [1, -2, 0, 0, -1, 1, -2, -2], [0, 0, -2, 0, 0, -2, 0, 0], [1, 0, 0, 1, 0, 0, 0, 2], [6, -2, 0, 3, -1, 2, -1, -1], [1, -3, 1, 2, -2, 2, 0, -1], [2, 0, 4, 2, 0, 4, 0, 0], [-3, 3, -2, -3, 0, -3, 1, 2], [2, 2, -2, 0, 2, 0, 0, 0], [6, 0, 0, 2, -2, 2, 0, -2], [1, -4, -4, 0, -1, -1, 0, 0], [2, 0, 0, 2, 0, -2, 0, 2], [2, -4, 0, 0, 0, 2, 0, -4], [0, -2, 0, 2, 0, 0, -2, -2], [7, -1, 0, 0, 0, -2, 0, 2], [-2, 0, -4, -2, 0, 0, 0, 2], [-6, 4, 0, 0, 0, -4, 2, 2], [-2, 4, -2, 0, 2, -4, 0, 4], [0, 4, 0, -2, 2, -2, 0, 0], [0, 4, 0, 2, 0, -2, 0, 4], [0, 0, 2, 0, 0, 4, -2, 0], [7, -5, 0, 2, 0, 0, -2, 0], [6, -2, 4, 2, -2, 4, 0, -4], [3, -1, 4, 2, 0, 4, -2, 0], [-2, -2, 0, 1, -1, 2, -1, -5], [2, -2, 0, 0, -4, 4, 0, 0], [-3, 1, 3, -2, 0, 2, 0, -3], [-8, 8, 2, 0, 4, -6, 2, 4], [8, -4, 2, 2, 0, 2, 2, 0], [4, 0, 2, 4, 0, 2, 0, -2], [6, -2, 6, 0, -2, 4, 2, -4], [4, 2, 2, 2, 2, 0, 0, -2], [9, 2, 4, 0, 3, 1, 2, -2], [0, 4, -2, -4, 2, -2, 2, 2], [1, 0, 0, 0, 0, 0, 0, 0], [8, -2, 4, 2, -2, 6, -2, 0], [-6, 0, 2, 1, -3, 0, 1, 1], [-14, 8, 0, -4, 2, -8, 6, 4], [6, -1, -1, -1, 1, -2, 0, 0], [2, 0, -4, -4, 0, 0, 0, 4], [-6, 8, -2, -2, 2, -4, 0, 4], [-11, 4, -4, -4, -1, -3, 0, 0], [0, -8, -2, 0, 2, -2, -2, -2], [-11, 5, -4, -6, 0, -4, 2, 0], [4, 0, -4, -6, 0, -2, 0, -2], [2, -2, 0, 5, 1, 2, -3, -3], [-1, 1, 8, 3, 0, 7, -3, -2], [-2, -4, 0, -6, 0, -2, 2, -2], [3, -1, -5, 0, 0, 4, 0, -1], [1, -4, -2, 1, -2, 2, 2, 0], [-6, 2, -2, -4, 2, -2, 0, 4], [0, 0, 0, -2, 0, 2, 2, 2], [8, -2, 2, 0, 0, 2, 4, -2], [3, 3, -3, 0, -2, -4, 0, 1], [-2, -4, 2, 0, 2, 4, -2, -6], [6, 6, 2, 0, 2, -2, 0, 0], [-4, -2, -2, 2, 0, 0, -2, -2], [5, -13, -4, 1, -4, 9, -5, -6], [5, 3, 0, -3, 0, 1, -1, -4], [0, 3, 3, 3, 3, -2, 0, 4], [1, 0, 0, 1, 0, 4, 0, 0], [6, 0, 4, 0, 0, 6, -2, -4], [-8, 4, -2, -4, 0, 0, 0, -2], [-2, -6, 2, 4, -2, 8, -4, -6], [-3, 7, -4, 2, 0, -6, 2, 6], [-7, 3, -6, -3, 0, -3, 1, 6], [-4, -4, 2, -2, 2, 2, -4, -4], [0, 4, 0, -2, 4, 0, 0, 2], [15, -4, 2, -1, 2, 2, 2, -2], [-6, -6, -4, -4, -2, 0, -4, -2], [16, -9, 5, 5, -5, 14, -4, -6], [8, -12, -4, -2, 0, 2, -2, 0], [4, 2, -2, 0, 2, -6, 2, 8], [-10, 4, -2, -2, -2, -2, -2, 0], [4, 2, 4, 4, 6, -4, 0, 6], [0, 2, 0, 4, -2, 4, -2, 0], [15, -6, 4, 8, -3, 9, -6, -2], [-1, -9, 5, 4, -2, 4, 0, -3], [-6, 0, 2, 0, -2, 0, 0, 8], [-8, 3, -5, -5, 7, -14, 4, 4], [-3, -7, -4, -4, 4, -2, 0, -2], [-20, 8, -2, 0, 0, -6, 6, 8], [-8, 4, 4, -2, 4, 0, -2, 0], [-4, -2, -2, 2, 0, -4, -2, 2], [-10, 10, -4, -12, 6, -8, 0, 4], [10, -14, -4, 5, -5, 10, -5, -5], [-9, 7, 1, 2, 4, -2, 0, -1], [8, -8, 0, 0, -2, 10, -6, -8], [-3, 1, 4, 10, 0, 4, -2, 0], [11, -12, 0, -1, 0, 8, -4, -6], [-8, 6, -8, -8, 6, -14, 6, 4], [11, -5, -1, 2, 2, -2, -4, 5], [7, -7, 0, 3, 0, 3, 1, -2], [14, 0, 6, 1, 3, 4, 3, 3], [-14, -2, -4, -2, 2, 0, 2, 0], [-8, 5, 5, 1, 1, 6, 0, -4], [-6, 2, 0, -2, 4, 0, 2, -2], [-8, -10, 2, 2, -10, 10, 0, -4], [-11, 4, -2, 1, -2, -6, 6, 2], [-9, -7, 2, 7, -4, 7, -5, 2], [-2, 0, 2, 5, 7, -4, -1, 3], [-6, 6, -4, -6, -2, -10, 4, 0], [8, 0, -2, 2, 2, 4, -2, 2], [2, -2, -4, -2, -2, 4, 2, -2], [0, -3, -3, 1, 1, 2, -4, 0], [3, 4, 2, -1, -2, -6, 2, 2], [16, -8, 4, 6, -4, 12, -4, -10], [-1, 8, -8, -1, 8, -12, -4, 8], [-6, 0, 8, 8, 0, 4, 0, -4], [-6, 4, -6, 0, -2, -4, -2, -2], [4, 8, 0, 6, 0, -2, 0, 6], [-1, -7, 12, 8, -4, 12, 0, -8], [14, -6, 0, 6, -4, 8, 4, -4], [-13, 15, 0, -10, 4, -8, 2, 8], [9, -5, 4, 5, 4, 13, -5, -4], [-5, 6, 12, 4, 5, -1, 2, 6], [6, 6, 6, 0, 4, 6, -2, -2], [17, -8, 2, 9, -2, 6, -6, 0], [-1, 6, -4, 0, 1, -3, 10, 6], [10, 3, -5, -5, 1, -14, 4, 4], [1, 3, 4, -6, 0, 2, 2, 6], [8, -4, 4, 10, -4, 0, 2, -2], [-17, 1, 4, 3, 0, 3, -3, -2], [10, -6, 0, -5, 3, 2, -1, -1], [-22, 6, -4, -7, -1, -6, 7, -5], [0, -2, 0, 10, -6, 4, -4, -4], [6, -4, 4, 10, 0, 6, -6, 0], [-1, 8, -4, -5, 8, -8, 4, 2], [-6, 10, -2, 0, 2, -2, 4, 6], [-4, 12, 10, 8, -2, 6, 0, 0], [23, -12, 6, 11, -2, 14, -2, -12], [7, 3, 5, 2, 4, -2, 4, 3], [12, -4, 0, 4, -2, 12, -6, -4], [-15, 9, -5, -2, 4, -14, 8, 9], [-3, 4, -4, -8, 3, 1, 0, 0], [18, -8, -2, 3, -3, 8, -3, -3], [22, -16, -2, 2, -4, 10, -4, 2], [-33, 8, -8, -5, 0, -16, 4, -4], [2, 11, 1, 5, 5, -6, 0, 6], [13, -8, 2, 1, 2, 2, -6, 0], [-12, -8, 4, 0, 0, 4, 0, -8], [18, -4, 8, 6, -4, 12, -2, -10], [-2, 4, 2, 8, -2, 4, -4, 4], [14, -4, 12, 4, 0, 10, -4, -6], [11, -8, -12, -8, 1, -5, -4, 0], [-6, 20, 6, -4, 4, -10, 2, 8], [-2, 4, 2, -4, 4, -8, 2, 0], [2, 2, -2, -4, 4, 0, 4, 2], [-10, -2, 4, 8, 2, 8, -10, -2], [-22, 0, 4, 2, 0, 2, 0, 0], [-5, 0, 10, -1, -2, -6, 2, 0], [-6, 0, 2, -2, 4, -4, 0, -4], [14, 6, -4, -2, 0, 0, 0, 2], [16, -6, 6, 8, -6, 14, -4, -2], [-8, -22, -10, -4, -8, 8, -4, -10], [8, -20, -2, 10, -6, 16, -8, -12], [-4, 12, 8, -2, 2, -2, 2, 0], [-8, 8, 8, -2, 2, -8, 6, -2], [-18, 6, -8, -5, 7, -10, 3, 3], [11, -11, 12, 3, -4, 7, -3, -8], [12, 2, -10, -6, 8, -6, -2, 2], [6, -10, -4, 4, -4, 6, 2, -6], [24, -18, 4, 14, -10, 22, -6, -6], [-7, -1, -8, -4, -4, -4, 4, -4], [12, 6, 2, 2, 0, 4, 4, 4], [20, -15, 5, 5, -7, 10, -4, 4], [-14, -4, -4, -2, -2, -2, 0, -8], [4, -16, -10, 2, -2, 4, -2, -4], [6, 14, 2, 2, 6, -4, 2, 4], [6, -4, -6, -4, -2, 2, 2, -10], [-18, 0, -6, 0, 2, -2, 2, 2], [10, 4, -14, -4, 2, -8, -4, 0], [2, 2, -2, -2, 4, -4, 2, 4], [26, 0, 10, 8, -4, 10, -6, 0], [40, -10, 4, 12, -6, 8, -4, -4], [-11, -8, 0, 0, -1, 1, 0, 0], [-1, 5, -10, -13, 8, -9, -1, 4], [-3, 4, -8, -8, 3, -9, -4, 4], [3, -1, -3, -8, 2, -12, 8, 1], [6, 20, 2, -2, 2, -4, -2, 6], [6, -4, -8, 2, -2, 6, -6, 2], [2, -4, 6, 0, 2, -2, -6, 0], [-8, 7, 5, 5, 3, 2, 4, -2], [-1, 4, 12, -4, 5, 3, 0, 0], [-26, 16, 2, -8, 6, -12, 6, 6], [-20, 2, 4, -10, 2, 2, 0, -8], [0, -18, -4, 8, -6, 12, -6, -14], [22, -8, 2, 7, -1, 12, -1, 7], [9, -5, 2, 5, 0, 1, -3, 0], [-15, 5, 5, 0, 0, -4, -4, 5], [14, -16, 4, 4, -8, 22, -6, -14], [18, -16, 12, 18, -10, 20, -6, -8], [-14, 2, -10, -2, 0, -6, 4, -2], [-5, 14, -8, 4, 5, -3, 2, 6], [21, -16, -2, 1, -6, 14, 2, -4], [-6, 6, 6, -10, 4, -6, 4, 2], [-9, 11, -11, -12, 10, -16, 8, 13], [4, 8, 6, 2, 0, -12, 0, 6], [5, -3, -19, -4, 0, -16, 0, 5], [-1, -10, -4, -12, 5, -5, 6, -2], [27, -17, 8, 4, -4, 10, -8, -10], [-18, 14, 0, -12, 2, -12, 4, 6], [-22, 4, -8, -2, 0, 0, 0, 6], [6, -4, 2, -4, 2, 0, 0, 4], [4, 6, 0, -4, 2, -4, -2, -8], [-9, 14, 0, -12, 5, -13, 6, 10], [-11, 5, 7, 6, -4, 6, -4, -3], [12, 8, 18, 2, 4, 2, 2, -4], [0, 2, -4, 0, -8, -2, 6, -4], [18, -6, -2, 4, 2, 2, -6, 0], [-6, 2, 0, 0, -6, -12, 2, 0], [-8, -8, -4, 4, -6, 2, 0, -6], [3, 8, 12, 3, 0, 0, -4, 12], [32, -4, -2, -4, 2, -2, 4, 10], [14, -13, -1, 11, -3, 18, -8, -12], [5, 6, -8, 0, 11, -15, -6, 10], [-9, 7, -1, -2, 2, -6, 0, 5], [8, 14, -4, -6, 2, -10, 4, 4], [-8, -16, -2, 0, -6, 6, 0, -4], [27, -2, 0, 4, 1, 3, 2, 6], [-10, -1, -3, -7, 5, -10, 0, -10], [-3, -16, -12, -3, 0, 0, -4, -6], [8, -4, 10, 6, -4, 18, 4, 4], [-12, -8, -2, 6, -10, 10, -6, 6], [-2, 8, 2, -1, -5, 0, 7, -5], [-3, -4, -14, -3, -2, -10, -6, 4], [50, -22, 8, 16, -6, 16, -8, -10], [1, -15, -9, 0, -2, 4, -4, -13], [9, -9, -8, 0, 0, 0, 4, -4], [-8, -14, -2, 8, -4, 8, -4, 0], [2, 0, -6, -12, -2, -6, 2, -6], [12, 2, 6, 6, 2, -8, 8, 2], [20, -2, -4, -6, 4, -2, 2, -10], [20, -6, 4, 4, 2, -4, -6, 2], [-18, 6, -8, -5, 1, -10, 1, 1], [6, -2, 0, 12, 4, -2, -4, -4], [-12, 18, 2, -12, 6, -12, 4, -2], [4, -4, 0, -8, -6, 2, -2, 2], [2, 0, 4, 0, 2, -4, 0, 14], [46, -10, 8, 8, -2, 20, -6, -4], [4, 6, 12, -2, 2, -2, -2, 6], [18, -12, -2, -7, -7, 4, -3, -3], [4, 22, 2, -8, 2, -8, 6, 4], [-4, 8, -6, -6, 6, -16, 8, 8], [28, -12, 4, 4, -6, 16, -4, -2], [-2, -14, 0, -4, -2, 0, 0, -4], [-8, 2, 10, -4, 6, 8, -2, -14], [-14, -2, 4, -8, -4, -6, 8, -4], [28, -14, 4, 12, -4, 12, -4, -10], [-3, 8, -4, -3, 4, -12, -4, -4], [-18, 20, -6, -8, 4, -14, 6, 16], [7, 17, 4, 7, 0, -1, 1, 2], [20, 12, 2, 0, 10, -4, 8, 8], [16, 0, 4, 10, 10, -4, -6, 8], [6, -4, 12, 4, -2, 18, 0, 2], [-24, -12, 8, 4, -6, 4, 2, -8], [16, -2, 0, 12, -4, 4, -2, -6], [21, 11, 12, 10, -4, 10, 2, -2], [-15, 4, -16, -19, 8, -20, 4, 6], [-2, 14, 2, -2, 12, -10, -4, 10], [14, 2, 8, 16, -6, 10, 0, 2], [-30, 6, 10, 2, 4, 2, -2, -8], [-28, -14, -12, -2, 2, 0, -4, 2], [11, 20, 4, -1, -4, 4, 4, 2], [2, 0, 6, -1, -5, 0, 7, -1], [6, 8, 10, 0, -4, 8, 4, -8], [1, 0, 10, 1, 2, -6, 6, 0], [-3, 3, -8, 9, 0, -7, 7, 8], [13, -18, 4, 8, -1, 3, -6, -10], [6, 4, 6, -4, -6, 0, 8, -4], [-15, 7, 0, 2, 12, -6, -2, -2], [-26, 26, 0, -8, 8, -18, 12, 8], [-14, 2, 6, -2, -2, 2, 4, -2], [-6, -3, 3, 11, 3, 14, 0, -6], [-9, -15, 8, 2, -4, 14, -6, -18], [-20, 10, -8, -8, 4, -2, 4, 2], [-2, -16, -14, 0, 0, -8, -4, 4], [23, -7, 12, 6, -4, 18, -10, -6], [-13, 23, 3, -6, -2, 6, 0, 5], [-38, 2, 2, -14, 4, -4, 2, -4], [-1, 7, -3, -4, 6, 0, 0, -3], [6, -4, -6, 5, 5, -4, 1, 5], [-11, 0, -6, -3, -2, -6, -2, -6], [-12, 6, 8, -6, 2, -10, 6, -12], [-30, 18, -8, -19, 13, -26, 13, 9], [19, -5, -7, -4, 2, -8, -8, -3], [-20, 20, -16, -18, 6, -16, 4, 12], [-18, -4, -8, -6, -4, -12, -4, -2], [-30, -6, 0, 0, -4, 0, 4, -10], [-4, 0, 16, 14, -6, 8, 0, 0], [-15, -9, -16, -4, 0, -8, -8, 4], [18, -10, 12, 12, 4, 18, -8, 0], [-16, -2, -18, -4, -8, -6, -2, 0], [14, 10, 0, -7, 7, -10, 7, 7], [-17, 1, -8, -6, 0, 6, -6, -2], [44, -8, 12, 16, 4, 0, -8, 4], [-6, -14, -4, -5, -9, 6, -5, -1], [16, 1, 21, 13, 1, 6, -4, 0], [-16, 2, 8, 8, -6, 12, 4, 6], [4, -13, 5, 5, -5, 14, -12, -18], [-2, 6, -6, -12, 2, 0, -4, 8], [-20, 10, -20, -14, 12, -24, 8, 12], [5, 15, 0, -3, 0, 1, 7, -6], [13, -11, 4, 8, -4, 14, 0, 6], [-14, 18, -10, -16, 14, -14, 4, 8], [5, 8, 4, -8, 7, -7, 4, -8], [-6, 0, 0, 0, -14, 8, 4, 10], [-8, 18, 10, -6, 6, -14, 10, 6], [-10, 15, -11, -3, 5, -14, 4, 10], [2, 4, 14, 9, -7, 24, -3, -3], [26, -19, 1, 5, 3, 10, -4, -16], [-15, 16, -4, -8, -5, -5, 8, 0], [22, 8, 4, 10, 4, -2, -2, 12], [-10, -18, 6, -2, -12, 12, -2, -14], [-16, 6, 8, 2, 0, -6, 6, 10], [-24, 12, 4, 0, 0, -12, 8, 8], [5, 1, 13, 12, -12, 8, -4, 1], [19, -20, -12, -9, 0, 12, -8, 0], [16, -4, 12, 10, 10, 6, -4, -2], [32, -17, 7, 3, 3, 14, -8, -4], [-33, 12, 8, 3, 4, 0, 0, 2], [33, -20, 4, 9, -4, 20, -8, -18], [-10, -8, 0, 8, -2, 6, 2, -2], [40, -18, 4, 10, -12, 24, -2, -8], [13, -13, 4, 9, -8, 25, -5, -10], [-20, 2, -4, -6, 8, -12, -6, 12], [-20, 12, 6, -2, -8, 6, 10, 6], [-34, 24, -2, -8, 2, -14, 8, -4], [-26, -4, -6, -14, 2, -6, -2, 8], [-23, -1, -4, -10, 0, -18, 2, -6], [47, -11, -8, 3, 0, -5, -3, -4], [3, 11, -3, -2, -4, -14, 0, 3], [20, -28, -12, 4, -10, 4, -8, -2], [26, -28, -6, -1, -1, 8, -13, -17], [7, -1, 3, 6, 2, 6, -4, 13], [19, 1, -10, -9, 8, -13, 3, -4], [22, -8, 18, 10, 6, 18, 0, -2], [-9, 17, 12, 0, 0, 8, 4, 0], [0, 12, 2, -2, -8, -10, 8, -2], [-7, 27, 8, 2, 16, -22, 2, 14], [-11, 19, -4, -7, 0, -15, 7, 28], [-22, 18, -8, -15, 9, -26, -3, 13], [16, -10, 2, 10, -6, 4, 2, 0], [-16, -16, -24, 0, -8, -4, -4, 4], [0, -6, -4, -6, -4, -6, 2, -6], [14, -8, -4, 6, 4, 10, -16, -2], [12, -2, -8, -14, -4, 0, -2, -6], [25, -15, -11, 8, -12, 8, -8, -7], [35, -15, 4, 3, -8, 15, 5, -2], [3, 2, 4, -4, 1, -15, 6, 6], [23, -8, -6, -5, -2, 6, -2, -4], [-16, 0, 6, 8, 2, 4, 4, -4], [24, 6, 8, 2, 4, -2, -4, 14], [33, -24, 2, 13, -6, 18, -10, 10], [-17, 1, -14, -5, 4, -9, 3, -4], [24, 9, -1, -1, 5, -14, 4, 2], [18, 30, 8, -2, 14, -16, 8, 16], [-28, 10, 12, 0, 10, -4, -2, -2], [28, -20, 12, 14, -4, 12, -4, -10], [-22, 10, -12, -13, 3, -6, 3, 15], [-34, 16, -18, -16, 10, -20, 8, 12], [1, -3, 0, -10, 8, -4, 6, 4], [4, -6, -2, 6, 4, 6, 0, -2], [10, 9, -11, -15, 11, -14, 12, 4], [10, -24, 2, 8, -10, 20, -2, -18], [-5, 1, -22, -13, -4, -13, -1, -2], [-2, -4, 10, 4, 8, 24, -4, -12], [21, 8, -2, -7, 6, -14, 14, -6], [2, -16, 6, 10, -16, 20, 2, -18], [-40, 0, -4, -6, -2, -6, 0, -4], [-32, 2, 10, -6, 0, 6, 4, -16], [-43, -5, 0, 0, 0, 0, 0, -8], [-65, 37, -2, -9, 8, -21, 11, 12], [59, -11, -6, 3, -8, 11, -5, -8], [-20, -2, 4, -8, 2, 14, -2, -4], [-2, 0, -4, 8, 0, -2, 2, 12], [-60, 18, -4, -20, 16, -30, 4, 14], [30, -6, -2, -2, -10, 4, 4, 0], [26, 2, -6, -8, 2, -4, -10, -4], [7, 1, 0, 12, 0, 0, 8, 12], [18, -6, 16, 18, -2, 20, -4, -4], [-28, -2, 12, 4, -10, 12, 6, -6], [20, -17, 1, -3, 7, 10, -4, -10], [1, -18, 0, 0, -13, 11, -2, -10], [46, 8, 8, 4, 4, -4, 8, -4], [22, 6, -12, -14, 4, -14, -4, -6], [44, -2, 0, 6, 4, -4, 2, -4], [-24, 4, -14, -22, 0, -12, 6, -4], [-14, 4, 4, 2, -2, -10, -2, 22], [-8, 2, -8, -6, 8, -6, 4, 8], [-38, 23, -7, -23, 9, -22, 0, 10], [-11, 9, -4, -6, 12, -16, 2, -8], [-21, 9, -16, -17, -8, -13, 5, -8], [4, 2, -20, -8, 10, -18, 0, 6], [16, -16, 6, 6, -2, 4, -10, -6], [12, -2, -6, -8, 2, 0, 4, 6], [16, 8, 6, -4, -4, 8, 4, 10], [0, 16, 16, 20, 0, 10, 4, 2], [37, -15, 15, 18, -4, 26, -16, -7], [17, -12, 10, 13, -14, 18, 6, -8], [28, -2, 0, -10, 2, -6, -2, -4], [12, 7, 5, -3, -9, 6, 0, -2], [-25, 3, -21, -4, 4, -16, 4, 11], [18, -4, 2, 0, -2, 0, 0, 0], [-14, -13, 7, 3, -3, 6, -12, -4], [12, -34, -4, 16, -6, 20, -14, -16], [-6, -7, 5, -3, 3, 2, 0, 4], [3, -1, 4, 10, 0, 4, -2, -12], [36, -24, -12, 4, -14, 24, -10, -10], [-41, -2, 0, 0, 1, -3, -2, -10], [-3, 7, 12, 12, 4, 8, -8, 0], [-6, -12, 4, -2, 8, 2, -4, -6], [6, 2, -12, 14, 2, 4, -8, 6], [2, -10, 2, -8, -10, 0, 10, -2], [-4, 10, -10, -14, -8, -2, 6, 14], [2, -2, 6, 12, 2, 8, -6, 10], [-21, 2, 4, -8, -7, 11, -6, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4014_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4014_2_a_z();".
// 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_4014_2_a_z(:prec:=8)
    chi := MakeCharacter_4014_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_4014_2_a_z();".
// 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_4014_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4014_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-942, -1596, 98, 788, 130, -106, -24, 4, 1]>,<7,R![-11312, 11712, -496, -2512, 456, 176, -40, -4, 1]>,<11,R![-126, -3600, -666, 1932, 650, -122, -50, 2, 1]>],Snew);
    return Vf;
end function;