
// Make newform 4014.2.a.x in Magma, downloaded from the LMFDB on 25 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_x();".
// 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_x();".
// 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 := [-28, 82, -23, -90, 43, 28, -14, -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], [116, 841, -466, 57, 260, -144, -22, 13], [-243, 251, 533, -341, -257, 111, 21, -8], [3258, -3297, -3532, 2693, 974, -626, -66, 39], [3370, -4425, -4196, 4387, 1506, -1240, -114, 85], [2764, -5181, -2862, 5091, 1004, -1344, -76, 89], [1355, -1889, -2234, 2222, 883, -692, -68, 49]];
    Rf_basisdens := [1, 1, 194, 97, 194, 194, 194, 97];
    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, 0, 0, 1, -1, 0], [-1, 0, 0, 0, 0, 0, 0, -1], [1, -1, 1, 0, 0, -1, 1, -1], [0, -1, 0, -1, 0, 0, -1, 1], [1, 1, 0, -1, -1, -1, 0, 0], [0, 2, 1, 1, 0, 1, -1, 0], [2, 0, -1, 0, 2, 0, 1, 0], [3, 1, -2, 0, -1, 0, -2, 1], [-1, -2, 1, 0, -2, 1, 0, -2], [-2, -1, 0, -1, 2, -1, 1, 0], [2, 1, 1, 1, 0, 0, -1, -1], [-4, 1, -1, -1, -1, 1, -2, 0], [1, -3, 1, -1, 1, 1, 2, -1], [2, -2, 0, 0, 0, 1, -2, -1], [2, -1, 2, 0, -1, 1, 1, -2], [1, 2, -1, 1, 3, -1, 0, 1], [-2, 2, -2, 2, 1, 0, 1, -1], [5, 1, -2, 1, 0, 3, 1, 1], [3, 1, -3, 1, 0, 2, 0, 2], [4, -2, -2, 1, -1, 1, -1, 0], [6, -2, -1, 2, 1, 2, -2, 1], [8, -1, 2, 2, 1, 0, 1, 1], [2, -1, 1, 0, 1, 3, -1, 0], [2, 5, 0, -1, 0, 1, -4, 2], [-2, -3, 1, -4, -1, -2, 3, -1], [1, 3, 2, -2, -3, -3, -1, 1], [0, 0, -2, -3, 1, 0, 1, 2], [2, 3, -1, -2, 0, 1, 1, 3], [0, 0, 1, 1, 0, -3, 2, -2], [2, -2, 1, 2, -1, -3, 1, -2], [3, -2, -1, -3, 0, -1, -1, 2], [-2, -1, 2, 1, -1, -3, -1, -2], [9, -1, -1, 2, 0, 2, 1, 1], [6, -3, 0, 2, 2, 0, 5, 0], [0, -1, 0, 1, 1, 0, 6, -1], [4, 0, -2, 2, -2, 2, 0, 2], [-1, -2, -4, 0, -1, 2, 1, 0], [-1, 2, -4, 1, 5, 0, 1, 1], [5, -1, 0, 0, 4, -1, 3, 2], [4, 3, 0, 0, -3, -1, -4, 2], [10, -3, 0, -1, 1, 1, 2, 1], [0, 6, 1, 1, 3, -3, 1, 3], [-1, -4, 3, -4, -4, -1, -1, -1], [1, 0, 2, 0, 2, 0, 2, -1], [-4, 2, -1, -2, 0, -6, 4, 3], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 3, -1, 0, 3, -4, 0, 0], [1, -6, -2, 1, 4, 3, 3, 1], [-3, 6, 1, 3, -1, -2, -5, 0], [3, 1, -2, 3, -2, 4, -3, -2], [5, 1, 1, 2, -1, 4, -7, -1], [7, 1, -3, 2, -1, -1, 3, 3], [4, 0, -1, 3, -1, -4, 2, -2], [-3, -1, -1, -3, -2, 3, 2, 1], [11, -3, 1, 1, 3, 1, 3, 0], [-3, -1, -1, -3, -4, 3, -3, -4], [-1, 5, 1, -2, -2, -7, 4, 0], [-1, 2, -1, 0, 3, 3, 1, -1], [-2, 5, -1, 0, -1, -2, -5, 5], [1, -4, 2, -2, 0, 1, -5, -2], [6, -3, 7, 0, 3, -2, 0, -1], [-4, 5, -1, 1, -7, 3, -5, -1], [11, 7, -2, -2, -1, 5, -2, 5], [3, 5, -3, 5, 4, 1, 1, 1], [1, -10, 3, -3, -4, -2, 3, -6], [-3, -1, -3, -1, -2, -4, 3, -2], [-7, -5, -2, -6, 0, -5, 1, -4], [6, 4, 2, 0, -1, 1, -4, 4], [2, 5, 0, 4, -1, 6, -7, -3], [6, 3, 1, -2, 1, -2, 1, 3], [0, 6, -4, -3, -4, 2, -2, -1], [0, -5, 1, -5, 0, 2, 1, 3], [12, 4, -6, 1, 0, -1, -6, 2], [-14, 1, 2, -3, -3, -3, -4, -3], [12, -9, 5, 2, -1, 2, -1, 0], [3, 2, 0, 2, 2, -5, 0, -7], [-2, 4, -1, -3, -5, -4, -4, -4], [10, -3, 2, 5, 4, 2, 4, -1], [5, -7, 8, 0, -2, 2, -2, -1], [4, -5, 7, 0, -1, 2, 1, -3], [11, 0, -7, 4, 9, -2, 4, 2], [6, 6, -2, 1, -2, -4, -3, -3], [9, -3, -4, -2, 2, -5, 5, 2], [5, -3, 3, 0, -1, 3, 3, 1], [8, -1, -4, -3, 3, -6, 7, 2], [13, 4, -1, 4, -4, 4, -5, -1], [6, 4, -1, 5, -1, 10, -7, 4], [-5, -6, 5, -3, 2, -9, 8, 1], [-1, -13, 4, -1, -3, -5, 5, -6], [-12, 2, 1, -6, 2, -1, 3, 4], [10, -8, 3, 2, -2, 1, 2, 1], [0, -1, -1, -6, 8, -1, 3, 6], [-1, -6, -1, 0, 3, 0, -1, 3], [-3, 0, -7, -6, -3, 0, -4, 6], [-7, 7, 3, 1, 2, 1, -3, -2], [-6, -1, 4, -4, 0, -4, 6, 0], [-2, -2, 6, 6, -1, -4, 5, -7], [0, 3, 3, 2, -1, -6, 5, -9], [0, 0, 2, -3, 0, 1, -2, 4], [3, 6, 2, 2, -6, -5, 1, 0], [2, -9, 3, 2, -4, 7, 3, 0], [6, -1, -1, 1, -2, 2, 0, 3], [-1, -3, -4, 1, 3, 2, 5, 3], [-2, -4, 4, 0, -3, 0, 2, -12], [-1, 5, -1, 2, 3, 4, -1, 6], [-2, -1, 6, 0, 1, -5, 8, -4], [10, 1, -2, -2, 3, -1, 1, 11], [4, 2, 3, -3, 5, 1, 3, 1], [-9, 7, -9, -4, 3, 4, -6, 8], [-5, -1, 9, -1, 2, -7, 5, -4], [-3, 0, 3, 2, 1, -9, 5, -1], [-5, 0, 3, 3, -5, -2, 2, 1], [-7, -5, -3, -5, 8, -7, 7, 2], [6, 2, 3, 4, 0, 3, -5, 7], [-13, 12, 1, 6, -5, -4, -5, 3], [-9, 2, -1, -1, -1, -6, 7, -9], [9, -7, -3, 3, -3, 7, 1, 0], [-8, 2, -6, 4, 0, -5, 9, -1], [-7, 3, -6, 0, -5, 5, -5, 2], [-9, 10, 0, -4, -5, -6, 0, -2], [-11, -3, 13, 1, -2, 2, 1, -4], [-5, -11, 0, -2, 4, -2, 8, -3], [6, -1, 3, -2, -2, 6, -13, 4], [7, 8, 0, 1, 3, 1, 2, -6], [7, -11, 6, -1, -7, -3, 4, -6], [7, -2, 1, 3, 2, 4, 5, 0], [1, -4, 0, 8, -2, 0, -2, -7], [-22, 14, 0, 8, -5, 1, -8, 0], [-5, 5, -8, 1, 7, -2, 3, 7], [2, 4, 6, 8, 1, 4, 4, -9], [-5, 4, 0, 5, 4, -2, 2, -2], [7, 4, -12, -2, 1, -1, 3, 3], [-14, 2, -1, -6, 1, -14, 7, 5], [-1, 2, 4, 3, -6, 4, -3, 3], [4, 11, 1, 1, -5, 0, -8, -4], [-11, 2, 7, 5, 4, 6, -1, 0], [-2, 1, 0, -7, 6, -2, -2, 9], [2, -2, -4, -3, 1, -6, 11, 2], [-2, 0, -8, 2, -1, -8, 5, 3], [-13, -8, -4, -5, -3, 0, 0, -2], [-17, -1, -5, 0, 3, 3, -3, 9], [-6, 1, -2, 6, 7, 7, 1, -6], [6, -6, 0, -5, -4, -8, 1, -2], [-3, -6, -1, -4, -3, 4, -8, 10], [9, -11, -1, -1, 10, 3, 9, 4], [-13, -2, 3, 2, 5, -3, 1, 1], [-13, 8, 1, 5, 4, 4, 1, 2], [-11, -4, -2, -2, -1, -4, -1, -10], [-12, -5, 10, 1, -1, 12, -4, -1], [1, 5, 3, 0, -2, -11, -2, 2], [-16, -7, 1, 0, -2, 12, -11, 4], [6, 6, -2, 2, -6, 10, -5, 6], [-18, 11, -3, -2, -1, -10, 1, 3], [10, -15, 3, 4, -4, -2, 7, -4], [9, 12, 2, 2, 6, -10, 5, 5], [-8, 4, -3, 5, -5, 0, 6, 2], [-3, -9, -3, -11, -2, -10, 5, 0], [2, -1, -2, 3, 10, -1, 0, -2], [-7, 3, -1, 0, 7, -8, 4, 8], [-10, 14, -6, 5, -4, -1, -11, 6], [-15, -6, 3, 0, 3, 10, -5, 2], [11, 9, -5, 7, 2, 5, -7, 6], [-5, 3, 3, 8, -1, -3, 10, -12], [-14, -6, 0, -8, -4, -2, 6, 6], [-13, 8, 3, 2, -3, -1, -2, 3], [4, 2, -4, 3, -2, 5, -3, 4], [16, -3, -7, -4, 3, -9, 3, -4], [-18, 12, -7, -3, -1, -8, 3, 3], [2, -14, 9, 2, -7, 6, -6, -5], [12, -14, 6, -1, 3, 4, -4, -6], [14, -7, -6, 0, 3, -3, 7, -1], [-23, -1, -1, 3, 2, -8, 9, 4], [4, 18, -4, 3, -3, 14, -18, 9], [7, 5, -9, -7, -6, -3, -3, 6], [-21, -3, -2, 5, 2, 6, 4, -2], [6, -16, 7, 1, -7, 1, 5, -13], [-2, 2, 2, -1, -12, -2, -8, -1], [-8, 1, -2, -6, -5, -9, -2, 0], [-2, 14, -2, 0, 2, -5, -5, 5], [1, -5, -5, -6, 1, -4, 4, -6], [-10, 0, 4, -7, 10, -3, 9, -4], [3, 7, 5, 7, 4, -3, -1, 0], [9, -1, 2, 5, -5, 4, -10, 5], [9, -17, 5, -1, -2, -1, 5, 1], [2, 13, -4, 3, -3, 8, -13, 4], [-6, -9, 0, -7, 1, 1, 5, 2], [-3, -7, 7, -6, -5, -5, 8, -15], [-3, 4, 2, 9, 7, 1, 7, -1], [8, -9, -1, -7, 4, 6, 6, 0], [8, -9, 7, -4, 13, -10, 13, -3], [7, -12, 7, 3, -5, 1, 4, -2], [-12, 2, 4, -2, 0, -6, -6, 2], [5, 11, 0, 0, -2, -1, -1, 4], [-23, 5, -3, -4, 4, -7, 6, 4], [-12, -5, -1, 1, -2, -2, 0, -2], [2, -2, 0, 7, 10, 3, 1, 4], [-24, 4, -9, -1, -1, 1, -2, 3], [-20, -1, 10, -3, 1, -11, 6, -1], [-1, 6, 6, 8, -10, 1, 7, -10], [-1, 8, -7, 12, 4, 3, 6, -6], [1, 7, 2, 1, 7, -1, 8, 6], [-8, 3, 4, -3, -2, -8, 0, -9], [4, -10, -8, -6, -4, 0, -8, 2], [-1, 2, 0, -11, -3, -10, 4, 8], [0, 2, 12, 4, 6, -2, -6, 2], [-3, -3, 1, 5, -1, -4, 3, -9], [-8, 5, -5, -5, -9, -5, 4, 5], [1, -13, 4, 1, 5, 12, -1, -13], [15, -11, 2, 4, -7, 9, -12, -9], [6, 13, 1, 11, -7, 16, -12, 6], [-1, 2, 0, 7, -2, 2, 4, -4], [3, -7, 11, 5, -4, -3, -5, -2], [-6, -14, -3, -8, -10, 2, -3, -12], [-27, 18, -7, -6, 4, -6, -7, 5], [-5, -8, 8, 6, -2, 0, 0, -1], [-8, 1, 9, 2, -1, -16, 5, -9], [11, 5, -6, -1, -1, -10, 12, -9], [3, 1, -9, -4, 12, -12, 11, 1], [0, -6, 0, -2, 4, 8, 2, 6], [-4, 0, 7, 4, -3, -7, 9, -18], [5, 7, -7, 1, -3, -6, -3, 6], [-1, -10, 1, 4, 3, 0, 7, 2], [5, 9, 2, -2, -16, -1, -1, -2], [-8, 2, -7, 8, 0, -2, 2, 4], [1, 0, 3, -3, 3, 10, -1, 8], [18, 16, 3, 8, -1, 7, -6, -1], [4, -10, -7, 1, 2, 21, -8, 0], [-2, 5, 5, 1, -8, -9, -1, 1], [-22, -4, -2, -2, -3, 3, 12, -2], [-15, 6, 1, -4, -3, 8, -8, 4], [-17, -8, -7, 6, 2, -1, 10, -2], [15, 3, -2, -1, 3, 1, 8, 8], [-15, -10, 9, 3, 0, -10, 9, -6], [-8, 1, 6, -9, 3, -3, -2, 3], [-1, 15, -13, -5, 4, -2, 1, 8], [-13, 4, 4, 6, 6, 1, 15, -6], [-13, 5, -8, 2, 5, -7, 9, -9], [-8, -4, 4, -14, -8, -12, -12, 0], [-33, -1, 12, 1, -5, -7, -3, -12], [1, 4, -7, 9, -6, 4, -7, 6], [-4, -4, -1, -7, -4, -3, 3, 1], [24, -7, 0, -6, 10, -2, 0, 2], [7, 12, -3, 6, 11, -7, 0, 12], [3, -8, -5, 10, 15, 4, 14, -2], [-15, -3, 17, -13, -8, -2, 0, -7], [7, 1, -7, -3, 11, -2, -1, 11], [-3, 6, -8, -5, 3, 3, -5, 18], [6, -15, 1, -5, 5, -3, 1, -3], [-2, -12, 0, 9, 8, 3, 3, -6], [-10, -9, 5, -1, 5, 1, 0, 3], [8, -1, 0, 3, 9, -3, 2, -5], [31, 2, 2, -1, -1, 5, -10, 12], [0, 5, -17, 0, 0, 7, -8, 11], [-2, -12, 0, -6, -6, -5, -3, -7], [-20, 2, 2, -8, -8, -2, 6, 0], [12, -10, -9, 2, 6, 6, 11, 2], [4, 17, -7, -6, 1, -5, -1, 8], [5, 5, 4, 6, 3, 13, -6, 5], [-2, -2, 7, -3, -6, -11, 3, 3], [19, 5, 3, 4, -3, 7, 3, -7], [6, 3, -1, -12, -5, -12, 3, 5], [-2, 5, -8, -12, -2, -10, -4, 12], [24, -1, 4, 10, 1, 8, 3, -7], [-8, -8, -7, -2, 14, 4, 11, -4], [9, 13, -2, 0, -7, -12, -12, 2], [-13, -9, -3, 6, 3, 17, -4, -6], [-3, -12, 6, -4, -3, -16, 0, -13], [-19, 20, 5, 10, -6, -7, -5, -2], [-10, 4, 1, 0, -10, 6, -4, 8], [-29, -6, -7, -9, 6, -4, 9, -2], [21, 2, -1, 0, 2, 6, -21, 9], [-8, 2, 5, 0, -9, -6, 6, -5], [1, 5, 2, -8, -5, -17, 10, -3], [-12, -12, 11, 7, 0, 15, -3, -7], [5, -2, -9, -1, 4, -11, 10, -1], [7, 1, 5, 5, -15, -1, 8, -14], [-11, -2, -4, -4, -3, -1, 3, 0], [-2, -12, -4, -5, 12, 6, 2, 11], [-6, 15, 6, 2, -6, 1, -8, -4], [-2, 4, -18, 0, -2, 12, -2, 2], [1, 5, 0, 11, -3, 0, -6, -11], [-1, -9, 3, 7, -4, 9, 5, -6], [9, 1, 7, -9, 0, -9, 2, 9], [5, 1, -3, 5, 0, 5, 4, -13], [5, -1, -5, -2, -9, -6, -5, 7], [-2, -6, -10, -8, 6, 0, 12, -6], [12, 20, -1, 3, 10, 1, -7, -3], [3, 6, -4, 13, -2, 7, -7, -9], [-5, 8, 14, -2, -12, -11, -1, -4], [-8, -3, 2, -7, 0, -4, 9, -14], [-28, 13, 1, 6, -1, -4, 11, 11], [19, 16, -11, 6, 2, 7, -12, -8], [1, -1, 3, -5, -7, 3, 8, -10], [9, -6, 14, 10, -8, 7, 5, -14], [6, -9, 6, -2, -6, -9, 10, 1], [-1, -2, -12, -7, -5, -8, 3, 11], [0, 9, -7, 1, 6, 17, -17, 19], [0, 19, -7, 3, 7, -7, 0, -8], [21, -2, 1, 12, 14, -2, 9, 1], [-15, -2, 3, 10, 0, 1, 7, -16], [4, 9, 11, 14, -3, -3, 6, -6], [5, 17, -4, 8, 2, 8, -3, 22], [-6, -8, 4, 4, -8, 10, -11, -17], [7, 8, 9, -4, -6, 4, -11, -5], [18, 18, 3, 6, 10, -1, -4, 3], [33, -1, 3, 11, 3, 9, -10, 10], [18, 2, -9, -6, -5, -3, 0, -1], [-17, 16, -1, 13, -5, -7, -10, -11], [-26, -4, -2, -12, 1, -14, -4, 5], [22, 1, 5, -8, -1, -6, -13, 3], [10, 0, 11, 1, 7, 2, 12, -10], [-35, 10, 4, -1, 11, 3, -9, 2], [19, -4, -4, 6, 13, 4, 11, 8], [23, 6, -2, 6, -1, 16, -16, -10], [-2, -7, 6, 1, 2, 8, -13, 10], [19, -8, -13, 0, 9, 6, 2, 2], [7, 6, -16, 3, 0, -7, 2, 5], [18, -13, 13, -7, -4, -6, -8, -3], [-1, -5, 1, 3, -4, -9, 9, 8], [13, 17, -3, 4, 7, 1, 0, 17], [18, 3, -19, 6, 2, 1, -6, 7], [-13, 8, -7, -8, 4, -13, 4, 2], [-13, 17, 2, 7, -5, 13, -18, 6], [-11, -3, -4, 2, -5, 5, 0, 7], [-11, 7, 4, -11, -8, -13, -5, 7], [0, -2, 9, 7, -4, -3, -5, -6], [-7, -12, -3, -15, 1, 4, 3, 11], [-11, 7, 5, -4, -10, -10, -11, 3], [-16, -2, -4, -1, 12, 18, -7, 4], [12, -2, 1, 1, 1, 9, -5, 1], [-11, -4, 12, -7, 4, 0, -4, -10], [-8, -19, -6, -1, 2, -14, 17, -4], [-13, -9, -6, -5, 0, 8, 0, -2], [18, 24, 8, 0, -4, -14, -6, 4], [4, 8, -8, 13, 7, 4, 11, 4], [-29, -16, 7, -10, -11, 5, 9, -1], [12, 2, -1, 5, 5, 13, -17, 17], [-9, 6, -5, 14, 11, 12, 3, -4], [-14, 24, -6, -2, 5, -1, -6, 4], [33, 0, 1, 12, 7, 5, 6, 2], [-15, 2, 18, -9, 9, -14, 9, -3], [-10, -4, -17, -8, 2, 1, -4, 7], [11, 11, -7, -4, -8, 12, -19, 5], [19, -5, 16, 4, -6, 11, -15, 4], [-12, 5, 2, 1, -4, -10, 3, -7], [38, 18, 2, -2, -4, 12, -18, 12], [-12, 12, 2, -2, -11, -8, -1, -19], [-11, -4, 12, -5, -9, -20, 6, -10], [-33, -10, -1, -3, -5, 4, -2, 7], [-8, 11, -14, -13, -6, -8, -16, 15], [-5, 5, 5, -6, 4, 4, 1, 9], [-1, 8, 9, 1, -4, 7, -2, 3], [-28, -8, -9, 1, 5, 2, -6, 16], [2, -17, 7, 1, -9, 3, 8, -11], [-10, -13, 1, -2, 10, 4, -2, -16], [15, -17, -8, -10, -4, -11, 17, 8], [-4, -15, -5, -10, -7, -2, 3, -5], [-25, 20, 5, 4, -3, -10, 2, -10], [16, 9, -12, 7, 9, 4, 12, 7], [7, 10, 8, 7, 6, -1, 3, -7], [25, -3, -14, -3, 1, 0, -10, 11], [-2, 6, -15, -7, 8, 9, -10, 14], [4, 3, -11, 3, 3, -7, 8, 4], [42, -21, 4, -9, -4, -4, 10, -7], [0, 10, -12, 10, 6, -2, 4, 8], [-14, -22, 15, 2, -4, -19, 20, -13], [15, 8, 8, 4, -9, 7, -13, -6], [18, 6, -22, -4, 6, -5, 1, 17], [-16, -6, 4, 10, -10, -6, 1, -25], [22, 3, 1, 5, 9, -7, 7, 7], [12, 13, -1, 4, 14, -1, -2, 3], [0, 8, 17, -6, -16, 9, -8, -1], [-1, -9, 2, -8, -4, -15, 0, -16], [-34, -4, -5, -8, 1, -2, 14, 1], [39, 3, -14, -7, 7, 6, -17, 11], [-14, -1, 10, 7, 2, -18, 12, -1], [21, -14, 8, 1, 14, -13, 23, -11], [25, -16, 2, -3, -18, -6, -1, -9], [2, 2, -12, -3, 3, 9, 6, -1], [2, -3, -2, -9, 0, 5, -3, -2], [-15, -2, 1, -10, 1, 9, -10, 3], [3, 1, 1, -2, 4, 6, -20, 10], [-19, 1, 4, 3, 3, -4, 17, -15], [17, 3, 12, 9, -5, 10, -10, 11], [-6, -6, 21, 6, -4, 5, 0, 3], [22, 7, -18, 6, 1, 21, -8, 16], [-34, 13, 0, 4, -5, 0, -8, -1], [-8, -15, 2, -13, -3, -1, -4, 7], [12, 9, -3, 18, -1, 5, 13, -10], [9, -1, -10, 2, -1, 0, 8, 3], [30, 3, 1, 17, 10, -3, 13, -9], [-12, -3, 6, -7, -7, 9, 2, 7], [8, -10, -1, 6, -4, -8, 9, -14], [-2, 13, -9, 3, 0, 4, 4, 10], [-2, 22, 0, 8, 6, 8, 4, 12], [22, -8, 2, -2, 8, 6, -6, -8], [-16, -15, 0, 0, -5, 2, 6, -9], [33, -7, 3, 9, 6, -5, 5, -2], [-22, 10, -4, 5, 11, 10, -9, 14], [4, 8, 5, 2, 4, 17, -9, 13], [-46, 4, -5, -7, 8, -20, 10, 8], [0, 2, 3, 9, -1, 8, -4, -16], [0, -22, 14, 3, -9, 7, 6, -23], [10, -10, 7, 5, -12, 1, -4, 6], [-22, -18, 12, 1, -22, 15, -15, -24], [27, 11, -3, -8, -12, -6, 2, 10], [35, -12, 14, 0, -3, 6, 4, -15], [-17, 9, 2, 8, -18, -1, -23, -4], [-29, -5, -14, -12, 1, 9, -6, 3], [-8, -15, 9, -1, 7, -20, 20, -2], [-4, 17, -3, 15, -8, 11, -7, -1], [-3, -8, -3, 3, 13, 5, 6, 2], [4, -3, -1, -14, -9, -5, -10, 5], [-14, 9, 5, -10, -12, 11, -24, -1], [14, 17, -7, -8, -5, -14, -3, 7], [39, -12, 1, 5, 6, 11, -2, 1], [-8, 4, 3, 0, -2, -7, 1, -3], [-15, -19, 4, -17, -7, -8, -4, 13], [4, -2, 8, -2, -16, -6, -2, 14], [-4, 0, 0, -7, 5, -4, -5, 14], [-18, 18, -7, -2, -4, -15, -9, 6], [-5, 17, 0, 7, -3, 19, -17, 6], [0, 0, -8, -6, 16, -6, 12, -2], [26, -12, -5, -1, 5, 16, -1, 7], [-25, -9, 10, -14, 6, -8, 8, -7], [2, 2, -8, -13, 7, 6, -15, 14], [-2, -16, 3, -9, 0, -3, 15, -4], [-10, -1, -3, 4, 3, -16, 17, 9]];
    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_x();".
// 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_x(: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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4014_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-48, 104, 189, -268, -34, 82, -7, -6, 1]>,<7,R![-844, -658, 515, 478, -56, -102, -9, 6, 1]>,<11,R![-981, 1227, 2664, -1017, -487, 200, 13, -11, 1]>],Snew);
    return Vf;
end function;