
// Make newform 4014.2.a.w in Magma, downloaded from the LMFDB on 24 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_w();".
// 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_w();".
// 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 := [18, -38, -36, 50, 12, -14, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-98, 703, -287, -388, 92, 37, -6], [-858, -703, 526, 388, -92, -37, 6], [817, -771, -151, 313, -89, -28, 11], [513, 1439, -1212, -476, 411, 38, -32], [985, -269, -2049, -12, 641, 11, -47]];
    Rf_basisdens := [1, 1, 239, 239, 239, 239, 239];
    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, 0, 1, 0, 0, 0], [1, 0, -1, 0, 0, 0, 1], [-2, 1, 0, 0, 1, 0, 1], [0, 0, 0, 1, 1, 0, -1], [0, 0, 1, 1, 2, 2, 0], [0, -2, 0, 0, 0, 0, 0], [-2, -1, -1, 0, 0, 1, 1], [-2, 1, 1, 0, 2, -1, -1], [0, -1, 0, -1, -2, -1, 1], [0, 1, 1, 2, 2, -1, 1], [6, 1, -1, 0, -2, -1, -1], [2, 2, 0, 0, 2, 0, 0], [2, 1, -2, -1, -4, -3, 1], [2, -1, -1, 2, 2, 1, 1], [0, 1, -1, -1, -4, -3, 1], [0, 0, 3, 2, 1, 1, 0], [4, -3, 1, 1, -1, 1, 2], [0, 0, -4, -2, 0, 0, 4], [-5, 1, 0, 2, 0, 1, 0], [4, -2, 0, 0, 0, 2, 2], [2, -2, 0, -2, -2, -2, -2], [4, 0, -3, -1, -4, 0, 2], [2, 2, -2, -2, -2, 2, 0], [4, -2, 2, -2, 0, -2, 0], [2, -1, -1, 0, 0, 3, -1], [0, 0, 3, 4, 4, 5, -1], [4, 2, -2, 0, -2, -2, -2], [4, 2, 2, 2, 2, 0, -4], [2, 2, 0, 0, 4, 0, 2], [0, 0, 2, 4, 4, 4, 0], [4, 2, -2, 0, -4, -2, 4], [2, 0, 2, 0, 4, 0, -2], [-6, 4, 1, -2, 2, -3, -3], [2, 1, -1, -2, 2, -1, -3], [6, -2, -1, 2, -2, -1, -1], [2, 4, -1, 4, 5, 1, 4], [0, -1, 3, 0, 2, 1, 1], [-8, -2, 4, -1, 3, 4, -3], [-4, -4, 6, 2, 4, 4, 0], [-2, 1, 1, 0, -2, 1, 1], [-2, -1, -1, -4, 0, -3, 5], [2, 2, -4, 0, -4, -4, 0], [-4, 4, -2, 0, 2, 0, 4], [2, -1, 4, -1, 0, -1, -5], [4, 0, 4, -4, -2, 0, -2], [-1, 0, 0, 0, 0, 0, 0], [4, -2, -6, -8, -4, -2, 2], [4, 0, 3, 0, 5, 1, 0], [8, -8, -2, 0, 0, 2, 4], [0, -1, 2, 1, -2, -1, -3], [-2, 3, -3, 0, 2, 3, 3], [4, -2, -6, -2, 0, 0, 4], [2, 8, -1, -1, 0, 0, 0], [0, -3, 3, 0, 4, 1, -5], [-6, -4, 3, 2, 6, -1, -1], [8, 3, -3, -2, -8, -5, 1], [4, -2, -5, -2, -1, -3, 2], [5, -1, 8, 2, 4, 3, -4], [12, 0, 0, 2, -2, -2, 4], [8, 4, 0, 3, -1, 0, 1], [-4, 3, 1, -3, 2, -1, -3], [14, 1, 1, 2, -2, 1, -1], [6, -2, 6, -2, -2, 0, -2], [0, -1, -3, 0, -8, -1, 3], [4, -7, 3, -3, 3, 3, 0], [4, 2, 2, 6, 6, 2, 2], [-6, -6, 0, 2, 8, 6, 4], [2, -1, -3, 4, 0, -3, 3], [11, 0, 1, -6, -6, -8, -5], [-5, 2, 3, 0, 0, -2, -3], [-14, 3, -2, -1, -6, -1, -5], [4, 2, -2, -5, -4, -6, -2], [10, 0, 4, 2, 4, 6, 0], [0, -7, -3, -2, 0, 3, 5], [-4, -3, 1, 2, 4, 1, 1], [-10, -6, 1, -2, 0, -3, 3], [3, 0, 1, -2, 2, 0, 7], [-4, 8, 4, 0, 4, 2, -4], [6, 4, 4, 0, 2, 0, -4], [4, -4, 0, -5, 2, 0, -2], [-4, 3, -3, 2, 8, 1, 3], [0, 3, -6, -3, -4, 1, -1], [10, -5, -3, -4, -2, -3, 3], [6, 2, 4, 10, 0, 0, -2], [0, 6, -2, 0, 0, -2, 4], [-2, 8, -4, 2, -2, -10, 4], [8, -1, 1, 8, 6, 3, 5], [-4, -13, 0, 1, 0, 3, 5], [0, -1, 9, -3, 3, 5, -6], [14, -4, -2, 0, -4, 2, -4], [-6, 5, 0, 3, 6, 5, 1], [8, -5, -2, -12, -6, -2, -4], [16, 2, 4, 2, 0, 6, -2], [-12, -5, 1, -6, -2, 5, 1], [-2, 6, -6, -4, -2, 0, -2], [0, 2, -2, 0, -8, 0, -2], [-6, 0, -1, 0, 5, 1, -4], [-4, 4, -6, -3, 5, 2, 1], [-16, -4, -6, -6, -8, -2, 6], [2, -4, -9, -6, -10, -9, -1], [-8, -2, 2, -3, 6, 6, -2], [-10, 0, 4, -6, -4, -4, 0], [0, 3, -1, -1, -3, -5, 2], [-11, -13, 4, -2, -4, 3, 0], [2, 1, 0, 6, 1, -4, 1], [-12, 8, 4, -2, 0, 4, -6], [-10, -3, -2, 1, -2, 1, 5], [8, 6, 4, 4, 6, 2, 2], [2, -5, 3, 0, 2, 3, 1], [4, -4, 0, -7, 6, 0, -2], [1, -5, 4, 2, -4, 3, 4], [2, -9, -6, 2, -3, 2, 7], [0, -5, 9, 0, 6, 5, -3], [-12, 8, 0, 0, 2, 0, 8], [-16, 0, 4, -4, -2, -4, 2], [6, -7, 2, 1, 10, 1, 1], [0, 2, 6, 7, 2, -2, 0], [2, -2, -2, 0, -6, -6, -2], [8, -4, -4, -3, -8, -8, 4], [6, 5, -1, -8, -10, -7, 1], [-4, 6, -2, 2, 2, -2, 2], [-16, -2, 10, 2, -2, -2, -8], [0, -4, -1, -12, -6, -3, -3], [2, -1, -5, -2, -2, -3, -3], [10, 10, 3, 2, 6, 3, -1], [-9, -2, -1, -4, -8, 2, -3], [-6, -7, 4, 1, 4, -3, -7], [0, -1, 5, 0, 8, 9, -3], [0, 2, -2, -7, -8, -6, -4], [-12, 7, -4, -3, 0, -5, -3], [-4, 7, 2, 5, 2, 3, -7], [-2, -2, 1, 6, 8, 5, 3], [8, -6, -10, -4, -2, 0, 8], [1, 1, -6, -2, 8, 1, 2], [8, -10, 5, 2, -3, 3, -2], [-6, -3, -2, 4, 7, 2, 5], [-16, 7, -3, -2, 2, -3, 3], [0, 8, 4, 4, 0, -6, -8], [-4, -6, -2, 1, -10, 2, 6], [-10, 3, -1, 4, 0, 7, -3], [-14, 0, 4, -2, -4, -6, 0], [0, 9, 3, 3, 2, 9, 3], [4, 6, 0, 9, 5, 8, 3], [10, 1, 9, 0, 6, -1, -11], [4, 6, -8, 3, -9, -12, 9], [14, 13, -6, -1, 2, 1, -3], [10, -4, -5, -6, -7, 1, 0], [-18, -3, 1, 2, 0, 7, 7], [16, -6, 6, 1, 0, -2, -2], [-14, 2, 7, 3, 18, 8, 4], [-16, -2, -2, -3, -8, -2, 8], [0, 0, -4, -4, 0, 4, 4], [-10, 2, 6, 8, 14, 8, 0], [6, -6, -4, 0, 4, 0, -2], [4, 12, 4, -4, -6, -4, -6], [10, 2, 3, -3, 4, -6, 2], [-18, -6, -4, -2, -8, -2, -2], [8, -5, -5, 8, -6, -5, 1], [4, -4, 0, -2, -6, -4, -2], [-6, -4, 4, -2, -2, -8, 2], [6, -10, -6, -8, -4, -6, 0], [-28, -9, 5, 7, 2, 3, -3], [8, 8, -2, 6, 0, 2, 0], [-10, -2, 0, 8, 6, 0, 4], [4, 2, -8, 4, -2, 2, 0], [-26, 0, 0, 6, 2, 4, -2], [-14, 4, -4, -6, -8, -10, -2], [-10, 13, 1, 4, 10, 7, 3], [-12, -6, 4, -4, -2, 4, -2], [8, 12, -1, 2, 5, 1, -4], [-7, 0, -5, -4, 0, 4, -3], [-8, 10, 2, -2, 2, 8, 0], [12, 1, -3, -8, 0, -3, -1], [-20, -4, 8, 2, 6, 0, -4], [-12, 7, -2, 12, 8, 10, 10], [-4, -3, 3, 6, 2, 1, 3], [-14, -9, 0, 1, -6, -5, 3], [-18, 1, -3, 4, 6, 13, 5], [24, 3, 1, 2, 2, -1, 7], [12, -6, 4, -10, -8, -12, -8], [-14, 1, 15, 12, 8, 5, -9], [6, -6, 2, -8, -6, 2, -8], [-6, 6, -4, -4, -16, -4, 2], [8, -12, -4, -2, -8, 6, 8], [-2, -9, -5, -2, -10, -1, 7], [-6, 8, -8, -4, -4, -6, 4], [2, 1, 10, -1, 10, 5, 1], [-7, 8, 3, -4, 4, 4, -3], [-12, 12, 1, 3, 0, -4, 6], [4, 2, -2, -3, -3, 4, 9], [22, -2, -6, -4, -6, 6, 6], [-12, -1, 11, 6, 8, 9, -7], [-22, -4, 4, -2, -12, -4, -6], [4, 7, -10, 5, -8, -11, 7], [10, 4, 9, 1, 0, -4, -8], [-16, 11, 1, 2, -4, -5, -1], [22, 4, -12, -8, -8, -2, -2], [-12, 8, -8, -2, -2, 12, -2], [-8, 0, -3, 12, 19, 11, 8], [-29, -8, 5, 0, 8, 8, 7], [4, -6, -10, -15, -7, -2, 13], [2, 4, -4, -4, -14, -6, -8], [-6, 4, -2, -6, 0, 6, 0], [16, -7, -1, 6, -8, 1, 5], [20, 3, 0, 1, -8, 3, -3], [-8, -13, 1, -3, -2, 3, 1], [24, -8, 8, -6, -2, 0, 0], [8, 1, 7, 1, 7, 3, -8], [10, -4, 6, -4, -2, 2, -8], [4, -8, 0, 1, -3, 0, -5], [14, -3, 0, -3, -12, -11, -3], [-8, 3, 8, 0, 6, -4, 2], [4, -7, -9, -14, -12, -9, -7], [12, 6, 6, -4, -2, -6, -10], [10, -8, -10, -8, -8, 2, 4], [4, -6, 6, 4, 0, 2, -8], [-10, -3, 20, 9, 16, 9, -3], [-12, 4, -6, -5, -1, 6, -5], [14, 2, -8, -8, 0, -6, -6], [2, -9, -13, -14, -10, -13, 9], [12, 0, 2, 10, 4, -6, -12], [-6, 7, -3, 4, 6, 7, 5], [-14, 4, 2, -4, -2, 4, 4], [4, -2, 6, 9, 16, -2, 2], [-6, 8, 0, 4, 14, -2, -4], [-4, -5, 6, -3, 2, 7, 5], [10, 1, -6, -13, -8, -15, -7], [12, 11, 3, 7, -3, -5, -10], [10, 19, -7, -4, -10, -3, 1], [10, -6, 2, 2, 0, 4, 6], [22, -7, -4, -3, -12, 1, 1], [-2, -11, 0, 3, 0, 15, 7], [-4, 9, -5, 3, -12, -3, -3], [4, 12, -2, -8, -8, -6, -10], [-6, 10, -2, 2, 10, 6, 0], [-12, -4, -3, -12, -1, -1, 12], [8, 13, -5, -7, -14, -11, 3], [-28, -5, 1, -2, 4, 5, 3], [4, 6, -10, -1, -9, 4, 11], [-4, -7, 0, -4, 12, 12, -4], [2, 2, 6, -4, -20, -6, -8], [18, -3, -5, 0, -12, -11, 11], [16, 8, -6, 4, 10, 4, 0], [-16, 9, -11, 10, 8, -3, 7], [-4, -1, -3, -2, -12, -5, 13], [38, 3, -14, -4, -3, -2, 7], [2, 2, -8, -2, 0, -8, -6], [-34, 7, 7, -2, 0, -13, -1], [8, -1, -9, -8, -12, -7, 9], [18, -10, -8, 2, -2, 6, 12], [-22, -10, 10, -4, 0, 6, -8], [8, -4, 16, 6, 6, 8, -12], [24, 6, 1, 16, 21, 11, 14], [4, 0, -10, 0, 4, 2, 2], [14, 9, -9, -10, -6, -13, 11], [4, -6, -4, 10, -6, 10, 8], [4, 13, -5, -4, -2, 3, -7], [0, 8, -2, -6, 6, -2, -6], [-26, -7, 1, 2, 14, 15, 9], [14, -12, -14, -2, -14, 6, 18], [0, -12, 4, -5, 4, 12, 0], [-22, 11, 11, 6, 14, 7, 3], [13, -3, -10, 2, -12, 1, 6], [32, -2, -18, -12, -16, -4, 0], [36, 6, 4, 2, 0, 4, -2], [0, -6, -4, -4, -18, -6, -6], [-2, -8, 6, 0, -12, -2, -4], [-20, 1, -7, -10, -8, -7, 13], [2, 0, -5, -2, -20, -9, 9], [16, 3, 1, 3, 4, -1, -1], [0, 0, -8, -6, -10, -4, 0], [-10, -1, -7, -16, -12, -7, 5], [30, -2, -18, -4, -16, -12, -4], [-16, -1, 5, -4, 8, 1, -13], [-20, 6, 2, -7, -2, -6, -10], [-24, -8, 5, 0, 3, 7, 8], [-14, -7, -1, 0, 6, 3, -5], [16, -6, 2, -11, 0, -10, 0], [-1, 10, 19, 8, 20, 6, -11], [-8, 9, 0, -9, 4, 5, -5], [-30, 8, 6, 0, -8, -2, 8], [30, -14, 1, -2, -8, 5, 3], [2, 0, 6, 2, -8, 6, -4], [0, 1, 5, 2, 0, -19, 7], [6, 7, 0, -11, -8, -15, -3], [-10, 6, -1, 2, -12, -5, -3], [-16, 5, -3, 0, 4, -1, 13], [26, -1, 9, 12, 6, 9, 1], [10, 12, -3, -10, -12, -23, -5], [-4, 2, -8, 7, 3, 10, 5], [24, -6, 2, 2, -8, -8, -14], [-20, 11, 5, 9, 3, 1, 10], [-20, -2, 9, -4, 5, 11, -2], [20, 0, 12, -3, 6, -4, -10], [-4, 8, 4, 4, 4, 0, 4], [4, 6, 11, -2, 11, 1, -2], [-28, -7, 3, 1, -9, -1, 8], [-10, 6, -16, -6, -8, -6, 0], [8, 0, 0, -12, -10, -12, -16], [-16, 7, -5, 6, 16, 9, 9], [2, -5, -1, 6, 18, 11, 1], [-8, -8, -11, -8, -6, -5, 15], [18, 0, -2, -10, -4, -2, 0], [-2, -4, -4, -6, 4, -2, 2], [18, -4, -5, -4, -15, -3, -12], [22, -8, -7, 2, -16, -11, 3], [28, 6, -14, 4, 0, -6, 2], [-8, 2, 17, 14, 13, 15, -18], [-6, 5, 10, 17, 10, 9, 5], [-14, 2, -2, -6, -6, -12, -2], [24, 3, 6, 9, 8, 5, -5], [-8, -4, 22, 8, 12, 12, -12], [-12, -3, 5, 2, 6, 3, -5], [7, 16, 5, -2, 10, 4, -9], [28, 0, 9, 8, 4, 3, -11], [-10, 11, -1, -2, -6, -5, -7], [-2, -11, 2, 15, 2, 9, 1], [2, -20, 10, 2, -18, -4, -10], [-4, 0, -2, -2, -2, -14, -8], [-2, -12, 5, -9, -6, -10, -2], [28, -14, -3, 4, -3, -1, -2], [4, -5, -12, -5, -10, -13, -7], [36, 2, -5, -21, -16, -18, -8], [-12, 6, -6, 4, 4, 16, 6], [0, -4, -8, 4, -6, 10, 0], [-16, -12, 14, 2, 18, 12, -2], [-24, -12, 20, 12, 24, 12, 4], [-28, 6, 10, -11, -3, -2, -3], [-4, 0, 0, -11, 8, -8, 8], [-16, 13, 9, -2, -12, -3, -11], [-18, -15, 8, -1, -10, -3, -3], [4, 2, -6, 9, 6, -2, 2], [0, -13, 17, -1, -4, 7, -17], [2, 6, -8, 20, 14, 4, 4], [-22, 2, 16, 14, 16, -2, 2], [-25, 17, 0, 6, 8, 13, 16], [18, 0, 2, 6, -12, 0, 8], [18, -3, -3, -2, 4, -7, -3], [-28, -3, 15, 10, 30, 15, 9], [12, 14, -2, 4, -8, 6, -8], [18, -4, -9, -6, 0, -1, 9], [9, 16, -5, -12, 4, 8, -3], [-28, -6, -4, -7, -19, -8, -9], [16, 12, 2, -10, -2, -16, -6], [32, 2, 15, 8, -9, -11, -18], [4, 3, 7, 11, 21, 11, -2], [5, -8, -9, -12, -20, 4, 1], [-26, 6, -2, -2, 2, -4, -18], [12, 3, 0, 0, -8, -12, 0], [40, -4, -2, -6, -10, -2, 6], [6, -6, -7, -2, 0, 5, -1], [-17, 4, 5, 4, 4, 12, -1], [16, -2, 7, 6, 15, 1, -6], [-22, -10, -12, -10, -8, 4, 12], [-40, -8, -8, -8, -12, -4, 0], [32, 2, 4, 2, 10, 2, 0], [18, -2, 6, 10, -2, 6, 8], [-18, 20, 6, 12, 6, 8, -4], [8, -18, 10, 1, -11, 2, -15], [-19, 8, 15, -6, 6, -4, -11], [-8, -1, 8, 13, 16, 3, -15], [36, -3, -9, 3, -6, -3, 3], [16, -8, -4, 12, -4, 14, 2], [10, -6, 8, 10, -14, -4, 2], [24, 4, -12, -11, -10, -16, -2], [-7, -16, -1, 12, 4, 0, -7], [40, -8, 11, -13, -2, -6, -12], [2, 8, 4, -2, 4, 10, -8], [4, 2, -10, -10, -6, 10, 8], [22, 9, -5, 0, 0, 9, -3], [12, 16, 7, 18, 9, -3, -4], [-18, -2, -4, -8, 4, 0, 14], [-26, 16, 11, -6, -6, -5, -17], [34, 4, -16, -6, -2, 8, -4], [4, 7, 8, 19, 2, 7, -7], [-4, 6, 6, 30, 18, 6, 10], [21, -16, -5, -10, -14, 0, 1], [14, -2, -10, 8, 12, 8, 16], [0, 14, -6, -7, -6, -10, -4], [22, 4, 6, 8, 10, -4, 2], [32, 2, 0, -8, -20, -2, 2], [10, -6, 6, 8, 8, 10, 14], [28, -4, 5, 0, -2, 3, 7], [-11, -6, -7, -20, 8, -6, -1], [-43, 20, -5, 4, 0, -8, 9], [14, 19, -19, -6, -14, -13, 7], [14, 8, -16, -10, -16, -22, -4], [12, -4, 2, 12, -10, -6, 8], [2, -12, -6, -26, -12, -12, 4], [38, -4, -2, 4, -20, 2, -10], [-16, 0, 3, 8, 14, 13, 9], [6, -2, -2, -18, -24, -16, -14], [12, 11, -15, -10, -12, -13, -3], [-36, 4, 5, 5, 18, 10, 0], [0, -11, 20, 7, 4, 9, -9], [-2, -15, 15, 4, -2, 5, -15], [4, -10, 8, 10, -10, 4, 14], [22, 17, 5, -2, -2, -3, -7], [-40, 3, 7, 10, 18, 19, 1], [8, -2, 8, 2, 14, 24, -8], [18, 3, -17, -4, -18, -7, 23], [10, 12, -19, -13, -6, -10, 6], [50, -8, -17, -2, -10, -9, 7], [-11, 14, 9, -12, -4, -6, -17], [-36, -1, 9, 4, 12, 13, -7], [-8, -4, -8, 0, 10, -8, 12], [-14, -20, 18, 2, -6, 4, -24], [8, -20, -2, -6, 14, 2, 8], [26, 2, -6, -8, -6, -10, 20], [-12, -4, 22, 15, 35, 14, -9], [12, 5, -9, -7, -18, -15, -1], [-28, -2, 14, 8, 4, -2, -6], [24, -5, -2, -7, -16, -19, -9], [-4, 16, 4, -1, 3, -12, -3], [10, -14, 8, -8, 0, 4, -14], [16, 13, -4, -7, 6, -3, -9], [8, -12, -4, 12, 0, 0, -10], [-20, -3, 10, -9, -10, 1, -5], [14, -4, -15, -2, -10, -11, 13], [-22, 9, -19, -2, 0, 3, 17], [0, 9, 6, -3, -4, -3, -13], [24, -3, 12, -12, -16, -16, -16], [-14, -8, -12, -14, -22, -4, -6], [-30, -6, 18, 8, 30, 24, -6], [-2, 4, 6, -6, -30, -14, -2], [20, -14, 2, 2, 2, 8, 16], [30, -12, -6, -18, -10, -2, -10], [-10, 9, -12, -3, -8, -15, 5]];
    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_w();".
// 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_w(:prec:=7)
    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_w();".
// 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_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4014_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![36, 174, 256, 92, -42, -22, 2, 1]>,<7,R![96, 80, -224, -48, 88, -8, -6, 1]>,<11,R![-3494, 394, 2028, -58, -254, -14, 9, 1]>],Snew);
    return Vf;
end function;