
// Make newform 4017.2.a.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4017_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_4017_2_a_e();".
// 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_4017_2_a_e();".
// 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 := [7, -10, -183, 243, 1113, -710, -2118, 639, 1771, -251, -763, 45, 177, -3, -21, 0, 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, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [34, 75, -513, -963, 936, 2215, 127, -2219, -853, 1224, 488, -382, -106, 62, 8, -4], [-2, -71, -10, 1142, 659, -3365, -1328, 3544, 955, -1773, -318, 458, 50, -59, -3, 3], [-19, 159, 468, -2598, -3558, 7997, 6497, -8881, -4832, 4773, 1723, -1349, -293, 193, 19, -11], [3, 58, 66, -1188, -995, 3800, 1667, -4091, -1039, 2035, 299, -512, -40, 63, 2, -3], [-59, 111, 1488, -2484, -7776, 9055, 12397, -10901, -8665, 6195, 2990, -1831, -499, 272, 32, -16], [32, -132, -1113, 3162, 6885, -10882, -11446, 12578, 8056, -6882, -2765, 1960, 457, -281, -29, 16], [82, 0, -1977, 2103, 9309, -9239, -13367, 11431, 8492, -6369, -2689, 1802, 416, -253, -25, 14], [-43, 207, 1341, -4104, -8709, 13583, 15428, -15745, -11453, 8817, 4108, -2600, -704, 388, 46, -23], [-115, 120, 2805, -3672, -13257, 14477, 18947, -17398, -11933, 9525, 3739, -2654, -572, 367, 34, -20], [-54, 101, 1269, -2233, -6387, 8018, 9793, -9445, -6579, 5207, 2187, -1484, -353, 212, 22, -12], [-100, 378, 2943, -8454, -17766, 28664, 29312, -32929, -20516, 17925, 7009, -5078, -1154, 724, 73, -41], [-155, 516, 4224, -11310, -23808, 39136, 37054, -44789, -24610, 23913, 8015, -6571, -1264, 902, 77, -49]];
    Rf_basisdens := [1, 1, 1, 1, 3, 1, 3, 1, 3, 3, 3, 3, 3, 1, 3, 3];
    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_4017_a();"
function MakeCharacter_4017_a()
    N := 4017;
    order := 1;
    char_gens := [1340, 1237, 1756];
    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_4017_a_Hecke(Kf)
    return MakeCharacter_4017_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 := [[0, -1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, -1, 0, 1, 0, -1, -1, 0, 1, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [-3, 2, 0, -1, 2, -1, 0, -1, -1, -1, -1, 0, 0, -1, 0, 1], [-1, 0, 0, -1, -2, 1, 1, 1, 0, 0, -1, 2, 2, 1, 0, -1], [-3, 1, 0, -1, 0, -1, 0, -2, 0, 1, -1, 1, 2, -2, 0, 0], [-3, 1, 0, -2, 2, -3, -2, -4, -2, 2, -2, 0, 1, -2, 0, 1], [0, -1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 1, 0, 0, -1, -1], [-3, 0, -2, 1, 0, -2, 1, -2, 0, 0, 0, -1, 1, 0, 1, 1], [-2, 1, -2, 0, -1, 2, 2, 1, 0, 0, 0, 0, 1, -1, -2, 0], [0, -2, -1, 3, -2, 1, -1, 3, 3, -1, 3, -1, -1, 0, 0, -1], [0, 2, 0, 1, 1, -2, -3, -3, 1, 0, 3, -3, -2, -3, 2, 2], [-8, 2, 0, -3, 2, -3, 1, -3, -3, 0, -1, 3, 3, -3, -3, 1], [-1, 2, -3, 1, -1, 2, 3, 0, -1, -1, 1, -2, -1, 0, -1, 0], [-4, 4, 0, 0, 0, 0, 2, -1, -1, -2, 0, 0, 2, -2, -1, 0], [1, -4, -1, 3, -3, 2, 1, 4, 3, -1, 2, 1, -1, 3, 0, 0], [-3, 0, -1, 3, 1, 0, -2, 0, 1, 0, 2, -3, -3, 0, 0, 1], [1, 3, -1, 1, 0, 0, 0, -2, 1, -1, -1, -2, 1, 1, 1, 0], [-1, -1, 5, -5, 3, -3, -2, -5, -2, 3, -3, 2, 1, -2, 1, 2], [1, -2, 0, 2, -1, 1, 0, 2, 1, 0, 1, -1, -2, 2, 1, -3], [-2, 0, 0, 2, 0, 2, 1, 3, 0, -2, 2, -2, -1, 0, -3, -2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 1, 1, -2, 3, -2, 0, -2, -2, 2, 0, 2, 1, -5, 0, 2], [4, -3, 4, -1, -3, 1, -1, 1, 3, 2, -2, 1, 0, 4, 2, 0], [-2, -1, 0, -2, -1, 1, 0, 1, 0, 1, -1, -1, 1, 2, 0, 0], [-8, 0, 1, -1, 2, -1, 2, 0, -1, 0, 1, 1, 1, -3, -3, -1], [1, -2, 3, -3, 0, 0, 1, 1, 1, 2, -3, 1, 0, 1, 1, -1], [3, -2, 2, -3, -1, -2, -1, -3, -2, 2, -2, 0, 1, 2, 4, 0], [-2, -4, -2, 1, -3, 5, 4, 5, 0, -2, 1, 2, 2, 4, 0, -3], [4, 0, 3, 1, -2, 7, 2, 6, 3, -3, -1, 1, -2, 5, 0, -3], [2, -1, 1, 2, 3, 3, -2, 4, 3, -4, 2, -1, -6, 1, -1, 0], [7, -5, 1, 0, -5, 4, -3, 6, 3, 3, -2, 1, -5, 6, 4, -1], [-1, -2, 4, -5, 3, 3, 2, 2, -2, -1, -6, 4, 0, 3, -2, -1], [5, -2, 2, 0, -1, -5, -5, -2, 1, 2, -1, -2, 0, 1, 2, 1], [-1, 0, 2, -4, 1, 0, 2, -2, -4, 2, -4, 2, 0, -2, -1, 2], [-1, -1, 4, -1, 2, -1, -2, 0, -1, -2, 4, 0, -2, -3, 1, 2], [-10, -3, 2, -1, -1, 1, 2, 0, -1, -3, 2, -2, -2, 0, 1, 1], [0, 1, -3, 4, -1, 2, 3, 4, 2, -3, 1, 3, 2, 1, -4, 0], [4, -2, 1, -2, -3, 1, -1, -2, 0, 1, 1, -2, -5, 1, 2, -1], [3, -1, 7, -5, -1, -2, -2, 1, -3, 3, -4, 2, 2, 1, 1, 1], [-4, -2, 1, 5, 0, 1, 2, 4, 0, -2, 2, 2, 1, 1, -3, -1], [-1, 5, 0, -1, 4, -4, -2, -4, -1, 0, -2, -1, 0, 1, 0, 0], [-7, 3, 0, -4, 2, -4, 0, -8, -3, -2, -3, 0, 2, -4, 2, 1], [6, -8, 3, 2, -2, 1, -6, 8, 4, 2, 2, 0, -4, 3, 4, -3], [-8, 2, -3, 0, 3, -4, 1, -6, -2, -1, 0, -5, -1, -2, 2, 1], [-3, 4, -3, 3, 1, -4, -1, -5, -3, 1, 0, 0, 3, -1, -2, 1], [-3, 3, -3, -1, -2, 4, 7, 3, -3, -4, 3, 3, 2, 0, -6, -1], [6, -3, 3, 1, -1, 3, -1, 1, 1, -1, 0, 0, -2, 1, 3, -1], [0, -1, -1, 3, -2, 2, 7, 6, 2, 0, -2, 4, 4, 3, -4, -4], [1, 0, -2, 1, -1, 1, 3, 3, -1, -3, 2, -2, -1, 4, -1, 1], [-9, 3, -5, 0, 2, -4, 3, -4, -3, -1, 3, 0, 2, -3, -3, 0], [3, -4, 4, 1, 0, 0, -8, 1, 3, 3, 0, 0, -2, 4, 2, 0], [-1, -1, 3, 2, 1, -4, -3, 0, -1, 3, -2, 1, 5, 2, 2, -1], [2, 0, 2, 1, -1, -4, -7, -5, 2, 5, -4, -4, -1, -2, 7, 2], [2, -5, -3, 3, -3, 0, 0, -3, 2, -1, 3, 0, 1, -1, 0, -2], [-1, -2, 1, -1, -4, 3, 4, 0, -2, 2, 0, 3, 0, -3, -3, 0], [4, 1, -6, 3, -3, 3, 3, 2, 0, 0, 1, -3, 0, 1, -4, -3], [1, 1, -7, 6, -1, 0, -4, 0, 2, 0, 2, -4, 1, 0, -2, 0], [1, -2, -2, 0, -4, 5, 4, 4, 1, 2, -1, 1, 2, 5, -1, -5], [1, 1, 1, -1, -3, 1, -4, 0, 0, 0, 4, -2, -4, -1, 2, 3], [-2, -5, -6, 9, -4, 3, 6, 8, 3, -3, 4, 4, 5, 4, -3, -5], [3, 3, 3, -1, 0, -1, 2, 2, 1, -3, 0, 0, -1, 0, 0, 3], [-11, 1, -1, -1, 1, -4, 4, -7, -3, -1, 0, -1, 3, -5, -1, 1], [-5, -4, 0, 2, -1, 0, -1, -1, -1, 0, 2, -1, -4, 2, 5, 5], [0, 2, 0, 1, 6, -3, -3, -6, -3, 2, -3, 0, -4, 0, 1, 3], [-1, -3, 0, -1, 0, -1, 4, 2, -3, -3, 1, 1, 5, 3, 0, -1], [-4, 2, -9, 1, -4, 3, 4, 1, 0, -2, -1, -2, 3, -1, -2, 0], [-7, -2, 8, -7, 6, -4, -4, 0, -1, 1, -2, 2, 2, 1, 4, 3], [-14, 4, -2, -3, 9, -4, 4, -7, -6, 1, -5, 0, 6, -10, -5, -1], [4, -4, -1, -1, -3, 5, 4, 7, 0, 4, -2, 4, 1, 2, -4, -5], [-8, 6, -6, -1, 2, 2, 3, 6, 1, -3, 1, 3, 0, 0, -5, -3], [9, -9, -7, 5, -6, 7, -3, 6, 7, -3, 3, -1, -7, 9, 4, -2], [-2, -8, 4, -5, 1, -1, 3, 2, -1, 1, -6, 7, 5, 2, 3, 1], [-5, 6, 1, 1, 7, -10, -9, -10, -6, 0, 3, -5, -2, -8, 1, 6], [12, -2, 4, -1, -1, 0, 0, 2, 1, 1, -2, 3, -5, 4, 1, -2], [1, 4, -4, 1, 3, 2, -4, 4, 5, -5, 4, 0, -3, -2, -3, -2], [7, -6, 0, 3, -8, 5, -3, 5, 8, 3, 1, -3, -1, 8, 5, -3], [6, -3, 5, 2, -3, -3, -5, -1, 0, 2, -2, 3, 2, 2, 6, 2], [-5, -2, 7, -4, 3, -2, -2, -2, -3, -1, -3, 1, 1, 1, -3, 3], [-2, 9, -6, 4, 1, 1, 0, -3, 0, 2, -4, -5, 3, 2, -1, 1], [-9, 2, -9, 1, -3, -9, -2, -8, 1, 3, 2, -3, 6, -9, 3, 4], [-4, -1, -3, 1, -2, 0, 1, -4, -3, 3, -5, 2, 7, 3, 0, 0], [1, 5, -6, 6, -4, 5, 0, -1, 4, -5, 7, -4, -4, -4, -1, 2], [-9, 6, -2, 3, 5, -5, 2, -5, -3, -5, 4, -3, 3, -7, -6, 0], [4, -6, 0, 3, 3, 2, -4, 3, 2, -1, 5, -3, -6, 3, 5, -2], [5, -6, -2, 7, -1, 9, 2, 11, 2, -5, 2, 1, -5, 8, -2, -10], [-4, -4, -2, 2, 4, -1, 1, 5, -1, -3, 1, 3, 0, -2, -3, -3], [-11, 6, -6, 7, -5, 1, 6, 4, -3, -4, 4, 0, 7, 0, -6, -5], [-3, 3, 6, -5, 9, -3, -9, -5, 2, 4, -3, -4, -5, -5, 2, 5], [-4, 1, -4, 1, 1, -1, 6, -3, -5, 3, -1, -1, 6, -1, -3, 2], [1, 2, -1, 3, -7, 2, -3, -2, 2, 5, 0, -6, 1, 0, 4, 4], [6, -3, 2, -5, -3, -3, -7, -2, -1, 5, -3, 0, -1, 5, 9, 3], [2, -6, -1, 5, -6, 4, 0, 5, 2, 2, 5, 0, 2, 4, -1, 0], [-3, -4, 1, -3, -4, 7, -2, 12, 6, 3, -1, 3, 0, 5, 1, -2], [1, -5, 3, -3, -4, 1, 3, 4, -4, 2, -3, 2, 3, 8, 2, -1], [-6, 0, -4, 3, 4, 2, -4, 6, 2, -3, 5, -2, -9, -5, 1, -1], [-1, -3, -5, 1, -7, 6, 6, 10, 3, -3, 0, 2, 1, 1, -7, -2], [16, -7, 12, 0, -5, 2, -8, 5, 5, 4, -4, -3, -8, 13, 7, 4], [-2, 5, 4, -4, 8, -4, -5, -8, -3, -2, -6, -3, 0, 1, 2, 4], [-4, 1, 0, -1, 1, -5, 3, 1, 2, -1, 1, 3, 3, -5, -5, -1], [-1, -5, 0, 2, -6, 3, -2, 10, 1, 3, 3, 1, 0, 3, -2, 3], [1, 2, -2, 1, 9, 1, -2, -4, -2, 2, 0, -7, -1, -7, -2, 0], [10, -1, 0, 5, 8, -2, -8, -2, 4, 0, 2, -6, -11, -3, 3, 7], [-12, -1, 3, -10, 11, -4, 5, -5, -6, 2, -10, 7, 2, -6, -3, 2], [-9, 3, 4, 0, 5, -4, -8, -1, -5, 3, -2, 2, -2, 1, 2, 2], [3, -5, 0, 9, 4, -5, -14, 0, 6, -4, 6, -9, -11, 1, 5, 5], [2, 9, -7, 4, 8, -2, 2, -7, -2, -7, 5, -5, -8, -8, -4, 3], [-4, 3, 4, -9, 0, -6, -2, -3, -2, 3, -4, 5, 2, 2, 3, 1], [6, -10, 4, -1, -6, -4, -8, -2, -1, 7, 0, 2, 3, 3, 8, 2], [-2, 4, 4, -1, 4, -1, 6, 4, -2, -1, 1, 0, 3, 0, -10, -1], [5, -10, -1, 3, -8, 6, 5, 11, 3, 1, 6, 4, 0, 3, 1, -2], [-6, -1, 4, 2, 2, -1, 2, -2, 5, 3, 1, -1, 0, -4, 1, 5], [-3, -4, -7, 10, -4, -1, 0, 6, 0, -2, 4, -5, -4, 6, 2, -3], [2, -2, 3, -4, -2, 4, 1, -2, 4, 2, -2, 2, -7, -1, 6, 1], [1, 4, 1, 1, 8, -3, -11, -5, -3, 0, 4, -9, -9, -2, 3, 3], [-12, 8, -6, 5, 8, -2, 3, 5, -2, -3, 2, -1, -6, -2, -7, 1], [-8, 12, 7, -12, 9, -7, -3, -6, -7, 1, -7, 2, 1, -6, 3, -1], [-6, 5, 3, -8, 7, 2, 7, -2, -5, -5, -5, 3, -1, -6, -6, -1], [-4, -5, 1, -5, 2, -2, -1, 1, -2, 7, -1, 5, 5, -3, -2, -2], [-12, 13, -3, -5, 9, 2, 12, -2, -5, -5, -6, 3, 3, -7, -11, -4], [8, 2, 7, -7, 0, -4, -4, -5, 2, 2, -7, 2, -1, -1, 3, -2], [5, -3, -7, 12, -4, 3, 0, 7, 1, -3, 9, -2, 0, 0, -4, -5], [-3, -6, 2, -3, -1, -3, -5, -3, 3, -3, 3, -1, 2, 2, 5, 4], [-5, -5, 3, -7, -1, -2, -6, -3, -2, 7, -5, 2, 2, -2, 2, 2], [-3, 13, -4, -9, 3, -5, 1, -8, -5, -1, -5, 0, 1, -5, -1, 2], [5, 0, -3, -6, -6, 4, 1, 3, 3, -2, 5, -2, -1, 4, 1, -1], [13, -6, 0, 5, -9, 13, 3, 11, 10, 0, 7, 0, -11, 8, 4, -2], [-2, 0, 8, -4, -1, -8, -4, -3, 3, 7, -2, 2, -1, 0, 11, 8], [12, 4, 11, -3, -4, 0, -8, -1, 1, 4, -1, -3, -6, 7, 5, 2], [1, 4, -3, 5, 0, -3, 8, -8, -5, -1, 2, -3, 1, -4, 1, 0], [0, 3, -5, 5, 7, 2, 3, 1, 3, -8, 1, 2, -3, -2, -3, -3], [1, -4, 5, 4, 2, -5, -9, 0, 4, 7, -2, -1, -5, 5, 4, 5], [11, -4, -4, 6, -5, 3, 0, 9, 6, -4, 6, -2, -1, -3, 1, -3], [0, 3, -9, 5, -1, -7, 3, -7, -1, 0, 5, -3, 2, -10, -4, 0], [9, 7, 0, -4, 5, 0, -3, -12, -6, 4, -6, -3, -6, -3, 10, 2], 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11, -11, -11, -6, 5, 4], [-27, 10, -12, -3, -6, 3, 17, 7, 2, -11, 5, 4, 13, -3, -3, -5], [2, -6, 4, -2, 3, -5, -8, -3, 0, 7, 1, -1, -11, 1, 11, 4], [-7, -12, -2, -1, -7, 9, 20, 5, -2, -5, -5, 14, 20, 2, -11, -10], [-39, 4, 8, -15, 7, -5, 4, -15, -10, 1, -9, 4, 2, -11, -1, 9], [-10, 5, 5, -1, 6, -24, -7, -28, -8, 17, -5, -5, 16, -17, 4, 4], [14, 6, -9, 20, -1, 5, 0, 2, 10, -8, 13, -15, -3, -4, -5, -6], [17, -7, -8, 9, -9, 14, 9, 19, 8, 1, 1, -2, 1, 9, -10, -13], [-24, 3, -15, 9, 4, -7, 6, 1, -8, -5, 12, -2, 8, -10, -12, 1], [-5, 6, 1, -4, -4, 8, 15, 1, -9, 3, -6, 6, 10, -3, -12, 4], [3, -6, -9, 13, 5, 13, 7, 18, 5, -13, 13, 2, -7, 4, -11, -6], [-17, 9, -8, -10, 3, 7, 19, -6, -12, -8, -6, 5, 9, -4, -9, 3], [27, -17, 0, 10, -19, 18, -2, 17, 13, -1, 10, -4, -6, 19, 5, -2], [17, -8, 13, -15, 5, -7, 1, -7, -3, 12, -9, 8, -5, -4, 10, 11], [-7, -2, 1, 8, 9, 8, -1, 2, 1, -8, -1, -4, -14, 5, 0, -4], [13, -11, 7, 3, -4, 6, -5, 11, 1, 7, -5, -5, -7, 16, 10, -4], [-16, -10, 9, -4, -6, -7, -4, 4, 0, 1, 0, 3, 5, 7, 0, 2], [-16, -9, -2, 1, -13, -6, 4, 0, 0, 2, 0, 9, 19, -8, -7, -4], [5, 6, -14, 4, -3, 3, 11, 3, -5, 2, 2, 0, 7, 2, -10, -5], [9, 3, -10, 4, -7, 14, 13, 13, 4, 3, 1, 0, 5, -8, -13, -7], [6, -1, -4, 8, -9, -3, 6, 3, 2, -1, 8, -6, 2, -1, -2, -8], [15, 14, -6, 5, -3, 12, 9, 4, 5, -6, -1, -6, -11, 4, 7, -3], [35, -21, 1, 9, -26, 14, 3, 17, 12, -3, 4, 4, -10, 22, 4, -16], [3, -1, -4, 11, -15, 14, 2, 17, 2, -1, 11, 1, 2, 15, -2, -7], [-18, 20, 6, -1, 12, -11, -7, -10, -6, -2, -4, 5, 17, -11, 0, 2], [-19, 4, -8, 7, 6, 2, 11, -2, -5, -11, 0, -1, -4, -2, -7, 2], [-13, 9, 8, -8, 6, -6, 3, -7, -10, -3, 3, 6, 0, -12, 1, 3], [-9, 2, -9, 9, -1, -11, -6, -8, 10, 2, 7, -7, 5, -7, 1, 3], [6, 12, -10, 1, 7, -12, -15, -13, 7, 5, 2, -6, -3, -13, 0, 2], [-1, -5, -3, -2, -9, 13, 4, 10, 4, 2, 3, -1, 7, 0, -3, -17], [22, -15, 6, 5, -25, 22, 3, 17, 16, 1, 10, 3, -12, 6, 9, -3], [-10, 3, 9, -7, 0, 2, -5, 9, -1, -1, -8, -4, 5, 7, 6, -6], [5, -13, -3, 19, -16, 19, 13, 28, 13, -12, 9, 8, -2, 23, -9, -4], [-7, -4, -3, 1, -17, 2, 3, 7, 4, 1, 7, 3, 4, 2, 1, -1], [2, -1, 18, -3, 5, -17, -7, -12, -3, 6, -1, 4, 2, -4, 10, 16], [11, -11, 7, 6, 3, -6, -1, -11, 3, 9, -1, -10, -5, -6, -1, 8], [-11, -13, -13, 7, -5, -1, 11, 6, -7, -2, -3, 9, 16, 5, -11, -6], [-16, 0, 3, -10, -4, 6, 26, 3, -15, 0, -16, 16, 19, 8, -7, -3], [-28, 14, -15, 6, -16, 12, 15, 13, -2, -6, 5, 0, 16, -4, -11, -7], [-17, 3, 1, 3, -3, 1, 10, -17, -1, 3, -1, -5, -4, -8, 0, 6], [-3, -3, 7, -10, 19, -14, -15, -12, 0, 3, -9, -7, -1, -8, 7, 11], [-4, -11, 21, -28, -1, -3, 1, -5, -14, 16, -25, 17, 11, 5, 6, 7], [-23, 12, 10, -23, 13, -23, -18, -18, -9, 12, -7, -2, 5, -18, 8, 15], [-13, 8, 2, -3, 11, -3, -7, -2, -3, -2, -4, 0, -6, -4, 1, 6], [-6, -9, 7, -11, -1, -4, -3, -4, -6, 6, -4, 13, 6, -10, 5, 11], [16, 2, -4, 8, -13, 9, -12, 6, 13, 12, 8, -17, -15, 9, 12, -1], [-11, 2, -18, -2, -2, 10, 15, 2, -5, -4, -9, 7, 0, 2, -12, -12], [-9, 14, -9, -6, 12, -19, -11, -20, -2, 3, -2, -5, 1, -23, 4, 8], [-30, 19, -4, 1, 11, -4, 5, -14, 3, -9, 10, -10, -2, -17, -3, 6], [8, 8, -10, 6, -3, 4, 10, -9, -4, 0, 6, -4, 2, -5, 2, -3], [24, -30, 21, -11, -8, 1, -9, 6, -4, 16, -7, 6, -4, 8, 10, -2], [2, -1, 16, -6, 3, -1, 1, 8, -6, 11, -1, 4, 17, 4, -1, 1], [-15, 2, 10, 1, 8, 14, -12, 18, 5, -3, 8, -4, -8, 3, 7, -2], [28, -6, 0, 3, -9, -2, -22, 3, 15, 9, 7, -5, -8, 0, 15, 5], [-13, 13, -6, 6, 4, 1, 21, 2, -5, -11, -2, 6, 12, 12, -4, -10], [-5, 0, 6, -5, 4, 12, 1, 6, 2, -8, -8, -2, -14, 9, 0, 3], [4, 0, -8, 6, -14, 12, 23, 13, -6, -9, 6, 7, 2, -1, -12, -11], [-34, -2, -1, 6, 5, -8, 8, 8, -14, -5, 5, -1, 5, 0, -13, -5], [19, 9, 12, 2, 16, -20, -22, -18, -5, 5, -3, -10, -6, -6, 3, 14], [8, -4, 3, -2, -9, 5, -12, 4, 8, 4, 1, -8, 0, 6, 4, -1], [9, 5, -16, 15, 0, 16, -9, 11, 11, -8, 7, -11, -15, 8, -4, -13], [1, 15, 25, -23, 21, -16, -13, -21, -12, 12, -13, -1, 4, -15, 0, 12], [-1, -4, -11, 17, -11, 8, 5, 11, 11, -9, 11, 6, 6, 2, -5, 0], [-3, -8, -8, 6, 0, -3, 4, -5, -4, -9, 14, 0, 13, -2, -4, -2], [-4, -4, 0, -4, -2, 8, 4, 11, 8, -8, -1, 10, 8, -1, -4, -11], [2, 10, 7, -8, 5, -10, 2, -15, -6, 7, -13, 1, 2, -9, 11, 6], [-22, 14, -20, 8, 4, 7, 15, 6, -11, -13, -1, -1, 11, 1, -8, -6], [1, -10, 1, 8, 2, -3, 4, 0, 8, 1, 1, 3, -1, -3, 4, 4], [16, -20, -4, 9, -18, 17, 5, 11, 5, 3, 7, -4, -8, 19, 3, -11], [4, 0, -11, 18, -5, 0, 5, 9, 6, -6, 11, -5, 0, 8, 0, 1], [21, -10, 1, 9, -5, 6, -8, 11, 9, 8, 0, -13, -18, 14, 11, -10], [-6, 35, -5, -12, 3, -7, 10, -26, -16, 1, -16, 3, 22, -17, -8, 3], [1, 11, -11, 4, -12, 18, 5, 9, 8, 1, -2, -2, -1, 13, -3, -6], [-19, 17, -20, 1, 2, -10, -1, -8, 3, -10, 7, 1, 1, -24, -12, 4], [-6, 4, 3, -7, 1, 4, 6, 13, 7, -9, -7, 12, -2, 9, -3, 3], [-20, 5, 3, -4, 15, -19, -17, -20, 10, 2, 11, -7, -7, -23, 9, 9], [-10, 5, -6, -5, 5, 8, 0, -1, 2, -2, -3, -3, -3, -6, -7, 4], [-13, -12, 14, 3, 0, -2, -5, 9, -3, -5, -2, 5, 3, 18, -1, -3], [25, 13, 9, -7, 4, 9, -4, 8, -2, 0, -7, -8, -10, 9, 6, 7], [10, -5, 9, -3, 11, -24, -20, -20, -2, 12, -2, -3, 0, -11, 20, 8], [-19, 14, -7, 0, 10, -7, 2, 4, -13, -16, 8, -4, -5, -13, -5, 2], [12, 5, 7, 0, 13, -5, -10, -6, -2, 1, 0, -1, -11, -9, -8, -1], [-24, 17, -19, -11, -11, 0, 7, -8, -10, 6, -3, -3, 10, -11, -4, 2], [-24, 13, 8, -10, 25, -13, 0, -14, -10, -9, 3, 5, 5, -20, -6, 4], [10, -9, 11, -8, 1, 10, 8, 7, 2, 5, -15, 21, -4, 2, -6, 1], [-11, 5, 4, -11, 12, -11, -9, -18, -1, 7, -8, -9, -22, -1, 10, 4], [7, -8, 3, 8, -6, 7, 22, 7, 0, -1, -3, 6, 9, 8, -4, -2], [-16, -3, -3, -2, 1, -20, 3, -11, -7, 9, -5, 4, 18, -18, -4, -4], [-14, 10, -5, 13, -17, 25, 8, 25, 10, -17, 19, 2, -10, 16, 1, -2], [-8, 15, 8, 2, 12, -11, -13, -17, -2, -8, 8, -9, -7, -13, 9, 5], [49, -25, 9, 12, -21, 11, -6, 12, 16, 3, 3, 11, 3, 16, 9, -7], [-7, -6, -5, 6, -6, -2, -4, 2, 8, -1, -7, 5, 13, 9, 3, -5], [19, -15, -3, 16, -10, 19, -17, 31, 22, 1, 17, -16, -35, 17, 10, 0], [3, 2, 1, -3, -12, 16, -1, 7, 4, 1, 1, 4, 2, 7, 1, -10], [14, -13, 10, -7, -17, 7, -1, 12, 12, 6, 5, 3, -2, 4, 5, 2], [12, -5, 3, -6, 6, -2, -5, -8, -4, 11, -11, 7, 3, -8, -3, 3], [10, 1, 7, 13, -9, 18, 4, 10, 11, -8, 1, 3, -8, 12, -2, -11], [-22, 17, -9, -8, -4, -2, 12, -4, -2, -5, -7, 3, 20, -13, -2, 5], [-4, -5, -4, 11, -11, 3, 22, 7, 0, -3, 0, 13, 24, 9, -16, -16], [-10, 15, -9, -1, 8, 5, 13, -1, 2, -13, -7, 4, 3, -2, -6, 8], [8, 7, -3, 15, 7, 3, -11, 12, 12, 2, -5, -9, -8, 11, 1, -8], [-4, 11, -8, 15, -8, -5, 8, -1, -13, -2, 11, 0, 5, -3, -10, -4], [-25, 16, 6, -19, 25, -25, -5, -33, -19, 3, -13, 3, 7, -14, 10, 12], [-22, -9, -17, 10, -4, 5, 7, 12, -1, -3, 4, -4, 14, -3, -10, -5], [10, 11, -3, -2, 3, -5, -4, -17, 4, -3, 9, -8, -21, -11, -1, 4], [-22, 22, 5, -11, 7, -18, -16, -23, -5, 4, -7, -5, 6, -5, 0, 3], [-21, 27, -1, -22, 19, -17, 8, -19, -6, -6, -4, 7, 2, -30, -14, 6], [-5, 3, 14, -8, 6, 4, -7, -10, 1, 2, -4, -14, -20, 3, 3, 7], [-12, 7, -1, 0, 10, -9, -7, -6, -3, -7, 1, 3, 2, -10, 7, 9], [15, -10, 10, -6, -4, 2, 1, 3, 5, -1, 3, 3, -3, 2, 0, 0], [-4, -6, -7, -7, -7, 18, 1, 22, 1, 0, -7, 10, -8, 17, 7, 2], [0, 3, -8, 7, -19, 10, 5, -4, 2, 5, -2, -6, 9, 3, 4, 10], [-1, -6, -17, -6, -2, 12, 19, 14, 0, 4, -12, 14, 11, 5, -11, -11], [-23, 6, -5, -10, 3, -2, 20, 0, -8, -5, -4, 6, 1, -10, -4, 1], [16, -11, -2, 23, -8, 19, -3, 19, 20, -11, 14, -8, -11, 14, -2, -17], [4, -2, 13, -3, 3, -1, -19, 7, 14, 11, -6, 4, 7, 7, 1, 5], [31, -15, -7, 0, -23, 7, -11, 4, 9, 10, -6, 3, -5, 12, 8, -1], [3, 1, 17, -8, 12, -19, -4, -21, -10, 7, -10, 0, 7, -5, -1, 2], [-7, -7, -6, 15, -3, 4, -1, 16, 9, -5, 15, -5, -18, 0, -16, -5], [11, -3, -12, 8, -12, -2, 4, -1, 7, 4, -10, 8, 10, 4, -3, -8], [13, 8, 0, 3, -1, -10, -11, -20, -1, -1, -2, -3, 3, -7, 17, 11], [-1, 11, -6, 22, 5, 3, 3, 8, -3, -10, 8, -9, 10, 4, -10, -2], [-1, 1, -8, 2, 6, -2, -3, 3, 2, -8, 7, -4, -10, -3, -9, 0], [21, -27, 10, -2, -7, -3, -16, 0, 6, 6, 19, -4, -21, -1, 8, 5], [0, -1, 18, -15, 4, 4, -6, -7, -8, 9, -9, -6, -13, -5, 3, 12], [15, -7, -9, 8, -14, 17, 10, 13, 10, -5, -1, 5, -3, 9, -9, 1], [-7, -8, -8, 8, 3, 0, 1, -6, -8, 3, -1, -3, 10, -5, -3, -5], [-11, 4, -4, 5, -11, 1, -1, 19, 1, -3, -2, -1, 2, 2, 3, 4], [14, 5, -6, 14, -5, 23, 5, 17, 18, -15, 15, -1, -11, 1, 0, -5], [25, 12, -11, 7, -17, 7, 1, 9, 7, 1, -11, 4, 7, 19, 10, -5], [2, -3, 2, -6, 10, -1, 0, -20, -10, 2, -8, -7, 0, 6, 6, -5], [14, -17, 8, 17, 3, -3, -13, 2, 5, -1, 13, -6, -9, 4, 5, -4], [19, 2, -12, 11, -18, 9, 6, 4, 13, -1, 7, 0, 5, -2, 2, -11], [12, 2, -2, 12, -4, 7, -8, 17, 6, 3, 7, 2, 2, 10, -8, 2], [-6, -9, 2, -12, 10, -4, -6, -5, 1, 5, 1, -1, -17, -7, 8, 5], [3, -4, -5, 4, -2, 12, 14, 9, 6, -13, 6, 0, 11, 11, 3, -6], [-19, 1, 7, -3, -14, -6, 12, 2, 2, -8, 1, 15, 13, -2, 1, 10], [21, 8, 5, -2, 10, -17, -3, -21, -7, 5, -15, -10, -5, -7, 7, 13], [-11, -24, 6, -2, -6, 13, 8, 17, 2, -9, 13, -9, -14, 1, 4, -1], [14, -3, -3, 7, 3, 10, 11, 8, 3, -5, -7, 16, 1, 12, -7, 4], [40, -33, -4, 17, -22, 17, 2, 33, 21, -7, 16, -3, -13, 30, 0, -11], [5, 0, 13, -11, -22, 8, 2, 7, 0, 17, -10, 15, 8, 15, 5, -3], [12, -4, -6, 14, -13, 2, 4, 1, 6, 0, -1, -3, -5, 10, 10, -10], [25, -4, 0, -3, -7, 3, -6, 6, 3, 9, 3, -4, -8, 9, 8, 6], [9, 1, -13, 0, -10, 8, -2, 1, -1, 10, -4, -5, 0, 2, -3, -5], [-13, 18, -13, -16, 11, -12, -7, -15, -12, -1, 1, 2, -3, -16, -1, 7], [-10, 5, -2, -8, 8, -17, -7, -30, -14, 3, -15, 1, 6, -3, 11, 10], [15, -5, 12, 3, -13, 3, -7, 3, 7, -1, -1, -4, -4, 6, 11, -3], [8, -11, 7, -3, 4, -21, -17, -14, 0, 19, -7, 0, 9, -7, 19, 4], [2, 15, -36, 19, -5, 8, 12, 14, -4, -10, 6, -6, 8, 1, -13, -5], [-12, -24, 7, -4, 1, -24, -4, -22, -13, 12, -8, 12, 15, -6, 2, 3], [-21, 2, 0, -12, 12, 0, 5, -10, -10, 0, -15, 14, -2, 3, -5, -4], [15, -9, 19, -3, -3, 7, -4, 14, 5, -3, 3, 7, 6, 9, 12, -3], [10, 1, -19, 31, -4, 1, 3, 10, 5, -15, 21, -9, 4, -3, -13, -8], [7, 21, -21, 22, -3, 12, 0, 9, 7, -8, 14, -21, -20, 7, 7, -6], [-12, -3, -20, 18, -3, 0, 17, 4, -1, -5, 9, 1, 6, 0, -12, 0], [10, 5, -2, -6, 6, 8, 1, 8, 9, -10, 2, -9, -11, -8, -5, 3], [13, -16, 10, -8, -3, 9, -8, 7, -2, 7, -4, 8, -4, 3, 15, 3], [29, 1, 20, -12, 9, -5, -26, -9, 1, 11, -11, -7, -26, 8, 14, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4017_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4017_2_a_e();".
// 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_4017_2_a_e(:prec:=16)
    chi := MakeCharacter_4017_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_4017_2_a_e();".
// 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_4017_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4017_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![7, 10, -183, -243, 1113, 710, -2118, -639, 1771, 251, -763, -45, 177, 3, -21, 0, 1]>,<23,R![-42466673, 207295534, -24410658, -345750359, -64550378, 116293552, 35744003, -13651920, -6162022, 361375, 425898, 28996, -11244, -1657, 52, 21, 1]>],Snew);
    return Vf;
end function;