
// Make newform 4016.2.a.k in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4016_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_4016_2_a_k();".
// 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_4016_2_a_k();".
// 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 := [256, 2240, 1920, -8800, -9400, 13752, 13680, -11921, -9216, 6186, 3178, -1867, -582, 317, 54, -28, -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, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4304, 34224, 59404, -78844, -231693, 28861, 276564, 19540, -156321, -14575, 46913, 3513, -7676, -374, 647, 15, -22], [-3200, -42336, 66416, 335368, -234500, -723064, 384541, 626264, -312422, -246626, 120979, 48558, -23617, -4694, 2264, 178, -85], [-27776, -183520, 55216, 779784, -54216, -1198330, 146661, 866052, -174414, -313888, 79979, 59424, -17157, -5638, 1752, 212, -69], [448, 4336, 4320, -13706, -16325, 14855, 18119, -7882, -9331, 2275, 2570, -363, -391, 30, 31, -1, -1], [23776, 190336, 170912, -587700, -694624, 619761, 783066, -323534, -401886, 91723, 109772, -14297, -16568, 1148, 1306, -37, -42], [19264, 220000, 508704, -368928, -1860528, -214814, 2278651, 517908, -1360330, -261584, 432413, 58072, -74651, -6066, 6608, 244, -235], [1184, 30688, 125672, -12012, -441435, -143716, 539429, 176682, -321823, -78608, 102092, 16544, -17575, -1676, 1551, 66, -55], [8384, 60736, -264000, -616832, 933932, 1475730, -1298289, -1330756, 908994, 539504, -326267, -109080, 61465, 10814, -5804, -420, 217], [-63040, -668096, -907248, 1879504, 3345112, -1609452, -3830305, 586448, 2071842, -92994, -601779, 3222, 96501, 670, -8052, -54, 273], [-3424, -24512, -132396, -58112, 496127, 322886, -642219, -344458, 409343, 147646, -136930, -30574, 24529, 3068, -2231, -120, 81], [3424, 25120, 132396, 58112, -496127, -322886, 642219, 344458, -409343, -147646, 136930, 30574, -24529, -3068, 2231, 120, -81], [17904, 173264, 273948, -411582, -1036154, 185079, 1242448, 72676, -715256, -68557, 219968, 17893, -36974, -2018, 3204, 85, -112], [-31296, -348832, -1002880, 418192, 3625052, 789678, -4436367, -1224636, 2647622, 576016, -840781, -124248, 144951, 12770, -12812, -508, 455], [98624, 940640, 1307936, -2477584, -4930848, 1718170, 5799123, -264940, -3259674, -107976, 982565, 44784, -162659, -5874, 13936, 268, -483], [80704, 714624, 903176, -1908228, -3494786, 1400433, 4115096, -307434, -2300544, -33845, 688604, 23535, -113202, -3364, 9636, 159, -332]];
    Rf_basisdens := [1, 1, 304, 1216, 1216, 16, 608, 1216, 304, 1216, 1216, 304, 304, 304, 1216, 1216, 608];
    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_4016_a();"
function MakeCharacter_4016_a()
    N := 4016;
    order := 1;
    char_gens := [2511, 3013, 257];
    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_4016_a_Hecke(Kf)
    return MakeCharacter_4016_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, 0, 0, 0, 0, 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, 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, 1, 0, 0, 0], [1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1], [1, -1, 1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-2, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1], [-1, 0, 0, 1, 1, 1, 1, 0, 1, -1, 1, 1, 0, 0, 1, 0, 0], [2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 1, -1, 1, -1, -1, 0, -1, 1, -1, -1, 1, -1, -1, 1, 0], [0, -1, 0, 0, 2, 3, 0, 0, 1, -1, 1, 0, 0, 0, 2, 0, 1], [1, 1, 0, 0, 0, -1, 1, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0], [0, 1, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, -1, 0, 1, 0, 0], [2, 0, -1, 0, 0, -1, 1, 1, 0, 1, -1, -2, 0, -2, -1, 0, 0], [0, 0, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1, -1, -1], [-1, -2, -1, 1, -2, 1, 1, 0, 0, -1, -1, 0, 0, 0, 1, -2, 1], [7, 2, 0, -1, 0, -3, -2, 0, -1, 3, -1, -2, 0, 0, -2, 1, 0], [0, -1, 0, -1, -1, 0, -2, 0, -1, 1, 0, -1, -1, 2, 0, 0, 0], [4, 1, 1, -1, 2, 0, 0, 0, 0, 2, 0, -3, 0, 0, 0, 1, -1], [0, -1, 2, -1, 3, 2, 0, 1, 0, 0, 0, -2, 0, -1, 1, 2, 0], [-2, 1, -1, 2, -2, -1, 0, 1, 1, -1, -1, 0, 0, -2, 0, -1, 0], [-1, -1, 1, 0, 0, 2, -1, 1, 0, 0, -2, -1, -1, -2, 0, 1, 1], [4, 3, -2, 1, 1, 0, 3, 1, 2, 0, 0, -2, -1, 0, 2, 0, -2], [-3, -4, 2, -2, 0, -1, -3, -1, -2, 1, -1, 2, 1, 0, -1, 0, 2], [3, 2, 1, 0, 1, -1, 1, 2, 0, -1, 1, -1, -1, -2, 0, 2, -1], [2, 1, 1, 0, 2, 1, 0, 1, 1, 1, -1, -1, 0, 0, 2, 0, 0], [1, 1, -1, 2, 0, 0, 2, 1, 2, -2, 2, 2, 1, 0, 2, -1, 0], [4, -1, 0, 1, -2, -1, -1, 1, 0, -1, -1, 3, 2, -2, -1, -1, 1], [0, 1, -1, -2, -1, 0, 2, -2, 0, -1, 2, 0, 1, 2, -1, 0, -1], [-3, -3, 1, 1, 0, 1, -1, 2, 1, -1, -3, 0, -2, -2, 0, -1, 1], [-1, -3, 1, 0, 3, 2, -1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0], [0, 0, 1, -1, 0, 0, -2, 1, -2, 2, -2, -3, 0, 0, 0, 0, -1], [5, 5, -1, -2, 0, -4, 0, -2, -3, 4, -2, -5, 0, -2, -3, 3, -2], [6, 2, -2, 0, 1, 3, 2, -2, 2, 1, 3, -3, -1, 2, 3, 1, -1], [0, 0, 0, 0, 0, 1, 0, 1, 3, -1, 1, 2, 0, 2, 2, -1, -1], [-2, -3, 1, -1, 0, 2, 0, -1, 1, -2, 2, 5, 2, 2, -1, -1, 1], [2, -1, 1, -2, 0, 1, -1, -1, 1, -1, -1, -1, -1, 2, 0, 1, 0], [-2, -3, 1, -1, 4, 3, 0, -3, -3, 1, 1, -1, 0, 0, 0, 2, 0], [3, 1, -1, 1, -2, 1, 2, -2, 1, -1, 1, -1, 0, 2, 2, -1, 0], [4, 3, -1, -2, -2, 0, 3, 1, 1, -1, 1, -1, 0, 0, 0, 2, -1], [4, -1, 1, -2, 0, -3, -6, -4, -6, 3, -1, -2, 1, 0, -3, 2, 0], [0, -1, 1, 0, -2, -1, -1, -3, -3, -1, -1, 3, 1, 2, 0, -1, 0], [-1, 0, 1, 0, 2, 4, 0, 1, 3, -2, 2, 3, 0, 0, 1, 0, 2], [-3, 0, 1, -2, -3, -1, -1, -1, -1, 0, -1, 0, 2, 0, -3, 1, -1], [2, -1, 0, 0, -2, -2, -2, -2, -3, 2, -2, -3, -1, 0, -3, 1, 0], [-5, -4, 1, -1, 0, 2, 0, -1, -2, 0, 1, -1, 0, 0, 0, 0, 0], [-2, -1, 0, 0, 0, 1, 0, -1, 0, 1, 3, 3, 1, 0, -1, 0, 1], [7, 2, -3, 2, 0, -1, 2, 2, 2, 0, -2, -3, 1, -2, 0, 1, -3], [2, 3, -2, 3, -8, -3, 1, 1, -2, 1, -3, -2, -1, -2, -1, -1, -1], [2, 1, -3, 2, -2, -1, 2, -1, 0, -1, -2, -1, -2, 0, -1, 0, -2], [6, 2, 1, -1, 2, 2, 2, 2, 1, -2, 0, -3, 0, -2, 1, 0, -1], [-1, 0, 1, -1, 0, -2, -3, -2, -4, 3, -1, -2, -1, 0, -1, 1, 2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 4, -1, 1, 2, 0, 3, 1, 1, -2, 2, -2, 0, -2, 1, 1, 0], [1, 3, -3, 2, -4, 0, 5, 0, 1, 0, 0, 0, 2, 0, 1, -2, 0], [-2, -3, 1, 1, -2, 2, 0, -2, -2, 0, -4, -4, -1, 0, 2, -1, 1], [1, 2, 0, 0, -2, -1, -5, -1, -2, 0, 0, 2, 0, 0, 0, 2, 2], [10, 2, 0, -2, 2, -1, 2, 3, 1, 1, 1, -4, -2, 2, -2, 1, -1], [-4, -1, 1, 1, 0, 3, 2, -1, 1, -1, 3, 0, -1, 0, 2, 0, 2], [3, 3, -1, 0, -2, -3, 2, -3, -2, 1, 1, -1, 0, 2, -1, 0, -3], [-1, 1, 0, -3, 4, 1, 0, -2, -2, 3, 1, -5, -1, -2, 1, 2, 0], [-1, -4, 2, -3, -2, 2, -2, -3, 0, 2, 0, 1, 1, 2, 0, 0, 3], [6, -1, 0, -6, 1, -2, -1, -2, -3, 2, 4, -1, 0, 3, -2, 3, 0], [-2, -2, 2, -1, -2, 2, -3, -3, 1, 0, 0, 1, -2, 4, 1, 1, 0], [-4, -3, 2, 0, 2, 1, -1, 1, -4, -1, 1, 1, 0, -2, -1, 1, 1], [7, 4, -1, -1, -3, -2, -4, 1, 1, 4, 0, -2, -1, 1, -2, 3, -2], [4, -1, -2, 2, 2, 2, 2, 1, 0, 4, -4, -8, -4, -1, 2, -1, -2], [-9, -9, 4, 0, 0, 4, -1, -2, 2, -2, 0, 6, 3, 0, 0, -2, 5], [10, 4, -1, -2, 4, 4, 1, -2, 1, -2, 6, 1, -1, 4, 3, 3, -1], [0, 0, -1, 0, 2, 0, 1, 2, 3, -2, -2, 1, 3, -2, 1, -3, 1], [2, 3, -3, 1, -4, 1, 4, 0, 2, 0, 2, 1, 1, 4, 0, -2, 0], [-2, 0, -3, -1, 0, 0, 2, -2, 1, 2, 0, 1, 0, 2, 1, -2, -1], [7, 0, -1, 1, -7, -2, 0, 1, 1, -2, 0, 5, 2, 2, 0, -4, 0], [0, 2, 0, 0, 2, -4, 0, 3, -1, 2, -2, -4, -2, 0, -1, 2, -2], [4, 0, 0, 1, -2, 1, -1, -1, -3, 3, -3, -3, -2, 0, 0, 0, -1], [2, -3, -1, 0, 4, 3, -1, -1, 3, 1, 5, 1, -1, 4, 4, 1, 0], [-6, 0, 2, 0, 2, 3, 2, 1, 1, -1, -1, 4, 2, -2, 2, 1, 3], [3, -1, 1, 0, 2, 1, -2, 1, 0, 3, -3, -5, 0, 0, 1, 2, -1], [5, 3, 0, -1, -6, -3, -2, -2, -4, 5, 1, -1, -1, 2, -1, 2, 0], [9, 3, 0, -1, -4, -9, -2, 0, -4, 1, -3, -2, -2, 0, -3, 2, -2], [8, 0, 0, 0, -4, 3, 4, -1, 5, -3, 1, 4, 2, 2, 0, -3, 1], [2, 4, 1, 2, -5, -6, 3, -1, -4, 2, -4, -3, -1, -2, -2, 1, -3], [2, 0, 4, -1, 2, -2, -3, 3, -1, 0, -6, -1, 0, -4, -5, 1, 0], [-2, -2, 2, -1, 0, 0, 1, 2, 0, -2, -2, 4, 3, -4, -2, -3, 2], [-7, -7, -1, 2, 0, 4, 3, -1, 0, -2, -4, 1, -1, 2, 2, -5, 1], [8, 4, -3, -1, -6, -2, 3, 1, 2, -2, 0, -3, -1, 2, 0, 1, -3], [-6, -4, -2, 6, -8, -1, -2, 0, -4, -1, -5, -2, -2, -2, 1, -5, 3], [5, 7, -5, 5, 0, -2, 1, 3, 2, 2, -2, -2, 0, -6, 2, 0, -2], [-3, -1, 2, 0, -2, 2, -1, 1, -1, -2, 0, -1, -2, -2, 1, 2, 1], [-3, -5, 3, -2, 0, 1, -4, -3, 0, 1, -1, 5, 2, 0, -3, 0, 3], [1, 2, -2, 3, -6, -2, 2, 2, 3, -4, -2, 0, 0, 4, -1, -2, -1], [8, 4, 2, -3, 2, -3, -1, -1, -3, 1, 1, 0, 1, 2, -2, 2, -1], [-3, 1, 0, 3, 4, 1, 3, 4, 4, -3, 1, 2, 4, -4, 3, -2, 0], [2, 1, -2, 6, 2, 4, 2, 4, 5, -4, 2, 4, 0, -2, 7, -3, 2], [-2, -1, 2, 0, 4, -1, -3, 4, 3, -1, -3, 1, 1, -4, 2, 0, 3], [-2, 3, -1, 1, 0, 4, 1, 0, 3, 0, -1, -3, 1, -4, 1, 0, 1], [9, 10, -1, 1, 0, -7, 1, 2, 2, 1, 3, 1, -1, -2, -1, 4, -1], [4, 2, -1, 2, 7, 5, 4, 5, 8, -1, 5, 1, 0, -2, 6, -1, -2], [11, 1, -2, 3, 1, 4, 9, 0, 2, -2, 0, -3, -2, 2, 5, -2, -3], [-6, -2, 0, 3, 3, 5, -2, 1, 5, -5, -1, 2, 0, 1, 3, -4, 3], [-4, 0, 1, 2, 0, 2, 1, 0, 3, -2, -2, 0, 0, 4, 3, -1, -1], [8, 3, -3, 1, -8, -9, -2, -5, -7, 5, -3, 0, 1, -2, -6, -2, 0], [8, -2, 2, -4, 0, -1, -5, -5, -5, 3, 3, 2, -1, 2, -4, 3, 0], [0, -4, 1, 0, -2, -3, -6, -2, -5, 5, -5, 1, 2, -4, -6, -2, 3], [-6, -2, 4, 0, 4, 1, -3, 1, 5, -3, 1, 8, 5, -2, 0, 1, 0], [-15, 0, 0, -5, 0, 1, -3, -6, -7, 3, 5, -1, -1, 2, -4, 3, 4], [-17, -11, 4, -2, 1, 4, -6, -5, -5, -2, 0, 7, 1, 1, -2, 0, 5], [-5, -2, 6, -1, 4, 5, -4, 1, 6, -3, 3, 9, 3, 0, -1, 1, 2], [7, 0, -1, 0, 0, -8, -6, 1, -3, 4, -4, -1, 4, -2, -3, -2, 0], [-2, -3, 1, -3, 8, 4, -1, -3, -1, 6, -2, -4, 4, -2, -5, 3, 0], [-3, 0, 1, -1, 4, 0, -1, 2, -4, 0, 2, -3, -1, -4, 0, 1, 2], [-11, -5, 2, 2, -9, 2, -3, -5, -1, -2, 0, 7, 2, 4, -1, -2, 3], [1, 0, 0, 4, -3, -6, -2, 4, -1, 3, -8, -5, 1, -4, -2, -3, -3], [-2, 1, 1, 2, -2, 1, 5, 3, 1, -1, -5, -5, 1, -4, 0, -1, -2], [-1, 0, 1, -4, 4, 4, 0, -1, -1, 2, -2, -5, 0, -4, -3, 4, 2], [1, 3, -4, 1, 0, 0, 8, -3, 3, 2, 2, 5, 3, 4, -3, -1, -1], [8, 2, -2, 1, 0, -3, -1, 3, -1, 1, -5, -4, -1, -4, -2, 0, -3], [0, 3, -2, 2, 4, -2, -2, -2, 1, 0, 2, 2, 0, 0, 1, 3, -2], [-1, 1, 2, -2, 0, -3, -9, -8, -8, 5, -1, -4, -1, 2, -3, 5, 2], [0, 4, 3, -3, 10, 5, 4, 1, 4, -3, 5, 1, 0, -2, 1, 5, 0], [2, 6, -1, -2, 2, -3, 1, 4, 3, 3, 5, -3, 1, 0, -2, 4, -2], [5, 8, -5, 2, -11, -4, 7, 2, 0, 0, 2, -1, -3, 3, 1, -4, -2], [2, 0, 3, -5, 8, 6, -1, 0, 1, 0, 6, 3, 4, 0, -3, 0, 3], [-4, -5, 4, -4, -6, -6, -5, -2, -8, 2, -7, 2, 5, 0, -8, -2, 2], [16, 1, 5, -2, 2, -1, -3, -2, -2, 3, -5, -5, 1, 0, -3, 3, 0], [0, 1, -2, 3, -12, -3, 1, 2, -3, 3, -5, -4, -1, -5, 0, -3, 1], [-23, -1, 0, 9, 0, 5, 7, 6, 6, -7, 3, 11, 0, -2, 5, -5, 0], [11, 6, -2, -1, 0, -2, 0, 4, 1, -1, 3, -2, -1, 0, 4, 0, -3], [-2, -4, 1, -1, 2, -2, -4, -2, -3, 2, -6, -2, -3, 0, -3, 2, 1], [1, 5, -4, -3, -4, -1, 5, -6, -4, 3, 1, -5, -2, 2, 1, 1, -2], [3, 3, -1, -1, 1, 6, -1, -3, -2, 0, 2, -3, -1, 0, 2, 0, 2], [-6, 2, -6, 2, -4, -3, 3, 0, 0, -1, 1, 3, 4, -4, -3, -6, 1], [5, 1, -1, 5, -7, -1, -2, -1, 0, 1, -3, 0, -1, 3, 4, 1, 1], [-8, 1, 0, 2, -2, 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2], [-21, -14, 4, 3, -2, 12, -7, -2, 4, -10, 5, 20, 7, 2, 8, -7, 9], [-19, -8, 8, -4, 4, 3, -7, 0, -5, -3, 7, 5, -2, 4, 2, 8, 4], [13, -5, 2, 0, -6, -3, -4, 1, -5, 7, -9, 0, 1, 2, -4, -1, 2], [7, 2, 0, -5, 6, -6, 0, -3, -2, 2, 4, 1, -1, 6, -4, 0, -1], [-22, -9, -2, 4, 0, 9, -1, -6, 1, -15, 7, 13, 1, 4, 6, -6, 1], [-5, -2, 3, -2, -4, -4, -7, 0, -5, 2, -9, -1, -3, 2, -5, -1, 0], [-15, -3, 1, 9, -4, -2, -5, 7, 2, -6, 0, 2, 0, -6, -2, -2, 0], [9, 0, 2, 1, 1, -4, 4, 6, -1, 0, -6, -1, 1, -8, -3, -2, 1], [19, 0, 2, -6, 10, 2, -6, 0, -7, 6, 0, -13, -1, -2, 5, 7, 0], [0, -3, 8, 1, 7, 8, -4, -3, 4, 0, 2, 5, -3, 2, 3, 5, 6], [7, 3, -3, -9, 8, 2, -1, -8, -3, 8, 8, -4, 2, -1, -7, 2, -2], [-10, -10, 12, -3, 4, 5, -13, -3, -7, 1, -5, 5, 2, -4, -2, 0, 5], [7, 11, -5, 2, -6, -4, 4, -5, -2, 2, 8, -11, -10, 4, 2, 9, -4], [5, 1, 2, -7, 8, 9, 3, -4, 8, -3, 11, 5, 0, 10, 3, 5, -2], [-7, -4, 5, -9, -2, -7, -5, -1, -1, -1, 3, 6, 2, -4, -4, 4, 1], [-16, 0, 4, 3, 2, 0, 3, -3, 3, -4, 0, 14, 3, 0, 1, -7, 4], [14, 16, 2, 0, 2, -1, 4, -3, 1, -5, 7, -1, 3, 2, 2, 5, -3], [17, -1, -1, -6, 0, -17, -12, -6, -13, 9, -7, -7, -6, -2, -6, 6, 1], [5, 1, 1, 4, 2, 1, -14, -4, -5, 1, 1, 0, -1, 4, -2, 2, 3], [13, 3, -3, 3, -6, -12, 7, 1, -2, 4, -10, -2, 4, -6, -8, -2, -8], [4, 2, 2, 0, 8, 2, -2, -8, -8, 2, -6, -8, -4, 5, 6, 2, -4], [26, 5, -1, -11, 0, -13, -8, 0, -8, 9, 1, -6, 1, 4, -7, 8, -2], [7, 1, -3, 3, 12, 5, 5, 11, 6, -5, 3, 3, 3, -8, 7, -3, 1], [-20, -9, 3, -1, -3, 3, 2, -2, 0, -7, 1, 8, 1, 4, -3, -4, 6], [-5, 4, -6, 9, -4, 2, 6, -1, 2, -6, -4, 0, -6, 2, 6, -8, 1], [16, 1, -5, -8, 0, -4, 1, -1, -1, 4, -2, -7, 3, 6, -3, -2, -5], [-7, -8, 1, 0, 11, 11, 9, 4, 10, -5, 1, 7, -1, 6, 4, 0, 1], [26, 12, -10, 4, -4, -1, 12, 4, 8, -1, 9, -4, -2, 4, 5, -3, -5], [9, 5, -1, -1, -4, 0, 13, 3, 2, 2, -2, -10, -4, -2, -2, 2, -10], [30, 3, -3, 3, -6, -4, -2, 2, 0, 2, -2, 2, -5, 8, -2, 1, -5], [7, 5, -5, -5, -2, -7, -3, -2, -5, 3, -1, -2, 6, 2, -8, 1, -1], [0, 3, 6, -1, 16, 5, -1, 1, -4, -3, 5, -4, -5, -2, 1, 7, 3], [8, 13, -9, 2, 0, -8, 12, 7, 2, 8, -1, -11, -2, -6, 0, -1, -1], [10, 8, 4, -2, -1, -2, 0, 10, 0, -2, -2, -4, -4, 0, 2, -2, 0], [-13, 3, -7, 8, -6, -5, -4, 0, -3, -3, -3, 10, 9, -4, -4, -10, 5], [11, 3, -1, 3, 2, -5, 1, 9, 4, -3, -1, 2, -2, -8, -1, 1, -3], [-3, 2, 3, -3, 15, 6, -6, 1, 5, 4, 4, -4, -1, -1, 0, 5, -4], [-9, -9, -2, 4, -5, 6, 0, 5, 5, -2, -10, -6, 0, -6, 0, -1, 3], [24, 8, -4, 1, 4, -4, -5, -5, -1, 10, 6, -6, -5, 8, -1, 3, -2], [0, 0, -5, 2, 4, 9, 15, 7, 10, -9, 7, -1, 3, 0, 11, -2, -6], [0, 0, 0, 4, 2, 2, -3, 8, 5, 0, -5, -3, -4, -2, 9, -5, 2], [-6, -1, 4, -2, 6, -2, -8, -3, -6, -2, 6, 9, 5, 0, 0, -1, 2], [4, 5, -8, -2, 8, 1, 7, -4, 2, 1, 4, -6, -1, -6, -1, 3, -3], [-32, -10, 4, 2, 2, 9, -4, -3, -1, -7, -3, 8, 4, -6, 8, -3, 7], [9, -8, -1, -1, -2, 4, -6, -3, 1, -4, 4, 12, 2, 2, -1, -5, 2], [-3, 12, -3, -2, -16, -14, -1, 0, 4, 0, 4, 8, -1, 4, -10, 2, -2], [9, 8, -7, 9, 4, -2, 9, -3, 11, -4, 8, 12, 2, 14, 5, -3, -4], [7, 15, -12, 12, -2, -2, 12, 4, 6, 0, 4, -2, -4, 4, 6, -4, -8], [3, 6, -2, 6, -4, -3, 6, 5, 5, -1, 4, 14, 4, -2, -4, -3, 0], [-4, -2, 3, -4, 4, -3, -13, 1, -2, 3, -3, 11, 7, 2, -3, 0, 8], [-5, -2, -8, -3, -3, 7, 5, -2, 5, -3, 7, 5, 0, 4, 3, -4, 6], [23, 2, 1, -2, -2, -5, -2, 0, -6, 1, -5, -3, 2, 0, -5, -3, -3], [20, 6, -5, 5, 8, 3, 2, -4, 1, 1, 1, -2, -1, 4, 8, -1, -2], [-24, -5, 4, 12, -11, 6, 1, 8, 5, -8, -12, 10, 1, -6, 8, -12, 4], [-15, -7, -1, 4, -2, 0, 4, 1, -6, -4, 0, 10, -1, 2, 6, -7, 2], [-4, -1, -2, 6, -2, 2, 6, -3, -6, -4, 0, 3, -1, -2, 2, -1, 6], [-8, 0, -1, 3, 8, 9, 2, -2, 5, -3, 3, 3, -2, -5, 4, -3, 4], [6, 1, -2, -7, 12, -3, -1, -12, -11, 6, 4, -11, -5, 4, 1, 5, 3], [6, 5, -10, 7, -18, 3, 3, -5, -6, 1, 3, 3, 0, 4, 1, -9, -1], [3, -2, 6, -2, 12, -1, -11, -4, -7, 10, 0, -11, -1, 6, 1, 2, 0], [20, 11, -2, -8, 7, -7, 1, -3, -8, 8, 9, -7, -2, -2, -6, 8, -5], [24, 12, 2, -11, 4, -11, 9, 6, -4, 8, 4, -10, 0, -6, -8, 8, -7], [9, -9, 6, -8, 4, -5, -15, -10, -18, 17, -1, -10, -5, 2, -13, 9, 4], [-25, -16, 4, 5, 2, 0, -14, 3, 0, -6, -2, 9, -5, 0, 4, 0, 11], [-7, -2, 3, -7, 14, 14, 0, -11, 3, 2, 6, -10, -2, 6, 7, 9, 0], [18, 4, 9, -6, 2, -7, -14, 1, 0, 3, -5, -2, -1, -10, -7, 7, 4], [-5, 8, -7, 3, -7, -8, 6, 5, 3, 6, -2, -10, -5, 1, 2, -1, -10], [-9, 0, 1, -2, -11, -4, -13, -6, -4, 4, -2, 7, -1, 3, -3, 0, 6], [16, 1, 6, -2, 4, 2, -12, -6, -7, 12, -10, -9, -5, -5, -5, 9, 4], [16, 3, 2, -4, 5, -4, -2, 2, 3, -2, -4, -7, -5, -2, -1, 3, -6], [-17, -7, 0, -8, 11, 0, -13, -7, -1, 0, 6, 3, -2, 4, -3, 4, 5], [6, -9, 8, -3, -4, -1, -15, -2, 3, 0, -10, 5, 3, -4, -7, -1, 3], [5, -3, 2, 0, -10, -8, -4, -9, -9, 0, -4, 6, 3, 8, -9, -4, -3], [-15, -2, -2, 11, 8, 6, -2, 4, 11, 2, 2, 3, -1, -2, 5, -2, 1], [2, 2, -1, 4, -4, 0, -1, -9, 4, 0, -2, 13, 5, 6, -2, -5, 3], [-12, -8, -3, -1, -2, 1, 9, 0, 2, -13, 12, 18, 4, 4, -1, -4, 6], [-7, 9, 0, 8, 6, 6, 7, 6, 2, -6, -4, -4, -5, -6, 6, -4, 1], [17, 11, 0, 2, 2, -13, 0, 15, 3, 5, -7, -12, -5, -2, -2, 1, -6], [21, -4, 0, 2, -2, 6, -3, 8, -1, 0, -6, -10, -5, -8, 3, -1, 3], [17, 0, 3, 5, 1, -1, 2, 4, 2, 0, -1, -2, 1, -6, 0, 4, 5], [-2, -7, 7, -3, 8, 11, 2, -3, 5, -5, -5, 3, 8, -6, 0, 0, 6], [17, 3, -5, 0, 2, 3, 2, -7, 0, 9, 1, -22, -7, -3, 7, 4, -5], [5, 0, 6, -2, -4, 1, -5, 1, 2, -1, 9, 2, 1, 6, 5, 6, 6], [-20, -12, 0, 10, -8, 6, -8, 5, 3, -6, -10, 7, -1, 0, 5, -14, 10], [9, -14, 7, 0, 8, 1, -16, -8, -10, 5, -15, 4, -1, 2, -3, -7, 3], [25, 6, -5, 3, 0, -4, 3, 3, -3, 4, 4, -9, -3, -4, -1, 3, -4], [4, -6, 5, -9, 21, 10, -7, -3, -2, 5, 4, -1, -1, -2, 1, 10, 7], [0, -3, 6, 0, 12, 7, 0, 9, 6, -1, -5, -9, -3, -8, -3, 4, 1], [-7, -13, 0, -2, -4, -2, -9, 0, -12, 6, -2, 5, 0, 0, -4, -2, 3], [8, -10, 5, -4, -8, -9, -3, -8, -11, 5, -3, 12, 8, 6, -8, 2, 0], [-11, 4, -8, 12, 1, 3, 12, -2, 3, -10, -3, -1, 3, 2, 3, -4, -6], [2, 12, 1, 2, 6, -3, 12, 0, -1, 1, 7, -7, -4, 0, 6, 2, -9], [14, 2, 3, -6, 0, -13, -9, 0, -13, 9, -13, -16, 2, -4, -14, 6, -4], [-22, 1, 3, -10, 2, 5, 4, -3, 0, -7, 4, 5, 8, -4, -3, 0, 2], [10, 3, -1, -9, 8, -9, -4, 2, 2, 7, -5, -12, -3, -4, -3, 2, 0], [-7, -15, 5, -11, 1, 4, -3, -11, -8, -2, 8, 9, 3, 16, 2, -6, 8], [-7, -6, 5, -1, 4, 10, 7, -2, 0, -5, 3, 8, 3, 0, 3, 1, 4], [5, -12, 7, -6, -2, 7, -3, -2, 3, -3, -6, 3, 3, 0, 0, 0, 1], [-7, -14, -2, 7, -12, -1, -6, 0, -3, -7, -9, 11, 3, 2, 4, -5, 8], [-12, -2, -2, 6, -1, 12, 4, 6, 10, -16, 4, 8, 2, 0, 14, -2, 2], [-2, 13, -2, -7, 6, -12, 2, -11, -10, 12, 0, -16, -2, -6, -8, 7, -4], [-19, -10, -1, 0, -2, 3, 2, 0, 0, -13, 5, 7, -4, 10, 5, -5, 3], [9, 20, -8, 2, -10, -16, 8, 6, -3, 4, 2, -7, -3, -2, -5, 3, -4], [-23, -9, 2, 4, 8, 16, 8, -2, 0, -10, 10, 16, 4, 6, 6, -5, 3], [-13, -6, -5, -3, -6, 9, 8, -10, -2, -11, 13, 14, 3, 4, 3, -6, 2], [5, 3, 0, 6, 6, -6, -2, 3, 3, 6, 2, 0, -1, 0, -3, 0, -3], [39, 3, 4, -7, 2, 0, -4, -1, -1, 0, 4, -4, -2, 4, -1, 5, -3], [-12, -15, 6, 1, 6, 5, -13, -7, 0, -3, 5, 14, 3, 0, 1, 1, 11], [-10, -8, 1, -3, 8, 9, 12, -6, -3, -7, 11, 11, 6, 3, 0, -1, 0], [-5, -8, -1, -6, -10, -3, 1, -14, -5, 3, 2, 11, 11, 16, -6, -2, 5], [-3, -6, -2, 3, -4, 2, -4, 5, -2, 0, 8, -2, -4, -2, 0, -2, 3], [-6, -5, 0, 0, 5, 4, -3, 8, 3, 0, 2, 1, 4, -9, 2, -1, 2], [13, 10, -2, 3, 10, 7, 8, 6, 11, 1, 5, -6, -4, -2, 4, 3, -2], [-10, -2, 2, 7, 8, 5, -5, 0, 4, 1, -1, -4, -1, -6, 1, 0, 3], [-35, -4, 0, 5, -2, -3, -6, -4, -3, 5, -15, 2, 8, -2, -6, -5, 4], [22, 3, -2, -8, -4, -10, -6, -13, -8, 16, 0, 1, 9, 8, -14, 5, -2], [-8, 0, 6, 3, 6, 4, -2, -3, -3, -2, 6, 16, 5, -6, -1, 1, 6], [8, 8, 2, -11, 0, -13, -3, -6, -4, 6, -2, 8, 6, 6, -14, 2, -1], [-16, 0, 4, 3, -1, -10, -9, -7, -11, 4, 2, 5, 2, 2, -1, 5, 2], [-1, 9, -5, 3, -8, 2, 9, 4, 7, -4, 0, -2, -4, -2, -1, -6, 0], [2, -7, 7, -8, 10, 1, -11, -2, -2, 3, 1, 5, -1, 10, 7, 1, 6], [21, 10, 1, 4, -3, -5, -7, 4, 4, 1, -3, 0, 0, 3, 6, 3, -1], [-10, -8, 5, 0, -8, 10, -9, 2, 3, 0, -4, 13, 5, 0, 3, -3, 9], [3, 15, -11, 2, 18, 10, 18, -2, 11, 0, 18, 2, -1, 2, 9, 1, -8], [24, 0, -4, 8, -1, 5, 10, 8, 6, 7, -7, -19, -11, -2, 5, 1, -7], [-26, -7, 4, 7, 18, 11, -5, -3, -2, -4, 4, 4, -6, 2, 8, 5, 9], [-6, 4, -5, 8, -4, -2, 9, 9, 8, 2, 2, -1, -3, 4, 4, -1, 1], [3, 8, -3, 8, -15, -13, 4, 2, -4, -1, -5, -2, -3, -6, -9, -1, -3], [11, 10, -5, 0, -12, -17, -2, -4, -10, 10, -8, -7, -3, -2, -6, 5, -1], [-12, -4, -5, 4, -12, 3, 12, -6, 7, -9, 1, 17, 6, 2, -8, -7, 4], [-6, -7, 2, -1, -8, 5, 3, 7, 2, -3, 3, -3, -2, 2, 5, 1, 1], [13, 14, -3, 4, -12, -1, 2, -4, 4, 1, 1, -4, -1, 8, 1, 3, -5], [-22, -8, -2, 1, 6, 8, -1, -11, -1, -2, 6, 9, -4, 12, 5, 1, 8], [9, -19, -1, 2, -13, 4, 5, -6, -3, -2, -10, -2, -4, -1, 0, -9, -2], [-15, -11, -2, -1, -2, -4, -4, -3, -5, -2, -2, -2, -2, 6, -1, -7, -1], [-9, 3, 11, -5, 4, -4, -5, -4, 3, 3, 1, 11, 6, 6, -8, 6, 2], [-12, 8, 5, 5, 8, 5, 6, 4, 5, -7, 7, 5, 2, -2, 0, 1, -2], [-17, -3, -3, -6, -8, 2, -3, 0, 5, -2, 2, 14, 4, 4, -5, -6, 8], [-5, -3, -1, -1, 3, 1, 0, 3, 2, 1, 3, -1, 4, 4, 6, 2, -1], [12, 21, -9, 5, 2, -10, 6, 10, 6, 10, -4, -18, -3, -10, 6, 5, -7], [-11, 0, -5, 7, -17, -14, -12, -1, -7, 4, 0, 18, 3, 3, -4, -7, 8], [17, -1, 8, -9, 4, -4, -2, 3, -3, 6, -4, -2, 4, -8, -9, 5, 1], [11, 13, -4, 1, 0, -9, -1, 12, -3, -1, 2, -9, -8, -10, 2, 7, -4], [-2, 5, -2, 10, -10, 5, 10, -2, -3, -3, -5, -12, -8, -2, 8, -4, -3], [12, 13, 8, 1, 6, -6, -6, 1, 4, -2, 4, 7, 0, -8, -8, 10, 2], [-8, -3, -8, -2, 4, -6, -2, -2, 3, 1, 13, 4, -1, 10, 4, 1, -2], [12, -3, -2, 2, -8, 3, 3, 12, 1, -9, -5, -2, -3, -10, 4, -3, -3], [-11, 2, 5, -9, 0, 5, -3, -1, -1, 3, 13, 3, 5, 6, -2, 4, 5], [0, -14, 6, -4, 0, 5, 1, -2, -2, -1, -1, -6, -6, 6, 3, -7, -1], [10, 0, 7, -8, 9, 4, -10, -5, 0, 0, 4, -4, -1, 2, 3, 10, -5], [-5, 5, -2, 5, 9, 2, -7, -2, -4, 2, -6, -6, -1, -5, 3, 1, 1], [0, -1, 1, -4, 14, 11, 5, -7, -5, 1, 9, -9, -3, 8, 8, 1, -2], [16, 13, -14, 1, -6, -2, 17, 2, 3, -2, -2, -17, -4, 6, 1, 0, -10], [-23, -3, -10, 9, -10, -2, -2, 1, -3, -4, -6, 1, -7, 1, 5, -5, -3], [-16, 1, 0, -2, 8, 5, 4, 14, 11, -7, 9, 2, -2, 2, 6, -2, -3], [-24, -7, 2, 1, -17, -7, -6, -7, -8, 3, -9, 0, -2, 6, 0, 0, 1], [6, 9, -9, 11, -22, -12, 5, 3, -3, -2, -10, 0, 0, -8, 1, -4, -1], [6, 9, -5, 1, 6, 9, 12, -6, 2, -3, 9, -9, 2, 4, 7, 4, 0], [17, -2, 3, 0, -7, -12, -14, 6, -4, 14, -16, 1, 10, -4, -14, -2, 2], [-5, -4, 3, -12, 2, 16, -2, -3, 7, -8, 10, 17, 2, 18, 3, -6, 4], [9, -5, 0, 3, 10, 1, 6, 1, 7, 1, 5, -3, 1, 0, -2, 6, -4], [-9, -13, 7, -7, -2, 9, 8, -2, 5, -5, -3, 3, 6, 2, -2, -1, 2], [-4, -2, 6, -2, 2, -6, -10, -5, -7, 2, -4, 6, 10, 4, -5, 2, 4], [-33, -21, 9, -3, 6, 2, -3, 1, -8, 0, -4, 8, 10, -6, -6, 0, 2], [-33, -13, -1, 5, 0, 3, -13, -3, -4, 1, -13, 3, 7, -10, -1, -5, 7], [-17, -6, 5, -4, 6, 7, -8, 12, 8, -9, 1, 2, 1, 2, 9, -5, 3], [-6, 4, 2, -5, 10, -14, 5, -3, -3, 6, 0, 0, 3, 2, -3, 3, -4], [4, 5, 5, 3, 6, -3, -4, 15, 3, 2, -10, -8, 0, -2, 7, 2, -4], [-1, 6, -6, 12, -12, 1, 14, 11, 9, -5, 2, 2, 0, -6, 0, -3, 0], [-24, -15, 9, -4, 8, 5, -21, 0, 0, -3, -3, 5, -1, 2, -1, 7, 6], [-8, -8, 5, -9, 14, -2, -8, 4, -5, 10, 0, -4, 3, -10, -5, 2, 7], [-17, -16, 6, -11, -3, -7, -9, -4, -13, 3, -3, -1, -6, -4, -9, 2, 2], [11, 3, 6, -2, 6, -7, 1, 4, -4, 5, -5, -6, -5, -2, -7, 7, -6], [-15, -8, 2, 12, -22, 10, 16, 12, 7, -14, -4, 11, 1, -10, 7, -9, 4], [-20, -5, 10, 0, 0, 16, 6, 1, 14, -10, 8, 15, 1, -2, 10, 3, 8], [4, 4, -8, 0, 2, 8, -2, -1, 7, 0, 2, -3, 3, 4, 5, -2, -2], [-15, 4, 4, 6, 1, 5, 6, 8, 5, -8, -5, -1, -3, -2, 11, -8, 4], [10, -11, 2, -5, 7, 3, 4, 2, 1, 2, -5, -3, 5, -8, 1, -9, -1], [7, 1, -6, 9, -16, -8, 10, 3, 3, 4, -4, -13, -9, 2, 3, 1, -3], [-23, -10, 0, -3, -15, 0, -7, -7, -2, 1, -6, 1, 2, -2, -5, 0, 7], [-4, 5, -6, 3, 8, 18, 11, -4, 11, -14, 10, 7, 0, 2, 7, 2, 2], [-5, 0, 0, -5, 20, 22, 10, -3, 14, -8, 20, 4, -6, 16, 20, 6, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4016_2_a_k();".
// 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_4016_2_a_k(:prec:=17)
    chi := MakeCharacter_4016_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_4016_2_a_k();".
// 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_4016_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3164, 10587, 5377, -41362, 3260, 63091, -10424, -48174, 6280, 20039, -1445, -4602, 142, 582, -5, -38, 0, 1]>],Snew);
    return Vf;
end function;