
// Make newform 4000.2.f.d in Magma, downloaded from the LMFDB on 16 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4000_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4000_f_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_4000_2_f_d();".
// 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_4000_2_f_d();".
// 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 := [1024, -1536, 1536, -640, -192, 640, -448, 192, 64, -120, 116, -60, 16, 24, -28, 20, -3, -5, 6, -3, 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, 0, 0, 0], [10752, -7168, 5120, 1856, -3200, 2944, -384, -224, 944, -404, 388, -16, -88, 200, -90, 23, 43, -32, 11, 1], [-2048, 3328, -1408, -832, -64, -768, 480, -128, -184, 212, -236, -16, -16, -44, 20, -19, -5, 6, -11, 1], [7168, -8960, 2432, 320, -3904, 1920, -672, -512, 536, -484, 124, -112, -160, 124, -100, -1, 25, -30, 7, -5], [-9728, 14336, -15104, 3904, 3456, -6720, 3648, -832, -1232, 1356, -860, 384, -24, -256, 278, -145, 3, 80, -61, 17], [8704, -7168, 14080, -3392, -768, 5184, -3840, 1568, 880, -684, 988, -288, 24, 152, -238, 185, -3, -40, 53, -25], [2560, -512, 3072, -2240, 0, 704, -1152, 640, 128, -276, 212, -192, -8, -24, -62, 47, -17, -12, 11, -11], [32256, -34304, 15872, -1344, -12288, 10048, -3136, 0, 1920, -2268, 1180, -672, -360, 600, -394, 101, 69, -132, 41, -25], [1024, 4992, -4608, 3072, 640, -1776, 1776, -144, -60, 424, -260, 220, -20, -30, 119, -52, 11, 25, -18, 9], [3072, -8448, 6656, -6080, -2176, 2656, -3296, 608, 184, -964, 316, -488, -32, 52, -188, 95, -31, -46, 23, -21], [10752, -15616, 11136, -3008, -4864, 5760, -2944, 368, 952, -1156, 732, -416, -32, 256, -268, 111, 17, -70, 39, -17], [-19968, 19712, -9216, 448, 8384, -6240, 2528, 272, -1208, 1628, -756, 424, 280, -368, 272, -57, -43, 86, -33, 19], [-26112, 37888, -19456, 7488, 13824, -10496, 6464, 192, -1584, 3084, -1148, 1200, 360, -528, 590, -193, -21, 168, -61, 41], [-8448, 12800, -7360, 704, 4960, -4304, 1920, 256, -760, 940, -452, 260, 104, -220, 192, -49, -35, 58, -25, 7], [-4608, 11776, -11520, 3776, 2816, -5248, 3328, -480, -576, 996, -676, 432, -8, -176, 262, -107, 5, 60, -39, 15], [-24064, 47616, -33792, 14016, 15872, -16704, 11136, -1152, -2400, 3668, -2164, 1568, 200, -696, 886, -335, -23, 212, -123, 51], [-13824, 23552, -20608, 4928, 6976, -9440, 5376, -608, -1504, 1852, -1340, 584, 8, -400, 450, -181, -29, 100, -73, 25], [13824, 31232, -26624, 25408, 3200, -10560, 13824, -2272, 448, 3164, -1500, 1664, -280, 32, 762, -405, 187, 148, -121, 65], [-21504, 35328, -27904, 10432, 11264, -12224, 8704, -1216, -1760, 2900, -1652, 1216, 136, -488, 646, -271, -7, 164, -91, 51], [-77312, 100864, -90112, 15296, 32768, -42816, 21504, -2624, -8320, 7268, -5924, 2528, 296, -1880, 1814, -811, -203, 444, -311, 103]];
    Rf_basisdens := [1, 3072, 1536, 1536, 3072, 3072, 1024, 3072, 768, 1536, 1536, 1536, 3072, 768, 1024, 3072, 1536, 3072, 1536, 3072];
    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_4000_f();"
function MakeCharacter_4000_f()
    N := 4000;
    order := 2;
    char_gens := [2751, 2501, 1377];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4000_f_Hecke();"
function MakeCharacter_4000_f_Hecke(Kf)
    N := 4000;
    order := 2;
    char_gens := [2751, 2501, 1377];
    char_values := [[1, 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, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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, 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, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 1, 0, 0, 1, 0, -1, 0], [0, 0, 1, 0, 0, 0, -1, 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, 0, 0, -1, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1], [0, 0, -1, -1, -1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1], [-1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, -1, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 1, 0, -3, -1, 1, -1, 0, 0, 0, 0, 0, -1, 0], [-3, 0, 0, 0, 0, 0, 0, 1, 0, -1, -1, 0, -1, -1, 0, 0, 1, 0, -1, 0], [0, 0, 1, 1, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0, -1, 0, 0], [-1, -2, 0, 0, 0, 0, 0, -1, 0, 2, 3, -1, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1], [0, 0, 0, -2, 0, -2, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, 3, -1, 0, 0, 1, 0, 1, 0], [-2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0], [3, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 1, 0], [3, -1, 0, 0, 0, 0, 0, -2, 0, 1, 1, 0, 1, -1, 0, 0, 0, 0, -1, 0], [-1, 3, 0, 0, 0, 0, 0, 1, 0, 1, 1, -1, 1, -1, 0, 0, 1, 0, -1, 0], [0, 0, -1, 0, -1, 1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2], [0, 0, 1, -2, -2, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0], [0, 0, -2, 0, 1, 0, -1, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 1], [-1, 5, 0, 0, 0, 0, 0, -2, 0, -2, 1, -2, -1, 1, 0, 0, 2, 0, -1, 0], [0, 0, 0, 2, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, -2, 0, 0, -1, 0, 0], [0, 0, 1, 0, -3, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2], [0, 0, 1, -3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, -2], [0, 0, 0, 0, -1, -1, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1], [0, 0, 0, 0, 3, -1, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, -4, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1], [0, 0, -3, -1, 0, -2, -1, 0, 1, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 1], [7, 0, 0, 0, 0, 0, 0, 1, 0, -2, -3, 4, -1, -1, 0, 0, -1, 0, -1, 0], [3, -1, 0, 0, 0, 0, 0, -2, 0, -1, 0, -2, 1, 1, 0, 0, -1, 0, 0, 0], [8, 2, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, -1, -1, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, -2], [4, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 2, -2, -1, 0, 0, -1, 0, 2, 0], [0, 0, 0, -2, -2, -1, 0, 0, -2, 0, 0, 0, 0, 0, 2, -2, 0, 2, 0, -2], [0, 0, 2, -1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1, 0, -2, 0, 1], [6, 2, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0], [0, 0, -1, 0, 1, -1, 1, 0, 3, 0, 0, 0, 0, 0, -2, 0, 0, 1, 0, -2], [3, 3, 0, 0, 0, 0, 0, -3, 0, 2, -1, 1, -2, 0, 0, 0, 0, 0, 1, 0], [3, 0, 0, 0, 0, 0, 0, -5, 0, -3, 0, -2, -3, 3, 0, 0, 1, 0, -3, 0], [0, 0, 1, -1, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 2, 4, -1, 3, 0, 1, 0, 0, 0, 0, 0, -2, 2, 0, 2, 0, 0], [0, -6, 0, 0, 0, 0, 0, -2, 0, -1, 1, 2, 0, 2, 0, 0, 2, 0, -2, 0], [0, 0, -3, 3, 1, -3, 1, 0, 2, 0, 0, 0, 0, 0, -3, -1, 0, 2, 0, -1], [0, 0, -3, -1, 1, 3, -2, 0, 1, 0, 0, 0, 0, 0, -1, -1, 0, 1, 0, -1], [-4, 1, 0, 0, 0, 0, 0, -3, 0, -3, 6, -2, 0, 2, 0, 0, 1, 0, -2, 0], [4, -2, 0, 0, 0, 0, 0, 2, 0, -1, -8, 4, -1, 1, 0, 0, -3, 0, 0, 0], [0, 0, -5, -1, 1, 2, 1, 0, 4, 0, 0, 0, 0, 0, -1, -4, 0, 3, 0, -2], [0, 0, -3, 1, 3, -1, 0, 0, 3, 0, 0, 0, 0, 0, -1, -1, 0, 1, 0, -1], [0, 0, -1, 1, -1, 1, -1, 0, 4, 0, 0, 0, 0, 0, -3, -2, 0, 1, 0, -2], [0, 0, -2, 3, -4, 2, -1, 0, 0, 0, 0, 0, 0, 0, -3, -3, 0, 0, 0, 3], [-3, 4, 0, 0, 0, 0, 0, 1, 0, 6, 4, 0, 3, -1, 0, 0, 1, 0, -1, 0], [3, -5, 0, 0, 0, 0, 0, 2, 0, -2, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0], [-2, 3, 0, 0, 0, 0, 0, -1, 0, -3, 7, -1, 1, -2, 0, 0, 2, 0, -1, 0], [-8, 5, 0, 0, 0, 0, 0, -3, 0, -2, 3, 0, 0, 0, 0, 0, 3, 0, 2, 0], [4, 2, 0, 0, 0, 0, 0, 0, 0, -1, -4, 2, 3, -1, 0, 0, 1, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, -3, -6, 2, -2, -2, 0, 0, -2, 0, 0, 0], [-2, 3, 0, 0, 0, 0, 0, -3, 0, 7, 8, -4, 6, 2, 0, 0, 3, 0, 0, 0], [0, 0, -2, -4, 0, -4, 0, 0, -2, 0, 0, 0, 0, 0, 2, -4, 0, 0, 0, -2], [7, 1, 0, 0, 0, 0, 0, -3, 0, 1, 1, 1, 1, -1, 0, 0, 3, 0, -1, 0], [0, 0, 3, -1, 0, 2, -2, 0, -4, 0, 0, 0, 0, 0, 1, -2, 0, -2, 0, 1], [0, 0, 3, 2, 1, 1, 2, 0, 4, 0, 0, 0, 0, 0, -2, 2, 0, 2, 0, 0], [3, 4, 0, 0, 0, 0, 0, -3, 0, 1, -1, -2, -1, 3, 0, 0, -1, 0, 1, 0], [0, 0, -1, -3, 2, 4, 3, 0, -1, 0, 0, 0, 0, 0, 3, -1, 0, 2, 0, -1], [0, 0, 0, 1, 6, 4, 2, 0, 3, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, -1], [-2, -3, 0, 0, 0, 0, 0, 7, 0, -3, -7, 2, 2, -2, 0, 0, -1, 0, 0, 0], [0, 0, -3, -1, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, -2, -4, 0, 0, 0, 1], [1, -5, 0, 0, 0, 0, 0, 3, 0, -1, -9, 4, -7, 0, 0, 0, -4, 0, 2, 0], [0, 0, -2, 2, 3, 3, 1, 0, 5, 0, 0, 0, 0, 0, -2, 2, 0, 1, 0, -1], [0, 0, -2, 2, -2, -1, 0, 0, -2, 0, 0, 0, 0, 0, 0, -4, 0, 2, 0, 2], [0, 0, 1, -5, 2, 2, -3, 0, 1, 0, 0, 0, 0, 0, -1, 1, 0, -2, 0, -3], [-4, 6, 0, 0, 0, 0, 0, -1, 0, -8, -1, -2, -4, -1, 0, 0, -1, 0, 2, 0], [5, 1, 0, 0, 0, 0, 0, -2, 0, 0, -2, -1, 0, -1, 0, 0, 3, 0, 3, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 7, 4, 0, 1, 1, 0, 0, -3, 0, 2, 0], [0, 0, 2, -4, 1, -2, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, -3, 0, 1], [0, 0, 1, 4, 1, 1, -3, 0, 1, 0, 0, 0, 0, 0, -4, 2, 0, -1, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 2, 0, 4, 0, 0, 0, -4, 0, 0, 0], [0, 0, -2, 2, 1, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 1, 0, 2], [3, -4, 0, 0, 0, 0, 0, 1, 0, 5, 3, 0, 3, 3, 0, 0, -1, 0, 3, 0], [-12, 10, 0, 0, 0, 0, 0, -2, 0, -4, 3, -2, 0, 2, 0, 0, 0, 0, -4, 0], [3, 1, 0, 0, 0, 0, 0, 3, 0, -7, -7, 2, -1, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -2, -4, -4, 0, -7, 0, 0, 0, 0, 0, -1, 1, 0, -1, 0, 1], [0, 0, -4, 3, -1, 1, -4, 0, -2, 0, 0, 0, 0, 0, -3, -3, 0, -1, 0, 5], [0, 0, -1, -5, -6, -1, -2, 0, -2, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, -1], [-1, -6, 0, 0, 0, 0, 0, -1, 0, -5, 0, 2, -1, 3, 0, 0, -1, 0, -1, 0], [6, -7, 0, 0, 0, 0, 0, -3, 0, 3, -1, 0, 4, 2, 0, 0, 1, 0, -2, 0], [0, 0, 2, -4, 3, 0, 2, 0, -2, 0, 0, 0, 0, 0, 2, 2, 0, -1, 0, -3], [0, 0, 0, 6, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, -4, 0, 0, 2, 0, 2], [0, 0, -1, -1, 0, -1, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 1, 5, -1, 2, 3, 0, 3, 0, 0, 0, 0, 0, 2, 2, 0, 1, 0, 2], [0, 0, 4, -3, -6, -2, -2, 0, -1, 0, 0, 0, 0, 0, -1, 3, 0, -1, 0, -1], [-3, 4, 0, 0, 0, 0, 0, -5, 0, -2, 7, -3, -2, 5, 0, 0, 1, 0, -1, 0], [-11, 1, 0, 0, 0, 0, 0, -2, 0, 7, 5, 0, -1, -3, 0, 0, -4, 0, 5, 0], [0, 0, -2, 3, 5, 3, 3, 0, 7, 0, 0, 0, 0, 0, -3, 1, 0, 0, 0, -3], [-11, 0, 0, 0, 0, 0, 0, -1, 0, -4, -7, 0, -3, -1, 0, 0, -5, 0, -1, 0], [-12, 9, 0, 0, 0, 0, 0, 1, 0, -5, -5, -3, 3, -2, 0, 0, 4, 0, -1, 0], [10, -10, 0, 0, 0, 0, 0, 2, 0, -2, 2, -4, 4, 0, 0, 0, 4, 0, -4, 0], [-4, -3, 0, 0, 0, 0, 0, 3, 0, -1, 3, 3, -5, 2, 0, 0, -2, 0, -3, 0], [0, 0, 3, 1, 4, 2, 1, 0, 7, 0, 0, 0, 0, 0, 1, 4, 0, -2, 0, -3], [0, 0, 7, -2, 0, -2, 6, 0, 1, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 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-5, -3, -10, 0, -8, 0, 0, 0, 0, 0, 2, 0, 0, -4, 0, 6], [0, 0, 4, -1, -4, 4, -8, 0, 1, 0, 0, 0, 0, 0, 1, -1, 0, -5, 0, 7], [4, 1, 0, 0, 0, 0, 0, 1, 0, -8, -13, 4, 0, -6, 0, 0, 5, 0, 0, 0], [0, 0, 1, -1, -1, 3, 0, 0, -8, 0, 0, 0, 0, 0, 6, -4, 0, -1, 0, 0], [0, 0, 8, 2, -17, 3, -5, 0, -6, 0, 0, 0, 0, 0, 6, -4, 0, -5, 0, 5], [8, 19, 0, 0, 0, 0, 0, -5, 0, -3, -3, 2, 3, -1, 0, 0, 5, 0, 2, 0], [0, 0, -3, 11, 2, -8, 7, 0, 6, 0, 0, 0, 0, 0, -5, 4, 0, -2, 0, 1], [13, 3, 0, 0, 0, 0, 0, 2, 0, -17, -7, 6, -7, 5, 0, 0, -4, 0, -1, 0], [6, -1, 0, 0, 0, 0, 0, -2, 0, 2, -2, -4, -2, 8, 0, 0, 8, 0, -8, 0], [-20, 9, 0, 0, 0, 0, 0, 7, 0, 2, -2, 2, -6, -4, 0, 0, -5, 0, 6, 0], [0, 0, 2, -2, 4, 0, 4, 0, 4, 0, 0, 0, 0, 0, 5, 4, 0, -4, 0, -4], [26, -5, 0, 0, 0, 0, 0, 5, 0, 5, -5, 4, -3, -4, 0, 0, -6, 0, 6, 0], [-11, 8, 0, 0, 0, 0, 0, -6, 0, -11, -12, -1, 1, -3, 0, 0, 1, 0, -7, 0], [0, 0, -2, 2, 3, 6, -2, 0, 1, 0, 0, 0, 0, 0, -6, 2, 0, 1, 0, -1], [-12, 19, 0, 0, 0, 0, 0, -3, 0, -8, -6, -2, 2, -6, 0, 0, 5, 0, -2, 0], [12, -9, 0, 0, 0, 0, 0, -1, 0, -4, -1, 2, -4, 6, 0, 0, -3, 0, -4, 0], [0, 0, 1, 1, -2, -3, -1, 0, 4, 0, 0, 0, 0, 0, -4, -2, 0, -2, 0, 9], [0, 0, 10, 0, 1, -8, -1, 0, 2, 0, 0, 0, 0, 0, -3, 6, 0, -11, 0, 7], [0, 0, -2, 9, -7, 1, 4, 0, 4, 0, 0, 0, 0, 0, -1, 3, 0, -1, 0, -1], [0, 0, 0, 2, 1, 7, 6, 0, 7, 0, 0, 0, 0, 0, 2, 2, 0, -2, 0, 0], [-5, -11, 0, 0, 0, 0, 0, 2, 0, -4, 6, 0, 3, -1, 0, 0, 2, 0, -7, 0], [0, 0, -10, 2, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -8, -4, 0, 0, 0, 4], [0, 0, 2, -4, -3, 10, 0, 0, 6, 0, 0, 0, 0, 0, 0, -4, 0, 7, 0, -7], [0, 0, -4, 4, 0, -6, 3, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2], [0, 0, 2, -2, -1, 2, 1, 0, 2, 0, 0, 0, 0, 0, 4, -2, 0, 1, 0, -3], [20, -1, 0, 0, 0, 0, 0, -1, 0, -5, 9, -2, 9, -2, 0, 0, 2, 0, 8, 0], [-5, 18, 0, 0, 0, 0, 0, 1, 0, -4, 5, 5, 10, -8, 0, 0, 4, 0, -1, 0], [0, 0, -1, 3, -7, 5, -7, 0, -1, 0, 0, 0, 0, 0, -1, -4, 0, -1, 0, 6], [-4, 2, 0, 0, 0, 0, 0, 0, 0, -9, -4, 4, -3, -5, 0, 0, 1, 0, -2, 0], [-4, 12, 0, 0, 0, 0, 0, 9, 0, 2, -5, 3, -2, 1, 0, 0, 3, 0, -3, 0], [0, 0, -1, -5, -6, 5, -7, 0, -1, 0, 0, 0, 0, 0, -3, -2, 0, -2, 0, -3], [0, 0, 3, -12, 0, -2, -6, 0, -9, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -4], [17, -6, 0, 0, 0, 0, 0, 3, 0, -3, -5, 4, -9, -1, 0, 0, -1, 0, 3, 0], [-8, 2, 0, 0, 0, 0, 0, -4, 0, 4, -6, -7, 4, 2, 0, 0, 6, 0, 1, 0], [0, 0, -2, -4, 6, -2, -2, 0, -2, 0, 0, 0, 0, 0, 2, 8, 0, 0, 0, -8], [0, 0, 1, 1, -11, 1, -7, 0, -5, 0, 0, 0, 0, 0, 7, -2, 0, -5, 0, 0], [-5, -10, 0, 0, 0, 0, 0, -7, 0, 13, 3, -2, 9, 7, 0, 0, 3, 0, -3, 0], [0, 0, -4, 5, 6, -10, 3, 0, 2, 0, 0, 0, 0, 0, -7, 11, 0, 0, 0, -5], [4, -17, 0, 0, 0, 0, 0, 4, 0, 15, 14, 4, 5, 0, 0, 0, -2, 0, 2, 0], [0, 0, 3, 6, 0, 8, -1, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, -5, 0, 2], [0, 0, 3, -1, 10, -9, 7, 0, 4, 0, 0, 0, 0, 0, -4, 10, 0, -4, 0, -7], [0, 0, 3, -2, 2, -8, 3, 0, -2, 0, 0, 0, 0, 0, 4, 4, 0, -3, 0, -4], [13, -4, 0, 0, 0, 0, 0, 7, 0, 4, -13, -2, 4, 2, 0, 0, 0, 0, 0, 0], [-35, -9, 0, 0, 0, 0, 0, -6, 0, 9, 5, -8, 5, 7, 0, 0, 4, 0, 5, 0], [-17, 13, 0, 0, 0, 0, 0, -2, 0, -2, 16, -6, -2, 1, 0, 0, 9, 0, -4, 0], [0, 0, 15, -5, -6, -10, -7, 0, -5, 0, 0, 0, 0, 0, 5, 3, 0, -8, 0, -1], [-25, -6, 0, 0, 0, 0, 0, -3, 0, 1, 5, -6, -3, 5, 0, 0, 1, 0, -5, 0], [0, 0, -3, 0, 4, 0, 4, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, -6, 0, 0], [-9, 17, 0, 0, 0, 0, 0, -8, 0, 2, 1, -8, 1, 1, 0, 0, 6, 0, -1, 0], [0, 0, 6, 11, 2, -6, -6, 0, -1, 0, 0, 0, 0, 0, -9, 5, 0, -3, 0, 7], [0, 0, -5, 3, -2, 4, -4, 0, -2, 0, 0, 0, 0, 0, -4, -6, 0, 10, 0, 1], [-12, 12, 0, 0, 0, 0, 0, -2, 0, 5, 12, 0, -2, 2, 0, 0, 4, 0, -2, 0], [0, 0, 3, -4, -9, 9, -10, 0, -4, 0, 0, 0, 0, 0, 2, -4, 0, 6, 0, -8], [9, 0, 0, 0, 0, 0, 0, 0, 0, 9, -2, 4, 5, 3, 0, 0, -1, 0, -8, 0], [0, 0, -1, -5, 2, -6, 1, 0, -5, 0, 0, 0, 0, 0, -3, -2, 0, -2, 0, -3], [0, 0, 6, 0, -1, 3, -1, 0, 5, 0, 0, 0, 0, 0, -6, 6, 0, -7, 0, -1], [21, 2, 0, 0, 0, 0, 0, 3, 0, 4, -6, 4, 3, -1, 0, 0, 1, 0, 5, 0], [0, 0, 6, -2, 3, -6, 3, 0, 4, 0, 0, 0, 0, 0, 4, 4, 0, -9, 0, 3], [0, 0, 7, 4, -11, 3, -3, 0, -1, 0, 0, 0, 0, 0, -2, 4, 0, -1, 0, 4], [14, 11, 0, 0, 0, 0, 0, 1, 0, -6, 5, 4, 6, 4, 0, 0, -1, 0, 0, 0], [3, 29, 0, 0, 0, 0, 0, 0, 0, -1, -4, 0, 1, 1, 0, 0, 6, 0, -5, 0], [0, 0, 7, -5, -7, -1, -6, 0, -9, 0, 0, 0, 0, 0, 9, 7, 0, -7, 0, 1], [9, 8, 0, 0, 0, 0, 0, 1, 0, -10, 1, 0, 6, 4, 0, 0, 8, 0, -2, 0], [-12, 20, 0, 0, 0, 0, 0, -6, 0, 7, 13, -4, -2, -2, 0, 0, 4, 0, 2, 0], [-8, 1, 0, 0, 0, 0, 0, -11, 0, 3, 11, -2, 3, 14, 0, 0, 6, 0, -6, 0], [0, 0, 5, -9, 5, -5, 0, 0, -5, 0, 0, 0, 0, 0, 5, -3, 0, -5, 0, -5], [-6, -7, 0, 0, 0, 0, 0, -5, 0, -2, -6, -4, 2, 6, 0, 0, 1, 0, -12, 0], [0, 0, 8, 1, -2, 4, -7, 0, 4, 0, 0, 0, 0, 0, -1, 3, 0, -6, 0, 3], [0, 0, 0, -8, -8, 1, -5, 0, -5, 0, 0, 0, 0, 0, 2, -8, 0, 0, 0, 2], [-4, 5, 0, 0, 0, 0, 0, 16, 0, 0, -22, 5, -6, -7, 0, 0, -5, 0, 3, 0], [0, 0, -9, -1, 7, -1, -4, 0, -5, 0, 0, 0, 0, 0, -5, -7, 0, 3, 0, -5], [-20, 8, 0, 0, 0, 0, 0, 0, 0, 0, -1, -4, 8, -4, 0, 0, 0, 0, 8, 0], [9, -22, 0, 0, 0, 0, 0, 11, 0, 2, -13, 6, -4, -7, 0, 0, -1, 0, 2, 0], [0, 0, -5, -3, -7, -5, -4, 0, -2, 0, 0, 0, 0, 0, 0, -12, 0, 1, 0, 4], [-10, 5, 0, 0, 0, 0, 0, 5, 0, 0, -4, 10, 0, 0, 0, 0, -1, 0, -2, 0], [0, 0, 2, -6, -2, 2, 1, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, -3, 0, 6], [-36, 22, 0, 0, 0, 0, 0, -6, 0, -7, 16, -3, 3, -5, 0, 0, 5, 0, 5, 0], [-25, 30, 0, 0, 0, 0, 0, -7, 0, -11, 5, -6, 1, -5, 0, 0, 3, 0, -3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4000_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4000_2_f_d();".
// 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_4000_2_f_d(:prec:=20)
    chi := MakeCharacter_4000_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_4000_2_f_d();".
// 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_4000_2_f_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4000_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, -76, -61, 292, -94, -168, 77, 32, -16, -2, 1]>],Snew);
    return Vf;
end function;