
// Make newform 2600.2.k.f in Magma, downloaded from the LMFDB on 21 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2600_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2600_k_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_2600_2_k_f();".
// 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_2600_2_k_f();".
// 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 := [64, 0, 0, 0, 3545, 0, 0, 0, 6924, 0, 0, 0, 1134, 0, 0, 0, 60, 0, 0, 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, 0, 0, 0], [-1179284, 0, 0, 0, 3176491, 0, 0, 0, 501937, 0, 0, 0, 25181, 0, 0, 0, 411, 0, 0, 0], [0, 0, 22790499, 0, 0, 0, -1813185, 0, 0, 0, -881097, 0, 0, 0, -59139, 0, 0, 0, -1102, 0], [6994696, 0, 0, 0, 10339669, 0, 0, 0, 1505811, 0, 0, 0, 75543, 0, 0, 0, 1233, 0, 0, 0], [5860080, 0, 0, 0, 28189727, 0, 0, 0, 4699877, 0, 0, 0, 246469, 0, 0, 0, 4055, 0, 0, 0], [8413200, 0, 0, 0, 34182331, 0, 0, 0, 6136961, 0, 0, 0, 344145, 0, 0, 0, 5971, 0, 0, 0], [0, -1721336, 0, 14354413, 0, 11921331, 0, 8204897, 0, 2315575, 0, 1075207, 0, 134301, 0, 50743, 0, 2369, 0, 788], [0, 0, 24430702, 0, 0, 0, 40432645, 0, 0, 0, 6564959, 0, 0, 0, 346431, 0, 0, 0, 5767, 0], [0, 6834912, 0, 55696316, 0, -36124371, 0, 44740919, 0, -6513515, 0, 6616403, 0, -355589, 0, 337273, 0, -6013, 0, 5521], [0, 0, -51538947, 0, 0, 0, -99088269, 0, 0, 0, -16161429, 0, 0, 0, -853487, 0, 0, 0, -14212, 0], [0, 0, 53140121, 0, 0, 0, 125194558, 0, 0, 0, 20718640, 0, 0, 0, 1099194, 0, 0, 0, 18335, 0], [0, -11595144, 0, 90203307, 0, 24203040, 0, 117401312, 0, 4197940, 0, 18671114, 0, 221288, 0, 979392, 0, 3644, 0, 16267], [0, -20074520, 0, 147622145, 0, -32176414, 0, 286925138, 0, -5201814, 0, 46978476, 0, -271650, 0, 2484918, 0, -4466, 0, 41403], [0, 23517192, 0, 176330971, 0, 8333752, 0, 303334932, 0, 570664, 0, 49128890, 0, 3048, 0, 2586404, 0, -272, 0, 42979], [0, 16631848, 0, 118913319, 0, 56019076, 0, 270515344, 0, 9832964, 0, 44828062, 0, 540252, 0, 2383432, 0, 9204, 0, 39827], [0, 15994616, 65773198, -114755357, 0, 50450727, 145320786, -256510575, 0, 8521263, 24109422, -42369821, 0, 456313, 1284238, -2248369, 0, 7657, 21492, -37526], [0, -15994616, 65773198, 114755357, 0, -50450727, 145320786, 256510575, 0, -8521263, 24109422, 42369821, 0, -456313, 1284238, 2248369, 0, -7657, 21492, 37526], [0, 32342576, 0, -236345626, 0, 64777083, 0, -476896779, 0, 10529011, 0, -78226127, 0, 557037, 0, -4141149, 0, 9301, 0, -69039], [0, -38824144, 0, -285207030, 0, -64777083, 0, -557762069, 0, -10529011, 0, -91356045, 0, -557037, 0, -4834011, 0, -9301, 0, -80573], [0, -84759672, 0, -620313397, 0, -161730580, 0, -1240718696, 0, -26259836, 0, -203430730, 0, -1385724, 0, -10767216, 0, -23068, 0, -179481]];
    Rf_basisdens := [1, 810196, 1620392, 810196, 1620392, 1620392, 3240784, 1620392, 3240784, 1620392, 1620392, 3240784, 6481568, 6481568, 6481568, 3240784, 3240784, 3240784, 3240784, 6481568];
    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_2600_k();"
function MakeCharacter_2600_k()
    N := 2600;
    order := 2;
    char_gens := [1951, 1301, 1977, 1601];
    v := [2, 2, 2, 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_2600_k_Hecke();"
function MakeCharacter_2600_k_Hecke( : Kf := false)
    N := 2600;
    order := 2;
    char_gens := [1951, 1301, 1977, 1601];
    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], [-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 := ISA(Type(Kf), Fld), 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, 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, 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, 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, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -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, -1, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 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, -1, 0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 1, -1, 1, 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, 2, 0, 0, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1], [-2, 0, 0, 1, -1, -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, 1, 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, -1, 2], [0, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0], [0, 0, -1, 0, 0, 0, 0, 4, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 2, 0, 1, 2, 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, -2, 0, 0, 1, -1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -2, 0, 0, 1, -1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, -3, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 2, 0, 0, -1, 1, -1, 0, 0], [4, 3, 0, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, -2, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0], [8, 1, 0, 1, 3, -1, 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, 3, -1, 0, 0, 0, 1, -2], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -2, 0, 0, 1, -1, 2, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 2, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, -2, 0, -4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, -4, 0, 0, -1, 1, 0, 0, 0], [-2, -1, 0, 1, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 2, 0, -2, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 6, 0, 1, -3, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 4, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 0, 1, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, -1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 4, 0, 2, -1, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 5, 3, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, -2, 2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -3, -1, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, -4, 0, 0, 0, 3, -1], [-2, 2, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 1, 3, 0, 0, -2, 2, -1, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -4, 0, 0, 1, -1, -1, 0, 0], [-4, -2, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -3, -2, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, -2, 0, 0, -1, 1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 3, 2, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 2, 0, 0, 1, -1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, -4, -3, 0, 0, 0, 0, -2, -2, 0, 0, 0], [-4, -2, 0, -2, -2, 2, 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, 2, -3, 0, 0, 0, 0, 3, 0, 0], [0, 0, 1, 0, 0, 0, 0, -7, 0, -4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, -4], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -1, 0, 0, 2, -2, 1, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -3, 0, 0, 4, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 1, -3, 0, 0, 0, 3, -1], [0, 0, 1, 0, 0, 0, 0, 1, 0, -2, 3, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 4, 4, 0, 0, 0, -2, 2], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, -2, -4, 0, 0, -3, 3, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, -3, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0, 3, -2], [0, 0, 0, 0, 0, 0, 0, 4, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, -4, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, -2, 0, 0, 0, -2, -2], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 4, 0, 0, 0, -1, 3], [4, -1, 0, -1, -3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 3, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -3, 3, 0, 0, 0, -2, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -5, 0, 0, 0, 0, -1, 0, 0], [6, 0, 0, -2, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, -5, 0, 0, 2, -2, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -5], [0, 0, 1, 0, 0, 0, 0, 8, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 3, 5, 0, 0, 0, -3, -1], [0, 0, -1, 0, 0, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -8, 0, 0, 4, -4, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -3, -2], [0, 0, 1, 0, 0, 0, 0, 10, 0, 0, -3, 0, 0, 0, 0, 4, 4, 0, 0, 0], [-6, -3, 0, -1, 3, 1, 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, -6, 1, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, -2, 8, 0, 0, 0, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -4, -7, 0, 0, 0, 0, -1], [0, -5, 0, 1, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, -1, -8, 0, 0, 2, -2, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 1, 5, 0, 0, 0, -1, 3], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 4, -3, 0, 0, -1, 1, -1, 0, 0], [0, 0, -4, 0, 0, 0, 0, -5, 0, 1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 5, 0, 0, -1, 1, -1, 0, 0], [-10, 1, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -1, 0, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, -1, 0, -4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 8, 0, 0, -1, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 4, 0, 3, -4, 0, 0, 0, 0, 2, 2, 0, 0, 0], [-12, -6, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -1, 0, 1, 1, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, -2, 0, 0], [0, 0, -1, 0, 0, 0, 0, 6, 0, 5, 4, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 10, 0, 4, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 6, 9, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 2, 3], [2, -4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -4, 2, 0, 0, 1, -1, 3, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 1, -1, 0, 0, 0], [18, 2, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, -5, -1, 0, 0, 0, 9, -2], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 2, -7, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 1, -4, 0, 0, -3, 3, -4, 0, 0], [0, 0, -1, 0, 0, 0, 0, 4, 0, -4, -7, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, -4, 0, 0, 0, 0, -1, 0], [4, 0, 0, 3, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, -2, 7, 0, 0, 0, -2, 3], [-10, 6, 0, 2, -2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -2, 2, 0, 0, -2, 2, -4, 0, 0], [0, 0, 3, 0, 0, 0, 0, -2, 0, -7, -5, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0, 3, 0], [0, 0, -2, 0, 0, 0, 0, 4, 0, -1, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [14, 2, 0, -2, -2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, -8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, -6, 0, 0, 0, 2, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 2, 0, 0, 0, 4, -2], [0, 0, 0, 0, 0, 0, 0, 9, 0, -9, -6, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, -3, 0, 0, 0, 0, 8, 0, 2, 7, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -6, 0, 0, -1, 1, 2, 0, 0], [-12, -1, 0, -1, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 5, 4], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -4, -9, 0, 0, 0, 4, -2], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 2, 6, 0, 0, -1, 1, 3, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, -2, -5, 0, 0, 0, 0, -4, -4, 0, 0, 0], [-6, 0, 0, -3, 5, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, -7, 0, -1, -2, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, -2, -8, 0, 0, -2, 2, 0, 0, 0], [-4, 8, 0, -2, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, -3, -2, 0, 0, 0, 0, 4, 4, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, 8, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -3, 0, 0, -2, 2, 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, -4, -11, 0, 0, 1, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -7, 0, 0, 2, -2, 6, 0, 0], [4, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -8, 0, -2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 1, 0, 0, 0, 0, 5, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 3, 0, 0, 0, 1, 1], [0, 0, 1, 0, 0, 0, 0, -20, 0, 4, 9, 0, 0, 0, 0, -4, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -15, 0, 0, 4, -4, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8, 0, 3, -4, 0, 0, 0, 0, 4, 4, 0, 0, 0], [18, -3, 0, -1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 7, 0, 0, -1, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, -4, 0, -1, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0], [-2, 8, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -3, 10, 0, 0, 1, -1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, -6, 0, 0, 0, 10, -6], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 5, 14, 0, 0, -1, 1, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -3, 0, 0, 0, -5, -3], [-20, -4, 0, 6, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 5, 0, 8, 1, 0, 0, 0, 0, -2, -2, 0, 0, 0], [-18, 6, 0, 4, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-20, -1, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 2, 5, 0, 0, 0, -6, 9], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, 5, 0, 0, 0, -6, 1], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 14, 0, 0, -6, 6, 2, 0, 0], [-2, 0, 0, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 3, 0, 0, 0, 0, 4, 4, 0, 0, 0], [-12, 4, 0, 2, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -3, 2, 0, 0, -1, 1, 6, 0, 0], [0, 0, -4, 0, 0, 0, 0, -2, 0, 2, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, -6, 0, 4, 3, 0, 0, 0, 0, 4, 4, 0, 0, 0], [24, 2, 0, 6, -1, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 14, 0, 0, 1, -1, 2, 0, 0], [0, 0, -3, 0, 0, 0, 0, 8, 0, -4, -1, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 5], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, -8, -10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, -4, -17, 0, 0, 2, -2, 5, 0, 0], [0, 0, 5, 0, 0, 0, 0, 0, 0, 10, 3, 0, 0, 0, 0, 2, 2, 0, 0, 0], [-10, 5, 0, 3, -9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, -5, -1, 0, 0, 0, -3, 2], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 5, 0, 0, 3, -3, -1, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -4, 4, 0, 0, 4, -4, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, -3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -10, 0, 0, 9, -9, -2, 0, 0], [0, 0, 3, 0, 0, 0, 0, -9, 0, 6, 5, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 2, -4, 0, 0, -6, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -3, -3, 0, 0, 0, -1, -5], [2, 4, 0, -6, 4, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 16, 0, 4, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [8, -1, 0, 1, 10, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -8, 0, 0, 0, 9, 0], [0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, -3, 0, 0, 0, 1, -1, 8, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, -3, 11, 0, 0, 0, 4, 3], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, 4, 0, 0, 6, -6, 5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, -4, 0, 0, 0, 2, 6], [2, -5, 0, 1, 5, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 0, 2, -2, -4, 0, 0], [18, 4, 0, 5, 3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, -2, -8, 0, 0, -4, 4, 2, 0, 0], [0, 0, 1, 0, 0, 0, 0, 4, 0, -8, 7, 0, 0, 0, 0, -6, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 1, 4, 0, 0, 0, -3, 6], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 3, -8, 0, 0, 0, 11, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 5, -5, 3, 0, 0], [-6, -1, 0, -2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 2, 0, 12, 7, 0, 0, 0, 0, 4, 4, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 2, 0, 1, 5, 0, 0, 0, 0, -1, -1, 0, 0, 0], [6, 0, 0, 3, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, -5, 0, 0, 0, -8, -1], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 14, 0, 0, 0, 0, 10], [0, 0, -4, 0, 0, 0, 0, 4, 0, -5, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 4, -4, -8, 0, 0], [22, -2, 0, 5, -1, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 3, 1, 0, 0, -2, 2, 1, 0, 0], [10, -4, 0, 2, -4, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -4, 12, 0, 0, -2, 2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, -2, 1, 0, 0, 0, 8, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1, -1, -6, 0, 0], [8, 0, 0, 6, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, -4, 4, 0, 0, 0, -4, -6], [-18, -11, 0, -2, -9, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 13, 0, 0, 2, -2, -2, 0, 0], [0, 0, -2, 0, 0, 0, 0, -12, 0, -9, 6, 0, 0, 0, 0, -4, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 7], [0, 0, 2, 0, 0, 0, 0, -11, 0, -3, 6, 0, 0, 0, 0, 4, 4, 0, 0, 0], [-28, -4, 0, 2, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 2, 0, 2, 14, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-10, 5, 0, -1, -5, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 6, 6, 0, 0, -2, 2, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 5, 5, 0, 0, 0, -1, 5], [0, 0, 0, 0, 0, 0, 0, 10, 0, 6, -4, 0, 0, 0, 0, 1, 1, 0, 0, 0], [-18, 0, 0, -2, 12, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, 8, 0, 1, -10, 0, 0, 0, 0, 8, 8, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -8, 2, 0, 0, 0, -3, -2], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -10, 0, 0, 0, 1, 2], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 4, -2, 0, 0, -1, 1, -3, 0, 0], [0, 0, -4, 0, 0, 0, 0, -13, 0, -7, 6, 0, 0, 0, 0, -6, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2], [0, 0, -6, 0, 0, 0, 0, 8, 0, -1, 2, 0, 0, 0, 0, -4, -4, 0, 0, 0], [-16, 3, 0, -3, 9, -5, 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, 6, -14, 0, 0, 3, -3, 6, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -2, -3, 0, 0, 0, -8, 1], [0, 2, 0, 4, 9, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, 4, 0, 14, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 4, 14, 0, 0, -5, 5, -2, 0, 0], [0, 0, 1, 0, 0, 0, 0, 11, 0, 8, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 11, 0, 0, 2, -2, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 5, 9, 0, 0, 0, -1, 5], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, -3, -6, 0, 0, 7, -7, -2, 0, 0], [0, 0, 1, 0, 0, 0, 0, -1, 0, 8, 3, 0, 0, 0, 0, 6, 6, 0, 0, 0], [30, 1, 0, -1, -5, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, -7, 3, 0, 0, 0, -9, 3], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 10, 12, 0, 0, 0, 2, 8], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 4, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -3, -2, 0, 0, -7, 7, -6, 0, 0], [28, 4, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 0, 0, 0, -12, 0, -8, 7, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, 4, 0, 0, 0, 0, -6, 0, -2, -4, 0, 0, 0, 0, 1, 1, 0, 0, 0], [-12, 6, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 7, -14, 0, 0, 0, 0, 4, 0, 0], [0, 0, -3, 0, 0, 0, 0, 18, 0, 4, -15, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 0, 12, 0, 0, 5, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -10, -5, 0, 0, 0, 2, 3], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, -1, 2, 0, 0, 0, -1, 11], [-16, -5, 0, 3, 1, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, -4, -16, 0, 0, 4, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 3, -13, 0, 0, 0, 6, -5], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 6, 6, 0, 0, -2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, -6, -1, 0, 0, 0, -2, 11], [0, 0, -1, 0, 0, 0, 0, -4, 0, 8, -7, 0, 0, 0, 0, 2, 2, 0, 0, 0], [2, 4, 0, 0, -12, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -12, 0, 0, 0, 0, -3, 3, 0, 0, 4, -4, 3, 0, 0], [0, 0, -4, 0, 0, 0, 0, -4, 0, 4, -13, 0, 0, 0, 0, 2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, -7, 10, 0, 0, 0, -1, 4], [12, -8, 0, -3, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 8, 0, -5, 11, -5, 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, -5, 0, 0, 9, -9, 1, 0, 0], [0, 0, 3, 0, 0, 0, 0, -12, 0, 3, -8, 0, 0, 0, 0, -2, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 4, -2, 0, 0, 0, -3, 6], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 8, 0, 0], [30, -6, 0, -4, 6, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -4, 0, 0, -4, -8, 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, -8, 0, 0, -7, 7, 4, 0, 0], [14, -1, 0, -3, -11, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -7, 0, 0, 0, 0, 0, 0, 8, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -3, 0, 0], [0, 0, 8, 0, 0, 0, 0, -4, 0, -4, -1, 0, 0, 0, 0, 6, 6, 0, 0, 0], [28, -4, 0, -2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2600_k_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2600_2_k_f();".
// 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_2600_2_k_f(:prec:=20)
    chi := MakeCharacter_2600_k();
    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_2600_2_k_f();".
// 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_2600_2_k_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2600_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-64, 0, 224, 0, -256, 0, 112, 0, -19, 0, 1]>],Snew);
    return Vf;
end function;