
// Make newform 6045.2.a.bi in Magma, downloaded from the LMFDB on 07 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6045_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_6045_2_a_bi();".
// 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_6045_2_a_bi();".
// 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 := [-112, -892, -746, 5064, 4744, -12832, -7839, 17347, 4398, -12159, -446, 4582, -411, -935, 156, 97, -21, -4, 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, 1, 0, 0, 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, 0, 0], [0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, -7, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [317132228, 727071450, -1874991824, -103549980, 4661942660, -5229581577, -3608265451, 6645499448, 1071437101, -3413604116, -59637650, 891219629, -31566349, -123630228, 6198769, 8567893, -336512, -228355], [152855836, 1777245698, 1183377184, -11484187796, -4515463168, 24385040543, 2831804025, -21628657114, 168689975, 9631313854, -639195438, -2313633925, 211957035, 299322856, -27974025, -19095959, 1332792, 446299], [222862852, 862479718, -728115164, -3873662520, 2731690702, 6955418053, -6769396543, -4581147060, 6315037203, 976237358, -2759586728, 151793377, 613552943, -98215806, -67180803, 14791069, 2872262, -740009], [-175204944, -1389743616, -939434104, 6955732814, 4235562551, -13698626968, -4696943709, 12312429494, 2463590886, -5830999753, -711950221, 1554418411, 116139789, -234498267, -10014793, 18681397, 354827, -610666], [-101408846, 323408598, 3275128816, 121132724, -10798785650, -1242628076, 14213130072, 383710967, -9283925003, 584398593, 3285967890, -440926905, -639146514, 119878466, 64014013, -14528758, -2572882, 658857], [39621994, 262624594, 1478840262, 246169214, -6833192315, -1412926912, 11465269759, 175276449, -8697601105, 942614544, 3385844207, -623550578, -701176099, 163062311, 73271914, -19402563, -3033149, 870813], [547325784, 4385523658, 4248411632, -16961256348, -18654446728, 27680230503, 25833441061, -24057580418, -17208963633, 12072754710, 6172190758, -3589511029, -1212202237, 620355732, 122228351, -57254415, -4934604, 2172011], [-238817406, -1638615510, -1274068280, 8225920740, 7692601359, -16275952098, -11972992943, 15559974451, 8553876469, -8101395364, -3215790521, 2422989002, 652347529, -415176825, -67272692, 37784959, 2759313, -1412139], [239590942, 1216991632, -1548008000, -6641795912, 7115561525, 12043662899, -15183703738, -7494401204, 13173455103, 1252562487, -5541738709, 436001482, 1204272526, -204157033, -129885278, 28955834, 5494307, -1412019], [-297472862, -1072984312, 3521009816, 8029654174, -13649055788, -15972796554, 23716389037, 10879951810, -18709447578, -2503879729, 7492954365, -250642325, -1582940924, 205502471, 167766567, -31883760, -7015797, 1613979], [-43868350, -973053192, -3418543606, 836788936, 14283007824, 1419906454, -23233156855, 400301490, 17479801312, -2108436645, -6795292397, 1283554233, 1408087416, -329077721, -147290343, 38885676, 6102749, -1740463], [194458614, 50031272, -3544809292, -2217653358, 13292663783, 4577755507, -21905743275, -1134599382, 16857338927, -1758453438, -6674822568, 1266225703, 1402238426, -336742404, -148177865, 40308086, 6186612, -1814482], [543688764, 4490857882, 7516525160, -13449627508, -32683891182, 18090271893, 49918468081, -17639487940, -35930753321, 11275251570, 13590072516, -4205477675, -2766904757, 868124882, 285997777, -91459471, -11750022, 3825863]];
    Rf_basisdens := [1, 1, 1, 1, 1, 17310212, 17310212, 17310212, 8655106, 8655106, 8655106, 17310212, 8655106, 8655106, 8655106, 8655106, 8655106, 17310212];
    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_6045_a();"
function MakeCharacter_6045_a()
    N := 6045;
    order := 1;
    char_gens := [4031, 4837, 1861, 2731];
    v := [1, 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_6045_a_Hecke(Kf)
    return MakeCharacter_6045_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, 0, 0], [1, 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, -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, 0], [-1, 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, -1, 0, 0, 0, 0, 0, 0], [1, 1, -1, 1, 0, -1, 1, 0, 0, -1, 0, 0, 1, 1, 1, 0, 0, 0], [2, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 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, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, 1, 0, 0, -1, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0, 0], [2, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0], [3, -1, 1, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0, -1, 0, 0, 0, 1], [2, 0, 1, -1, 0, 0, -1, -1, 1, 0, 0, 0, 0, -1, -1, 1, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, -1, -1, -1, 0, 0], [1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 1, 0, 1, -1, 0], [-1, 0, -1, 1, 1, 0, 0, 0, 0, -1, 1, 0, -1, 1, 1, -1, 0, -1], [0, 1, -1, 1, 0, -1, 0, 0, 0, -1, 0, 0, 1, 2, 0, 0, 0, 0], [1, 1, -1, -1, 0, 0, -1, 0, 0, 1, -2, -1, 0, -2, 0, -1, 1, 1], [1, -1, -1, -1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0], [1, -1, 1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1], [2, 0, -2, 1, 1, -1, 1, 0, 0, -2, 0, 0, 1, 3, 1, -1, 0, -1], [-1, -2, -4, 2, 1, 0, 0, -2, -1, 0, -2, -1, -1, 0, 1, -2, 1, 0], [1, -1, 0, 1, 0, 0, -1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, -1, -2, -1], [1, 1, 0, -1, -2, 2, -2, 0, -1, 2, 1, 0, -1, -3, -1, 1, 0, 0], [4, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [1, 1, 0, -2, -1, 1, -1, 3, -1, 2, 2, 2, 0, -1, -1, 0, 1, -1], [1, -1, 1, 0, 0, -1, 0, -1, 0, -1, 2, 0, 0, 1, 1, 0, 0, 1], [2, -2, 1, -1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, -1, -1], [2, 1, -2, -1, 0, 1, 0, 1, 0, 0, -1, -3, 0, 0, 0, 0, 0, 0], [0, 1, -2, 2, 1, -2, 1, -2, 1, -1, -1, 1, 1, 1, 1, 0, 0, -1], [1, 0, 0, 1, -1, 1, 1, 3, 0, 0, -1, -1, 0, 0, 0, 1, -1, -1], [2, 0, -1, -1, 1, -1, 0, 0, 1, 0, -1, 0, 1, 0, 1, -1, 0, 0], [0, 0, -1, 0, 2, 0, -1, 0, 0, 0, 2, 0, 0, 1, 0, 1, -1, 0], [3, -2, 1, -1, 1, -1, 1, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, 1], [3, -1, 0, 0, 0, 1, -2, 0, 1, 2, -1, 0, -1, -4, -1, 0, 1, 0], [3, -1, 1, -2, 0, -1, 1, 2, 1, 1, -1, 1, 1, 0, -1, 0, 1, 1], [4, -4, 0, 1, -1, 0, 0, 0, 0, 1, -1, -1, 0, -1, 0, 0, 0, 0], [3, -3, 2, 0, 0, 1, -2, 0, 1, 0, 1, -1, -1, -2, -2, 0, -1, -1], [-3, 0, -1, 1, 1, 0, 2, 0, 0, -2, 1, -1, 1, 3, 2, 0, -2, 0], [0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 2, 1, 0, 1, 1, 1, 1, -1], [0, 1, -3, -1, 1, 1, 1, 1, 1, 0, 0, -1, -1, -1, 1, 1, 0, 0], [0, 0, 2, 0, -1, 2, -1, 0, 1, -1, 0, -1, 0, -1, -1, 2, -1, 1], [2, 0, 1, -2, 1, 1, -2, 1, 0, 0, -1, 1, -1, -2, -1, 0, 0, 0], [2, 1, -2, 3, 1, -2, 1, 1, -1, -2, 1, 1, 0, 2, 1, -1, 0, -3], [1, 0, 0, -1, 1, -1, 1, 2, 0, 1, -1, 0, 0, 0, 2, 1, 1, 1], [5, -2, 1, 0, 0, 2, 1, 1, 0, -1, 3, 0, 0, 1, 0, 1, -1, -2], [-3, -2, -1, -1, 1, -1, 0, -2, 0, 0, 1, -1, -1, 1, 0, 0, 0, -1], [0, -3, 2, 0, -1, 1, -2, 0, -1, 2, 0, 1, -1, -2, -3, 0, -1, 0], [-1, -1, 0, 1, 0, 3, -2, 3, -1, 3, 1, 1, -2, -2, 0, 1, -1, -1], [6, 1, 1, 0, -1, 0, 0, 2, 2, -1, 3, 2, 1, 1, -1, 2, -1, -1], [3, 0, -3, 2, -1, 1, 0, 1, -1, 1, -3, -1, 0, 0, 0, -1, 0, -1], [-1, 1, 1, 0, -1, -1, 0, 0, 0, -1, 2, 1, 0, 0, 0, -2, 2, 0], [-4, 0, -1, 3, 2, -1, -1, -2, -1, 0, 1, 0, -1, 0, 1, -1, 0, -1], [-2, -1, -2, 3, 2, -1, 1, -1, -1, -3, -1, 1, 2, 4, 1, -2, 0, 0], [0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, 2, 0, -1, -1, 1, 1, 0], [4, -2, -4, 0, 1, -2, 0, 1, -1, -1, 0, -1, 1, 2, 2, -1, 1, -1], [3, 2, -3, 4, 0, -2, 1, 2, -1, -2, 0, 3, 2, 3, 1, -1, 0, -2], [1, -1, 4, -1, -3, 4, 0, 1, 1, 2, 2, 2, -1, -2, -2, 3, -1, 0], [1, 0, 1, -2, 0, 1, -1, 0, 0, 0, 3, 1, -1, -2, -1, 2, -1, -2], [-2, -3, 1, -3, 0, 0, 2, 3, 0, 0, 0, 0, -2, 2, -1, -1, -1, 1], [1, 1, 2, -2, -3, 4, -3, 1, -1, 4, -2, 0, -1, -4, -1, 2, 1, 2], [-2, 1, -1, -1, -2, 2, -3, 1, -1, 2, -2, -1, -1, -1, -1, -1, 2, 1], [-3, -1, 2, 1, -1, 0, 1, 0, 0, 0, 1, -1, 0, 0, -1, 1, -1, 0], [5, -3, -2, 3, 0, -1, 0, 0, -3, 0, -1, -1, 0, 2, 2, -2, 1, -1], [-3, 3, -5, 1, 2, 1, 1, 0, 0, 0, -2, 0, 1, 1, 1, -1, 0, 0], [3, -2, -4, 1, 0, 1, -1, 0, -1, 2, -2, -1, -1, 0, 0, 0, 0, 0], [-2, -2, -1, 4, 0, 0, -1, -1, -2, 0, 0, 0, 0, 1, 0, -3, -1, -2], [-2, -5, 4, -1, 0, 2, 0, -1, -1, 2, 1, -1, -2, 0, 0, 1, -1, 2], [-1, 0, -1, 1, 0, -1, 1, -1, -2, 0, -1, -1, -1, 0, 2, 1, 1, 1], [-2, 3, -4, 1, 1, 0, 1, 1, 0, 0, -2, 0, 0, -1, 0, -1, 0, 0], [3, 3, -4, 0, 0, 1, 0, -1, -1, 1, 2, 1, 1, -1, 0, 0, 0, 0], [4, -4, 2, -2, -3, 2, 2, 1, 1, 1, 0, -1, -1, -3, -1, 2, -3, 2], [1, -3, 5, 0, -2, -1, 2, 0, 1, 0, 1, 4, 1, -2, -1, 0, 1, 2], [0, -1, -3, 1, 2, -2, 3, -1, 1, -1, 0, 0, 0, 1, 2, -1, 1, 1], [-2, 4, 2, -3, 0, 3, -1, 0, 0, 1, 4, 3, -1, -2, -3, 0, -1, -2], [4, 3, 0, 2, 0, -4, 2, -1, 2, -2, 1, 0, 2, 0, 1, -1, 1, -1], [-1, 0, 4, -2, 0, 1, 0, -3, 2, 1, 4, 2, -1, -1, 0, 1, -1, 0], [3, -1, 1, 0, 0, -1, 0, -1, 2, 1, 0, 3, 1, -1, -2, 1, 1, 1], [0, -1, -2, -1, -1, 0, 2, 0, 0, -1, 3, -1, 1, 1, 1, 1, -1, 1], [0, 3, 2, -2, 1, -1, -1, 2, -1, 0, -2, -1, 0, -1, 0, -1, 1, 0], [4, -2, -4, 5, 3, -2, 4, -1, 0, -6, -1, -4, 1, 5, 2, -3, -1, -1], [-1, 2, 1, 1, 0, -1, -2, -4, 1, 1, 1, 2, -1, -3, -1, 1, 0, 0], [-2, 5, -2, 3, 0, -3, 0, 0, 1, -2, -3, -1, 1, 2, 2, 0, 0, -2], [-4, 1, 2, -2, 2, 2, 1, -1, 2, 0, 4, 1, -1, 1, 0, 1, -1, -3], [6, -2, 3, 0, 0, -2, 1, 2, 2, -2, -3, 0, 3, 2, 0, 0, 0, 1], [11, 1, -4, 1, 0, -2, 0, -1, 0, -1, -6, -5, 2, -1, 2, -4, 2, 3], [1, 5, 4, 0, -1, -2, -1, 3, -1, 2, 1, 2, 2, 0, 0, 3, -1, 1], [-5, 0, -3, 1, 1, -3, -1, 2, -1, 1, -3, 1, 0, 2, 0, -3, 4, -2], [7, -3, 2, -2, -1, 0, 1, 3, 1, 0, -2, 0, 2, -1, -2, 2, 0, 2], [5, 7, -2, -4, -1, 2, 0, 2, 3, 2, -4, 0, -1, -6, -2, 1, 3, 0], [4, 1, 0, 3, 0, -2, 3, 1, 0, -3, -1, 2, 4, 2, 1, -2, -1, -2], [-7, 0, 1, -3, 2, 2, 1, -1, 1, -2, 6, 3, -1, 2, -2, 3, -3, -2], [-1, -2, -2, 3, 2, -1, -1, 1, -2, 1, -4, 2, 0, 5, 1, -4, 1, -2], [1, -3, 0, 2, 3, 0, -4, 0, -1, 1, 1, 1, -2, 0, 0, -2, 0, -3], [0, 1, -1, 1, 2, 0, -1, 0, -4, 0, 3, 1, 1, 2, 4, 0, 2, -2], [-2, -1, 1, -1, 1, -1, 0, 0, 0, -1, 4, 0, 0, 0, 0, 1, 1, -1], [-4, 1, -1, 1, -2, 0, 0, 0, -1, -1, -3, -3, 0, -2, 0, 2, -1, 0], [2, 3, -2, 4, -1, 3, -2, 2, -1, 0, 0, -3, 0, -2, 2, 2, -2, 2], [10, -4, 3, -3, 1, 2, -2, 1, 1, 1, -6, -1, -2, -3, -1, 0, 1, 2], [-4, -6, 1, 1, 1, -1, 1, -4, 2, 1, 0, 1, -1, 2, -1, 0, -1, 0], [-6, -2, 3, 4, 0, 0, -1, 0, -1, -1, 3, -1, -2, 2, 1, 0, -3, -3], [3, 4, -2, 0, 1, 1, 2, 3, -1, 0, 3, 1, 1, 3, 2, 1, 1, 0], [4, -7, 2, 2, -1, -2, -1, 1, -1, 0, -3, -1, 1, 1, -1, -3, 2, 1], [-2, 3, 2, -2, -3, 4, -5, 0, 0, 3, 1, 1, -3, -5, -5, 1, -1, 0], [-2, 0, 6, -3, 0, 0, 0, 0, 0, -2, 3, 0, 0, -1, -2, -1, 1, -1], [7, -4, 9, -2, 0, -2, 3, 0, 3, -1, 0, 2, 0, 0, -1, 1, 0, 2], [2, -3, 0, -1, 0, -2, 0, -4, 0, 0, 3, 1, 1, -1, 0, 3, 1, 2], [4, -3, -1, 3, 1, 3, -2, 1, 0, 1, -1, -2, -4, 1, -1, -2, -1, 0], [5, 2, 4, -4, 1, -1, 2, 3, 2, -1, 1, 4, 1, -1, 1, -2, 3, 1], [-7, -5, 1, 3, 0, -4, 1, -1, -1, -4, 2, 2, 1, 4, 0, -5, 1, -1], [0, -3, -2, 2, 0, -5, 5, 0, 1, -3, 3, 2, 5, 5, 2, 1, 1, 0], [6, 4, 1, 2, -3, -1, -2, 0, -1, 2, 3, 2, 0, -3, -1, 0, -1, -1], [8, -3, 3, -3, 2, 4, -2, 2, -1, 2, 1, 0, -1, -2, 0, 1, 0, 1], [6, 1, -1, 0, -2, 2, 0, 2, 0, 3, 4, 0, -3, -4, -2, 2, 0, 1], [3, 2, 0, 2, -1, -2, 0, 2, -1, 2, 1, -1, 2, -1, 1, -1, 2, 1], [-1, 0, 1, -1, 2, 2, 2, -2, 1, -4, 2, 1, -1, 0, -2, -1, -1, -1], [-4, 0, 3, -3, -2, -1, -1, 2, 0, -2, 4, 4, 2, -1, -2, 0, -2, -2], [-1, -2, 3, 1, 2, -1, 1, 0, 0, 0, -3, 1, 2, 2, 1, -2, 1, 2], [3, -3, 2, -2, -1, 0, 2, -3, 1, -1, 5, 1, 2, -2, 0, 3, -3, -2], [0, -5, -1, 0, 1, -2, 0, 0, -1, 0, 2, -1, -1, 5, 2, -4, 2, -1], [3, -3, -2, -2, 0, 0, 0, 0, -1, 2, -6, -4, 0, -2, 1, -2, 1, 3], [1, 1, -1, 2, 3, -3, 6, -4, 4, -5, -2, -1, 3, 6, 2, -1, -1, -1], [7, -4, 0, -1, 2, -2, 4, 1, 3, 0, -3, -1, 1, 0, 1, 0, 1, 0], [7, 3, 4, -5, -2, 3, -3, 4, 2, 5, 1, 3, 2, -6, -4, 4, 0, 1], [-2, -1, 1, -5, 0, 3, -1, -1, -1, 5, -4, 0, -1, -3, -2, 1, 3, 2], [0, -2, -1, 0, -2, 3, -2, 4, -2, 0, 1, -4, -1, 0, 0, 1, -1, 0], [5, -1, -1, -1, -4, -1, -1, -1, 3, 0, -4, 0, 0, -3, -4, -1, -1, 0], [-4, -1, -7, 4, 1, 0, 4, 2, -2, -3, -1, -2, 2, 2, 6, -1, -1, 0], [-7, 1, -1, 2, -2, 0, 2, -3, 3, -4, -4, -1, 3, -1, 0, 0, -1, 0], [4, -2, -6, -1, 3, -3, -1, 0, -2, -2, -2, 0, 3, 1, 4, 0, 2, 0], [-7, 1, 6, 2, -2, 3, -5, 0, -2, 4, 1, 0, -4, -2, -2, -1, -1, 0], [0, -3, 0, -3, 1, 3, -4, -3, 0, 5, -2, -5, -6, -2, -1, -1, 1, 2], [3, -5, 2, -4, -2, 5, -1, 5, 2, 1, -1, -4, 0, -3, -1, 4, -2, 2], [-4, -4, 2, -1, -1, 4, -4, 0, 0, 2, 6, 0, -3, -1, -4, -1, -3, -1], [-10, 9, 1, -1, -3, 3, 0, 2, -3, 0, 3, 1, 1, 0, 0, 0, 0, 0], [-6, 2, 8, -1, -3, 2, -1, 2, 0, 2, 3, 4, 0, -1, -1, 4, -3, -1], [-4, 0, -2, 0, -2, -3, 0, -4, 1, 1, 0, -1, 1, -1, 2, 0, 2, 3], [1, -4, 3, 3, 1, -2, 3, 5, 1, -2, 0, 1, 1, 0, 3, 2, 3, 1], [-2, 1, 0, 3, 1, -3, 0, 1, 1, 0, -4, -3, 0, 1, 0, -5, 3, -1], [1, 2, 2, -1, 2, 0, 0, 2, -1, -2, 1, -1, 0, 1, -1, -3, 2, -3], [1, 4, -5, 3, 2, -1, 1, -3, -2, -3, -2, -4, 0, 1, 1, -2, 1, 0], [3, 0, 5, -2, -4, 3, -1, 3, 2, 6, 1, 1, -2, -2, -3, 5, 0, 1], [2, 5, 4, -3, -3, -1, 3, 3, 4, -1, 5, 4, 2, 1, -3, 4, -1, 1], [2, 4, 4, -3, -1, 0, 1, 3, 2, 0, 0, 2, 4, 0, -2, 2, 0, 3], [1, -3, 2, 1, 3, -6, 2, -1, -1, -2, -1, 2, 4, 4, 3, -2, 3, 3], [0, 0, -4, 2, 1, -3, 4, 3, 4, -3, 0, -3, 3, 5, 2, 0, -2, 1], [1, -2, -2, 3, -1, 2, -5, -1, -5, 4, -5, -3, -1, -2, -2, 0, 0, 3], [3, -4, 1, 4, -2, -3, 1, 0, -1, -2, -5, 1, 2, 4, 1, -3, 0, 1], [4, 10, 1, -6, 0, 1, -2, 2, 0, 0, -1, 2, -1, -1, -1, 1, 2, -2], [-6, -7, 0, 3, 3, -3, -1, -3, 1, -2, 0, -1, 0, 4, 0, -6, 2, -1], [-5, -1, 1, -1, 3, 1, -2, 0, -1, 2, -1, -2, -2, 3, 2, 1, -1, -1], [-2, 2, -5, 6, 0, -1, 0, 0, -2, -2, 0, 2, 0, 2, 1, -2, -3, -3], [0, 5, 1, -2, 1, -1, 3, 0, 0, -2, 3, -3, 2, 2, 3, 4, -3, -1], [-3, -9, -5, 0, 1, 3, 4, 0, 2, -3, -3, -7, -1, 3, 0, 1, -1, 3], [0, 0, -1, -2, 1, -3, 0, 5, -1, 1, 0, 1, 3, 1, 0, 0, 0, -1], [9, 1, -1, 1, -2, 1, -1, 1, -1, 2, -2, -2, 1, -2, 1, -3, 0, 0], [-3, 1, -2, 4, -3, 4, -6, 0, -1, 5, -2, -5, -5, -4, -3, -1, 0, 0], [-6, 1, 1, 6, 3, -2, -3, -2, -3, 0, 0, -2, 0, 1, 3, -1, 1, -2], [0, -1, -3, 0, 1, 2, 1, 0, 0, -1, 6, 0, -2, 3, 4, 0, 0, -2], [-4, 2, -3, -1, -1, 0, 2, -1, -1, -1, 5, 1, -1, 1, 1, -2, 1, -2], [-2, 0, -2, -2, 1, 1, 1, 0, 1, -1, 6, 5, 0, 0, 0, 5, -5, -3], [-12, -4, 2, -1, 3, 1, 1, -1, -2, -1, 1, 3, 0, 3, 3, -1, -2, -1], [0, 0, -3, 5, -2, 1, -1, -6, -2, -2, 1, -4, 0, 1, 3, 1, -4, -1], [-4, -3, 6, 4, 1, -1, -2, 3, -3, -1, 3, 2, 1, 2, 2, 1, -1, -3], [1, 0, 1, 2, 3, -4, 0, 5, -1, -2, -2, 1, 2, 5, 1, -1, 2, -1], [-8, -1, 1, -1, 0, 0, -2, -4, -3, -1, 7, 2, -2, -2, 0, -2, 0, 1], [-2, -6, -4, -3, -2, 4, -6, 0, -1, 6, -3, -2, -4, -10, -2, 1, 1, 1], [-9, -2, -7, 0, 0, 0, 2, -4, 3, -2, 1, 0, 0, -3, -2, 0, -3, -1], [2, 4, -2, -2, 0, 4, -3, 0, 1, 1, -2, -3, -2, 2, 0, -2, -2, -1], [6, -2, 0, 4, -4, 2, 0, -1, 1, -2, 3, 0, -3, -1, -1, 1, -2, 1], [5, -5, 5, 2, 4, -3, 1, 0, 4, 0, -3, 0, -2, 5, 0, -3, 3, 2], [-10, 8, 4, -1, 0, -1, 3, 4, 0, 0, 3, 3, 2, 3, -2, 4, -1, -1], [-4, -2, 5, -2, 1, -4, -1, -7, 2, -2, 0, 1, -1, 0, -2, -4, 1, 0], [8, -1, -2, 3, 0, -5, 5, 3, 0, -2, 2, 5, 1, 1, 3, 2, 1, 1], [-6, 1, 1, -2, 1, 4, 0, 1, -3, -1, -3, -4, -2, 1, 1, -2, -1, -3], [0, -2, -4, 4, 5, -7, 3, -5, -3, -3, -1, 1, 0, 5, 5, -5, 4, 1], [5, -1, -6, 3, -2, -4, 2, 1, 3, -1, 1, 3, 4, 0, 1, 3, -2, 0], [-5, -1, -3, 5, 3, 3, -4, 0, 0, 1, -6, -3, -2, -3, 0, 1, 1, 3], [-4, -1, 3, -4, 4, 4, 1, -2, 3, -2, 4, 1, -2, 1, -1, -3, -2, -1], [5, -7, -3, -1, 0, 2, 3, -3, 3, -3, 3, -3, -1, -2, 2, 1, -3, -1], [-9, 0, -5, 2, 2, 0, -1, -3, -3, 3, -6, 2, 1, 0, 1, -3, 2, 0], [-2, -1, 4, -3, -2, -2, 0, 2, 3, 5, -8, 0, 1, -5, -3, 0, 3, 3], [-8, 0, 6, -2, 4, 2, 1, -2, -2, 1, 2, 1, -3, 1, 0, 0, 0, -3], [-19, 2, -2, 4, 3, -1, -2, -4, -4, 0, -1, 3, -4, 3, -1, -5, 2, -2], [-18, -1, -5, -2, -1, 2, 0, -2, -2, 2, 0, -1, 3, 1, 0, 3, 1, 2], [-1, -7, 1, 3, 3, 1, -2, -3, -3, 3, -1, -4, -4, -3, -1, -3, 1, 0], [7, 9, -1, -3, -1, 4, 1, 1, -1, 2, 3, -2, -1, -2, 2, 6, 2, 0], [4, -4, 0, -2, -3, 1, 3, 2, 3, 0, -4, -4, 3, -3, 3, 3, 2, 3], [-12, 6, 1, 4, 0, -2, -2, 0, -2, -1, -1, 3, 2, 1, 3, 1, 0, -1], [9, 2, -9, -1, -1, 2, -3, 6, 1, 2, -4, -3, -1, -4, 0, 1, -1, 2], [10, 4, -3, 0, 1, 0, 3, 0, 2, -5, 2, 1, 2, 5, 2, 2, -2, 1], [-14, 5, -4, 10, 4, -5, 3, -3, 0, -6, 2, -1, 3, 7, 7, 0, -1, -3], [-10, -7, 2, -3, -2, 5, 0, -2, -3, 2, 2, 1, -1, -3, -2, 3, -1, -1], [3, 6, 1, -4, -5, 2, -2, 0, 2, 1, 4, 2, 3, -8, -3, 4, -1, 0], [-1, -4, 2, 2, 0, 1, 3, -3, 0, -3, 3, 0, -2, 8, 5, -1, 0, 1], [-4, 1, 4, 2, -3, 3, -4, -1, 0, 1, 7, 2, -3, -3, -3, 4, -5, -4], [-3, 9, -5, -5, -2, 5, 1, -1, 1, 1, -2, -3, 0, -6, 3, 6, 3, 3], [-3, 6, 3, -2, -1, 1, 0, 1, 0, 1, 1, 4, -2, -6, -1, 0, 3, 1], [0, 8, -2, 3, 0, -6, 0, 2, 1, -2, -2, 1, 5, 4, -1, 0, 2, 1], [1, 2, 8, 4, 1, 1, 0, 0, -4, 0, 1, 1, -5, 4, 1, 2, 1, -1], [3, 15, 1, -2, 0, -4, 3, 2, -1, 0, -3, 0, 3, 0, 1, 2, 3, 2], [-16, 3, 3, -2, -2, 5, -3, -2, -3, 1, 1, -1, -3, 0, -2, -1, -2, -3], [11, 1, -1, 3, 2, -8, 2, 0, 1, -3, -2, 4, 2, 2, 0, -2, 1, 0], [-8, -9, 3, 0, -1, 0, 5, -4, 3, -2, -2, -1, 3, 5, 1, -3, 0, 1], [-10, -12, 1, -6, 2, 2, -5, -6, 0, 4, -1, 0, -5, -6, -2, 0, -2, 1], [-10, -6, -8, 7, -2, -1, 1, -2, -3, -1, 0, -3, -2, 2, 4, -3, 3, 1], [1, -4, 2, -7, 0, 5, -4, 0, -2, 6, 0, -1, -1, -7, -4, -3, 3, 2], [4, 3, 10, -9, -6, 9, -1, 2, 6, 6, 4, 1, -3, -10, -4, 8, -2, 1], [8, 2, 6, -3, 1, -4, 4, 4, -1, 1, 0, 3, 1, 3, 1, -3, 2, 1], [2, 1, 1, -1, 4, -1, -2, -3, 2, 1, -9, -2, -1, -1, -1, -5, 2, 3], [-3, 0, 12, -7, -6, 6, -1, -1, 4, 5, 4, 2, -4, -8, -6, 1, -1, 0], [-9, -6, -3, 3, 3, 1, 1, 2, -2, 3, 1, 2, -1, 4, 3, 3, 0, -1], [0, 3, -3, 2, -4, 3, -4, 4, -2, 5, -5, -2, -2, -4, -2, 0, 0, 2], [6, 3, 9, -4, 1, 0, 4, -1, 3, -3, 7, 2, 1, 3, 0, 3, -5, -2], [5, -3, 3, -5, 0, 2, 1, 5, 3, -1, 3, -1, 2, -1, 2, 3, -1, -1], [-11, -3, 8, -3, 2, -3, 3, -1, 3, -2, 2, 0, -2, 1, 2, -3, -1, 4], [5, 2, 0, 2, -2, 1, 1, 6, -2, -3, -1, 1, 1, 4, 4, 2, -1, 1], [1, 8, 1, 0, -2, 1, -4, 5, -4, 0, 4, 0, 0, -1, 0, 2, -1, -2], [0, 9, -2, -4, 0, 0, 3, -4, 2, -3, -4, 2, 4, 1, 1, -3, 0, 1], [6, 13, -4, 3, -1, -1, 0, 2, -5, 1, -7, -2, 3, -1, 2, 0, 4, 2], [-1, 8, -1, -3, -3, 2, 1, 2, -1, 0, 2, 7, 3, -2, -1, 0, 1, -3], [7, 9, 7, -10, -7, 5, -4, 4, 0, 7, 2, 7, 0, -7, -6, 4, 0, 0], [5, 0, -3, 3, 2, -1, -1, -2, 0, -2, -4, -1, -1, 0, 0, -1, 1, 1], [-12, 10, 0, 0, 0, -2, 1, 5, 0, -2, -2, 2, 4, 2, -1, 3, -2, 0], [-7, 1, 5, 0, -4, 1, -1, 1, 3, 0, 2, 4, -1, -5, -4, 1, -3, 0], [-3, 0, -5, 0, 0, 2, -2, 0, -3, 1, 2, -1, -1, -6, 2, -2, 3, -3], [4, 13, 4, -10, -5, 7, -1, 5, 1, 5, 4, 4, 0, -6, -5, 8, -1, 3], [-13, 8, 0, -4, -4, 7, -3, 1, 1, 6, -2, -6, -2, -4, -4, 2, -1, 1], [6, -1, -7, 2, 8, -6, 1, 0, -2, -4, 1, 1, 5, 10, 7, -2, 1, -2], [-14, 0, -12, 3, 7, 0, 1, -4, -2, -2, -4, -2, 0, 6, 6, -2, 2, -1], [-5, 2, 1, 3, 4, -5, 1, -2, -4, -5, 2, 4, 2, 8, 3, -3, 0, -3], [0, -1, 10, -5, 2, 1, -3, -5, 1, 5, -7, 2, 1, -5, -4, -2, 2, 3], [-6, 9, 5, 1, -2, -1, 2, 6, 1, -3, 0, -2, 1, 3, -5, 1, -6, -1], [11, -8, 1, 0, 1, -2, 2, 0, -2, -1, -3, -2, -3, 1, 2, -6, 0, 2], [-4, 1, 11, 1, -1, -1, 0, -3, 1, 1, 4, 4, -4, 4, -3, -2, -3, -2], [0, -7, 0, 6, -4, -3, -1, 2, -3, 0, 4, 1, -1, 0, -1, 0, 1, -3], [-8, -10, 3, 4, 2, 1, -1, 0, -3, 0, 6, -2, -3, 4, 0, -1, -1, 1], [1, 8, -7, -2, -3, 2, 1, -2, 0, 5, -3, 2, -2, -4, -1, -4, 3, 1], [-5, 5, 0, 1, 2, -2, 2, -4, -1, -5, 0, 5, 1, 0, 0, 0, 0, -1], [-11, 11, -6, 8, 1, -4, 1, 3, -5, -3, -4, 3, 4, 8, 0, -4, -2, -4], [3, -7, 2, 0, 1, -1, 2, 0, 2, -2, -5, -2, 1, 1, 3, 3, 0, 3], [-4, -6, 0, -1, 0, 1, 0, 2, 1, -1, 2, 2, -2, -1, -4, -2, 1, -2], [-18, 3, 1, 0, 6, 3, -6, -5, -4, 0, 8, -5, -7, 1, 1, -2, -1, -4], [18, 0, 0, -2, 1, -6, 0, 2, 3, -1, 4, -1, -1, 3, 2, 1, -1, 3], [5, 5, -6, 1, -1, 3, -5, -6, -7, -1, 7, -3, -4, -1, 3, -1, -2, -2], [-1, -10, -2, -2, 3, -1, 2, 2, -4, 0, -2, -1, 2, 4, 1, 1, 2, 2], [0, -3, 4, 1, 1, -1, 5, -2, -2, -4, 5, 6, 1, 5, 2, 2, -1, -2], [-4, 0, 0, -1, -5, 1, -2, -3, 2, 2, -1, 2, 0, -3, -5, 3, 0, 4], [-5, 8, -6, 1, 4, -1, 1, 1, -1, -1, 3, 1, 3, 5, 7, 1, 1, -4], [-7, 0, 3, -1, 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0, 2, -2, 3, -3, 2], [5, -9, 4, -7, -2, 6, -2, 4, 0, 2, 6, 1, -2, -6, -2, 0, 1, 0], [3, -8, -1, -1, 0, -4, 2, -4, 3, 0, 0, -3, -2, 0, 0, -5, 1, 1], [-17, -1, -2, 10, 7, -4, -3, 0, -2, -3, 8, -1, 0, 9, 3, -3, 0, -7], [3, 9, -1, 0, -3, -1, 3, 3, 0, -2, 5, 2, 5, 2, -2, 0, 0, -1], [1, 7, -6, 2, -1, -3, 4, 3, -2, 1, -3, 1, -1, -2, 6, 0, 6, 2], [13, 2, -1, 6, -2, -2, 0, 1, -1, -1, -4, 0, 1, 2, 2, 2, 2, 2], [7, 7, 1, 3, 1, -9, 0, 4, 1, -4, 7, 8, 4, 9, -1, -5, 0, -5], [-5, -1, -4, -4, 0, -2, 2, -3, 4, -2, 1, -2, 0, -5, -3, -1, -2, -4], [-7, -1, 3, 1, 5, -4, 3, 3, 4, -7, 2, -1, 2, -1, 0, -4, -2, -4], [4, -7, -2, -4, 1, 5, 2, -1, -1, 0, -2, 3, -2, -3, 0, -1, 3, -1], [-1, 13, 6, -4, -3, 7, -6, 4, -1, 6, 7, 2, -4, -7, -6, 5, -1, -1], [-5, -4, -7, 2, 2, -1, 0, 0, -1, 3, -16, -1, -1, -2, -1, -6, 2, 4], [7, 1, -4, 9, 2, -2, 0, -2, 0, -4, 4, 0, 2, 7, 0, 1, -5, -8], [-2, 1, 8, -3, 0, 1, -8, -4, 1, 8, -6, -7, -3, -8, -4, -2, 2, 2], [6, -4, 3, 0, -5, 6, -4, -1, -1, 7, -2, -1, -3, -12, -2, 6, -2, 2], [3, 1, -9, 6, 3, -2, 2, -2, -4, -8, 1, 0, 4, 11, 4, -1, 1, -6], [-16, -8, -5, 1, 0, 1, -6, 1, 0, 4, -1, 2, -1, -5, -1, -3, 2, 2], [0, 4, 0, 0, -1, 4, -2, 3, -2, -1, 2, 3, -1, 0, 2, 1, 1, -3], [-6, 14, -1, 1, -3, -5, 0, 4, -5, -2, 6, 6, 6, 5, -1, 2, -3, -5], [2, 3, -1, 8, 2, 0, -1, 0, -3, -1, 0, 6, 1, 5, 4, 1, -1, -1], [8, 9, -7, 8, 3, -1, -3, -1, -2, -2, -4, -1, 4, 0, 2, 1, -1, -2], [15, 1, 3, 3, 2, -1, 1, -6, 1, -1, -3, -3, 4, 2, 2, -1, -1, -2], [-12, 5, -4, 2, 0, 1, -9, 4, -5, 2, -1, 0, -1, -2, -1, -3, 3, -1], [1, -2, 9, -10, -1, 5, -7, 2, 3, 10, 0, 9, -1, -8, -9, 3, 4, 3], [-4, 19, -5, -3, 0, -3, 0, -3, -2, 0, -2, 0, 2, -4, 2, -3, 8, 2], [-10, -3, -10, 8, 0, -5, -1, 2, 1, -4, 4, -1, 1, 4, 1, -4, -3, -7], [8, -5, -2, 2, -3, -1, 2, 6, -1, -3, 2, 0, 6, 1, 0, -1, -5, -1], [29, 4, -5, -7, 2, -1, 5, 2, 3, -2, -2, 5, 2, 1, 1, -2, 3, 0], [-3, 4, 9, 2, -1, -2, -1, 2, 0, 0, -2, -7, 0, 3, 2, -3, 1, 0], [5, -9, -6, 0, 4, 1, 1, -1, 0, -1, 4, -1, -6, 1, -1, 0, -1, -1], [-6, -2, 6, -2, 1, -4, 3, -2, 4, 1, -6, -3, 1, -2, 3, 0, 3, 5], [-1, -4, 0, 1, 3, -1, 1, -9, 2, -2, -2, -1, 1, -1, 0, 2, -2, -2], [-1, 0, -3, 2, 7, -3, 4, 2, 0, -3, 5, 0, 2, 8, 2, -1, -3, -4], [13, 10, 3, -3, -3, 0, 1, -3, -1, 3, -2, -4, 1, -2, 0, 2, 0, 7], [1, 0, -3, -9, -1, 3, -2, -4, 1, 10, -5, -5, -2, -4, -3, 1, 0, 8], [5, 10, 4, -9, -2, 5, -2, 8, 0, 6, 11, 9, 1, -8, -4, 8, 0, 1], [7, 7, -2, -3, 1, 2, 0, 4, -2, 1, 0, 7, 1, 2, 0, -1, 1, -2], [-4, 7, -5, 0, 2, 1, -2, 2, -5, 4, 6, 2, -3, 0, 0, 0, 3, -3], [-1, -1, 1, -5, 4, 3, 4, -1, 6, -4, 2, 0, 1, 6, -1, -3, 0, 1], [-4, 6, 2, 3, -1, 2, 1, -1, 1, -3, 6, -5, -1, 1, 7, 8, -3, 1], [3, 2, 4, 2, -2, 0, -4, -3, -1, 5, 1, -2, 1, -8, 0, 4, 0, 2], [-16, -12, 5, 2, 4, 5, -8, -7, -3, 5, -1, -5, -7, -7, 0, -2, 2, -1], [-2, -9, 0, 3, -1, -1, 10, 1, 1, -5, 2, 4, 3, 5, 3, 1, -2, -3], [-5, 7, 4, 3, -2, 5, 5, 10, -1, -1, 3, 3, 0, 4, 1, 4, -5, -1], [0, 4, 6, -1, -2, 2, -4, 1, -6, 4, 4, 0, -3, -7, 3, 3, 2, 1], [12, 2, 1, -9, -5, 6, -1, 6, 2, 0, 3, 4, 1, -5, -4, 11, -3, 2], [1, -2, -1, -4, -3, 3, 2, 3, 1, 3, 1, 0, 1, 0, -2, 5, -7, 0], [-20, 9, 2, -8, -1, 9, -3, 0, 0, 6, -2, -5, -3, -5, 0, 4, -2, -1], [-3, 5, 2, -2, -1, 4, -4, 2, -1, 0, -10, -4, -4, -3, -3, -3, 0, 4], [-2, 15, 11, -4, -5, -2, 9, 4, 9, -4, 4, 5, 5, -1, 1, 4, 1, 2], [11, -8, 5, -1, -4, 3, -1, 2, 3, 4, -6, 3, -3, -4, -4, 3, 3, 3], [6, 5, 0, -3, -3, -3, 1, 2, 6, -2, 2, 5, 4, -2, 0, 7, 1, 0], [1, 3, 0, 1, 5, 2, 3, 4, 4, -1, 0, -7, -1, 6, 1, -2, -1, -2], [17, -10, -4, -3, 2, 2, 4, -5, 3, -6, 3, 1, 2, -3, 5, 3, -2, -3], [0, -12, 6, -3, -5, 2, -1, 2, 1, 3, -6, -7, -3, -3, -2, -5, -1, 6], [-14, -2, -11, 10, 7, -3, 3, 1, -1, -5, -8, -3, 0, 11, 7, -5, 0, 1], [11, -12, 2, 0, -3, 2, -3, 0, 1, 0, 2, 3, -2, 2, -4, -3, -5, -4], [9, 8, -2, -6, 0, -1, 0, -3, 0, 4, -4, 3, 1, -9, -3, 1, 2, 0], [13, 12, -7, 7, 0, -8, 6, 3, -1, -6, 3, 3, 3, 3, 5, 1, -4, -1], [2, 9, 9, 0, -5, -2, 2, 4, 4, 2, 0, 5, -1, -3, -5, 2, -1, -1], [5, -3, 0, -10, -4, 4, 1, -4, 3, 1, 6, 5, 1, -10, -2, 2, -4, 0], [7, -2, -7, 2, 5, 0, 1, -2, 3, -3, -3, -2, 0, 0, -2, -5, 3, -4], [8, 2, -8, 4, -2, -4, -1, 2, -3, 5, -7, -1, 4, 0, 2, 2, 6, -1], [16, 10, 5, 1, 0, -2, -4, 2, -3, 4, -1, 2, -4, -7, 1, -1, 0, 1], [-10, 2, 3, 5, 1, -3, 0, -5, -5, -1, 1, 3, 3, 11, 2, -2, 4, -3], [3, -13, 7, 4, -3, -3, 2, -8, 2, 0, 1, 0, -2, -3, -2, 3, -1, 5], [6, -1, -2, -1, -4, 2, -5, -2, -4, 4, 0, -3, -5, -1, 0, 3, 5, 3], [6, -6, 1, -4, -1, 0, -2, 14, -1, -2, 2, 0, 0, 3, -2, -4, -2, -4], [19, -9, 1, 6, -2, -1, 5, -5, 2, -6, 0, 2, 5, 7, 0, 0, -4, -5], [15, 0, 3, -6, 1, 6, -1, 4, -2, 5, 4, 2, -9, -4, 3, -2, 3, -1], [-10, 11, -3, 8, 3, -7, 2, -6, -2, -6, 5, 1, 5, 13, 6, 0, 0, -4], [23, 9, 6, 3, -4, 2, 0, -1, 2, 0, -2, -2, -1, -6, -1, 3, 0, -2], [-4, 9, 5, 1, 0, -2, -2, 2, 0, -3, 10, 6, 2, 4, 1, 6, 1, -5], [-2, -5, 2, 1, -3, 4, -9, 3, -1, 4, -1, -7, -3, -7, -5, -3, 2, 5], [4, 10, -12, 2, 4, -8, 2, -1, -5, -5, -1, 3, 9, 6, 8, 0, 4, -4], [1, -11, -2, 5, 3, 2, 3, 2, -2, -4, 3, -5, 2, 2, 7, -1, -3, -4], [15, -22, 4, 3, -3, -2, 0, 6, 2, 3, -6, -3, 0, 0, -2, 3, 2, 4], [-14, 11, -5, 2, 0, -3, -3, 3, -3, 2, 2, 3, 1, 8, 5, 1, 2, 0], [-5, -6, 12, 0, -4, -3, -1, 5, 3, 1, -3, -4, 0, 1, -2, -1, -1, 1], [-5, 0, 6, 0, -2, -3, -4, -7, -3, -2, 5, 1, -1, -1, 0, 0, -2, -7], [3, -1, 7, -3, -2, -2, 3, -3, 1, 0, 0, 6, 2, -1, -1, 3, -2, 5], [-16, -3, -3, 6, 3, -5, -1, 0, -1, -1, -2, 8, 5, 6, 0, -3, 7, 2], [4, 1, -1, 11, 2, -6, 1, 3, 0, -4, 3, -6, 3, 12, 2, -2, -1, -5], [-6, 6, 10, 2, -2, 2, -6, -7, 4, 1, 1, 3, -3, -10, -7, -4, -1, -4], [3, 10, -5, -1, 3, 1, -2, -4, 2, 1, -1, -3, -3, -6, 0, 0, -1, -2], [3, -2, -1, 6, 9, -3, -1, -1, -5, -4, 7, 7, -1, 9, 5, 0, 1, -6], [7, -7, 7, -2, -3, 1, -4, 4, -1, 6, 1, -4, -5, -3, -3, -1, 0, -1], [8, 7, 9, -8, -1, 0, -6, 0, 6, 4, -6, -1, -3, -12, -7, -3, 6, 0], [8, 11, -7, 4, 5, -5, -4, 3, -2, -3, -1, 0, -2, 3, -1, -4, 1, 0], [-5, 5, 2, 2, 2, -7, -4, 5, 0, 2, 1, 4, 4, 0, 0, -4, 4, -2], [10, -3, 6, 5, 1, -4, 7, -1, 3, -11, 5, 2, 0, 12, 2, 0, -6, -2], [9, 3, 10, -3, -2, -2, -6, 6, -2, 0, 8, 6, -1, 4, -6, 1, -1, -6], [13, 8, 1, 4, -2, 2, 4, 7, -4, -2, 4, 3, 1, 6, -4, -2, -2, -6], [-5, -11, 0, 1, 2, 3, 2, -5, 0, 0, -7, -4, -2, -2, 1, 4, 1, 4], [5, 2, 0, 6, 2, -5, 0, 1, -1, -5, 6, 4, 6, 9, 5, -2, 1, -5], [-4, 6, 4, -3, -4, 2, 3, 8, 3, -1, -1, -4, 3, 6, -3, 1, -7, -5], [2, 11, 6, 0, -4, 1, -2, 4, 0, 3, 1, 4, 2, 2, -1, -3, 4, 1], [-15, 1, -4, 6, 3, 2, 6, -8, -1, -4, -3, -3, 2, 9, 1, 1, -7, -3], [18, -16, 3, -2, 1, 2, 2, 8, 3, 3, -1, 0, -3, 0, -3, -1, 1, 4], [6, 0, 4, -9, -4, 0, 0, 0, 9, 0, 7, 4, 1, -5, -5, 3, -4, 1], [11, -8, 1, 2, -3, 1, -2, 0, 1, -1, -8, -2, 1, -11, 1, 3, 0, 6], [-10, 0, -7, 3, 11, -4, 1, -6, -2, -8, -2, -4, 6, 5, 8, -2, 2, -1], [-4, 9, -9, -5, -1, -6, 8, 2, 2, -6, -1, 1, 8, 6, 5, -3, 2, 2], [-12, -4, -3, 3, -1, 0, -5, -1, -7, 3, -1, -5, -2, 7, 1, 0, 3, 1], [-5, 10, 2, -2, -3, 5, -8, -2, 1, 8, -4, 2, -2, -4, -7, -5, 1, -1], [14, 9, -7, 0, 0, 2, 7, 6, 0, -4, 3, 5, 4, -1, 6, 3, 1, 3], [-7, 4, 1, -3, 1, 0, 3, 1, 2, 1, 3, 1, 5, 2, -1, 4, -4, -2], [13, 9, -7, 7, -2, -4, 1, -4, -1, -2, 1, -2, 3, 4, 0, 0, 2, 0], [-11, 0, -2, 5, 4, 2, -3, 4, -4, 10, -5, 3, -3, -3, 1, 4, 5, -1], [-5, -3, 10, -7, 2, -3, 2, 2, -1, 4, 7, 3, 2, -2, 3, 5, 3, 2], [-8, 6, 6, -5, -1, -8, 12, 1, 8, -8, 3, 2, 9, 6, -1, 5, -4, 5], [5, 3, -10, 0, 2, 1, 1, 7, -2, 2, -3, -3, -1, -2, 0, -7, 2, -3], [7, -24, 3, -2, 2, 2, -2, 2, 0, 5, -11, -3, -4, 2, -3, 0, 2, 9], [0, -7, 12, -8, -3, 7, 0, 5, 5, 1, -2, -5, -6, -8, -8, 5, -4, 2], [15, 4, 5, 4, 0, -1, -1, 7, 5, -2, -7, 7, 0, -5, -2, 1, 1, 1], [0, 8, -1, -5, 0, 7, -5, 3, 0, 6, -9, -2, -5, -9, -2, -3, -1, 1], [-2, -3, -4, -2, -1, 0, -1, 5, 1, 5, -9, 1, 3, -9, -7, -6, 0, -1], [-3, 12, -4, 6, 0, -1, -2, -3, -4, -6, 8, 6, 2, 10, 3, 4, 0, -6], [-10, -6, 16, -9, -2, 9, -14, -4, -4, 11, 5, 0, -13, -11, -11, 0, 0, 2], [5, 2, -1, -2, -1, 5, -6, -5, 2, 7, -4, -5, -8, -3, -1, 4, 4, 2], [-3, -7, -6, -1, -1, 0, 5, 12, -1, 1, 5, 1, 0, 6, 3, 4, 1, 0], [-2, 1, 5, 3, 2, -7, 7, 6, 7, -7, 0, -1, 5, 3, 1, 3, -1, 1], [-4, 12, -12, 0, -1, 7, 2, 1, -3, 1, -4, -5, -4, 0, 0, 5, -7, 4], [15, 2, -1, 3, 1, 1, -5, -1, 7, -2, -6, -9, -4, -7, -3, 0, -3, -2], [-1, 2, 8, -15, -9, 11, 0, 10, 5, 6, 3, -1, -1, -8, -4, 6, -2, 5], [9, 10, -4, -5, 0, 5, -3, 6, 0, 7, -1, 4, 1, -3, -4, 7, -1, -1], [-26, -14, -6, 6, 1, 6, -5, -3, -6, 5, -1, -1, -3, 4, -1, -2, -1, -1], [-8, -3, -3, -4, 1, -2, -3, 12, 0, 6, 0, 3, 2, -1, -2, -3, 5, 1], [2, -4, 5, -1, 2, -3, 0, 1, -3, -1, 7, 9, -1, 1, 2, -1, 1, 0], [12, -2, 5, -3, -4, 3, -4, 13, 0, 8, 6, 6, -6, -8, -5, 2, 1, 1], [-3, 7, -1, 0, 2, 3, -7, -2, -9, 6, -2, -2, -5, -8, 1, -2, 5, -1], [13, 16, -9, -2, -4, -4, 5, 6, -5, -5, 11, 8, 7, 8, 0, -2, 2, -3], [13, 0, 2, 1, -3, -1, -2, -3, 5, -1, 0, 4, 2, -8, -1, 0, 3, 7], [7, -6, 9, 2, 2, -9, -1, 2, -4, 1, -1, 12, -1, 5, -1, -8, 3, -2], [5, 4, -1, -1, 7, -6, 8, -8, 2, -5, -4, 0, 4, 8, 7, -4, -1, 4], [-6, 4, -7, -2, 0, 3, 2, 3, 0, -1, 4, 1, 1, -7, -1, 0, -3, 3], [-17, 15, 0, -6, 0, 10, -6, 6, -4, 6, 6, -4, -8, -5, -3, 5, -4, 2], [8, 4, 4, 4, -5, -3, -5, 0, -3, 6, -3, 4, 0, -10, 3, 1, 4, 1], [4, 0, -10, 1, 2, 2, 5, 4, 0, 0, -2, -7, 2, 5, 1, 4, 0, 4], [18, -6, 1, -3, -3, 4, -3, 0, 1, 1, -7, -1, 0, -12, -5, 2, 1, 1], [-16, 2, 5, -11, -1, 4, 0, 8, 3, 6, -1, -3, -3, -3, -4, 2, 5, 7], [2, 16, 4, -13, -2, 4, -3, -4, 0, 7, -3, 1, -2, -11, -3, 0, 5, 5], [-11, 0, -12, 4, 5, 2, -1, -7, -8, -4, -4, -10, 1, 9, 6, -5, 0, -5], [-6, 6, 3, -4, 0, 7, -3, -3, 1, -2, -2, -6, -3, -2, -3, 4, -5, 2], [-4, 2, 5, 1, 6, -5, 4, 0, 2, 0, 7, 14, 1, 2, 0, -3, 3, -4], [19, -15, 13, -11, -6, 7, -6, 7, 1, 9, 5, 1, -7, -5, -6, 2, 0, 4], [-4, -3, 6, -6, -2, 1, 0, 0, 4, 5, -3, 12, -3, -12, -7, -1, 2, 3], [19, -7, 7, -3, -1, -4, 2, 0, 4, -6, -2, -4, -1, -2, 1, -4, -5, 1], [17, -5, -2, 0, 4, -6, -2, -4, -3, 2, -5, 0, 1, -3, -1, -5, 4, 0], [0, 1, 11, -3, 0, 0, -5, 1, -4, 5, 3, 4, -3, 4, -2, -1, 1, 1], [-1, 3, -1, 10, -3, -5, 1, -6, 0, -11, 6, 3, 1, -1, 1, -6, -3, -1], [-8, 3, -15, 10, 2, -8, 4, 0, -3, -6, -1, 1, 5, 7, 6, -4, 2, -9], [8, -2, 7, 5, 1, 3, 2, 6, -6, -1, 3, 1, 1, 6, 4, -6, 0, -1], [6, 8, 2, 3, -2, -1, 6, 1, 0, -4, 2, 9, 6, 1, -4, 2, -5, -2], [1, 7, -8, 3, -3, -4, 5, 4, 2, 3, -14, -7, 5, 0, 0, 0, 6, 6], [3, 13, -7, 1, 0, -2, 3, 2, 4, 0, -14, 3, 6, 1, -1, 2, 0, 1], [7, -8, -1, -9, 4, 1, -1, 2, 2, 5, -5, -6, -5, -6, 2, -6, 4, 6], [-7, -3, -5, 2, -1, 0, -3, 3, 1, 3, 2, -1, -2, -3, 2, 9, -5, -3], [-13, -10, -1, -5, -2, -3, -6, -5, 1, 3, -7, -4, 2, 3, -2, -2, 2, 3], [0, -15, -6, 6, 4, -6, -2, -4, 0, 4, -6, -1, 0, 7, 3, -6, 3, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6045_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6045_2_a_bi();".
// 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_6045_2_a_bi(:prec:=18)
    chi := MakeCharacter_6045_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_6045_2_a_bi();".
// 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_6045_2_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-112, -892, -746, 5064, 4744, -12832, -7839, 17347, 4398, -12159, -446, 4582, -411, -935, 156, 97, -21, -4, 1]>,<7,R![-11392, 11008, 586104, -2030612, 1170986, 3080409, -4070968, 289187, 1602728, -523667, -228161, 118380, 10537, -11282, 368, 491, -45, -8, 1]>,<11,R![693248, 3864832, 4871408, -8323264, -23369928, -10818984, 11089859, 8596104, -2364908, -2376168, 364513, 318802, -45733, -20204, 3069, 580, -93, -6, 1]>],Snew);
    return Vf;
end function;