
// Make newform 2013.2.a.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2013_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_2013_2_a_h();".
// 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_2013_2_a_h();".
// 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 := [-1, -15, 74, 386, -802, -745, 1123, 497, -627, -148, 167, 20, -21, -1, 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, 1, 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, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-13953, -27224, 307840, -83942, -411132, 140254, 186606, -62411, -35539, 10454, 2838, -586, -71, -1], [-22401, 24637, 253645, -233864, -713699, 125703, 556827, -4527, -179663, -7732, 25486, 1503, -1317, -82], [-1068, 43281, -209250, 306853, 435843, -466936, -273749, 245494, 74721, -59981, -9372, 6964, 444, -311], [-17712, -67706, 618105, 76842, -1428903, -232439, 1032859, 180436, -320701, -54664, 44412, 7221, -2254, -349], [885, -2743, -92020, 158118, 158677, -248794, -100390, 146488, 29344, -40303, -3957, 5183, 199, -251], [-20941, 113882, 883850, -1595379, -1915614, 2048843, 1308832, -1050382, -388563, 260758, 52066, -31082, -2582, 1423], [-12056, 52752, 607560, -1048784, -1286659, 1511263, 893382, -858752, -271938, 232803, 37201, -29727, -1872, 1433], [3337, -5623, -77090, 162984, 115493, -277112, -67311, 170845, 18455, -48107, -2343, 6246, 111, -303], [21502, 14111, -424595, 591393, 604778, -1177831, -360249, 786304, 104816, -233541, -14292, 31524, 729, -1576], [36556, -176207, -1064330, 2324484, 1876454, -3477208, -1203902, 2019747, 352898, -551558, -47406, 70487, 2367, -3393]];
    Rf_basisdens := [1, 1, 1, 1, 4505, 4505, 4505, 4505, 901, 4505, 4505, 901, 4505, 4505];
    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_2013_a();"
function MakeCharacter_2013_a()
    N := 2013;
    order := 1;
    char_gens := [1343, 1465, 1222];
    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_2013_a_Hecke(Kf)
    return MakeCharacter_2013_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], [-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, 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], [1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, 0, -1], [-1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0], [1, 0, 0, 0, -1, 0, -1, 1, 0, 0, -1, -1, 1, -1], [1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, -1], [1, 0, 0, 0, -1, 0, -3, 1, 1, -1, -1, 0, 0, -1], [2, 0, 0, 0, 0, 1, 0, -1, 1, 0, 1, 0, 0, 1], [1, -1, 0, 1, 0, 0, 2, -1, 0, 2, 0, 0, 0, 1], [-3, 0, 0, 1, -1, 0, 0, 1, -2, 0, -2, -1, 0, -1], [2, 0, 1, 1, 0, 1, 0, 0, -2, 0, 0, 0, -1, 1], [4, 1, 0, 0, 2, -1, -1, -1, 2, 0, 2, 3, -1, 0], [0, 1, 1, 0, 1, 1, 0, 0, 0, -2, 2, 0, 1, 1], [0, 0, 0, 1, 1, -2, 2, -1, -1, 1, 0, 1, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 1, 1, 1, 0, 1, 1, -2, 0, 1, 0, 1, 2], [-2, 0, -2, 0, 0, 1, -1, 0, 0, -2, 1, -1, 3, -2], [2, -1, -2, 0, 0, 1, 2, -2, -1, 1, 1, 0, -1, 1], [2, -3, 0, 0, 0, 1, 1, -1, 0, 2, 1, 1, 1, 0], [3, 1, -1, -1, 0, 0, -1, -1, 1, 0, 2, -1, 0, 0], [-1, -2, 1, 1, -1, 0, -1, 2, 0, 1, -1, -1, 0, 2], [2, 4, 1, 0, -1, 0, -4, 2, -1, -4, -1, -3, 2, -1], [2, 0, 0, 0, -2, 1, 1, -2, -2, 0, 0, 0, -3, 2], [3, 1, 1, 1, 1, 0, -2, 2, -1, -3, -2, -3, 1, -1], [4, 3, -2, 1, -2, 0, -2, 2, 1, -1, -2, -2, 0, -2], [4, 0, -2, -2, 1, -1, 2, -2, 3, 2, 0, 3, -4, 1], [1, 2, 1, 0, -1, -2, -1, 2, 0, 0, -2, -2, 0, 1], [3, 5, -1, 0, -1, -1, -2, 1, 2, 1, -1, -1, -1, -2], [2, 0, -2, -1, 0, -1, 0, -1, 1, 1, 0, -2, 1, 1], [-3, 0, 0, 1, 0, 0, 2, -1, -2, 3, 2, -2, 3, 2], [-1, -5, 4, 1, 1, -1, 3, 1, -4, 2, 0, 0, 2, 3], [0, 3, -2, -1, 0, 1, -1, -1, 3, -3, 3, 1, 0, -2], [5, 0, -2, -3, -4, 1, -2, 0, 1, 0, -1, 0, 0, -1], [4, 2, -2, 1, -1, 0, 0, 0, -1, 0, 1, 1, 1, 0], [6, 3, 2, -1, 2, -3, 0, -1, 3, 2, 1, 2, -2, -2], [4, -4, 0, -3, 2, 1, 2, -1, 2, 1, 3, 4, -1, 3], [1, 1, 0, -1, 1, 3, -2, 0, -1, -3, 3, 0, 4, 0], [-1, 1, 1, 1, -1, -1, 2, -1, -3, 4, -2, -1, 0, 2], [-1, 0, -4, -1, 0, 1, -1, -1, 0, 3, -1, -3, 2, 0], [0, 2, 3, 3, 1, 0, -3, 0, -1, -2, -1, -3, 2, 2], [-8, 0, 1, 1, 1, 0, -1, 2, -4, 0, -3, -4, 5, -1], [1, 8, 1, 0, 0, 2, -2, -1, -1, -4, -1, -2, 0, 0], [4, -1, 1, 0, 0, -3, -1, -1, 3, 2, 0, 4, -4, 1], [5, -2, 1, 1, 1, 3, 3, -2, -1, 0, 3, 1, 1, 0], [1, 2, -4, -2, -2, 2, -2, 2, 2, -1, -1, -3, 1, -1], [1, 3, 4, 3, 4, -1, 0, -1, -2, 0, -1, -1, -1, 0], [-1, 4, 0, 0, 0, 2, -1, -2, -3, 0, 0, -2, 3, -1], [-5, 0, 2, -1, -1, -2, -1, 3, -1, -1, -5, -2, 1, -4], [-3, 1, -4, 0, -2, 3, 2, 0, -2, 1, 1, -1, 2, 1], [4, 2, 1, 2, 5, 0, -2, -3, 1, -1, 2, 4, -2, 1], [-2, 1, 2, -1, -2, 1, 1, 2, -2, 1, -4, -3, -1, 0], [-2, 3, -2, 1, 1, 0, -2, 0, -1, 1, -1, -1, 2, -3], [-5, 5, -2, -2, 1, 0, 1, 1, 1, 0, 0, 1, 3, -2], [3, 1, -1, -2, 2, -3, -2, 0, 5, -3, -2, 0, -2, -1], [-2, -2, -2, 0, 1, 2, 2, -2, 1, -1, 3, -3, 1, -1], [0, 0, -1, -1, 4, 1, -1, 0, 1, -4, 1, 0, 2, 1], [-7, -4, 5, 2, -1, 0, 3, -1, -2, 5, -5, -1, -2, 3], [10, -3, 3, -3, 2, 1, 0, -1, 1, -1, -2, 0, 0, 1], [-1, -2, 2, -2, 1, -2, -2, -1, 3, 0, 0, 0, 2, 1], [3, 1, 2, 1, -2, -2, -2, 4, 0, -1, -5, -2, 1, 0], [2, -2, 1, 2, 3, 3, -1, 0, -1, 0, 1, 0, 2, 4], [-4, 2, 1, 2, 0, 2, 5, -1, -4, 3, -1, -1, 0, 3], [0, -1, 4, 0, -5, -1, -1, 3, -3, 2, -5, -1, -1, 3], [0, -1, -1, 1, 3, 1, 4, -2, 1, -2, 5, 6, 0, 0], [-8, -4, -1, 2, 0, 0, 4, -2, -2, 4, 0, -1, 5, -1], [-1, 2, 0, 5, 0, 1, 2, -1, -4, 1, 1, -1, -2, 3], [3, 1, -1, -3, -3, 1, 3, 1, 3, 2, -3, 3, -2, -3], [-3, -1, 9, 3, 5, 1, 3, -1, -7, 2, 1, -4, 4, 3], [-3, -1, -1, -1, -3, 0, -4, 1, -2, -4, -4, 1, -1, -3], [7, -1, 0, 2, -2, 1, -3, 2, -1, -3, -2, -7, 4, 0], [3, -3, 2, 0, 0, 2, 4, -4, -3, 1, -1, 2, -4, 5], [5, 4, 3, 3, 2, -2, -3, 3, -2, 1, -3, -2, 4, 0], [1, -4, 3, 0, 0, 0, 0, -1, -1, -1, -1, -2, -1, -2], [2, -1, -2, 1, 2, -2, 0, 0, 2, 0, 3, 4, -4, 3], [2, -2, -6, 1, -3, 0, -5, 0, 1, -2, -3, 0, 2, -4], [-7, -1, -1, -2, -4, 1, -3, 2, 3, -2, -7, 0, -2, -3], [0, -3, -3, 2, 2, 2, -4, 0, 3, -3, 3, -3, 0, 2], [-2, 0, 0, -2, 2, -1, 3, -1, -2, 1, 0, -2, 4, 1], [4, -5, -4, -1, 1, 5, 2, -3, -2, -2, 3, -1, 0, 3], [1, 0, -6, 0, -3, 1, 0, -2, 5, 0, 4, 3, -2, 0], [6, 1, -1, -6, 0, 1, 1, -4, 0, -3, 4, 2, -4, 1], [6, 1, -1, 4, 2, 1, 2, -3, -1, 1, 1, 1, 0, 3], [-12, 1, 3, 1, -1, 0, 7, 1, -4, 2, 1, -2, 5, -1], [-6, 4, -4, -3, -1, 1, -5, 0, 1, -4, 1, -6, 2, -2], [-10, -2, 1, -1, -1, -2, -1, 2, -2, 2, 0, -4, 6, 0], [1, -3, -2, -1, -1, 0, 2, -2, 2, 2, 4, 2, -4, 1], [5, -4, 1, -1, -3, -2, -3, -1, -2, 3, -2, -4, 3, 2], [7, 1, 3, -2, -3, 0, -6, 3, 0, 0, -8, -1, 2, -2], [3, 8, -6, -4, -1, 2, -2, -1, 6, -5, 2, 5, -4, -4], [6, 1, -3, -1, 0, 1, 0, -3, -2, 0, 5, 2, -3, 1], [-6, 3, 6, 1, 4, 2, 3, -5, -5, -2, 3, -3, -1, 6], [5, -6, -3, -2, -5, 2, -1, 1, -1, 2, 1, 1, -1, -1], [12, -3, -5, -3, 2, -1, 2, -5, 1, -2, 3, 5, -6, 0], [8, 0, 1, -2, -1, 0, 7, -3, 3, 4, 5, 7, -9, 2], [-6, -4, 2, 1, 4, 2, 8, -2, -5, -1, 7, 4, -1, 6], [1, 4, 1, 0, -1, 3, -2, 0, 1, -4, 1, -2, 1, 1], [-1, -5, 7, 1, 4, -3, 1, 0, -2, -3, 3, 5, 2, 0], [9, -2, 3, 1, -2, -1, -3, 1, -2, 1, -8, -5, 0, 0], [-4, -2, -2, -1, -4, -3, -1, -1, 6, 5, -6, 0, -2, -5], [5, 4, 4, 3, -3, 4, -4, 5, -7, -3, -5, -5, 3, 1], [0, -2, -1, 0, -3, -1, 3, -3, 0, 6, 3, 3, -2, 1], [6, -5, -1, 3, 0, 0, 2, 0, -1, 4, -2, 2, -4, -3], [18, 0, -6, 0, -5, -3, -4, -1, 4, 1, -2, 2, -5, -2], [4, 3, -2, 0, 1, -2, -1, -2, 3, 1, -1, 1, -1, 1], [-11, 2, 0, 2, -2, 1, -2, 7, -3, -1, -3, -2, 1, -2], [-7, 1, 0, -2, -3, 1, 1, -1, -1, 4, -4, 0, 4, -1], [8, -1, 4, -3, 7, -2, 1, -2, 2, -1, 4, 3, -2, 4], [3, -10, 2, 3, 2, -2, 10, -4, -1, 5, 3, 6, -8, 0], [-2, 1, 3, 1, 2, 0, 2, 1, 1, 2, -4, -4, -1, 1], [13, 1, -2, -2, 0, -6, 2, 0, 5, 8, -2, 3, -6, 0], [0, -1, 1, 3, 6, 0, -1, 3, -2, -4, 1, -5, 6, -5], [0, 0, 0, -3, -1, 2, -4, 3, 4, -4, 2, -3, 3, -3], [-5, 6, 4, 1, -1, 0, 0, -1, 0, 2, -3, 0, -2, 1], [8, 0, 10, 0, 5, -2, 1, 3, -2, 0, 0, 2, 3, 4], [8, -6, 1, -4, 6, -1, 5, -6, 1, 0, 4, 7, 0, 1], [-9, -3, -4, -2, -3, 2, 1, 3, 1, 1, -1, -2, 2, -2], [2, 6, -1, -1, -1, 5, -2, 3, -7, -5, 1, -6, 9, 0], [4, -9, 3, 1, -7, -2, -2, 4, -6, 1, -7, -6, 0, -3], [-10, 3, 4, 0, 1, 2, -6, 0, -1, -3, -6, -6, 3, 0], [4, 5, 5, 2, 0, 3, -1, 0, -3, -5, 0, -1, -2, -1], [16, 3, -2, -6, -5, -1, 2, 1, 6, -2, 0, 8, -6, -3], [6, 4, 1, -2, 0, 2, -6, 1, 1, -1, 0, -5, 4, 0], [4, -2, -1, 2, -1, -2, 3, -2, 7, 4, 7, 6, -3, -1], [1, 5, -3, 2, 7, -1, 4, -2, 3, 5, 8, 8, -3, 1], [5, 4, -3, 0, -6, -1, -1, 3, 1, -1, -4, -1, 0, -4], [2, -2, 2, 2, -1, 0, 6, 1, -4, -3, -2, -3, 0, -1], [12, -3, -1, 0, -5, 3, -8, 3, 0, -2, -8, -7, 4, -5], [2, -7, -1, -2, 6, 4, 6, 0, 1, -1, 5, 2, 5, -1], [-6, 4, 3, 1, -1, -4, 1, 1, 1, 7, -9, -4, -3, 1], [5, -1, -4, -3, -3, -2, -3, -1, 6, -4, 3, 5, -6, -1], [-2, 2, -4, -2, 3, 2, 0, -1, 6, 1, 1, 3, 3, -4], [-5, -6, -1, -1, -1, 0, -4, -1, -2, 1, -3, -3, -3, -1], [7, -2, 3, 2, 0, 0, -4, 2, 0, -4, -8, -4, 0, -7], [0, -4, 0, -3, 2, 2, 0, -2, 2, 0, -2, -2, 7, 3], [12, -11, -2, 0, 0, 3, 3, -6, 1, 3, 2, 0, -4, 5], [8, 3, -2, 1, 0, 2, 2, -1, -2, 0, 3, -5, 1, 3], [-17, -2, 3, -5, 0, -4, 0, 3, -1, 0, -3, 2, 1, 1], [10, 9, 3, -6, -4, -1, -4, 4, -3, -1, -4, 0, 1, -3], [-3, -4, 7, 5, -2, -1, 2, -2, -3, 0, -1, 0, 1, 2], [-4, 4, 4, -4, 1, 1, -6, 2, -1, -3, 5, -3, 4, 2], [10, -6, -3, 1, -2, 1, 1, -2, 4, 4, 0, 4, -2, 3], [-5, 0, -8, 4, -3, 1, -7, 3, 2, 3, -3, -3, 7, -8], [-7, 11, 1, 0, 5, 3, 2, 1, -7, -4, 0, -3, 7, -1], [2, 1, 7, -2, -1, -1, -5, 5, -1, -6, -1, 0, 1, -5], [-16, 0, 9, -1, 6, -4, 10, -1, -3, 6, 4, 6, -3, 10], [7, -7, 5, 0, -6, 0, -1, 5, 1, 2, -7, -8, 4, 2], [6, 0, 5, 4, -2, -3, -4, 6, -2, 0, -10, 1, -2, 0], [8, -1, -8, 1, 0, 6, 0, -3, -3, -2, 6, 0, 3, 3], [3, 5, -1, -1, -1, 2, -6, -3, -3, -6, 3, -1, 1, -3], [1, 2, 8, 2, -1, 0, -4, 0, 2, 1, -2, 3, -2, 5], [-3, -3, 4, -2, 0, 1, -6, 0, 3, 3, -5, 1, 0, 3], [3, -4, -7, -3, 4, -1, 5, -4, 5, 3, -2, 4, -6, 1], [-7, 4, 2, 1, 1, -3, -5, 1, 8, -1, -3, -3, 0, -5], [0, 8, -9, -2, -7, 0, -5, -1, 0, -2, 4, -7, 2, -3], [1, 3, 1, -1, -2, 5, -3, 1, 0, -6, -4, -4, 3, 1], [-3, 2, -2, 0, 5, 0, -4, -1, 2, 1, 0, 1, -4, 0], [-17, 4, 0, 3, -2, -2, -8, 6, -3, 0, -12, -11, 7, -2], [14, -2, 6, -1, 2, 4, -3, 1, -2, -3, -5, -1, 2, 5], [-19, 1, 1, 4, 2, 0, 1, 5, -1, 4, -6, -1, 4, -2], [4, -5, 4, -2, 4, 2, -1, 3, -7, -5, 0, -1, 5, 3], [-5, -3, 5, -2, 0, -2, 9, -2, -7, 3, -5, -1, 1, 5], [-2, 1, -4, -2, 1, -3, 0, -3, 1, -2, -5, -3, 1, 0], [11, 5, -7, 1, -2, 1, 0, -1, -3, 4, -4, -5, 4, 1], [-14, -1, 2, 4, -2, 2, 1, 4, -6, 1, 2, -6, 11, -2], [4, -4, -5, -5, -4, 6, 1, 1, -1, -1, 1, -1, 0, -1], [-17, 10, 9, -2, 5, -3, 8, 1, -7, 1, 1, -4, 2, 4], [5, -6, -2, -7, -1, 3, 6, -2, -5, 1, -1, -2, 2, 3], [-15, -3, 6, -6, -7, -2, 5, -2, -5, 7, -6, -1, -2, 5], [-5, 5, 6, 1, -3, -1, -4, 9, 1, -2, -5, 2, 0, -2], [18, -8, -2, 2, 0, -6, 7, -4, 5, 4, -4, 13, -14, -1], [-7, 5, -3, -1, -2, 0, -5, 8, 2, -3, -5, -1, 4, -10], [-16, 0, 3, 5, -3, 2, 4, 2, -8, 1, -7, -10, 7, 1], [-10, 1, 2, 1, 5, 2, -3, 6, -9, -5, -2, -10, 7, -1], [-1, 0, 10, 3, -2, -3, -4, 5, 2, 0, -4, 3, -3, -1], [-2, 11, 1, 1, 6, 3, -8, 4, -9, -14, -2, -13, 8, -5], [-1, 5, -6, -1, -9, 3, -4, 0, -6, -5, -5, -6, 0, 2], [2, -1, -3, 0, 0, 3, -3, -1, 4, -2, 1, 0, -8, 5], [-4, 9, -4, 0, 0, -2, -4, -1, -1, -1, 5, 2, -3, 0], [-12, 0, -1, 4, -2, -1, 4, 2, -6, -2, -5, -1, -1, 1], [-1, 5, 5, 2, 7, 1, 6, -6, 1, 3, 10, 10, -2, 2], [-2, -6, -3, 3, 4, 5, 9, -8, -2, 5, 10, -3, 2, 6], [8, -6, 0, -1, 6, 4, 2, -2, -2, -2, 3, -4, 3, 1], [-1, 3, -1, 1, -1, -1, -1, 3, -4, 3, 1, 4, 2, 4], [8, -3, 0, 3, 5, -1, 3, -4, -3, -1, 10, 5, 2, 4], [12, -3, 1, 5, 3, 1, 4, -7, 2, 4, -1, 4, -4, 1], [-5, 0, 0, -1, 12, 0, -2, -4, 0, -8, 12, 5, -1, 4], [-13, -8, 0, 3, -1, 0, 3, 6, 2, 2, -4, -3, 2, -1], [10, 4, -12, 2, 0, -3, 2, -1, 5, -1, 5, 7, -3, -1], [7, -1, -6, 3, -5, -2, -3, 5, 1, 0, -1, 3, -2, -3], [-12, -3, -2, 1, -5, 1, 3, 1, -9, 4, -10, -13, 8, 0], [-4, 1, -6, -3, -1, 0, -1, 4, 6, -7, 4, 4, -5, -7], [15, -9, 0, 4, -1, -4, 3, 3, -1, 4, 0, 4, -3, -2], [-3, -4, 0, 4, 6, 4, 7, 2, -3, -2, 1, 0, 2, 0], [-18, -5, -1, 2, -4, -1, -2, 3, 0, 0, 1, -7, 5, -1], [2, -11, -9, -1, 2, 3, 4, -7, 9, 4, 7, 8, 1, 1], [1, -7, 9, 3, -1, 0, 2, -6, 3, 2, -1, 6, -11, 4], [5, 7, 12, 2, 11, -2, 1, 0, 2, -7, 10, 10, -2, 3], [5, 0, -4, -1, -4, -3, -4, 1, 4, 2, 0, 7, -3, -5], [13, -11, 3, 4, -2, -5, 6, 4, -1, 5, 1, 6, -3, 6], [6, -3, -5, 2, 5, 1, 4, -4, 7, 1, 8, 2, 1, -4], [-6, 0, 7, -1, 4, -3, 5, 0, 6, -6, 6, 12, -6, 1], [16, 8, 2, -1, -6, -4, -3, -3, 4, 7, -6, 2, -3, 0], [-7, -2, 2, 3, -3, 5, -11, 7, -9, -9, -7, -15, 13, -2], [16, -2, -4, -6, 2, 0, -3, -5, 13, -6, 7, 10, -8, -4], [4, -5, -6, -4, -3, -2, -5, -5, 6, 0, -8, 0, -4, 0], [-15, 10, 6, 6, 1, 8, 1, 0, -13, -2, -1, -4, 2, 3], [-3, -10, -8, 0, 1, 1, 3, -5, 7, 2, -4, 3, -10, -4], [-6, -6, 5, 5, -3, -7, -2, 2, -1, 2, 0, -3, 0, -1], [10, 3, -7, 2, -3, 4, -7, 1, 5, -4, 6, -7, 2, -2], [-2, 7, 5, 1, -4, 7, 4, 0, -8, -1, 3, -13, 2, 6], [1, -5, -2, 3, 4, -1, 1, 2, -6, 1, -4, -2, 0, -2], [-6, 5, 7, 0, 7, 0, -1, 2, -7, -5, -3, -9, 13, 1], [-14, -2, 5, 3, 11, -4, 8, -1, -3, 5, 1, 1, -2, 7], [5, -3, -9, -2, 0, 0, 4, -8, 0, 6, 2, 5, -4, 2], [0, 2, -2, -2, 3, -3, -8, 4, 9, -9, 5, 2, 0, -4], [-2, -10, 2, -1, -7, 2, -8, 3, -5, -3, -14, -11, 0, 1], [5, -9, -2, 0, -1, 3, -3, -1, 2, 3, 3, 5, -1, 3], [-8, -5, 13, 0, 6, -1, 3, -2, 0, -1, 2, 0, 2, 7], [3, -3, 0, -5, -5, 2, 2, -3, -6, -3, 1, -4, 0, 1], [0, -5, -2, -2, 0, 1, -3, -3, 5, -5, -7, -2, 1, -3], [7, 7, -4, 1, -9, 5, -3, 2, -6, -3, -5, -6, 1, -6], [19, 7, -9, -3, 2, 5, -9, -1, 8, -2, -4, -1, 3, -5], [6, -3, 1, 3, 6, 0, 14, -12, -3, 7, 7, -4, -2, 4], [-6, -4, 1, -5, -10, -1, -7, 6, -1, -7, -6, -7, -1, -5], [11, 2, -8, 3, 3, 5, -6, -4, -1, -8, 11, 5, -1, -1], [8, 5, -6, 1, -1, 3, -4, 5, 4, -4, 5, 7, -7, -4], [22, -6, -7, -3, -5, 4, -5, -1, 5, 3, -2, 0, -3, 1], [13, 15, -8, -3, -10, 1, -14, 4, 5, -7, -5, -6, -4, -7], [4, 14, 9, 2, 2, 1, -12, 7, -10, -8, -8, -17, 17, 4], [6, 6, 6, 4, 3, 3, -3, -3, 2, -1, 6, 6, 0, 4], [-2, -16, 0, -1, -4, 7, 0, -2, -2, 2, -7, -4, -2, 4], [-6, -11, -5, -4, -2, 0, 5, -4, 4, 3, 3, 6, 4, 1], [-7, -3, 0, 0, -5, -6, -1, 5, 0, 4, -2, 6, -3, -4], [-11, -1, 16, 5, -1, -2, 3, 4, -16, 5, -8, -8, 2, 7], [1, 1, 3, -2, -1, 2, 1, 3, 0, -2, 9, -2, 0, 1], [-13, -1, 0, 2, 4, -4, 0, 4, -3, -7, 0, 1, -2, -5], [-16, 1, 4, 2, -2, 7, 6, -3, -12, 0, 1, -6, 4, 10], [25, 2, 1, 4, 2, -2, -8, 2, 7, -8, -7, 4, -11, 0], [-5, 3, -2, -1, -9, 0, -12, 6, 6, 1, -12, -6, -2, -8], [-7, -1, -5, 4, -4, -2, -2, -1, 12, 3, -3, 0, -2, -8], [-7, 1, 4, 4, 3, -1, 14, -2, -3, 7, -1, -1, -5, 5], [-2, -2, -4, -1, -3, -2, -11, 5, 1, -7, -4, -9, 10, -6], [-5, -5, 11, 6, 5, 1, 0, 4, -8, 3, -6, -10, 15, 3], [0, 8, 4, 1, 2, -1, 6, 2, -4, 4, 3, 8, -3, 3], [1, 14, -4, 1, 3, 6, -2, -6, 0, -6, 3, 1, 4, 4], [14, -4, 2, -1, 0, -5, 1, 2, -2, -2, 3, 7, -11, 4], [2, 1, -14, 4, -4, 3, -4, 4, 1, -5, -2, -2, 7, -7], [4, 5, 3, -4, 0, -4, -6, 1, -9, -4, -1, -4, 1, -1], [-23, 5, 10, 7, 4, -7, 2, 5, -3, 9, -7, -3, 0, 2], [-1, -15, 6, -3, 3, 1, 10, 2, 2, 3, 2, 8, 1, 3], [7, 4, -6, 1, 3, -2, 4, -4, -4, -4, 11, -3, -3, 7], [-10, -9, -4, 1, -11, 4, 0, 1, -4, -1, 0, -1, 3, -3], [4, 10, -12, -5, -13, 3, -5, -3, 8, -5, 2, 0, -1, -7], [-4, -7, -2, -3, 1, 5, 5, -1, 2, -2, 5, -1, 0, 4], [11, -6, 9, 4, 2, -6, 7, 4, -3, 7, -10, 0, -2, 7], [4, -7, -14, 1, -8, -2, 1, -1, 8, -3, 2, 0, 0, -6], [-10, -4, 6, -2, 3, 4, 1, 8, -4, -6, -3, -1, 4, -2], [-8, -14, 5, -1, 3, -10, 3, -3, 1, 8, 1, 5, -3, 0], [-16, 1, 2, 0, 5, -1, 0, 2, -3, -1, 3, -4, 10, -1], [-10, 1, 6, -3, 10, 9, 3, -3, -3, -6, 7, 11, -1, 2], [-1, 3, 0, -1, -2, -2, 4, 2, -4, 5, -2, -5, 5, 6], [-4, -21, -3, 2, 7, 1, 12, -10, -2, 8, 12, 7, 2, 2], [1, 6, 5, 2, 9, 4, -6, 2, -5, -6, -7, -4, 8, 2], [18, 13, 1, -8, 0, -3, -1, -2, 8, -1, -5, 11, -9, -4], [12, 12, 0, -2, -7, -2, -2, 1, 7, 3, 5, 10, -11, -4], [9, 5, 1, -6, 2, -2, 2, -2, 11, -4, 6, 6, -5, -3], [-15, 6, 0, -1, 12, 0, -2, -1, 2, -4, 5, -1, -1, 3], [-3, 6, 4, 5, 3, -7, 3, 3, -7, -1, 0, -3, 1, 0], [-3, 5, 12, 6, 10, -6, 6, 0, -1, 1, -7, 2, -5, -2], [1, 11, 6, -10, 2, 0, -8, 4, -5, -15, -4, -7, 5, 2], [3, -8, 0, -5, -9, 3, -4, 2, 6, 6, -2, -1, 0, 4], [-3, 10, -12, -7, -5, 0, -8, -5, 2, 3, 2, 2, -5, -1], [19, -2, 0, -5, -2, -1, 3, -7, 9, -3, 16, 15, -10, 0], [4, -3, 3, 5, 3, 7, -4, 6, -2, -5, -2, -7, 9, -4], [11, 11, 4, -4, -2, 1, 1, 0, -8, -7, -5, 0, 1, 3], [-12, 5, -4, -3, -4, -4, 2, -4, -3, 9, 6, 4, -4, 4], [-8, -1, 1, 10, 11, -1, -6, -5, -3, 0, 7, 2, -1, 4], [-4, -6, -4, 2, 12, -1, 14, -12, -1, 10, 10, 7, 1, 7], [6, 6, -3, -4, -5, -4, -9, 10, 10, 0, -17, -2, -2, -14], [3, -8, 9, 0, 0, -1, -1, 0, -3, -4, -3, -4, -6, 0], [-21, 13, -5, -3, 3, -5, -7, 3, 10, -8, -2, 3, -1, -6], [18, 12, -8, -5, 3, -1, -1, -4, -4, -10, 3, 4, -1, -3], [-4, -13, -3, 2, 2, 1, 5, 4, -6, 1, 2, -4, 4, -3], [4, 5, 5, -9, -6, 8, -14, 4, 3, -13, -4, -4, 1, -7], [-2, -1, -7, 1, -2, -6, 1, 4, 10, 8, 3, 7, -7, -7], [10, 3, 2, 6, 10, 0, 3, 4, 2, -3, 8, 1, 2, -5], [-12, 1, -6, 5, 3, 5, 11, -13, -12, 10, 13, 3, 3, 13], [-10, -6, -6, 1, -6, 5, 5, -7, 0, -2, 9, 5, 2, -6], [16, 4, -11, -10, -10, 0, 0, -1, 6, -11, 3, 0, -2, -8], [-26, -7, -5, 2, 0, 1, -1, 6, 4, 6, -5, -2, 1, 1], [-9, 13, 1, 6, 4, 5, -9, 3, -4, -10, 6, -6, 11, -3], [-11, 3, -1, -5, -9, 3, -4, 11, 6, -4, -10, 1, 1, -11], [5, -9, -1, -2, -1, -7, -2, 1, 7, 15, -2, 6, -2, 3], [-6, -1, 0, -3, -8, 0, -7, 5, 7, 6, -18, -9, 3, -4], [-6, -8, -2, 1, 2, 3, 15, -8, -7, 7, 16, 2, -1, 5], [6, -10, -4, -6, -17, 3, -9, 6, 9, 4, -11, -4, 3, -9], [5, -6, 3, -6, 9, 2, 8, -4, 10, 4, 13, 5, 2, 12], [6, 9, -7, 4, -4, -2, 2, -8, 7, 5, 7, 6, -4, 2], [-33, -10, 3, 2, 0, 2, 9, 6, -4, 7, -6, -6, 10, 3], [-17, 29, 5, 1, 1, 0, -2, 5, -2, 1, 0, -1, 2, -1], [14, 6, -14, -4, -2, 5, -7, 1, 0, -9, 4, -3, -5, 0], [-7, -11, 6, -3, -4, 2, 7, -2, 1, 6, 3, 0, 2, 9], [1, 1, 4, -4, -2, 3, -5, 6, -4, -7, -9, -6, 15, -3], [11, 5, -7, 1, 9, 0, -4, -8, 11, -5, 3, 8, -7, 1], [10, 1, -8, -7, -9, 0, -14, 1, 3, -10, 9, 1, 4, -11], [-9, -9, 8, -3, 3, 0, 6, 5, 0, 0, -4, 7, 0, 4], [3, 12, -13, -6, 4, -1, -6, -6, 9, -2, 7, 5, -8, -1], [-14, 1, -2, -6, 3, -2, -2, 1, 2, -8, 7, 0, -4, -5], [15, 5, -12, -7, 0, -4, -11, -6, 10, 3, 5, 5, -1, -1], [-8, 3, 6, 3, -1, 0, 5, -3, -4, 6, 3, -8, 6, 2], [-36, 1, 4, 3, 4, 1, 4, 3, -13, -6, -6, -14, 8, 0], [16, -1, -6, -2, 5, 6, 8, -10, -3, 1, 5, 2, -8, 13], [11, 0, 5, -3, -5, 3, -3, -3, 12, -3, 1, 16, -11, 0], [-10, -2, 1, -2, -4, -4, -3, 5, -2, 5, -10, -8, 6, 1], [7, -7, 4, 6, 7, 0, 10, -5, 4, 6, -1, 6, 3, 7], [13, -5, -1, -2, 1, 3, 10, -4, 7, 11, 1, 11, -12, 0], [-3, 7, 7, 2, 4, 6, -11, 4, 0, -11, 6, -13, 14, 8], [6, 1, -9, -9, -6, 4, 1, -1, -2, 3, -2, -9, 3, -1], [-11, -1, 3, 2, -10, -4, -12, 7, -2, -6, -10, -15, 1, -8], [4, -8, 4, 6, -5, -4, -7, 3, 1, 9, -10, -4, 0, 2], [-18, -2, 7, 3, 6, 6, 13, -6, -15, 7, 6, 4, 0, 18], [6, -6, 8, -8, -2, -4, 7, -1, -4, -5, -2, 9, -4, 1], [30, 9, -3, -5, -14, -2, -6, -4, 7, 1, -4, 4, -5, -5], [15, -5, -5, -3, -4, -10, -2, -5, 7, 7, -3, 10, -11, -1], [3, -8, 7, -4, -6, 0, 3, -3, -1, 8, -5, -1, 8, 1], [-11, 6, -3, 0, 8, 6, 5, -13, 2, -5, 16, 11, -5, 7], [9, -9, 14, 6, -4, 2, 5, 0, -13, 3, -5, -4, -6, 16], [12, -3, 4, 10, 11, 0, 3, 2, -4, -3, 1, -4, 2, 2], [10, 10, 14, 1, 10, 0, 10, -4, -3, -8, 4, 4, -9, 3], [-3, 18, -2, -2, -11, -5, -17, 12, 3, 2, -18, -7, 5, -8], [1, 18, 1, 0, 2, 4, -12, 5, -4, -15, -2, -5, -4, -7], [16, 3, -3, 6, 0, -4, 1, 0, -1, 9, -10, 4, 1, -6], [19, -15, -5, 9, 1, -1, 17, -7, -1, 14, 8, 4, -6, 5], [-13, -7, 1, 5, 14, -6, 8, -3, 1, 0, 4, 4, 7, 0], [-3, -4, -6, -5, -9, -4, -4, -7, 4, 10, -10, -2, -3, 2], [6, 0, -10, -2, 0, -1, -5, 8, 10, -2, 3, 5, -1, -8], [-18, 9, -10, 1, 6, 2, -5, -5, -4, -12, 6, 3, 6, -1], [26, 12, 0, -5, -8, 3, -15, 3, 3, -13, 11, 1, -3, 0], [3, 2, 6, -2, 6, -6, -4, -6, 9, -3, 15, 8, 4, 2], [-8, 7, 3, -2, -1, -3, 11, -2, -3, 0, 3, -2, 2, -1], [7, -7, 4, 0, 16, -2, 9, -5, 14, 1, 19, 15, -14, 7], [12, -5, -8, 5, -3, 6, 12, -11, 3, 8, 17, 5, 0, 11], [-4, 5, -5, -3, 0, -1, -4, -1, 5, -4, 9, 7, -8, -4], [28, 4, -8, -11, -2, 5, -2, -3, 10, 3, -3, 9, -8, -2], [-6, -10, 6, 1, 13, 0, 15, -3, -6, 8, -7, -3, -1, 12], [-3, 13, -1, -5, -2, 4, -10, 5, 6, -10, 6, 5, 4, -3], [-2, -4, 0, 1, 4, 0, 11, -3, 5, 9, 8, 0, 7, 0], [-12, 13, -7, 0, -5, 4, -8, -1, -4, -6, -2, -11, -1, -2], [3, -12, 0, 1, -4, 9, -2, 4, -15, -8, 2, -8, -1, -2], [-11, -15, 8, -1, 6, -6, 4, -4, 5, 2, 4, 17, -5, 9], [7, 11, 11, -2, -8, -5, -2, -3, -3, 5, -10, 1, 0, 4], [-11, -6, 3, 8, 3, 0, 4, -1, -2, 7, 7, 0, -5, 7], [-1, -13, 14, 1, 9, -5, 13, -4, -4, 10, 5, 7, 3, 7], [27, 2, -14, -7, 0, 5, 7, -3, 2, -1, 7, 7, -8, 0], [18, 5, 6, -5, -1, 0, -11, 0, -6, -9, -2, 0, 8, 0], [17, 2, -7, -4, -9, 7, -12, 0, 2, -6, -7, -14, -3, 5], [-18, -4, -8, 1, 15, -1, 11, -7, 11, 3, 13, 17, -5, -3], [-6, 7, 6, -1, 0, 8, -2, -3, -9, -8, -7, -8, 6, 4], [-17, -5, 11, 3, 0, -5, 11, -2, -11, 12, -6, 6, -6, 17], [-4, 1, -9, -4, -11, 0, -1, 6, 4, 5, -11, -5, 0, 0], [-3, -8, 1, -2, 10, 1, 8, -11, 5, 13, 9, 12, -7, 6], [-10, 0, 4, 12, 12, 6, 18, -3, -13, -7, 19, 0, -2, 10], [28, 12, 4, 5, 12, -2, 1, -10, 3, -2, 2, 7, -10, 5], [5, 8, 2, 2, 8, 5, 2, 4, -3, -6, -4, -3, 8, 3], [-20, 2, 4, 1, -3, -2, 0, -3, 1, 5, -9, -3, -8, -5], [17, 4, 8, 3, 1, -6, 4, 0, 6, 10, -7, 3, 1, 5], [5, -14, 4, 5, 4, 2, -1, 0, -3, -2, -4, 4, -9, 5], [-1, -22, 3, 0, 10, -4, 14, 0, 8, 9, 12, 6, -5, 5], [-11, -19, -5, -6, -13, 3, 2, -3, 1, 2, -1, -1, 4, 2], [-11, 4, 1, 4, 1, 9, 9, -5, -7, 8, 7, -6, 5, 11], [1, -13, -1, 0, 5, 1, 17, -7, -2, 6, 8, 15, -11, 4], [18, -5, -15, -8, -12, -3, -3, 5, 8, 8, -3, 8, -13, -10], [-22, 3, 3, 2, 2, 1, -4, 11, -6, -4, -6, -8, 8, 2], [-24, 11, 5, 6, 6, -1, -2, -6, -4, 3, 12, 1, -5, 3], [2, 3, -4, -3, 5, -4, 1, -5, 6, -4, 14, 12, -15, 9], [-11, 1, -6, -4, -3, -4, 7, 5, 7, 9, -10, -6, -1, -3], [14, -6, 0, -1, 6, 4, 10, -4, 6, -4, 16, 22, -14, 4], [11, -15, 5, 6, 3, 5, 10, -5, -8, 9, -8, 1, -12, 3], [6, -7, -5, -4, -10, -7, -15, 2, 5, 8, -11, 0, 4, -8], [-12, -2, 1, -4, 5, 8, 3, 2, -1, -10, 3, 4, 1, 1], [-11, 0, -3, 4, -1, -2, 4, -7, -13, 6, -4, -5, -2, 1], [-12, -8, 3, -4, -4, -2, 14, 1, -1, 10, -4, 7, -3, -2], [17, 6, 4, 2, -3, 3, -8, 3, 3, -3, -5, -6, -1, 3], [-2, 3, 2, -4, 3, -3, 17, -7, -5, 2, 10, 11, -1, 5], [28, -22, -8, 1, -10, -2, 1, -7, 3, 5, -3, 6, -15, 4], [22, -6, -2, 0, 11, 0, 17, -14, 3, -5, 22, 18, -8, 7], [-30, -2, 1, 2, 3, -3, -3, 5, 0, 3, -2, 3, 1, 7], [-1, 2, -8, 0, -14, 3, -11, 4, 6, -4, -3, -2, 2, -10], [11, -14, 11, 3, -13, 2, 2, 6, -4, 8, -17, -4, -1, -5], [10, 22, 9, 2, 8, -3, -1, 2, -9, -4, 0, -7, 8, -5], [-4, 4, -3, 2, -2, -9, 3, 0, -2, 7, 1, -5, 6, 4], [-4, -1, -2, 8, 0, 0, -4, -5, -2, -2, 14, -6, 1, 5], [-1, 10, 0, 8, 8, -5, -1, 0, 0, 2, 6, -6, 14, -6], [2, -1, 6, 4, -2, 12, -1, -3, -5, -4, -1, 0, -8, 5], [7, 15, -9, 6, 9, 4, -3, -8, 0, -9, 7, -1, -2, 1], [-30, -3, 11, -11, -2, -1, -4, 4, -3, -1, -8, -4, 3, -3], [3, -13, -8, 3, -1, -5, 8, -4, 16, 4, 6, 10, -2, -3], [22, -2, -1, -9, -4, -2, -12, 6, 4, -8, -13, -3, -9, 3], [-8, -6, 10, 1, 10, -6, -1, 4, 13, -3, 6, 14, -5, -5], [31, 3, -7, -2, 5, 2, -14, 2, 12, -20, 15, 1, -5, -2], [10, 5, -10, -4, -12, 6, 4, -7, 2, 1, -3, 3, 6, -3], [8, 5, -9, -1, 0, 1, 1, 4, -3, -9, 11, -7, 5, -8], [-5, 16, -19, -10, -13, -2, -9, 5, 5, -10, -4, -7, 4, -12], [34, 7, -14, -4, 5, 2, -11, -4, 17, -5, 7, 6, 2, -2], [-8, 6, -4, 0, -11, 1, -6, 12, -2, -3, -19, -13, -5, 1], [12, -3, 6, 1, 3, -3, -7, 1, 4, -2, 3, -4, 3, -7], [17, 7, -8, -6, -5, 0, -2, 0, 0, 5, -4, -8, 5, -1], [-11, 0, 9, 7, 10, -3, 6, 6, -10, 8, -15, -8, 12, -1], [-4, 9, 8, -1, -2, -3, -13, 4, -4, -1, -8, -2, -5, -4], [19, 1, -16, 3, -8, 3, -4, -10, 5, 1, -3, -9, -3, -6], [-5, -2, 2, 8, 7, -9, 13, -4, -9, 9, 2, 10, -6, 0], [11, 15, -5, -4, -7, -3, 1, 2, 0, 5, 3, -1, 1, 2], [11, 2, -5, -4, -7, -1, -3, -3, 6, -2, 3, -7, 3, -5], [14, 8, 3, -3, 1, -5, -22, 7, 10, -9, -16, -7, -8, -7], [5, -10, 8, -6, 2, -2, 6, -9, -1, 13, -7, 3, -7, 16], [2, -10, 6, 12, 1, 6, 7, 1, -17, 14, 1, -2, -1, 14], [18, 2, -11, 2, -5, -4, -5, -2, 11, 0, 2, 3, 1, -9], [-17, 10, -19, -2, 6, 1, -8, -10, 12, 0, 2, -6, -7, -2], [4, -7, 4, -3, 9, -3, -4, -2, -3, -1, -4, -3, 0, 3], [-11, 14, 8, 12, 8, 1, -10, 2, -6, 2, 5, -7, 7, 0], [34, 14, -9, -3, -2, -2, 0, -5, 11, -5, 12, 12, -14, -4], [18, -27, 1, 0, 5, 4, 6, 1, -2, 0, -5, 3, -5, 4], [22, -3, -3, 5, 7, 2, 1, 1, 9, 5, 2, 11, -5, -4], [6, -13, -18, 1, -2, 2, -2, -7, 10, 4, 1, 8, -10, -4], [-2, 0, 4, 2, 6, 0, 0, -11, 6, 2, 9, 5, 4, -1], [-4, 9, -3, 1, 3, -3, 6, -5, -3, 7, 9, 9, -3, -5], [-13, 4, 4, -1, 11, 2, 6, 2, 9, -6, 9, 1, -1, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2013_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2013_2_a_h();".
// 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_2013_2_a_h(:prec:=14)
    chi := MakeCharacter_2013_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_2013_2_a_h();".
// 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_2013_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2013_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 15, 74, -386, -802, 745, 1123, -497, -627, 148, 167, -20, -21, 1, 1]>],Snew);
    return Vf;
end function;