
// Make newform 4140.2.i.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4140_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4140_i_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_4140_2_i_b();".
// 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_4140_2_i_b();".
// 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 := [9604, 0, 91366, 0, 207293, 0, 170834, 0, 65359, 0, 12842, 0, 1303, 0, 62, 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], [5081910751692, 0, -34954230313558, 0, -33272888607081, 0, -8576884945810, 0, -618906405792, 0, 37688713508, 0, 5531647309, 0, 124491880, 0], [-1200536614236, 0, -20333608753130, 0, -67594251968055, 0, -43285810004227, 0, -10996984617426, 0, -1281439225700, 0, -65729686391, 0, -1099532275, 0], [-14661309337056, 0, -35488539307768, 0, -55244088167692, 0, -39670880077709, 0, -12170363351342, 0, -1666130880928, 0, -95894237026, 0, -1705812243, 0], [-1780511035288, 0, 41015309989992, 0, 92942764199040, 0, 60303704828537, 0, 16453894067170, 0, 2068772711892, 0, 112819382586, 0, 1953768963, 0], [-5607240199924, 0, 27686662202602, 0, 94014051245831, 0, 65524796371428, 0, 18185274766792, 0, 2294781870908, 0, 125161237541, 0, 2166758938, 0], [48526335472260, 0, 241360095702266, 0, 279225699778015, 0, 127544892361671, 0, 27878689600934, 0, 3033785819160, 0, 151124599275, 0, 2498489531, 0], [52453166710812, 0, 244563637180770, 0, 273705813014059, 0, 131178809583478, 0, 30510475697496, 0, 3498253609372, 0, 180777077505, 0, 3048577016, 0], [0, -1997529039534652, 0, -7000105675934470, 0, -6317749502223477, 0, -2219445285553531, 0, -364715172446186, 0, -29349714587204, 0, -1093972349025, 0, -14724018943], [0, -547253380343924, 0, -1672152861614734, 0, -1968758978806087, 0, -865698837976091, 0, -181475071162522, 0, -19163955374068, 0, -937621248815, 0, -15363299351], [0, 1846410077627688, 0, 1450924642416948, 0, -548764502515512, 0, -599990441146187, 0, -163184490956034, 0, -19272328078224, 0, -991263848858, 0, -16590392249], [0, -5858744937772, 0, -21242011460354, 0, -18607349404921, 0, -7051373895279, 0, -1356493811342, 0, -134946581276, 0, -6326074969, 0, -101161287], [0, 55390175420520, 0, 1073723905442576, 0, 2044584049834664, 0, 1414243296007969, 0, 411579426630098, 0, 54206097656324, 0, 3044902710698, 0, 53520056427], [0, -501097207085760, 0, -1739076827603624, 0, -1707650283886842, 0, -713514820478621, 0, -148299700666470, 0, -15657214632828, 0, -765686789820, 0, -12538314823], [0, -3101677908030864, 0, -6990887942311960, 0, -7565559169607386, 0, -3754796963037281, 0, -897509336354710, 0, -104585770514544, 0, -5452625870940, 0, -92337525767], [0, 1698349145805196, 0, 15230830747461262, 0, 17475742704325579, 0, 7840122814493578, 0, 1689896750254768, 0, 181689302841464, 0, 8962991653737, 0, 147338037348]];
    Rf_basisdens := [1, 3306361853624, 3306361853624, 3306361853624, 3306361853624, 3306361853624, 3306361853624, 3306361853624, 162011730827576, 162011730827576, 162011730827576, 839439019832, 162011730827576, 23144532975368, 162011730827576, 162011730827576];
    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_4140_i();"
function MakeCharacter_4140_i()
    N := 4140;
    order := 2;
    char_gens := [2071, 461, 1657, 3961];
    v := [2, 1, 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_4140_i_Hecke();"
function MakeCharacter_4140_i_Hecke(Kf)
    N := 4140;
    order := 2;
    char_gens := [2071, 461, 1657, 3961];
    char_values := [[1, 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], [1, 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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 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, -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, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0], [1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, -1, 0, 0], [-1, 0, 0, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, -1, -1, -1, 1], [-1, 1, -1, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, -1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, -1, 0, 1, -1, -1], [2, 0, 0, 0, -2, 1, 1, -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], [-1, 2, 1, 0, -2, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -1, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, -2, -1, 1, -5, 2, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, -1, 0, -1, 1], [-1, -1, 0, -1, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, -2, -1, 0, 0], [0, -1, 1, 1, 2, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, -2, 0, -2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, -1, 2, 1], [1, 1, 0, -1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, -1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 2, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 1, 2, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -3, 3, 3, 0, -1], [2, -2, 2, -2, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 1, 0, -2, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, -3, -1, -1, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -4, -2, 3, 2, 1, -2], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0], [5, -1, -2, -1, 0, -3, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 2, -2, 0, -1, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -4, -2, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, -1], [-4, -1, -2, 0, 2, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 4, 0, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 3, 2, -3, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, -3, -1, -1, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 1, -1, 1, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, -1, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -3, 2, 3, 0, -2], [5, 2, 2, -1, -4, 5, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -3, 1, 0, 0, 1, 1], [-2, -1, -1, -1, -2, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, 5, -2, -2, -1, 3], [-2, -1, -1, 1, -2, 1, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 2, 2, 0, -2, 4, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 1, 0, -1, -4, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -1, -2, 0, 0, 1, 0], [-7, -2, 0, -1, 2, -3, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, -4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 4, 6, -5, -5, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, -2, -4, -3, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -2, 0, -1, -1], [1, 2, -3, -2, 0, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 4, 1, 3, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -4, -7, 3, 7, 0, -4], [-1, 1, -1, 2, 0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -5, 2, 1, 5, -1, -3], [-1, -1, -4, -1, 4, -3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 1, -1, 3, -3, -2, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 7, -2, -3, 1, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 2, -2, 2, 0], [-4, 2, 0, 2, -2, 4, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -4, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 4, 0, -2, -2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [3, 2, -2, -1, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 3, -4, 1, -2, -3, -1, 5, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 1, 0, -2, 0, -4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 1, -4, -3, -2, 0], [-8, 2, 0, 0, -4, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 5, -3, 1, -1, -1, 2], [0, -3, 4, 0, 2, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, 0, 0, 3, -1], [0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 2, -5, -2, -2, 1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 2, 4, 0, -3, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 9, -4, -2, -2, 3], [-2, 2, 2, 0, 2, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -1, 1, 3, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [12, -1, 4, 0, 0, 6, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-6, -2, -4, 0, 0, -7, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 4, 3, -4, 2, -2], [-5, -2, -1, -2, 4, -1, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 1, -2, -2, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 7, -2, -3, -1, -1], [-6, -1, 0, 4, 0, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 0, -1, 3, 2, 4, 0, -3], [3, 3, -2, 1, -2, -2, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, -4, 0, 2, 2, 4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-8, -1, 0, 4, 2, 3, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 3, 0, -2, 0, -1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 2, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 2, -1, -14, 1, 0, -2, -1], [8, -2, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -8, -2, -2, 4, 2, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, -2, -6, 1, -8, 3, -1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -2, 0, 1, 3, 2, 0], [-6, -2, 1, -1, -2, 6, -2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -4, -2, 4, -3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 10, 1, -1, 2, -1], [5, -2, 1, -4, 2, 0, 2, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, -6, -5, 1, 1, 1, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, -7, -4, -2, 5], [7, -1, 2, -1, 0, -1, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 3, -8, -11, 3, 4, 3, -2], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, -5, 10, -3, -1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -7, 0, 1, 3, -1, -3], [-8, -1, 0, 0, 0, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 9, -5, -5, 2, 3], [-4, 4, 2, -4, 2, -3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-13, 1, 1, -2, 4, -5, -1, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 13, 0, 0, 1, 1], [-12, -1, 0, 0, 6, -5, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-7, -3, 0, -1, -4, -1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 6, -3, -6, 1, -4, 4, -1], [0, 0, 0, 0, 0, 0, 0, 0, 3, 1, -4, 2, -3, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 1, -6, -4, 0, 2, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, -2, 4, 1, -1, -2], [-9, 4, 4, 1, -4, 4, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -4, -3, 0, -1, 5, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -5, 6, 1, 0, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0, 5, 1, 1, 1, -1, -3, 1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 5, -6, -2, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 5, 6, -3, -4, -4, 1], [14, 4, 4, -2, 0, -2, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -1, -7, 11, 1, 0, -1, -1], [-17, 1, 0, -3, 0, -2, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 4, -5, -4, -1, 1], [8, -1, 1, 1, -2, 3, -4, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -6, 4, 3, -2, -3, -3, 0], [6, -2, -2, 0, 8, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [13, 1, -1, 0, 4, -7, 7, -3, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 2, 0, -1, 2, -6, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-14, -4, 0, 2, 4, 0, 4, -6, 0, 0, 0, 0, 0, 0, 0, 0], [-10, 6, 0, 0, 0, 4, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -2, 4, -6, -2, -2, 2], [-7, 1, -3, 0, 4, -2, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, -2, 0, 0, -2], [-4, 0, -2, 2, 2, -6, 5, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 1, 0, 0, -4, 4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0], [-10, 2, 0, 2, -2, 1, -3, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 8, 2, 1, -3, -7, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -1, -7, -1, 1, 2, 1, -1], [9, -2, 0, 1, -4, -1, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -6, 2, -2, -2, -2, -4, -2], [0, 0, 0, 0, 0, 0, 0, 0, -2, -1, -1, -10, 3, -1, 4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 17, -4, -5, -1, 0], [11, 1, 0, 1, 0, -5, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -2, -4, 5, -1, 1, -4, -2], [-5, -6, -4, 3, 2, 1, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-16, -2, 0, -4, 6, 1, 1, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6, -2, 2, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -6, 0, -1, 6, -5, 3, -3, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -6, 0, 0, -2, 4, 6, -3, 0, 0, 0, 0, 0, 0, 0, 0], [15, -1, 4, 3, 0, 1, -5, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, 4, -2, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, -6, -1, -5, 5, 5, 4, 1], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -5, -4, 1, 3, 3, 1], [11, 4, 6, -3, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [10, -5, -2, 0, 4, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [15, -4, 0, 3, 0, -3, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 3, -3, -5, -1, 5, -3, -4], [-9, -2, -2, -5, 2, -1, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0], [8, 3, -6, 0, -8, 2, -4, 6, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 3, 2, 1, 6, -4, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 3, 6, 6, 1, -2], [0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 3, 3, -5, -2, 6, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -7, 12, -3, -2, 2, -1], [4, 4, -2, 0, 0, -3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 5, 0, -1, -8, -1, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [8, -2, 2, -4, 0, 4, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 1, -11, 8, 4, -1, -3], [2, -1, 5, -3, -2, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -7, 4, -5, 4, -1, 6, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-8, -2, -2, 2, 0, 1, -2, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -3, 1, -11, -7, -8, -3, 3], [0, 2, -1, -1, -2, 8, 0, -7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -3, 0, 3, 6, 4, -5], [2, -6, -4, 2, 0, -5, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, -3, -5, -1, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 9, -4, -7, -6, 2, 3], [0, 0, 0, 0, 0, 0, 0, 0, 2, 7, 4, 0, -5, 0, 1, 1], [2, -3, 4, -2, -2, 7, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-12, -1, 4, 4, 6, -1, -5, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 2, -4, -1, 8, 9, 2, -4], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -10, 17, 6, 5, -3, -4], [2, 6, 4, 2, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 4, 5, 3, -1, -2, 0], [13, 1, 6, 1, -8, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-14, -7, 0, 0, 4, -5, -4, -2, 0, 0, 0, 0, 0, 0, 0, 0], [2, 8, 0, -2, -4, 5, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0, -11, 2, 5, 2, -2], [-29, -5, 0, -1, 8, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -8, -7, 5, 11, -2, -2], [0, 0, 0, 0, 0, 0, 0, 0, 3, -4, 5, -10, 1, 2, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1, -14, 3, 3, 1, -3], [0, 0, 0, 0, 0, 0, 0, 0, -3, -1, -5, 8, 1, 2, -1, -2], [-5, 3, 2, 1, 6, 4, -1, -4, 0, 0, 0, 0, 0, 0, 0, 0], [10, 2, 2, 2, -8, 4, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, -4, 0, -3, 8, 2, 4, -4], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 20, -3, -6, 1, 2], [7, 1, -4, -5, 2, -6, -2, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -3, 5, 10, -1, -9, -1, 3], [0, 0, 0, 0, 0, 0, 0, 0, 2, -10, 2, 11, -2, -7, 4, 4], [2, -6, 6, 2, -2, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 5, 4, -8, 5, 2, 1, 3], [-6, 3, -4, -4, 0, -1, 1, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 16, -1, -1, -3, -1], [-9, 4, 1, -2, -4, 3, -3, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 4, -9, -9, 3, -1, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0, -3, 5, 7, -7, -2, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 7, 5, -1, 4, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -3, -5, -2, 10, -1, -3], [-4, -7, -2, 2, 12, -4, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -8, -4, 4, -8, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, -4, -10, 1, 2, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -4, -5, 1, 6, 6, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 9, 5, 0, -7, -8, 2, -1], [8, 0, -2, 4, 6, -9, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 4, 9, 0, -3, -6, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 5, -3, -9, -13, 0, 5], [0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 2, -12, 3, -5, 0, 3], [0, 0, 0, 0, 0, 0, 0, 0, 5, 3, 8, 27, -15, -10, -5, 6], [-20, 3, 0, 0, 2, -5, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 15, 0, -1, 2, 0], [9, 0, 8, -5, -6, 9, -3, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 3, -2, 2, -4, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 9, 4, -6, -5, -5, -1, -4], [-11, 10, -4, -3, -8, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [6, -2, 2, 0, -6, 1, -2, 5, 0, 0, 0, 0, 0, 0, 0, 0], [2, -4, 2, 4, 8, -4, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 4, 2, -4, 3, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 6, -6, -19, 5, 3, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 4, -4, 8, -5, -3, 0, 2], [-4, 8, 0, -4, -10, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, -4, -1, -4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 13, 5, -1, 2, -5], [12, 6, -9, 1, -8, -1, -3, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 4, 4, 13, -10, -5, -1, 6], [17, -2, 4, -1, -4, 5, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-15, -4, 4, 1, 4, 5, -1, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, -10, 5, -33, 7, -1, 2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -7, -14, 3, 5, 3, 1], [0, 0, 0, 0, 0, 0, 0, 0, 5, 3, -7, 9, -6, -4, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, -4, -2, 2, 7, 0, 5, -6, -2], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 1, 15, 0, 2, 6, 3], [0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 5, 15, -12, -12, -3, 3], [8, 0, 6, 6, -6, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -7, 0, -23, 5, 4, 0, 3], [-24, -3, -1, 1, -2, -6, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [-5, -4, -4, 3, 2, 0, 1, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 7, 8, -7, -9, -5, -2, 8], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 6, -5, -5, -1, 0, 6], [0, 0, 0, 0, 0, 0, 0, 0, 6, -4, -6, 21, -4, -3, 2, 0], [15, -9, 2, -1, 8, -7, 6, -2, 0, 0, 0, 0, 0, 0, 0, 0], [9, -8, -6, -3, 4, -3, 2, -10, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 7, -4, 0, 0, 0, -4, -4], [0, 0, 0, 0, 0, 0, 0, 0, 6, 4, 0, 21, -5, -13, -2, 6], [13, -6, -3, 0, 0, -1, -1, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 10, -15, 1, -2, 2, 5], [0, 0, 0, 0, 0, 0, 0, 0, 9, -3, -1, 22, -2, -3, 1, 5], [0, 0, 0, 0, 0, 0, 0, 0, 3, -6, -3, 17, -1, 3, 4, -1], [10, -6, -2, 2, 2, -3, 1, -4, 0, 0, 0, 0, 0, 0, 0, 0], [18, 4, 2, 4, -10, 1, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 8, -7, -8, 5, 7, -1, 1], [3, -6, 4, -1, 6, 2, -1, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -2, 5, -10, 7, 2, 2, -3], [-20, 3, 0, -4, 0, 8, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -4, -2, 12, -2, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, -3, 2, -7, -14, 1, 8, -2, -5], [28, -4, 0, 2, 2, 0, -2, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 2, 3, -5, -1, -9, -2, 5], [2, 6, 2, 6, -8, 3, -8, 10, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -7, 9, 3, -10, -4, -4, 8], [0, 0, 0, 0, 0, 0, 0, 0, -7, -7, -5, 7, 9, 3, 1, -8], [29, -2, 0, -1, 2, 3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 7, 2, -3, 0, 4, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, -4, -2, -7, -2, 6, -4, -2], [-11, 2, -2, 1, -8, 5, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [9, -8, 0, -3, 4, 1, 5, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -6, -6, -4, 4, -2, 4], [15, 5, 4, 3, 0, 3, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0], [-12, -6, -2, 0, 10, -3, -3, 8, 0, 0, 0, 0, 0, 0, 0, 0], [-12, 4, 0, -2, -12, 6, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 4, -25, 4, 3, -2, -4], [0, 0, 0, 0, 0, 0, 0, 0, 4, 14, 3, -11, -6, -8, 1, 3], [0, 0, 0, 0, 0, 0, 0, 0, -7, -3, -2, -8, 7, -1, -3, 2], [12, 6, 2, 0, 2, -4, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 3, 19, -3, 1, 3, 4], [7, -4, 1, 2, 2, -10, -2, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, -10, 11, 17, -7, -3, -6, 5], [0, 0, 0, 0, 0, 0, 0, 0, 2, -8, -5, 6, 1, 9, -1, 5], [0, 0, 0, 0, 0, 0, 0, 0, 4, -2, 4, -12, 0, -2, 4, 2], [12, -6, 0, -6, -2, -1, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 4, 3, 5, -8, -2, -7, -1], [0, 0, 0, 0, 0, 0, 0, 0, -7, 4, 1, 1, 9, 7, 0, -3], [-29, -3, 4, 7, 0, 1, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0], [-24, -2, 2, 2, -8, -1, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -10, -6, 4, 7, 3, 3, 0, -2], [18, -1, -8, 0, 0, -8, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0], [-25, -5, 0, -3, 4, 1, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0], [-8, 0, -2, -2, 4, -6, -6, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-25, 6, 0, -1, 0, -5, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 16, 7, -6, -7, -6, -4, 5], [0, 0, 0, 0, 0, 0, 0, 0, -5, -9, -8, -8, 7, -1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, -1, -5, -1, 10, 7, 7, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 3, 3, 3, 3, -2, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, -2, 10, 3, -6, -1, 3, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -4, 17, -2, 5, -2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 11, -6, -10, -7, 1, 5], [0, 0, 0, 0, 0, 0, 0, 0, 3, -12, 3, -3, 0, -8, 2, 5], [-16, 0, 2, -4, -6, 6, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [17, 7, 1, -4, 4, -5, -5, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 12, 11, -16, -9, -8, -2, 3], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 7, 22, 1, 2, 2, -1], [22, -4, 8, 2, -6, 3, -3, 2, 0, 0, 0, 0, 0, 0, 0, 0], [14, -2, 4, 2, 4, 4, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-22, 2, -8, -2, 0, 2, 6, -5, 0, 0, 0, 0, 0, 0, 0, 0], [-14, -7, -6, 0, 0, 6, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 1, 3, -6, -4, 1, -1], [4, 6, -3, -5, -6, 2, 4, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -6, -1, -8, 5, 5, -1, -7], [0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 13, 19, -1, -9, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -8, -2, 0, 5, 5, -1, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 6, 20, -2, -8, 2, 2], [-4, 4, -4, -2, 4, 8, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, 3, 8, -4, -12, 9, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -5, 12, 5, -7, -4, -2, 1], [-22, 2, 0, 2, 10, 2, 3, -8, 0, 0, 0, 0, 0, 0, 0, 0], [6, 6, 0, 2, -8, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, -4, 8, -13, 4, 6, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, -1, 10, -7, 22, 1, 2, 0, -5], [-16, -9, 0, 0, -2, 3, 1, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 3, -9, -12, 1, 8, 6, -3], [-3, 2, 5, 0, 4, -3, 5, -9, 0, 0, 0, 0, 0, 0, 0, 0], [6, -8, -8, 6, 6, -5, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-20, -2, -4, 4, 0, 2, -10, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-23, 4, 4, -1, 2, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -2, 1, -7, -2, 0, -2, -5, 0, 0, 0, 0, 0, 0, 0, 0], [18, 0, -4, 2, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 9, -8, 6, -3, 6, -1, -5], [0, 0, 0, 0, 0, 0, 0, 0, -2, 7, 11, -13, -1, -8, 7, 5], [0, 0, 0, 0, 0, 0, 0, 0, -7, -10, -7, 2, 5, 6, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, -8, 10, 8, -25, -1, 1, -2, -4], [21, 7, 8, -3, -4, -3, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [43, -4, 2, 1, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -8, 7, -1, 2, 0, -3, 3], [0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 10, 3, 3, -7, 4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -10, -30, 4, 5, -1, -4], [0, 0, 0, 0, 0, 0, 0, 0, -6, -3, -5, -5, 4, 11, 2, -7], [0, 0, 0, 0, 0, 0, 0, 0, -6, -6, 8, 0, 4, 2, -4, 4], [-11, 4, -8, 1, 0, 1, -6, -2, 0, 0, 0, 0, 0, 0, 0, 0], [12, -6, -8, -2, 2, -4, 3, -6, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 6, 4, -4, 8, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -12, -5, -8, 5, 1, 3, 3], [0, 0, 0, 0, 0, 0, 0, 0, 0, -7, 7, -33, 7, -2, 5, 1], [0, 0, 0, 0, 0, 0, 0, 0, 4, -4, -8, -6, -4, 2, -8, 2], [0, 0, 0, 0, 0, 0, 0, 0, 4, -7, 8, 9, -8, -10, -4, 1], [-6, -2, 5, -3, -10, -2, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 6, 0, -16, -2, -4, 4, 2], [16, -2, 14, 0, 0, 4, -2, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -2, -4, 8, 0, 2, 2, 6], [-2, 0, 2, -2, 2, 3, 5, -12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 7, 7, 8, -1, -7, -1, 1], [-6, 2, 0, -6, -4, -4, 4, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 7, 7, -1, 3, 6, -1], [-11, -4, 8, -5, -4, 13, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-12, -6, 0, -2, 4, -12, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [32, 6, 2, 0, 0, 4, -2, 10, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 2, 8, 0, 4, 2, 6, -8, 0, 0, 0, 0, 0, 0, 0, 0], [20, 2, 0, -6, -6, 4, -3, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, -8, 7, 10, -5, -1, -3, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -4, 0, 4, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 8, -2, -21, -1, 5, 0, 0], [-1, -9, -1, 0, 0, -4, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0], [6, -6, 4, 2, 0, -5, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, -3, 2, 16, 2, 6, -6, -2], [20, 2, -7, 3, 0, -1, -1, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, -2, -6, -25, 8, 1, 4, -4], [0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 27, -13, 3, -10, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, -7, -4, -9, -6, -3, 3, 3], [10, 6, 2, 2, -4, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-16, 3, -2, 2, 4, -6, -2, 12, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 11, -1, -33, 5, 3, -2, -1], [0, 0, 0, 0, 0, 0, 0, 0, 6, 10, 4, 24, -12, -8, -4, -2], [-10, 0, 6, 2, -4, 10, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, -2, 5, 37, -15, -5, -6, 1], [0, 0, 0, 0, 0, 0, 0, 0, 11, 1, -2, 5, -4, -3, -2, 2], [10, -1, 5, -3, 6, -7, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, 4, -6, 4, 6, 12, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 7, 7, -32, -6, 2, -4, -1], [-23, -8, -1, 2, -2, -4, -2, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -7, 14, 3, -4, 1, -2], [2, 4, -5, -1, -6, 4, -6, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -7, -10, -22, -2, 5, -6, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 8, -1, 9, 0, -2, 6, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 4, -7, 18, -1, -4, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -8, -5, -4, -3, 5, 2, -1], [12, 2, 4, 0, 4, -5, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 10, 1, -12, 1, 1, 3, 5], [3, 12, 8, 1, -18, 9, -3, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -8, -22, 4, 10, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, -5, -4, 15, -7, 7, -3, -1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 4, -11, -17, 4, 14, -4, -5], [0, 0, 0, 0, 0, 0, 0, 0, -7, 0, -10, 6, 3, -1, 2, 0], [-22, -2, 8, -6, -2, 13, -5, -4, 0, 0, 0, 0, 0, 0, 0, 0], [33, 6, 3, -2, -10, -4, -6, 7, 0, 0, 0, 0, 0, 0, 0, 0], [-7, -6, 0, -7, 8, -5, 5, -10, 0, 0, 0, 0, 0, 0, 0, 0], [-7, 3, -2, -3, -4, 3, 12, -8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 12, -2, -23, 4, 1, 1, -8], [0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 9, 3, -8, -4, -7, 1], [0, 0, 0, 0, 0, 0, 0, 0, -2, 4, 9, 4, -9, 1, -4, 1], [-8, -8, 4, 2, 14, -11, 9, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, -8, -10, 15, 3, 5, -2, -2], [31, 4, -12, -1, -4, -3, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, -15, -3, 4, 7, 4, -1, 2], [10, -5, 0, 4, 0, -3, 10, -10, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 17, 1, 9, -6, -4, -2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4140_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4140_2_i_b();".
// 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_4140_2_i_b(:prec:=16)
    chi := MakeCharacter_4140_i();
    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_4140_2_i_b();".
// 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_4140_2_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4140_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-256, 5024, -4164, -636, 930, 16, -56, 0, 1]>],Snew);
    return Vf;
end function;