
// Make newform 640.2.j.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_640_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_640_j_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_640_2_j_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_640_2_j_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [512, 0, 256, -256, -160, -224, -80, 24, 74, 70, 37, 6, -10, -14, -5, -4, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-28416, -21696, -31296, -9920, 848, 13500, 15406, 10894, 4379, -583, -2676, -2550, -1487, -375, 6, 248, 89, 75], [41344, 30784, 44224, 12880, -2112, -19260, -21274, -14616, -5591, 1062, 3714, 3440, 1963, 460, -14, -342, -111, -100], [38912, 4736, 8896, -36256, -40656, -42352, -24152, -3462, 10426, 14845, 11894, 6294, 1614, -997, -1252, -1110, -380, -191], [58368, 37056, 54816, 5568, -14264, -35404, -32678, -18820, -3819, 5692, 8162, 6148, 2911, 286, -418, -762, -271, -182], [135168, 103488, 148288, 47200, -6464, -65260, -72618, -50632, -19867, 3464, 12878, 11920, 6791, 1610, -78, -1174, -407, -330], [134656, 97664, 136864, 31408, -19848, -73024, -73606, -46732, -14271, 8081, 15520, 12748, 6607, 1101, -506, -1406, -493, -357], [-71232, -43104, -61744, -752, 23236, 46806, 40462, 21455, 2381, -8848, -11038, -7827, -3459, -134, 672, 1013, 359, 228], [-46208, -32320, -45504, -8592, 8624, 26316, 25600, 15672, 4196, -3435, -5762, -4568, -2280, -321, 232, 524, 186, 129], [-528640, -381952, -540992, -125312, 77840, 288616, 291940, 185642, 56748, -32027, -61646, -50762, -26220, -4409, 2000, 5582, 1966, 1413], [588032, 405376, 575296, 103264, -113296, -337152, -326212, -198282, -51964, 45185, 74354, 58634, 29124, 3963, -3072, -6790, -2410, -1671], [670720, 466944, 661184, 127424, -119760, -376472, -368520, -226774, -62446, 47679, 81842, 65414, 32910, 4833, -3140, -7454, -2632, -1861], [-372864, -258560, -364800, -67376, 69792, 211368, 205144, 125222, 33350, -27653, -46182, -36566, -18238, -2547, 1844, 4198, 1484, 1039], [218944, 158976, 225296, 53272, -30132, -118616, -120714, -77418, -24294, 12619, 25220, 20982, 10958, 1929, -774, -2294, -812, -593], [418432, 290560, 412160, 76048, -78416, -238424, -231512, -141626, -37710, 31259, 52186, 41338, 20614, 2901, -2092, -4754, -1672, -1177], [807680, 542976, 767296, 110976, -179280, -476008, -448020, -264246, -61024, 70181, 106298, 81686, 39440, 4527, -4920, -9706, -3438, -2339], [1253888, 893824, 1258944, 268896, -203344, -692848, -689368, -431518, -125726, 82065, 149166, 121006, 61766, 9767, -5268, -13550, -4780, -3419], [677504, 473920, 669120, 125776, -125952, -385988, -375754, -230002, -61705, 50363, 84512, 66986, 33453, 4727, -3394, -7688, -2719, -1899]];
    Rf_basisdens := [1, 640, 640, 1280, 640, 640, 640, 320, 128, 1280, 1280, 1280, 640, 320, 640, 1280, 1280, 640];
    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_640_j();"
function MakeCharacter_640_j()
    N := 640;
    order := 4;
    char_gens := [511, 261, 257];
    v := [2, 1, 3];
    // 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_640_j_Hecke();"
function MakeCharacter_640_j_Hecke(Kf)
    N := 640;
    order := 4;
    char_gens := [511, 261, 257];
    char_values := [[-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, 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]];
    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, -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, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0], [0, 1, 1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0], [0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, -1, -1, 1, 1, 0, 0, 1, 0, 0, -1, 1, 0, -1, 0], [0, -1, 1, 0, 1, 0, 1, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1], [1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 1, -1, -1, 0, 0, 0, -1, -1, 1, 0, 0, -1, 0, 0, -1, 1], [-1, 0, -1, -1, 0, 0, -1, 2, -1, 1, -1, 2, 0, 1, 0, -1, 0, 1], [-1, 0, -1, 1, 0, 0, -1, 2, -1, 1, 1, 2, -1, -1, 1, -1, 0, 0], [3, -1, 1, 1, 0, 0, 1, -3, 1, -1, 0, -1, 0, -2, 0, 1, 0, -1], [-2, 1, 0, 0, -1, -1, 1, -1, -2, 0, 0, -1, 1, 1, -1, 2, 0, 0], [1, 0, 1, -1, -1, 0, 0, -1, -1, -1, 0, -1, 1, 0, -1, 0, -1, 0], [0, 0, 0, 0, -1, 1, -1, 1, 0, 2, 0, 1, 0, 1, 1, -1, 0, 0], [0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0], [-1, 1, 3, 1, 0, 0, 1, -1, 1, -1, 0, -1, 0, 0, 0, 1, 0, -1], [1, 0, 1, -1, 1, 2, -2, 2, -1, 1, 0, 1, 0, 1, 0, -1, 0, 1], [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1], [1, -2, -1, -1, -1, 0, 0, 2, -1, 1, 0, 1, 0, 1, 0, -2, 1, 1], [-1, 0, -1, 1, 0, 0, -1, 1, -3, 1, -1, 0, 1, 0, -2, 0, 1, 1], [0, -4, -2, 0, 1, 0, -1, -1, 0, 0, 2, -2, 0, -1, 0, 0, 0, 0], [2, -2, 1, -1, -1, 0, 1, 1, -2, 0, 3, 0, 2, -1, 0, 0, -2, 1], [-1, -2, -1, 0, 1, 2, -3, 3, -1, 2, 0, 0, -2, -1, 2, -2, 0, 1], [0, 1, -1, 0, -3, -2, -1, 1, 0, 0, 0, -3, 1, 3, -2, 2, 0, 1], [2, 0, 2, -2, -2, 0, 0, -2, 2, -2, 0, -2, 3, 0, -4, 2, 0, 0], [0, -3, -1, 0, 0, -1, -1, 1, 0, 0, 0, -2, -3, 0, -1, -2, 0, 1], [-1, -1, -1, 0, 1, 1, -1, 1, -1, 0, 0, 0, -1, -1, 1, 0, 0, 1], [2, 0, 1, 0, 2, 1, -2, 2, 2, 0, 0, 1, 0, -2, 1, 0, 0, -1], [0, 3, 0, 0, -3, -1, -3, 3, 0, 0, 0, 1, 3, 3, -1, 1, 0, 0], [-1, -1, 1, 0, -1, 1, -1, 1, -1, 0, 0, 0, -1, 1, 1, 0, 0, -1], [-4, -1, -4, 0, 2, 2, -8, 8, -4, 4, 0, 7, -1, -2, 2, -5, 0, 4], [-2, -3, -3, 4, 0, -1, -1, -1, 2, 0, -2, 0, 3, 0, 1, 0, 2, -3], [-4, 4, 2, 0, -2, 0, 2, -2, 0, 0, 0, 0, 0, -2, 0, 1, -1, 0], [0, 0, 0, 0, 2, 0, 2, -1, 2, 0, 0, 0, -2, 1, 2, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, -2, 8, 0, 0, 0, -1, 2, 2, 0, 0, 0], [-4, 1, -1, 0, 5, 2, -1, -1, 4, 0, -3, 0, -1, 5, -2, 0, 2, -1], [-3, 2, -1, 1, 2, 2, -1, 0, 1, -1, -1, 0, 0, 1, 0, -1, 2, -1], [-2, -1, -2, -2, 2, 0, 2, 2, 2, 0, 3, 0, 1, 2, 0, 0, 1, -2], [-1, 3, -2, 0, -1, 1, -3, 3, -1, 0, 0, 6, 3, 1, 1, -2, 0, 2], [0, 0, 0, 0, -2, 0, -2, -2, -4, 0, -1, -1, 4, 2, -2, 2, 2, -2], [1, 1, -1, 0, 2, 1, -3, 3, 1, -1, 0, 2, 1, -2, 1, -1, 0, 1], [2, -7, 3, 0, 0, 3, 0, -3, 0, 0, -1, 1, 0, -3, 0, -1, 1, 0], [-1, 0, -1, 1, 1, 0, 0, -2, 1, 1, -1, 0, -2, 3, 2, 1, 2, -3], [-4, 1, 2, 0, 1, -1, 3, -3, -4, -2, 0, -3, 1, -1, -1, 1, 0, -2], [0, 0, -1, 2, -2, -3, 0, 0, 0, 0, 1, 0, 0, -2, 3, 0, 0, -1], [-6, -1, -2, -2, 4, 4, -6, 2, -2, 2, -1, 3, 0, 0, 0, -1, -1, 2], [3, 3, 0, 0, 0, 1, -2, -2, -3, 0, 2, 0, -3, 0, -1, 0, -2, 0], [0, -1, -2, -1, -1, 1, 1, 1, 0, 0, -3, 0, 1, -1, -1, 0, 0, -2], [5, 2, -3, -1, -1, 2, 0, 2, -1, 1, 0, 1, 0, 1, 0, 0, -1, 1], [1, 5, 4, -1, -3, -1, 2, 1, -1, 1, 1, 0, 0, 0, 0, 1, -2, 1], [-2, -2, 4, -2, 0, 0, 0, 0, 2, 0, 5, 0, 2, 0, 0, 0, -1, 4], [1, -4, 0, -2, 1, 2, -5, -5, -1, 0, -4, 0, 4, 1, -2, 0, 2, 0], [4, 0, 0, 0, -2, 0, 0, 0, 4, -2, 0, 3, 0, 2, 0, -4, 0, 0], [-2, -4, -2, 0, -1, 0, 0, 0, -2, 2, 0, 0, -4, 1, 0, 2, 0, 2], [1, 0, 1, -1, 1, 0, 2, -2, 3, -1, -1, -2, -4, 1, -2, 2, 1, 3], [-3, 4, 1, -1, -1, -4, 0, 3, -1, 1, 3, -2, 0, 2, 0, -1, 0, 1], [-1, 0, -1, 1, 1, 0, 0, 7, 1, 1, 3, 4, -1, -6, 5, -5, -4, 2], [-4, -1, 2, 0, -4, -2, 4, 0, 0, 0, 3, -3, 0, 0, 0, 1, -1, 0], [2, 0, 2, -2, 2, 0, 4, -1, 8, -2, 4, 2, 2, -1, 2, 0, -2, -2], [9, -1, -3, -1, 0, 2, -1, -1, -1, 1, -4, 5, 0, -2, 0, -1, 0, 1], [-6, -8, -2, -2, 2, 2, -4, 0, -2, 2, -2, 4, 0, -2, 0, -3, 1, 2], [-3, 6, 0, 0, 4, -2, 2, 2, 3, 0, 4, 0, -6, 4, 2, 0, -4, 0], [2, 0, 2, -2, -6, 0, -4, -1, -8, -2, 3, 1, 1, -1, -5, 1, -1, 7], [0, 3, 2, 0, -1, -3, 1, 1, 0, 0, -1, 0, -3, -1, 3, 0, -3, 2], [3, 3, 1, 2, -4, -1, 0, 0, -3, 0, 0, 0, -3, -4, 1, 0, -2, 1], [-1, 0, -1, 1, 0, 0, -1, -3, -11, 1, -5, -4, 3, 4, -2, 2, 3, -3], [1, 7, 0, 0, -2, -3, 0, 0, 1, -2, 0, -2, 7, 2, -3, 4, 0, 0], [-1, 2, 0, 0, 0, 2, 2, -2, -1, 4, 0, 0, 2, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, -8, 0, -1, -1, 2, 2, -2, -1, -1, -2], [-2, -4, 3, 0, 0, 1, 6, -6, -2, -2, 0, -3, -4, 0, 1, 2, 0, -3], [-3, 0, -3, 3, -1, 0, -4, 5, -1, 3, -1, 2, -4, -2, 0, -3, 0, 5], [4, -1, -4, 0, 0, 0, 0, 0, 4, -2, 0, -1, -1, 0, 0, -1, 0, 4], [6, 2, 3, 0, 0, 1, -2, -2, -6, 0, 3, 0, -2, 0, -1, 0, 0, 3], [-2, 2, -2, -2, 2, 2, -1, -1, 2, 0, -6, 0, -2, 2, -2, 0, 4, -2], [0, 0, 0, 0, 1, 0, 1, -2, 0, 0, 3, 3, 3, 2, 1, -3, -3, 1], [1, 4, -1, 3, 3, 0, 0, -2, 3, -3, -1, -2, 0, 1, 0, -1, 4, -3], [-5, -1, -2, 1, 1, 1, 0, -1, 1, -1, -7, 6, 0, 0, 0, -3, 4, -1], [0, 3, 4, -4, 0, 0, 0, 0, 0, 0, 5, 0, -3, 0, 0, 0, -5, 4], [2, -1, 5, 0, 2, 1, 9, -9, 2, -4, 0, 2, -1, -2, 1, 2, 0, -5], [-3, 0, -3, 3, -3, 0, -6, 8, 3, 3, 5, 8, 6, -5, 4, -6, -3, 7], [-3, -1, -1, 0, -3, -3, 0, 0, -3, -1, 0, 0, -1, 3, -3, -3, 0, 1], [0, 0, 0, 0, 4, 0, 4, 2, 8, 0, -1, -1, -2, -2, 2, -3, -3, 2], [-2, -1, 4, -2, 0, 4, -2, 0, -2, 2, -1, 3, 0, -2, 0, 1, -3, 2], [4, 0, 4, -4, -4, 0, 0, -1, 2, -4, 6, 2, -3, -3, 3, 0, -4, 1], [1, -2, -2, 0, -6, -2, 0, 0, 1, 0, 0, 0, -2, 6, -2, 0, 0, 2], [-7, 0, -3, 2, 1, 0, -1, -1, 7, 0, -6, 0, 0, 1, 0, 0, 4, -3], [-4, -5, -6, 2, 5, -1, -1, -1, 4, 0, 1, 0, 5, 5, 1, 0, 0, -6], [12, -1, 2, 0, -4, -6, 4, 0, 0, 0, 9, -9, 0, 0, 0, 5, -5, 0], [-14, 4, -4, 2, 2, -2, 0, 4, 2, -2, -4, 2, 0, 6, 0, 0, 2, -2], [-4, -2, 4, 0, -6, 0, -4, -4, 4, 0, 9, 0, 2, -6, 0, 0, -6, 4], [6, -3, -2, 2, -1, 1, -1, -1, -6, 0, 3, 0, 3, -1, -1, 0, 3, -2], [-4, -1, 4, 0, -4, 2, -4, -4, 4, 0, 5, 0, 1, -4, -2, 0, -1, 4], [-4, 7, 7, 0, -3, -4, 7, -7, -4, -4, 0, -7, 7, 3, -4, 8, 0, -7], [-3, 5, 2, 0, 0, 3, 0, 0, -3, 0, 0, 2, 5, 0, 3, -2, 0, -2], [0, 0, 0, 0, -2, 0, -2, -6, 0, 0, -6, -6, -4, 6, -4, 2, 2, 0], [6, -3, -4, -2, 2, -4, -4, 6, -2, 2, 5, -3, 0, 4, 0, -1, -1, 2], [7, -2, 3, -2, 1, 6, -3, -3, -7, 0, -6, 0, 2, 1, -6, 0, 0, 3], [-10, -7, 0, -2, 0, 2, -2, 2, -2, 2, -1, 3, 0, 0, 0, -5, 3, 2], [-2, 0, -2, 2, -1, 0, -3, 6, 6, 2, 0, 2, 0, -4, 4, -4, -2, 6], [-2, 0, -2, 2, 4, 0, 2, -4, -2, 2, -6, -4, 1, 6, 0, 4, 6, -8], [-1, -3, 4, -3, 3, 7, -6, 5, -3, 3, 1, 2, 0, 2, 0, -3, 0, 3], [-12, 4, -6, 4, 2, 0, 2, 2, 12, 0, -5, 0, -4, 2, 0, 0, 1, -6], [-5, -1, -3, 4, -3, -3, -5, -5, 5, 0, 4, 0, 1, -3, 3, 0, 0, -3], [-3, 0, -3, 3, 4, 0, 1, 1, 19, 3, -3, 0, -5, 2, 2, -4, -1, -3], [-7, -1, -3, 0, 0, 3, -10, 10, -7, 6, 0, 10, -1, 0, 3, -4, 0, 3], [-1, 0, -1, 1, -3, 0, -4, 0, 5, 1, -3, -2, -4, 1, -4, 3, 4, 5], [3, 0, 3, -3, 1, 0, 4, 1, 1, -3, 3, 0, 5, -4, -1, 3, 0, 4], [2, 3, -2, 0, -1, -1, -7, 7, 2, 8, 0, 3, 3, 1, -1, -4, 0, 2], [-3, 0, -3, 3, 0, 0, -3, 0, 17, 3, -8, -5, -5, 3, 1, 0, 3, 0], [-3, 0, 4, 0, 5, -2, 6, -6, -3, -5, 0, -6, 0, -5, -2, -1, 0, -4], [-4, -2, 0, 0, -3, -3, 1, -1, -4, -2, 0, -11, -2, 3, -3, 3, 0, 0], [-10, -6, 6, 6, -6, -4, 12, -6, 6, -6, 2, -8, 0, 0, 0, 5, 1, -6], [-1, -1, 0, 5, 5, -3, 0, -7, 5, -5, -3, -2, 0, -2, 0, 5, 0, -5], [1, 0, 1, -1, 0, 0, 1, -7, 7, -1, 1, 0, 7, 6, -6, 4, 3, 1], [-4, -4, -2, 4, -4, 2, -2, -2, 4, 0, -3, 0, 4, -4, -2, 0, 4, -2], [7, -3, 0, -1, 0, -1, -1, 6, -1, 1, 0, 1, 0, 5, 0, -4, 3, 1], [-2, -3, 2, 2, 2, 4, -4, -4, 2, 0, -11, 0, 3, 2, -4, 0, 1, 2], [2, 4, 4, 0, -1, 0, 2, -2, 2, -6, 0, -8, 4, 1, 0, 6, 0, -4], [3, 0, 0, 0, -5, -6, 12, -12, 3, -5, 0, -14, 0, 5, -6, 7, 0, 0], [0, 1, -7, 2, -4, -9, 6, 1, 2, -2, 3, -5, 0, 3, 0, 3, -1, -2], [3, 3, -3, 1, -4, -2, 5, 3, 1, -1, -6, 5, 0, 4, 0, 1, 0, -1], [-4, 1, -2, 0, 1, -3, -1, 1, -4, -4, 0, -1, 1, -1, -3, 5, 0, 2], [-6, 3, -3, -2, 0, -3, 3, 3, 6, 0, 0, 0, -3, 0, 3, 0, -6, -3], [-7, 4, 7, -6, 0, 0, 2, 2, 7, 0, 6, 0, -4, 0, 0, 0, -8, 7], [13, -4, 1, 3, 3, 4, 0, -8, 3, -3, -8, 5, 0, -5, 0, 0, 3, -3], [12, -4, -4, -4, 6, -2, 4, 4, -12, 0, 1, 0, 4, 6, 2, 0, 0, -4], [-9, -4, -1, -1, 4, 4, -5, -4, -1, 1, 1, 0, 0, -5, 0, -3, 2, 1], [-2, 3, -8, 6, 6, -2, 6, 6, 2, 0, -9, 0, -3, 6, 2, 0, 5, -8], [12, 4, 6, 0, 2, 0, 8, -8, 12, -8, 0, -7, 4, -2, 0, 6, 0, -6], [2, 0, 2, -2, -4, 0, -2, -8, 14, -2, -7, -9, 10, 6, -8, 2, 0, 0], [7, -4, 3, 1, -5, -4, 6, -7, 1, -1, 3, -4, 0, -6, 0, 3, -2, -1], [-1, 0, -1, 1, 3, 0, 2, -6, 11, 1, 1, 2, 0, 7, 0, 3, 4, -1], [3, 0, 3, -3, -8, 0, -5, -1, -3, -3, -5, -8, 6, -2, -6, 3, 0, -1], [1, 0, 1, -1, 2, 0, 3, 0, -11, -1, 4, 3, -3, -1, 3, -4, -5, 2], [11, -1, -3, 1, 8, 2, -7, -1, 1, -1, 0, -1, 0, 0, 0, -5, 6, -1], [-1, 0, -1, 1, 6, 0, 5, -8, 11, 1, -4, -3, -5, 9, -3, 4, 5, -6], [-4, 2, 4, -2, -8, -6, 6, -2, -2, 2, 2, 0, 0, -4, 0, 2, -4, 2], [6, 4, 4, 2, 2, 0, -2, -2, -6, 0, 1, 0, -4, 2, 0, 0, -1, 4], [2, -1, 3, 0, -5, -7, 6, -6, 2, -6, 0, -10, -1, 5, -7, 4, 0, -3], [-8, 1, -7, 0, -1, 6, -11, 11, -8, 12, 0, 5, 1, 1, 6, -10, 0, 7], [3, 0, 3, -3, -2, 0, 1, -1, -11, -3, 5, 2, -3, -2, 4, 2, -1, -1], [4, 0, 6, 0, 3, 0, 10, -10, 4, -6, 0, -8, 0, -3, 0, 10, 0, -6], [2, 0, 2, -2, 6, 0, 8, 6, -6, -2, 3, 1, -10, -8, 4, -1, -3, 6], [0, 0, 0, 0, 4, 0, 4, 1, -2, 0, -1, -1, 1, -1, -1, -3, -3, -7], [5, -3, 5, 0, 5, -1, 9, -9, 5, -2, 0, -4, -3, -5, -1, 2, 0, -5], [4, -2, 0, 0, 1, -7, 5, -5, 4, 0, 0, -5, -2, -1, -7, 7, 0, 0], [10, -1, 4, 0, -1, -5, 7, 7, -10, 0, 9, 0, 1, -1, 5, 0, -6, 4], [-8, 0, -8, 8, -2, 0, -10, 9, 6, 8, -4, 4, 0, -1, 4, -4, 4, -2], [-7, 7, -6, -3, 0, -3, -3, 8, -3, 3, 3, 0, 0, 5, 0, -3, 0, 3], [-2, 0, -2, 2, -4, 0, -6, 6, -10, 2, 2, 4, -3, -4, 6, -6, -4, 0], [8, 8, -4, 8, -2, -4, 0, 0, -8, 0, -3, 0, -8, -2, 4, 0, -2, -4], [4, 0, 4, -4, -4, 0, 0, -10, 12, -4, 0, -4, 7, 6, -2, 6, 2, 0], [-1, 0, -1, 1, -5, 0, -6, 6, 1, 1, -1, 0, -12, -5, 4, -4, -3, -1], [-3, 0, -3, 3, 5, 0, 2, -2, 7, 3, -5, -2, -12, 5, 0, -1, 2, -5], [4, 0, 4, -4, 10, 0, 14, -11, 2, -4, 0, -4, 5, 7, -5, 6, 2, -11], [1, 2, 0, 0, 3, -4, 0, 0, 1, -1, 0, -2, 2, -3, -4, 7, 0, 0], [5, 5, 2, 2, -3, -5, 3, 3, -5, 0, 8, 0, -5, -3, 5, 0, -10, 2], [15, 3, -1, 1, 2, -4, -1, 5, 1, -1, -2, 1, 0, 6, 0, -1, 2, -1], [3, 4, -2, -6, 8, 0, 2, 2, -3, 0, -10, 0, -4, 8, 0, 0, 4, -2], [-16, -3, -7, 0, 5, 4, -7, -7, 16, 0, -13, 0, 3, 5, -4, 0, 10, -7], [16, -3, -4, -4, 5, 3, 1, 1, -16, 0, -11, 0, 3, 5, -3, 0, 5, -4], [2, -1, 0, -3, 1, -3, 1, 1, -2, 0, 7, 0, 1, 1, 3, 0, -4, 0], [-8, -6, 2, 0, 0, 4, 0, 0, -8, -4, 0, 3, -6, 0, 4, -2, 0, -2], [-2, 0, -2, 2, 4, 0, 2, 8, -18, 2, -3, -1, -14, -6, -2, -2, 0, 6], [11, -7, -4, -3, 7, -1, -10, 5, -3, 3, 0, 3, 0, 2, 0, -4, 1, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_640_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_640_2_j_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_640_2_j_d(:prec:=18)
    chi := MakeCharacter_640_j();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_640_2_j_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_640_2_j_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_640_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![256, 0, 2704, 0, 10624, 0, 19744, 0, 18624, 0, 9464, 0, 2656, 0, 408, 0, 32, 0, 1]>,<7,R![288, 4128, 29584, 108160, 239360, 315488, 244448, 85184, 17328, 11648, 14648, 4160, 352, -120, 200, 32, 2, -2, 1]>],Snew);
    return Vf;
end function;