
// Make newform 784.2.x.k in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_784_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_784_x_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_784_2_x_k();".
// 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_784_2_x_k();".
// 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 := [256, -256, 256, 128, -208, 256, -16, -68, 121, -34, -4, 32, -13, 4, 4, -2, 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], [70080, -5280, 17760, 60728, -12840, 18090, 25377, -7980, 9600, 6738, -1905, 3150, 992, -180, 465, 46], [166912, -64000, 54208, 146096, -72848, 63096, 56620, -36939, 33430, 12164, -9008, 10439, 1012, -900, 1526, -99], [-20608, 11648, -11424, -8176, 11064, -9520, 1244, 4141, -4130, 1724, 528, -1057, 684, 4, -26, 117], [-16224, 48, -2896, -15052, 1484, -2899, -7041, 1098, -1540, -2213, 263, -520, -438, 23, -79, -44], [-31232, 45248, -14240, -18080, 42936, -17256, -1574, 19519, -9590, 2308, 4994, -2687, 1276, 688, -336, 243], [-1952, -4880, 1648, -3480, -4484, 1953, -2748, -1838, 1028, -1301, -434, 258, -402, -51, 26, -70], [-167808, 94848, -70464, -125232, 98472, -74700, -38052, 45765, -38706, -3768, 10584, -11145, 1380, 1080, -1474, 429], [-44960, 26720, -17616, -33752, 27626, -18863, -10312, 12863, -10076, -989, 3022, -2939, 392, 317, -392, 119], [-17920, -26752, 5760, -24520, -22944, 7432, -16719, -8848, 3896, -7108, -2085, 826, -1980, -254, 53, -262], [15872, -41536, 13376, 2112, -38028, 16912, -6660, -15917, 9062, -5124, -3800, 2361, -1872, -476, 274, -293], [243072, -135296, 101920, 181968, -140776, 107180, 55660, -65511, 55318, 5760, -15128, 16211, -1900, -1536, 2182, -607], [79360, -102400, 32288, 41392, -97456, 43576, 1292, -42643, 23878, -7196, -10352, 6447, -3532, -1252, 806, -619], [49536, -113792, 34784, 8144, -104616, 42084, -15508, -44359, 22646, -12952, -10856, 5843, -4860, -1384, 630, -767], [-26560, -8496, -4784, -28238, -3980, -4253, -14330, -330, -2276, -4791, -68, -926, -1058, -29, -164, -108], [-145920, 188672, -82656, -75744, 186216, -97080, 2382, 81219, -52110, 15924, 19662, -14283, 7404, 2296, -1764, 1311]];
    Rf_basisdens := [1, 3008, 6016, 6016, 1504, 6016, 1504, 6016, 1504, 3008, 3008, 6016, 6016, 6016, 752, 6016];
    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_784_x();"
function MakeCharacter_784_x()
    N := 784;
    order := 12;
    char_gens := [687, 197, 689];
    v := [12, 3, 8];
    // 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_784_x_Hecke();"
function MakeCharacter_784_x_Hecke(Kf)
    N := 784;
    order := 12;
    char_gens := [687, 197, 689];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, 1, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, -1, -1, -1, 1, 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, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, 0, 0, -1, 1, -1, 0, 0, 2, 0, 0, 0, 1, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 0, 1, 0, -4, 1, 1, 1, 0, 0], [0, -1, -2, 0, -2, 2, -2, 2, 2, 1, -2, 1, 1, 0, -2, 0], [0, -1, -1, 0, 0, -1, 1, 0, -1, -1, 0, 1, -1, 0, -1, 1], [-1, -2, 0, 0, 1, 2, -2, 0, 0, 2, 0, 0, 0, 2, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, -3, -1, -1, 0, 0], [0, 2, 0, 0, -3, 2, 2, 3, -2, 2, 3, -2, 2, 0, 0, 4], [0, 1, 0, 4, 0, 3, -1, 0, 0, 3, 0, 0, 0, 3, -4, 4], [-4, -2, 0, -4, -4, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -4], [0, 3, 0, 0, 0, -1, 3, 0, -3, 1, 4, -3, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, -2, 0, 0, 0], [2, 0, 0, 2, 0, 0, 0, 2, 3, 0, 2, 0, 0, 3, 0, 0], [0, -1, 0, 0, 3, 0, 0, -3, 0, 1, 3, 1, 1, 0, 0, 0], [-2, 0, -4, 0, 0, 0, 0, 2, 4, 0, -2, 4, 4, -4, 0, 0], [0, 0, 0, -2, 10, -6, 0, 0, 0, -6, 0, 0, 0, -6, 2, -2], [0, 0, 2, 2, 0, 0, 0, -6, -2, 0, 0, 2, -2, 2, 0, 0], [0, -2, 2, 0, 0, -4, -2, 0, 2, 4, -6, 2, 4, 0, 2, 2], [0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 4, 0, 6, -6, 0, -6, 0, -6, 0, 0, -6, 0, 4, -4], [-4, 3, 0, 2, 0, -3, 3, 0, 0, 3, 0, 0, 0, -3, 2, 2], [1, 0, 0, -4, 0, 0, 0, 1, 0, 0, 1, -5, 5, 0, 0, 0], [0, 1, -6, 0, 4, 3, -1, -4, 1, 3, 0, -1, 3, 0, -6, 6], [0, 0, 0, 0, 2, -2, -2, -2, 2, 0, -2, 0, 0, 0, 0, -2], [5, 0, 4, 0, 0, 0, 0, -5, 0, 0, 5, 0, 0, 0, 0, 0], [6, 4, 0, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 1], [4, 1, 0, 1, 0, -5, 1, 0, 0, 5, 0, 0, 0, -5, 1, 1], [0, -2, -8, 0, -4, 1, -1, 4, 1, 2, -4, 2, 2, 0, -8, 0], [0, 0, 2, 2, 0, 0, 0, -6, 0, 0, 0, 0, -4, 4, 0, 0], [2, 0, 0, 2, 2, 5, 5, 0, 0, 0, 0, 0, 0, 5, 0, 2], [0, -2, 0, 0, 1, 4, 4, -1, -4, -2, -1, 2, -2, 0, 0, -2], [0, 0, -1, -1, 0, 0, 0, 4, 1, 0, 0, -1, -3, 3, 0, 0], [1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 1, -8, -8, -1, 0, 0], [0, -4, 2, 0, -6, 0, 0, 6, 0, 4, -6, 4, 4, 0, 2, 0], [0, -5, 0, 0, -4, -3, 5, 0, 0, -3, 0, 0, 0, -3, 0, 0], [0, 0, 8, 0, -1, -1, 1, 1, -1, 0, -1, 0, 0, 0, 8, 0], [2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2, -8, -8, 0, 0, 0], [-5, 2, 0, -4, -5, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, -4], [0, 2, 0, 0, 0, 2, 2, 0, -2, -2, 4, -2, -2, 0, 0, 0], [0, 0, 1, -1, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0], [1, 4, 0, -4, 1, 2, 2, 0, 0, 4, 0, 0, 0, 2, 0, -4], [0, 0, -4, -4, 0, 0, 0, -4, 5, 0, 0, -5, 1, -1, 0, 0], [-4, 0, 0, 0, 4, 4, -4, 0, 0, 0, 0, 0, 0, 4, 0, 0], [0, 2, 0, 0, 0, -4, 2, 0, 0, 4, 0, 0, 0, -4, 0, 0], [-2, 0, -6, 0, 0, 0, 0, 2, -2, 0, -2, 7, 7, 2, 0, 0], [0, -3, 0, 0, -5, -2, -2, 5, 2, -3, 5, 3, -3, 0, 0, -4], [0, 0, -2, 0, -14, -4, 0, 14, 0, -4, 0, 0, -4, 0, -2, 2], [-4, -3, 0, 3, 0, -5, -3, 0, 0, 5, 0, 0, 0, -5, 3, 3], [0, 0, -2, 2, 0, 0, 0, 0, 5, 0, 0, 5, -3, -3, 0, 0], [4, -1, 0, -4, 4, -2, -2, 0, 0, -1, 0, 0, 0, -2, 0, -4], [0, 2, 0, 0, 0, 0, 2, 0, -2, 0, -6, -2, 0, 0, 0, 0], [0, 0, 4, 4, 0, 0, 0, -6, -2, 0, 0, 2, -8, 8, 0, 0], [5, 0, 4, 0, 0, 0, 0, -5, -9, 0, 5, -10, -10, 9, 0, 0], [0, -4, -8, 0, 0, 2, -4, 0, 4, -2, 8, 4, -2, 0, -8, -8], [-1, 0, 0, 2, 0, 0, 0, -1, 0, 0, -1, 6, -6, 0, 0, 0], [0, 10, 0, 4, -4, 10, -10, 0, 0, 10, 0, 0, 0, 10, -4, 4], [8, 0, 0, -4, 0, 0, 0, 8, 0, 0, 8, 3, -3, 0, 0, 0], [-9, -7, 0, -8, -9, 4, 4, 0, 0, -7, 0, 0, 0, 4, 0, -8], [4, -6, 0, 0, -4, 7, -7, 0, 0, 6, 0, 0, 0, 7, 4, 0], [0, 0, 6, 6, 0, 0, 0, 4, -4, 0, 0, 4, 6, -6, 0, 0], [0, 9, 0, 0, 12, 7, -9, -12, 9, 7, 0, -9, 7, 0, 0, 0], [0, 0, 0, 0, 3, 2, -2, -3, 2, 0, 3, 0, 0, 0, 0, 0], [0, 2, 0, 0, -8, 4, 4, 8, -4, 2, 8, -2, 2, 0, 0, 0], [2, 8, 0, 3, 0, -12, 8, 0, 0, 12, 0, 0, 0, -12, 3, 3], [6, 0, 0, -4, 0, 0, 0, 6, -8, 0, 6, 2, -2, -8, 0, 0], [1, -3, 0, 0, -1, -4, 4, 0, 0, 3, 0, 0, 0, -4, 0, 0], [18, 0, 2, -2, 0, 0, 0, 0, -6, 0, 18, -6, -4, -4, 0, 0], [0, 3, 1, 0, 8, -13, -3, -8, 3, -13, 0, -3, -13, 0, 1, -1], [-12, 0, -6, 6, 0, 0, 0, 0, -4, 0, -12, -4, 0, 0, 0, 0], [0, -6, 0, 0, 9, -4, -4, -9, 4, -6, -9, 6, -6, 0, 0, -2], [8, 0, 0, 8, 8, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 8], [0, 1, 2, 0, 0, -5, 1, 0, -1, 5, 4, -1, 5, 0, 2, 2], [-1, 0, 0, 0, 0, 0, 0, -1, -4, 0, -1, 4, -4, -4, 0, 0], [0, -1, -8, 0, -9, -6, 6, 9, -6, 1, -9, 1, 1, 0, -8, 0], [0, -10, -7, 0, 0, -2, -10, 0, 10, 2, 4, 10, 2, 0, -7, -7], [0, 0, 2, 2, 0, 0, 0, 4, 3, 0, 0, -3, 7, -7, 0, 0], [8, 8, 0, 0, -8, -5, 5, 0, 0, -8, 0, 0, 0, -5, 0, 0], [11, 2, 0, 6, 11, 4, 4, 0, 0, 2, 0, 0, 0, 4, 0, 6], [0, 0, 1, -1, 0, 0, 0, 0, 9, 0, 0, 9, -1, -1, 0, 0], [8, 1, 0, 4, 0, 3, 1, 0, 0, -3, 0, 0, 0, 3, 4, 4], [0, 4, 6, 0, 8, -12, -4, -8, 4, -12, 0, -4, -12, 0, 6, -6], [0, 6, 0, 0, -8, 14, 14, 8, -14, 6, 8, -6, 6, 0, 0, 4], [-12, -4, 0, 0, 0, 8, -4, 0, 0, -8, 0, 0, 0, 8, 0, 0], [0, 0, 3, 0, -8, 8, 0, 8, 0, 8, 0, 0, 8, 0, 3, -3], [-11, -3, 0, 0, 11, 8, -8, 0, 0, 3, 0, 0, 0, 8, -8, 0], [-4, -8, 0, -8, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, -8, -8], [0, 5, 10, 0, -2, -8, 8, 2, -8, -5, -2, -5, -5, 0, 10, 0], [4, 0, -10, 10, 0, 0, 0, 0, 9, 0, 4, 9, -7, -7, 0, 0], [0, 0, -3, -3, 0, 0, 0, -8, 9, 0, 0, -9, 1, -1, 0, 0], [-2, 6, 0, 8, -2, -2, -2, 0, 0, 6, 0, 0, 0, -2, 0, 8], [0, 4, -2, 0, 18, 2, -2, -18, 2, -4, 18, -4, -4, 0, -2, 0], [0, -4, 0, 6, -4, 0, 4, 0, 0, 0, 0, 0, 0, 0, -6, 6], [0, -3, -4, 0, -9, 12, -12, 9, 12, 3, -9, 3, 3, 0, -4, 0], [0, 0, -6, -6, 0, 0, 0, 8, -7, 0, 0, 7, 7, -7, 0, 0], [0, -8, 0, 0, 6, -9, -9, -6, 9, -8, -6, 8, -8, 0, 0, -6], [0, 6, 14, 0, 11, -8, 8, -11, -8, -6, 11, -6, -6, 0, 14, 0], [8, 2, 0, 0, -8, -6, 6, 0, 0, -2, 0, 0, 0, -6, -2, 0], [-13, 0, -4, 0, 0, 0, 0, 13, 0, 0, -13, 6, 6, 0, 0, 0], [-10, 0, 2, 0, 0, 0, 0, 10, 2, 0, -10, -3, -3, -2, 0, 0], [0, -2, 5, 0, 0, 6, 2, 0, -2, 6, 0, 2, 6, 0, 5, -5], [-20, 0, 0, 8, 0, 0, 0, -20, 10, 0, -20, 12, -12, 10, 0, 0], [-2, 0, -10, 10, 0, 0, 0, 0, 16, 0, -2, 16, 2, 2, 0, 0], [-2, 4, 0, -2, -2, 1, 1, 0, 0, 4, 0, 0, 0, 1, 0, -2], [0, 10, 4, 0, 0, -4, 10, 0, -10, 4, 6, -10, 4, 0, 4, 4], [0, 0, -2, -2, 0, 0, 0, 0, -10, 0, 0, 10, -2, 2, 0, 0], [0, 0, 0, 4, -2, 6, 0, 0, 0, 6, 0, 0, 0, 6, -4, 4], [0, -16, -6, 0, 0, -2, -16, 0, 16, 2, -12, 16, 2, 0, -6, -6], [-3, 0, 0, -2, 0, 0, 0, -3, 0, 0, -3, -8, 8, 0, 0, 0], [0, -12, 0, -5, -6, 4, 12, 0, 0, 4, 0, 0, 0, 4, 5, -5], [-4, 0, 0, -4, 0, 0, 0, -4, 6, 0, -4, -1, 1, 6, 0, 0], [0, 0, 0, -6, 0, -8, 0, 0, 0, -8, 0, 0, 0, -8, 6, -6], [4, 0, -7, 7, 0, 0, 0, 0, 6, 0, 4, 6, -2, -2, 0, 0], [6, -9, 0, 0, -6, 2, -2, 0, 0, 9, 0, 0, 0, 2, 2, 0], [0, 0, -4, -4, 0, 0, 0, 20, 3, 0, 0, -3, -1, 1, 0, 0], [0, 0, -10, 0, 5, 0, 0, -5, 0, 0, 5, 0, 0, 0, -10, 0], [-8, 2, 0, 0, 8, -2, 2, 0, 0, -2, 0, 0, 0, -2, -2, 0], [9, 0, 0, -12, 0, 0, 0, 9, -8, 0, 9, 3, -3, -8, 0, 0], [-10, 0, 0, 10, 0, 8, 0, 0, 0, -8, 0, 0, 0, 8, 10, 10], [0, 14, 0, 0, -15, 11, 11, 15, -11, 14, 15, -14, 14, 0, 0, 4], [22, 0, 0, -14, 0, 0, 0, 22, -8, 0, 22, -10, 10, -8, 0, 0], [0, -6, -8, 0, 4, -1, 1, -4, -1, 6, 4, 6, 6, 0, -8, 0], [-1, 2, 0, 0, 1, 10, -10, 0, 0, -2, 0, 0, 0, 10, 2, 0], [0, 2, 0, 0, 3, 2, 2, -3, -2, 2, -3, -2, 2, 0, 0, 16], [0, 1, 10, 0, 0, 7, 1, 0, -1, -7, 4, -1, -7, 0, 10, 10], [0, 11, 0, 0, -8, 1, -11, 0, 0, 1, 0, 0, 0, 1, 0, 0], [0, -9, -8, 0, -1, 12, -12, 1, 12, 9, -1, 9, 9, 0, -8, 0], [-2, 0, 2, 0, 0, 0, 0, 2, -4, 0, -2, -8, -8, 4, 0, 0], [0, -3, 0, 9, 8, 1, 3, 0, 0, 1, 0, 0, 0, 1, -9, 9], [0, 5, 3, 0, 0, -5, 5, 0, -5, 5, -20, -5, 5, 0, 3, 3], [1, 0, 0, -10, 1, 8, 8, 0, 0, 0, 0, 0, 0, 8, 0, -10], [0, 3, -8, 0, -12, 11, -3, 12, 3, 11, 0, -3, 11, 0, -8, 8], [0, 7, 0, -2, 0, 7, 7, 0, 0, -7, 0, 0, 0, 7, -2, -2], [1, 0, 0, -16, 0, 0, 0, 1, 1, 0, 1, -6, 6, 1, 0, 0], [6, 0, -10, 0, 0, 0, 0, -6, 14, 0, 6, -3, -3, -14, 0, 0], [-5, 10, 0, 0, 5, -1, 1, 0, 0, -10, 0, 0, 0, -1, 16, 0], [0, 0, 5, 5, 0, 0, 0, 0, -8, 0, 0, 8, 4, -4, 0, 0], [0, -3, 0, 12, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 12], [0, 0, 0, 0, 9, -2, -2, -9, 2, 0, -9, 0, 0, 0, 0, 2], [0, 0, 6, 6, 0, 0, 0, -2, 10, 0, 0, -10, -4, 4, 0, 0], [-10, -2, 0, -2, -10, 12, 12, 0, 0, -2, 0, 0, 0, 12, 0, -2], [-1, 0, -12, 0, 0, 0, 0, 1, 10, 0, -1, 11, 11, -10, 0, 0], [0, -5, 0, 8, -4, 19, 5, 0, 0, 19, 0, 0, 0, 19, -8, 8], [-3, 7, 0, 4, -3, -10, -10, 0, 0, 7, 0, 0, 0, -10, 0, 4], [0, 0, -2, -2, 0, 0, 0, -8, 11, 0, 0, -11, -15, 15, 0, 0], [0, -7, 0, 0, 4, -2, -2, -4, 2, -7, -4, 7, -7, 0, 0, 8], [0, 8, 10, 0, 0, 8, 8, 0, -8, -8, -8, -8, -8, 0, 10, 10], [0, 0, 8, 0, -3, -20, 20, 3, -20, 0, -3, 0, 0, 0, 8, 0], [10, -20, 0, -2, 0, -2, -20, 0, 0, 2, 0, 0, 0, -2, -2, -2], [16, -4, 0, 0, -16, -2, 2, 0, 0, 4, 0, 0, 0, -2, 2, 0], [0, -9, -4, 0, 16, -3, 9, -16, -9, -3, 0, 9, -3, 0, -4, 4], [-2, 0, 0, 16, 0, 0, 0, -2, -2, 0, -2, 6, -6, -2, 0, 0], [-20, 5, 0, 1, 0, 7, 5, 0, 0, -7, 0, 0, 0, 7, 1, 1], [0, 12, 0, 0, 10, -10, -12, -10, 12, -10, 0, -12, -10, 0, 0, 0], [0, -15, 0, 0, 0, -5, -15, 0, 15, 5, -20, 15, 5, 0, 0, 0], [0, 6, 0, -2, 10, 4, -6, 0, 0, 4, 0, 0, 0, 4, 2, -2], [13, 0, 16, 0, 0, 0, 0, -13, -21, 0, 13, -12, -12, 21, 0, 0], [0, -2, 2, 0, 8, -6, 6, -8, -6, 2, 8, 2, 2, 0, 2, 0], [0, 0, 0, -12, 8, -4, 0, 0, 0, -4, 0, 0, 0, -4, 12, -12], [0, -5, 0, -3, 24, 7, 5, 0, 0, 7, 0, 0, 0, 7, 3, -3], [0, -8, 0, 0, -18, -9, -9, 18, 9, -8, 18, 8, -8, 0, 0, -2], [-16, 0, 12, -12, 0, 0, 0, 0, 10, 0, -16, 10, 10, 10, 0, 0], [0, 0, 4, 4, 0, 0, 0, -24, -1, 0, 0, 1, 7, -7, 0, 0], [18, 0, -8, 8, 0, 0, 0, 0, -14, 0, 18, -14, -12, -12, 0, 0], [-3, 0, 0, 16, 0, 0, 0, -3, 5, 0, -3, -2, 2, 5, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_784_x_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_784_2_x_k();".
// 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_784_2_x_k(:prec:=16)
    chi := MakeCharacter_784_x();
    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_784_2_x_k();".
// 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_784_2_x_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_784_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![10000, -4000, 800, -6720, -3056, 4176, 832, 1344, 1708, 136, 32, 32, -44, -8, 0, 0, 1]>,<5,R![16, 32, 32, 128, 80, -208, -64, -240, 108, 248, 32, 184, 100, -56, 8, -4, 1]>],Snew);
    return Vf;
end function;