
// Make newform 1764.2.x.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1764_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1764_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_1764_2_x_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_1764_2_x_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 := [6561, -4374, 3645, -4131, 1782, -837, 558, -156, 52, -52, 62, -31, 22, -17, 5, -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], [19787976, 4573017, 11094408, -2054484, 900801, -1174032, 84090, -181946, -7210, -113602, 77018, 2716, 49267, 824, 5068, -1307], [46300977, -33241671, 20620980, -18766647, 9488205, -2586978, 844320, -711356, 298928, -187876, 360098, -215537, 99223, -51544, 20725, -3695], [63188991, -11411037, 15850404, -21798153, 683775, -2201310, 1316016, -384796, 95200, -220796, 422782, -42655, 79853, -58592, -4165, -6613], [1850931, -5662872, 3505194, -2415663, 1801458, -864450, 146582, -73522, 78114, -27966, 29320, -39051, 21044, -8834, 4365, -1730], [50810571, 2373624, 28867428, -12408957, 3575664, -4479570, 777042, -462590, 100154, -294706, 271028, -38759, 146632, -27196, 13039, -8480], [20080305, -23810841, 4704480, -17595171, 3259071, -2641242, 1227580, -205752, 273436, -68192, 233886, -112105, 49233, -62076, 929, -9197], [-61443765, -7676370, -29851092, 18020475, -63018, 5698206, -1370658, 284726, -100070, 361210, -343040, -15775, -164782, 45760, -1333, 13862], [23210631, -13830831, 9058068, -15639345, 1996173, -3459822, 1197116, -166912, 240180, -117624, 211882, -66663, 71471, -52928, 435, -10991], [23648031, -21030192, 16524972, -20396205, 5124384, -5939262, 1677798, -282422, 398066, -166462, 247964, -128807, 123904, -73804, 9175, -16952], [-20594979, 11879784, -5306634, 19855935, 1317006, 4583574, -1730010, -105946, -302282, 129910, -227228, 24959, -74620, 72850, 11843, 17318], [-52673895, -820854, -30627234, 13168089, -3932226, 4717278, -843498, 484742, -122306, 321118, -284234, 48131, -153874, 29326, -13891, 8906], [65179161, -19369530, 32737446, -29583873, 5845284, -7868934, 2170158, -589198, 414682, -364490, 470224, -134761, 200162, -91538, 11675, -20668], [39766950, -2638737, 17811576, -12954546, 1569195, -3277620, 882698, -288162, 112070, -207322, 242874, -29234, 96837, -34392, 3790, -7981], [67869171, 13260510, 35643726, -24510843, -1362096, -9228186, 2242098, -223118, 251534, -437878, 389576, 48469, 225742, -72430, -1811, -25616], [77064777, -8267589, 44697096, -26067339, 6647859, -8655978, 1851660, -709136, 355172, -472072, 469334, -112385, 245365, -71044, 19537, -19601]];
    Rf_basisdens := [1, 621108, 1242216, 1242216, 207036, 621108, 414072, 621108, 414072, 621108, 621108, 310554, 621108, 207036, 621108, 414072];
    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_1764_x();"
function MakeCharacter_1764_x()
    N := 1764;
    order := 6;
    char_gens := [883, 785, 1081];
    v := [6, 5, 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_1764_x_Hecke();"
function MakeCharacter_1764_x_Hecke(Kf)
    N := 1764;
    order := 6;
    char_gens := [883, 785, 1081];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -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, 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, -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, 0, 0, 0], [1, 0, 0, -1, 0, 1, -1, 1, 0, 0, -1, 0, 0, 0, 0, -1], [0, 0, -1, 1, 1, 1, 0, 1, 0, -2, 0, 0, 1, -1, 0, 0], [1, 1, 0, -1, 1, 1, -1, 1, 0, 0, -1, 1, -1, 1, 0, 0], [0, -1, -1, 3, 1, 1, -1, -1, 2, -1, 0, 1, 1, -2, -1, 1], [0, -1, 0, 0, -1, 0, -1, -1, 1, 2, -1, 0, -1, 1, -1, 1], [-1, 0, 0, -1, -1, -1, 2, -1, -1, -1, 1, 1, -1, -1, 1, 2], [0, 0, 0, 1, -1, 0, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, -1, -1, 1, 0, 0, 0, -1, 0, 0], [0, -2, -2, 3, 1, -1, 1, -1, 1, -2, 0, 1, 1, -2, 0, 1], [1, -1, 0, -1, -2, -1, 1, -1, 0, -1, 1, -2, -2, 0, 0, 1], [-1, -2, 0, 1, 1, 1, 2, 1, -1, -2, -1, 1, 1, -1, 1, 0], [-1, 1, -1, 2, 0, 2, 0, 0, -1, -1, 0, 1, 0, -1, 0, 2], [-3, 4, 2, 0, 1, 3, -3, 2, -2, 3, -2, 1, -1, 2, -2, -1], [-1, -1, -1, 1, 0, -1, 1, -1, 0, -1, 0, 0, 0, -2, 0, 1], [0, -1, 0, 0, -1, -1, 1, -3, 2, 0, 2, -1, 2, -1, -1, 1], [-1, 2, 2, 1, -1, 2, 0, 0, -1, 0, -2, 0, -1, -1, 0, 0], [2, -2, 2, 2, -1, -2, 0, -1, 2, -1, 0, -2, -1, -2, 2, -2], [-1, 1, 0, -2, 0, 0, 0, 1, -2, -1, 0, 0, 0, 1, 0, 0], [-1, 1, -3, 1, 1, 4, -1, 1, -1, 0, -1, 1, 1, -1, -2, 3], [3, 0, -2, 0, 0, 1, -1, 0, 0, 2, -2, 0, -2, 1, -2, 1], [0, -2, -2, 3, 1, -3, 2, -4, 1, -1, 1, -1, 1, -2, 1, 2], [-3, 0, -1, -3, 3, 2, -2, 3, -3, 1, -1, 3, 2, 2, -1, -1], [1, -3, -1, 5, -1, -1, 2, -1, 2, -3, 1, -2, 2, -2, 1, 1], [-4, 7, -2, -1, 0, -1, 3, -1, 0, -3, 1, 1, 0, 0, 1, 0], [4, -2, -1, 4, -2, -1, 1, 1, 2, -3, 1, -2, 4, -1, 0, 0], [-1, 0, -2, 2, -1, 1, -1, 1, -2, -1, 0, -1, 5, -2, 0, 1], [-1, -2, 0, 2, -5, -4, 4, -7, 0, 5, 2, -2, 1, 0, 0, 2], [-9, 8, -3, 3, 2, -1, 1, 0, -1, -2, 1, 2, 0, 0, 1, 3], [-2, -1, -2, 0, 1, -2, 2, -2, -1, -3, 0, 1, -2, -4, 2, 3], [0, -2, 0, 2, -2, -1, 0, 0, 1, -2, -1, -1, 1, 0, 0, 2], [-5, -2, 0, -5, 2, 0, 2, 3, -3, -3, 0, 0, -4, 0, 0, 0], [2, 1, -1, -2, -1, -2, 2, -2, 3, -2, 3, -1, -4, 0, 0, -1], [2, 1, -1, 8, 3, 2, 0, 1, 2, -6, 0, 1, 9, -4, 3, -1], [1, 2, 0, -2, 2, 2, -2, 1, 0, 2, -2, 2, -4, 3, -1, 0], [5, -1, 2, -5, -1, 1, -1, 5, -3, 2, -3, -1, -3, 4, 2, -4], [1, -6, 0, -1, 0, 0, -3, -1, 2, 0, 0, 0, -1, 1, -2, 0], [0, 2, 1, 5, 0, -2, -2, 1, 1, 0, 1, -2, 0, -2, 3, -1], [-2, 2, -3, 5, 3, 0, -2, 1, 1, -1, 1, 2, 3, -1, 0, 1], [11, -5, 2, 0, -1, -1, 1, -3, 1, 1, 3, -1, 2, 1, 1, 0], [5, -3, -2, -6, -4, -3, 6, -3, -1, -2, 2, -1, -1, 2, 4, 1], [1, -1, 0, 4, 0, -2, 0, 1, -1, -2, 1, -1, 0, -1, 4, -1], [-2, 4, -3, 1, -2, 1, 3, 1, -3, -2, 3, 0, -2, -1, 3, 4], [4, -4, 1, -1, -1, -3, 0, 2, 4, -1, 1, -4, 2, -2, 1, -3], [-1, 1, 2, -3, -3, 2, 0, -3, -1, 3, 0, 1, 0, 0, -1, -1], [-2, -5, 2, -4, 1, 0, -3, 2, -3, 3, -2, 1, 1, 3, -1, -1], [-3, 4, 0, -5, 2, 2, -2, 3, -5, 1, -2, 2, 2, 0, 0, 0], [-3, 8, 1, -6, -1, -1, -1, 2, 0, 6, 1, -2, -1, 3, -2, 0], [-4, -3, 3, 1, 0, -2, 1, -1, 0, 0, 1, 1, 3, 0, 3, -2], [0, -2, -1, 2, 2, -2, -1, 2, 4, -6, 1, -4, 5, -5, 3, -2], [0, 4, 3, -6, 1, 1, -1, 2, -2, 0, -1, 4, -3, 3, 2, 0], [-7, 6, 3, 1, 0, 3, -3, -5, 1, 3, 0, 0, 1, -1, -4, 0], [3, 1, 2, -6, -2, 3, -2, -1, 0, 1, -1, 0, 1, 0, 2, -3], [7, 0, 0, -1, 6, 3, -3, 3, -1, -1, 0, 0, -4, -2, -2, -3], [1, 0, 0, 0, -2, -1, 1, -1, 2, 0, 2, -2, -2, -2, 1, 1], [2, -3, -4, 7, 1, 5, -2, -2, 5, 0, -1, 1, 1, -4, -3, 4], [-1, -3, 1, 3, 1, -2, 0, -5, 3, -2, 3, -3, 4, -6, 1, -1], [1, -3, 4, 5, -5, -9, 9, -3, -1, -5, 4, -5, 4, -3, 7, 1], [4, -5, 0, 2, 1, 1, -1, -2, 0, 2, 2, 1, 2, 1, -3, 3], [0, 3, 5, -3, -2, 1, -3, -1, 0, 4, -2, -3, -2, 1, -2, -1], [-8, 6, 0, 5, 3, 3, -3, 0, -1, 0, 0, 3, -1, -2, -2, 3], [-6, 4, 4, -1, 0, 0, 1, -2, -2, 2, 4, 2, -3, 2, 0, 2], [10, -2, -4, -1, 1, 1, 0, 0, -3, 4, -5, 3, -5, 6, -1, 2], [-1, 1, -5, 1, 5, 6, -6, 1, 3, 3, -1, 5, 2, 0, -5, 1], [-1, 2, -2, 0, 1, -4, 1, -3, 1, 1, 0, -2, 4, 4, 0, -2], [-2, -4, -2, 9, 1, 2, -1, 0, 2, -4, 0, 0, 10, -4, 2, 0], [1, 4, 3, -6, 2, 7, -8, 7, 0, 1, -4, 0, -1, 2, 0, -7], [0, -13, -2, 3, 2, 6, -3, 0, 2, 2, -4, 2, 1, -2, -4, 4], [-3, 11, 5, -2, -2, -3, -3, -1, 1, 3, 0, -5, -2, 1, 1, -3], [10, -3, -1, -6, -3, 4, -2, 0, -1, 2, -5, 1, 0, 0, 0, -3], [-5, 5, -3, 0, 0, 0, 3, 0, -3, -3, 0, 3, -3, 0, 3, 3], [5, -1, -1, 5, 5, 4, -4, 2, 4, -3, 0, -1, 5, -2, 0, -4], [-13, 17, 3, -5, -1, 2, -2, 9, -3, 1, -5, -1, -5, 2, 3, -6], [5, 0, 1, -2, 3, 1, -2, 9, -1, -4, 0, 1, -3, 2, 2, 0], [0, 0, 2, 5, 3, 4, -10, -2, 3, 4, -2, 0, 3, -1, -6, 2], [-6, -8, 1, -2, 0, 0, -1, -1, 3, 2, 0, 1, -6, 3, -2, -1], [-4, -8, -3, 4, -4, 1, -3, -6, 2, 11, 0, -2, 2, 0, -3, 2], [-8, 6, 1, 2, 0, 4, -1, -6, -1, 2, 1, 3, 0, -1, -4, 4], [2, -3, -6, 4, 0, 4, -4, -2, 4, 6, 2, 0, 0, -2, -4, 4], [-2, 4, -3, 3, 0, -4, 4, -3, 1, -8, 3, 0, 0, -2, 5, -1], [5, -7, 2, -5, -2, -2, 0, -2, 0, 1, 1, -3, -2, 2, 0, -1], [-4, -1, -2, 0, -4, -4, 6, -8, 0, -1, 2, 0, 2, -4, 2, 4], [17, -14, -2, 6, -1, -6, 4, 3, 5, -9, 4, -5, 8, -7, 2, 1], [4, -4, 0, 0, -2, -9, 5, -4, 6, -6, 4, -4, -2, -3, 6, -5], [-4, 2, -1, -2, 0, -1, 4, 4, -8, -2, 2, 0, 3, 1, 4, -1], [-2, 15, -1, -4, 3, -1, -2, 2, -1, -1, 2, 3, 1, 3, 0, -4], [0, -1, -1, 1, 0, -1, 1, -5, 1, -2, 5, 0, 0, -2, 1, 5], [13, 4, 3, -4, 1, 1, -1, -1, 0, 2, -1, 4, -1, 4, 3, 0], [-3, 0, -1, 0, 3, 5, 1, 3, -3, -4, -4, 3, 2, -1, -1, 2], [11, 1, 1, 1, 4, 3, -3, -2, 2, 4, 0, 1, -2, 2, -1, -3], [-6, -3, -3, 3, 6, 9, -9, 5, 3, -4, -3, 6, 6, 0, -3, -3], [-2, -4, -6, 9, 3, -6, 3, -12, 6, -6, 7, 0, 6, -6, 0, 7], [-6, 0, 2, -3, -6, -7, 7, -5, -3, 3, 2, 0, -4, 2, 2, 5], [-12, 15, 2, -6, -3, 2, 1, 2, -3, 7, -4, 0, 2, 8, -4, -4], [14, 0, 1, 4, 0, 1, -1, -1, 4, 1, 1, 0, 2, 3, 0, -2], [2, -1, 1, -4, -1, -9, 5, -4, 3, -6, 2, -3, -1, 1, 5, -7], [3, -2, -1, 2, 1, 0, 0, 2, 0, -6, 1, -2, 7, -6, 2, -1], [-3, 1, 2, 0, 0, -4, 7, 3, -6, -4, 2, 0, 3, 0, 7, -4], [-1, -1, 1, -3, 0, 1, 5, 5, -2, 2, -4, 3, 0, 1, 0, -3], [-7, 12, 0, -11, -6, 0, 6, 4, -5, 0, 0, 0, -5, 5, 5, 0], [9, -7, -3, 3, 2, 1, -1, -3, -2, 3, -4, 2, -4, -2, -4, 1], [4, -3, -8, 0, 1, -4, 7, 3, -5, -2, 0, 4, 1, 7, 3, 4], [-6, 8, -6, 2, 3, 3, -1, -5, 2, -5, 2, 4, 3, -1, -4, 5], [3, -5, 6, -3, -5, -8, 8, -3, 1, -1, 2, -5, -6, 0, 5, -3], [13, 1, -1, 2, -2, 0, 0, -6, 3, 5, -2, 1, 4, 3, 2, 2], [2, 2, 1, 0, 0, 3, -7, -4, 4, 11, -4, 4, 0, 4, -3, 0], [-4, -2, 2, 8, 2, -1, -1, 2, 2, 0, 3, -2, 2, -2, -5, -1], [0, -9, -1, 3, -10, -8, 7, -5, 0, 1, 3, -9, -4, -1, 0, 6], [-7, 8, 5, -14, -1, 6, -6, 4, -6, 6, -3, 8, -7, 6, -2, -3], [6, -3, 0, 3, -6, -6, 6, -4, 3, -1, 0, -6, -6, 0, 0, 6], [-13, 4, 2, 6, 4, -2, -1, -1, 2, -1, 4, -2, 1, -2, -1, 0], [-3, 2, 5, -4, 2, -2, 2, 5, -2, -9, 3, 2, 2, -2, 6, -1], [-5, -3, -4, 9, 2, -4, 3, -3, 0, 0, 0, 0, -1, -3, -4, 4], [6, 1, -6, 5, -2, 2, 0, -3, 1, 4, 1, -4, 1, -3, -3, 3], [4, -2, 3, 2, -8, -8, 8, -10, 0, 5, 5, -2, 2, 1, 3, 3], [-10, -3, 4, -11, 0, -1, -2, -2, -4, 12, 0, 3, -6, 5, -1, -3], [12, -6, -2, 0, -6, -1, 6, -8, -4, 8, -1, 4, -3, 5, -4, 5], [2, 9, 1, -5, 8, 7, -7, 3, 1, 3, -4, 8, -1, 3, -2, -4], [-1, -1, 9, -14, -4, -8, 5, 1, -6, 13, -1, -4, -7, 6, 3, -4], [1, -4, -12, 5, 5, 11, -2, -1, 7, -2, 1, 5, -1, -5, -7, 6], [-2, 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-3], [1, 1, 2, 1, 6, 4, -10, 5, 6, -2, -2, -6, 6, -2, 1, -10], [-24, 5, 1, -12, 2, 3, -3, 12, -7, -3, -4, 5, -14, 4, -4, 1], [8, -20, -7, 9, 6, 2, -6, 0, 5, 0, 3, 4, 6, -3, -4, 5], [-2, 32, 1, -19, -1, -2, -7, 2, -5, 15, 1, -1, -7, 10, -6, -1], [11, -1, 7, 4, 5, 11, -15, 13, 3, 9, -4, -3, 2, 9, -11, -14], [9, 19, 3, -11, 5, 5, -8, 3, -2, 7, -12, 8, -4, 6, -3, 4], [-5, 12, -2, -2, -3, 4, -2, 5, -10, 5, 4, -2, -3, 4, 2, 8], [-14, 8, -7, 5, -1, -2, 8, -4, 1, -4, 0, 5, 4, -3, -2, 5], [0, 2, 6, -7, 5, 7, -6, 14, -7, -4, -1, 7, -13, -2, -1, 0], [1, -4, -1, 13, 12, -5, -15, 0, 9, 2, -1, 2, 12, 0, -4, -6], [1, 9, -8, 26, 3, 7, -13, -1, 5, 2, 1, 3, 18, -7, -1, 4], [-5, 4, -7, 1, 0, -7, 7, -2, 1, -5, 7, 0, 3, 1, 4, 7], [11, -2, 5, 2, -15, -13, 6, -18, 5, 23, 0, -8, -9, 5, 0, 8], [-2, 1, 2, -6, 1, 1, -3, 2, 0, 1, -4, 3, -5, 4, -2, 1], [-1, -4, -6, 10, 14, 6, -6, 19, 6, -24, -2, -4, 8, -9, 1, -4], [-30, 12, -1, -4, -12, -8, 17, -8, -6, -2, 7, 6, -3, -1, 3, 14], [1, -4, 2, 1, -2, 0, 0, -2, 5, -2, 7, -2, 1, -7, -1, 5], [8, 20, 6, -20, -2, 0, -8, 1, -6, 15, 0, 6, -11, 11, -5, -6], [1, 5, 4, -12, -5, -3, -1, -1, 7, 11, 3, -7, -5, 1, -5, 0], [8, 0, 3, 10, 6, 1, 1, -6, 12, 1, 1, 4, 0, 0, 3, -5], [4, -3, 10, -4, -7, -14, 2, -4, 1, 9, 4, -7, -4, 4, 6, -7], [-24, 5, 5, -13, -1, -3, 3, -10, -8, 17, 0, 5, -19, 7, -2, 3], [-4, 10, -5, 10, 7, 8, -5, 4, -1, 0, -3, 7, 11, -3, -1, 1], [24, -23, -10, 0, 1, -3, 9, 7, -8, -6, -2, 7, -4, 5, 7, 5], [-24, 13, -11, 14, 3, -2, 8, -7, 2, -10, 0, 9, 3, -10, -1, 9], [-2, 7, 2, 5, 7, 0, 6, 2, 3, -17, -2, 7, 4, -3, 5, -7], [-34, 21, 3, -5, 5, 4, -4, -6, -5, 6, -1, 5, -10, 3, 1, 1], [16, 5, 6, 9, -8, -13, 12, -13, 4, 7, 11, -7, 1, -2, 8, -4], [-7, 6, -2, 3, 4, 11, -3, -4, 3, -1, -6, 8, 4, -3, -11, 5], [-12, 9, 7, -8, -3, 13, -13, 1, 1, 12, -2, 9, 3, 10, -4, -11], [10, 20, 0, -3, 1, -2, 5, 3, -12, -2, -2, -2, 10, -6, 6, 4], [37, -28, -3, 12, 1, -15, 11, -1, 7, -16, 12, -7, 10, -11, 7, 8], [-1, 4, 2, -5, 1, 6, -6, 0, -1, 4, -2, 4, 3, 1, 0, -4], [7, -14, 7, 12, 6, 2, -11, 1, 12, -6, 5, -12, 15, -18, -5, -14], [1, 4, 12, -4, 4, 1, -1, -3, 0, 4, 8, 4, -4, 4, 3, -9], [15, -6, 6, 5, -20, -26, 26, -15, 1, 1, 18, -20, -14, -2, 8, 2], [-28, 36, 1, -5, 3, 1, -4, 2, -8, 5, -5, 0, -4, 12, 3, -5], [10, -5, -7, 20, -7, -3, 6, -10, 13, -13, 10, -7, -1, -11, 4, 4], [0, 4, -2, 0, 4, 2, -2, 4, -8, 1, 0, 4, -5, 7, 3, 4], [-4, 5, -1, -8, -8, 0, 7, -8, -6, 6, -1, 6, -5, 5, 1, 6], [4, -6, -6, -10, -3, 0, 12, -4, 5, -8, 6, -3, -16, -2, -2, 3], [-7, -3, 2, -2, 3, -10, 10, -3, -5, -1, 5, -3, -15, -2, 1, 5], [11, -9, 3, 6, -1, -5, 5, 1, 1, -3, 8, -1, 2, -1, -1, 4], [-4, -1, 1, -11, 0, -1, 1, -4, -3, 14, 4, 0, -12, 4, -3, -4], [9, -17, 8, 1, 3, 5, -13, 1, -4, 8, -7, -1, 3, 5, -6, -2], [33, -9, 4, 14, -15, -20, 20, -13, 5, -7, 13, -9, 15, -4, 11, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1764_x_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1764_2_x_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_1764_2_x_b(:prec:=16)
    chi := MakeCharacter_1764_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(1999) - 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_1764_2_x_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_1764_2_x_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1764_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![324, 4698, 62289, 89424, 143289, 45738, 71361, 21573, 23103, 4167, 3600, 423, 405, 24, 24, 0, 1]>],Snew);
    return Vf;
end function;