
// Make newform 496.2.bg.c in Magma, downloaded from the LMFDB on 08 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_496_bg();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_496_bg_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_496_2_bg_c();".
// 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_496_2_bg_c();".
// 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, 0, 44, 0, 410, 0, 956, 0, 979, 0, 511, 0, 140, 0, 19, 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], [255, -360, 2839, 2409, 3382, 6431, 691, 5467, -641, 2196, -358, 458, -65, 48, -4, 2], [-255, -360, -2839, 2409, -3382, 6431, -691, 5467, 641, 2196, 358, 458, 65, 48, 4, 2], [-463, 439, -7953, 10699, -17926, 21308, -16144, 16835, -7260, 6619, -1732, 1374, -209, 144, -10, 6], [93, 1125, 0, 7255, 0, 14844, 0, 14865, 0, 7940, 0, 2250, 0, 315, 0, 17], [463, 439, 7953, 10699, 17926, 21308, 16144, 16835, 7260, 6619, 1732, 1374, 209, 144, 10, 6], [354, -532, 6074, -10699, 18145, -21308, 21730, -16835, 12516, -6619, 3653, -1374, 517, -144, 28, -6], [-143, -1347, -412, -6329, 6617, -6857, 14550, -1382, 10969, 1282, 3713, 716, 575, 130, 33, 8], [354, 532, 6074, 10699, 18145, 21308, 21730, 16835, 12516, 6619, 3653, 1374, 517, 144, 28, 6], [538, -2510, 5719, -22747, 19734, -47899, 26249, -44466, 16025, -21362, 4827, -5485, 695, -709, 38, -36], [61, -225, 303, -7000, 1337, -26132, 2430, -35213, 1770, -21738, 587, -6619, 89, -962, 5, -53], [36, 1788, -926, 27117, -7987, 62350, -13483, 59919, -9222, 29263, -2966, 7575, -445, 983, -25, 50], [126, 1489, 3765, 15383, 11369, 41009, 14174, 46581, 8581, 26161, 2608, 7535, 380, 1058, 21, 57], [597, 658, 7093, 14464, 26132, 32677, 35213, 31009, 21738, 15200, 6619, 3982, 962, 524, 53, 27], [470, 2510, 8467, 22747, 32530, 47899, 44177, 44466, 27451, 21362, 8411, 5485, 1229, 709, 68, 36], [359, 3724, 2730, 48574, 11336, 121504, 17671, 128230, 12098, 67940, 3942, 18763, 599, 2558, 34, 135]];
    Rf_basisdens := [1, 186, 186, 186, 186, 186, 186, 186, 186, 186, 186, 186, 186, 186, 186, 186];
    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_496_bg();"
function MakeCharacter_496_bg()
    N := 496;
    order := 15;
    char_gens := [63, 373, 65];
    v := [15, 15, 13];
    // 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_496_bg_Hecke();"
function MakeCharacter_496_bg_Hecke( : Kf := false)
    N := 496;
    order := 15;
    char_gens := [63, 373, 65];
    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], [0, 0, -1, 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 := ISA(Type(Kf), Fld), 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]];
    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], [2, -1, 0, -1, -1, 0, -1, 0, 1, 0, -1, 0, 2, 0, -1, 0], [2, -1, 0, 0, -1, 0, -1, 0, 2, 1, 0, 0, 1, -2, 0, 0], [0, 0, 1, 0, -1, -1, 0, 0, 1, -1, 0, 1, 0, 0, 0, 0], [1, -1, 2, 0, 1, 0, 1, -1, -2, 1, -1, 3, 2, 0, 0, -1], [1, -3, 0, 0, -3, 0, -2, -1, 1, 0, -1, 1, 3, 0, 1, -1], [-2, 1, 1, 2, -3, 0, 2, -4, -4, 1, -2, 3, 3, 1, 2, -1], [0, -2, -2, 0, -4, 1, 0, -1, 1, -1, 0, -1, 0, 1, -1, 1], [-2, 2, 3, 2, 1, 0, 2, -1, -2, 0, 0, 0, 1, 0, 1, 0], [2, -2, 2, 0, -2, 1, -2, 0, 2, 1, -1, 1, 4, 1, -1, -1], [0, 0, -4, 0, -1, 3, -1, 2, 2, -1, -1, -3, -2, 0, -1, 0], [-2, 1, -2, 2, 0, 2, 3, 0, -1, -2, 1, -2, -3, -2, 2, 1], [1, 0, 1, -3, 3, 0, 2, 3, 1, -1, 1, -1, -1, 0, -2, 2], [1, -3, -1, 0, -6, -1, -6, -1, 1, 4, -3, -1, 2, -2, 4, -2], [-2, -1, 0, 0, 2, 3, 5, 2, -1, -5, 1, 0, -3, 4, -6, 4], [1, -6, -1, 0, -3, 0, -3, 1, 1, -1, -1, -5, 1, 1, -2, 0], [0, 3, 5, -3, 6, 0, 5, 3, -3, -2, -1, 2, -1, -1, -2, 1], [-4, 4, -2, -3, 4, 2, 2, 3, 0, -5, 2, -1, -4, 0, -4, 2], [-2, 4, 1, 2, 1, -3, 2, -2, -2, 3, 3, 2, -2, -4, 5, 0], [3, -3, -1, 0, -4, 0, -4, 2, 2, -1, -1, -2, 1, 0, -1, 0], [3, 0, 2, -2, 4, -5, -5, 5, 3, 5, 3, -1, 0, -5, 5, 0], [-7, -3, 4, 8, -4, 0, 7, -8, -9, 4, -4, 6, 1, 4, 2, 2], [2, 2, 9, -5, 6, -1, -3, 6, -5, -3, 1, 2, 0, 1, -4, 4], [-6, 4, 5, 5, 3, 2, 8, -1, -12, 0, 0, 3, -4, 5, -1, 3], [1, -7, -3, -3, -2, -1, -5, -1, 7, 0, 0, -3, 2, 3, -4, 3], [5, -1, 8, -4, 0, 3, 2, -1, -4, -1, -6, 1, 8, 1, -4, 0], [4, 3, -4, -5, 3, 0, 3, 3, 6, -1, 3, -5, 0, -2, 0, 0], [-6, -1, 3, 7, 1, 0, 4, -2, -9, 2, -1, 4, 0, 5, 3, 2], [6, 2, 4, -2, 0, -1, 2, 3, 2, -3, 4, 6, -1, -1, 0, -1], [-11, 9, -2, 1, 11, 1, 9, -1, -13, -3, 0, 3, -18, 2, -1, 2], [-1, -1, -3, -2, 6, 3, 1, -1, -6, 1, 1, -1, -5, 0, -2, 3], [8, -7, -1, 1, -3, 6, 0, -1, 4, 2, 0, -5, 3, -1, -4, 2], [-2, -9, 3, -2, -3, -5, -7, -1, 5, 0, -1, 2, 11, 5, 1, -3], [4, 9, 1, -1, 7, 0, 1, 3, 2, -6, 7, 4, 1, -1, -1, -1], [4, -7, 0, 2, -10, 2, 2, 0, -2, -2, -4, -4, 1, 3, -3, 6], [-10, 0, 1, 3, 2, -2, 3, 0, -3, 0, 0, -4, 1, 6, -1, -1], [-4, -1, 6, 1, 0, -2, 3, -4, 4, -1, -2, 0, -2, 2, 0, 1], [-1, 1, -1, -1, 5, 3, -4, -2, -2, 2, -5, -1, -1, -2, -1, 0], [14, -14, -2, -4, -16, 0, -4, 1, 12, -3, -1, -2, 10, 2, -3, -2], [6, -8, -1, 5, -8, 4, -5, -2, -1, 5, -2, 1, 3, -4, 7, -2], [-4, 3, 0, 4, -4, 2, 2, -4, -6, 2, -8, 0, 2, 4, 0, 0], [2, 3, -2, 4, 6, -4, -1, 0, 0, 7, 4, 2, -3, -10, 8, -4], [5, -8, -1, 2, -4, 4, 4, -4, -1, 2, -4, 5, 4, -2, -2, 2], [8, -2, 2, 2, -3, 8, -4, -2, 2, 4, -1, -6, 8, -1, -4, 2], [-2, -7, 3, 0, 2, -2, -6, -6, -1, 6, -6, 15, 14, 0, 2, -4], [7, -7, 2, -6, 3, 3, -3, 3, 13, -5, 3, -5, 7, -3, -8, 2], [-7, 1, 20, 1, 9, 0, 9, -1, -11, -1, -6, 7, 3, 4, -3, 6], [5, 2, -6, -1, -7, -1, -8, 0, 4, 1, 0, 1, -4, -2, 2, -1], [-1, -5, -4, 6, -10, -4, -5, -10, 5, 7, -7, 5, 5, -4, 7, -7], [10, -12, -4, -5, -20, 3, -8, -7, 6, 2, -10, 5, 15, 2, 2, -3], [-1, -2, 2, -3, 10, 5, 6, 8, 0, -10, 2, 1, -1, 8, -13, 8], [8, -3, -6, 0, -5, 3, 3, 1, 6, -7, -1, 3, 6, 7, -14, 3], [-6, 7, 0, 3, 9, -3, 4, -3, -4, -1, -2, -5, -2, -1, 2, -1], [-2, -8, -7, -1, 0, -7, -18, -1, -1, 5, 0, -3, -2, -4, 3, -3], [15, -6, 7, 0, -11, -12, -12, 0, 11, 5, 0, 5, 11, -5, 10, -12], [10, 1, 6, -6, -3, -9, -15, 2, 5, 10, -2, 11, 7, -8, 12, -8], [0, -5, 7, -3, 5, -2, -6, 5, 0, 0, -4, -5, 5, 2, -7, 7], [-2, 0, 0, 11, -12, 2, 10, -4, -2, -1, 3, 0, 0, 1, 4, 0], [1, 6, 2, -5, 0, 3, 4, 3, 9, -5, 0, -9, -8, 0, -10, 5], [5, -3, -4, 4, -8, -2, -9, -1, -7, 7, 1, 10, 14, -1, 9, -7], [-2, -7, -10, 3, -14, 4, -5, -2, 3, 0, 0, -11, -2, -2, 3, 1], [-2, 3, 12, 1, 10, 0, 12, -2, -16, -1, -1, 19, -4, 2, -3, 5], [11, -12, -3, -1, -10, -2, -6, -12, 10, 7, -13, 8, 19, 5, -6, -7], [1, -3, -7, 2, -14, 1, -7, -7, 7, 6, -4, -3, 14, 0, 3, -4], [-13, 2, -3, 2, 8, 5, 1, 6, -8, -11, -3, -6, -16, 5, -3, 0], [4, 6, -7, -4, 8, 3, 10, 4, 6, -7, 2, -10, -9, 2, -10, 5], [-8, 0, 4, -17, 16, -9, -9, 16, 9, -12, 4, -10, -1, -1, -3, 3], [13, -9, -17, 1, -26, -1, -14, -5, 7, 0, 0, 3, 3, -1, 8, -9], [-6, 4, 12, 1, 4, -6, 12, -7, -7, -1, 0, 8, 8, 3, 4, -6], [-6, 4, -7, 1, 17, -1, 6, 6, -13, -2, 13, 5, -16, 2, 0, 0], [-1, -9, -9, 1, -22, 5, -6, -1, 18, -5, 1, -17, 5, 1, -5, 2], [-2, -2, -4, 12, -8, 1, -2, -2, -16, 12, -2, 6, -5, -1, 14, -2], [0, 3, -5, -13, 14, -12, -13, 14, 17, -6, 12, -8, -12, -1, -1, -1], [2, -5, -2, 7, -15, -3, -8, -12, -2, 19, -8, 5, 15, 0, 15, -8], [3, 6, 0, 3, -10, -2, -7, -3, -1, 10, -4, 3, 4, 1, 12, -6], [6, -4, 21, 0, 16, -2, 15, 6, -8, 0, 6, 14, 8, -3, 2, 2], [-1, 0, 2, 4, 10, 0, 6, 8, -6, 7, 4, 5, 0, -3, -2, -1], [-13, 7, 4, -3, 17, -3, 5, 0, 1, 3, -1, -5, -3, 0, 0, -2], [-25, 20, 7, 9, 15, -9, 16, -3, -19, 0, 6, 10, -21, -6, 24, 0], [15, 7, 2, -2, -4, 0, 4, 2, 1, 2, 6, 0, 0, -7, 5, -6], [11, -10, 3, 8, -22, 0, -8, -8, 0, 17, -9, 2, 6, -8, 8, -8], [-6, 2, 1, 3, -1, -12, -1, -2, -7, 4, -2, 4, -8, 12, 6, -3], [-2, 0, -10, 11, -15, 3, -2, -5, 2, 1, 6, -5, -1, 4, 7, -1], [-11, 24, -3, -9, 25, 0, 6, 13, -18, -8, 4, 8, -25, 0, 1, 4], [-8, 13, -5, 3, 16, -4, -3, 7, -11, 3, 4, -7, -13, -3, 7, -10], [-1, -7, 5, 0, 5, -9, -2, 0, 2, 4, -3, -2, 5, -1, 5, -9], [11, -2, 8, -3, 0, -4, -16, 2, 3, 10, -6, 7, 15, -4, 5, -4], [-1, -8, 15, 13, -19, -3, 13, -9, -11, 2, -5, 11, 12, -2, 9, 0], [2, 7, 16, -9, 15, 1, 4, 4, -8, 0, 0, 4, -4, -5, -9, 10], [9, 0, -18, 0, -13, -3, -9, 1, 4, 0, 4, -10, -5, -4, 4, 0], [12, -17, 2, -1, -17, 12, -11, 6, 20, -12, -6, -18, 0, 6, -8, 12], [7, 8, 3, -7, 11, 0, -5, 8, -4, -8, 4, -4, -13, 1, -9, 10], [13, -4, 7, 0, -7, -10, 2, -12, 7, 9, 5, 10, 14, -2, 11, -10], [-5, 8, 3, 6, 1, -2, 17, -8, -20, 8, 2, 12, -4, -10, 8, 2], [7, 21, 0, -5, 26, 2, 25, 16, -13, -10, 9, 2, -19, -6, 1, 8], [-3, 10, -7, -13, 22, 1, 3, 13, 1, -9, 12, -7, -11, 1, -10, 5], [21, 0, -11, -3, -11, 11, -10, 0, 3, 7, 0, 0, 3, -11, 7, -10], [6, 6, 10, -3, 6, -3, 13, 0, -2, 3, 4, -2, 6, -6, 6, -7], [5, 0, -3, 8, -8, 1, -11, 3, -10, 0, 0, 6, -17, 1, 3, -2], [3, -12, 9, 4, 11, 0, 2, 2, -1, 0, 1, 7, -1, 4, -7, 11], [-15, 11, -21, 4, 13, -8, -14, 9, 12, 11, 13, -7, -22, -10, 8, -3], [-2, -4, 1, 1, -9, 6, -2, -13, -10, 6, -6, 10, 9, 0, -1, -6], [-4, 4, 8, -10, -8, 5, -1, 3, 0, -11, -5, -9, 5, 12, -15, 7], [-8, 10, -19, 13, -5, -1, 3, -7, -19, -7, 6, -8, -20, 7, 1, 8], [-3, 4, 5, -2, 1, 4, -9, -4, -5, 7, -4, 7, 18, -4, 11, -13], [-6, -2, 2, -8, -8, -5, -14, -1, 21, 1, -4, -9, 20, 5, 1, -3], [-6, 6, 13, 2, -9, 0, 4, 0, -1, -11, 0, 8, 2, 0, 4, 11], [-22, 16, 3, 2, 4, -2, 3, -12, -4, 0, 0, -3, 0, 3, 13, -15], [-16, -4, -6, -10, 14, -3, -7, 14, 10, -13, 6, -13, -11, 4, -5, 5], [13, -5, -7, 2, -3, 0, -4, 6, -1, 2, 0, 5, -5, -8, 2, -6], [-10, 7, -7, -4, -26, -12, -6, -15, -3, -2, -15, 11, 12, 12, 13, -17], [1, 12, -8, -8, 5, -13, 5, 8, 10, -5, 14, 8, -17, -6, 8, -4], [-5, -2, 10, -5, 7, -6, -4, -3, 3, 11, 5, 4, 2, -8, 3, 2], [-24, 19, 4, -5, 21, -2, 11, 5, -21, -7, -2, 18, -21, 0, 0, -2], [-3, -9, -4, 4, -14, 4, -5, -5, 13, -7, -1, 0, 18, 6, -4, 3], [-12, 3, -8, 14, -13, 3, -2, -13, -9, -1, 3, 2, 2, 1, 13, -13], [8, 3, 10, -5, 2, -8, -4, 6, 11, 7, 7, -3, 10, -7, 5, 0], [18, -16, -13, -10, -13, -2, -7, -4, 23, 0, 0, 1, 18, -1, -3, 1], [-9, 9, 13, -2, 1, -1, 11, 0, -6, -13, 1, 11, 1, -1, -3, 13], [-11, -5, -5, 0, 1, 3, -19, -3, 10, -1, -6, -4, 4, 1, -2, 0], [6, 4, -6, -8, 8, 4, -9, 8, 7, -3, 7, -15, 1, 0, -5, 5], [17, -24, -5, 0, -27, 3, -13, -6, 5, 4, -13, 0, 24, 15, -11, 3], [20, -18, -11, -3, -7, 0, -14, 5, 21, 8, 0, -5, -2, -19, -3, -2], [-21, 5, 0, 9, -1, -10, 6, -5, -2, 6, -6, 10, 1, 0, 7, -6], [3, -1, 18, 0, -2, 11, 13, -4, 0, -8, 3, 0, 1, 9, -17, 11], [-14, 8, -2, 13, -1, 11, 10, 1, -3, -7, 9, -26, -22, -5, -7, 8], [-11, -5, 4, -5, 26, -3, -6, 13, 23, 0, 0, -22, -2, -6, -9, 6], [8, -7, -18, -9, 14, 8, -15, 17, 18, 9, 5, -24, -5, -12, -5, 3], [15, -7, 15, -3, 11, 0, 4, -4, -5, 1, -2, 25, 14, -4, -9, 5], [17, -9, 6, -14, -16, 2, -15, 2, 10, -4, -12, -9, 18, 1, -2, 2], [10, -18, -12, -12, 3, 6, -12, 11, 12, -7, -6, -13, -7, 2, -18, 1], [2, 6, -3, 4, -1, -5, -6, -3, -10, -1, -4, 6, 1, 0, 0, -4], [17, -6, 16, 10, -23, -1, -9, -10, 16, 3, -12, -6, 17, 2, 4, -2], [3, -14, 12, -7, -4, -13, -10, -6, 16, -7, -6, 1, 11, 13, -1, -6], [0, -22, 5, -4, 1, 0, 4, -4, 21, -5, -2, 2, 13, 1, -7, 8], [8, -5, 24, -5, 7, -2, 3, 3, -7, 5, -5, 8, 14, -3, 0, 3], [13, -4, -6, -5, -17, 10, -8, 5, 1, -5, -5, -4, 14, 5, -15, 5], [24, -15, 14, 6, -14, 7, 7, -10, 7, 10, -11, 9, 29, -7, 10, -1], [0, 20, -10, 0, 2, -23, -10, -5, 6, 11, 6, 25, 5, -12, 23, -23], [1, 13, -16, 4, 16, -3, 6, 9, 0, -2, 15, -4, -24, -7, 11, 4], [1, -21, -12, -5, -28, 7, -4, -1, 6, -7, -7, -8, -9, 9, -12, 6], [-12, 7, 1, 15, -12, 7, -5, -22, -14, 7, -13, 18, 11, 6, 1, 6], [14, -9, 7, -2, 15, 0, 2, 8, 4, 7, 4, 13, -1, -9, -10, 1], [-5, -6, -14, 9, -11, 4, 2, -11, 5, 6, 0, 2, -3, -2, 7, -7], [-16, -2, 10, 8, 7, 7, -2, -4, -11, -5, 0, 3, 9, 5, -2, 0], [-32, 11, 18, 8, 22, -12, 24, -18, -38, 9, -6, 36, 10, 6, 10, -9], [-2, 6, -14, 20, -15, -1, 8, -25, -4, 14, 3, 12, 5, -17, 22, -17], [-31, 24, -12, 13, 19, 3, 28, 9, -4, -13, 22, -14, -48, 2, -3, 10], [-7, -10, -11, 0, -15, 2, -18, -4, -9, 9, -10, -5, 5, 5, 4, 2], [6, -1, 3, 5, -23, 2, -25, -4, 27, 6, -2, -22, 8, -7, 3, -1], [24, 3, 18, 0, 2, 0, 6, 9, 13, -2, 6, 1, -2, -13, 11, 0], [5, -9, -17, 10, -14, 4, 3, 5, 4, 5, 5, -12, -12, -4, 0, 10], [9, -9, 24, 7, -7, 14, 9, -4, -11, -7, -11, -16, 17, 18, -21, 18], [-2, 7, 16, 0, 18, 3, 23, -3, -16, -9, 6, 3, -8, 9, -9, 9], [-9, -4, 7, -3, 2, 3, 5, 2, -9, -16, 1, 5, 6, 16, -11, 0], [6, -11, -18, 8, -22, 9, 1, 6, 37, -10, 9, -19, 11, -5, -10, 3], [-7, 9, 9, -13, 37, -1, 4, 21, -11, -13, 8, 7, -18, -4, 1, 14], [3, 5, 0, 0, 9, -7, 11, 11, 15, 6, 4, -6, -9, 0, 13, 4], [-1, -19, 8, 16, -18, 10, 0, -16, -10, 8, -26, -7, 17, 10, -2, 1], [7, 18, -13, -15, 22, 20, 10, 30, 3, -30, 15, -26, -36, 0, -20, 10], [-23, -4, -4, 16, -14, 0, 1, -10, -8, -2, -5, -17, 14, 18, 9, 9], [16, -1, -7, -17, 4, 4, -4, 12, 11, -5, 7, -7, 1, 5, -13, 0], [-14, 32, 16, 5, 15, 3, 29, -2, -13, -5, 0, -6, -27, 8, -6, 3], [32, 9, 9, -11, 3, 4, -6, 3, 0, 10, 2, -2, 16, -8, -5, 5], [-6, -2, 10, 6, -11, 0, 6, -17, -11, 4, -9, -15, 9, 5, 6, 5], [11, 7, -23, 0, -10, -5, -7, -5, 3, 10, 4, 11, 4, -9, 19, -5], [-7, 3, 9, 0, 25, -1, 10, 14, -12, 8, 13, -2, -25, 0, 9, 13], [-8, 2, 6, -13, 5, 0, 9, 4, -5, -16, 0, 4, 1, 19, -13, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_496_bg_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_496_2_bg_c();".
// 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_496_2_bg_c(:prec:=16)
    chi := MakeCharacter_496_bg();
    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_496_2_bg_c();".
// 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_496_2_bg_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_496_bg();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![961, -7068, 21989, -38859, 45837, -41628, 33043, -24459, 17306, -11361, 6433, -2967, 1092, -321, 74, -12, 1]>],Snew);
    return Vf;
end function;