
// Make newform 336.2.bc.f in Magma, downloaded from the LMFDB on 06 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_336_bc();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_336_bc_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_336_2_bc_f();".
// 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_336_2_bc_f();".
// 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, -13122, 13851, -10206, 5265, -1296, -846, 1332, -962, 444, -94, -48, 65, -42, 19, -6, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-52488, 28431, 21627, -14985, 2160, 378, -1272, 854, -252, -266, 384, -110, 48, 71, -15, -1], [19683, 34992, -38880, 41067, -28890, 846, 7392, -8138, 3546, -1288, -258, 662, -363, 142, -30, -11], [131220, -147987, 126846, -70227, 23193, 5787, -14727, 11651, -5787, 1483, 429, -737, 489, -238, 75, -22], [69984, -131949, 133893, -88047, 39366, -2826, -13818, 13268, -7812, 2656, -66, -770, 606, -295, 111, -25], [85293, -131463, 110970, -73926, 29421, -513, -11899, 11435, -6443, 2131, 145, -653, 500, -274, 95, -19], [599238, -888651, 857547, -587979, 239166, 3510, -97758, 92296, -53232, 16736, 1302, -5794, 4116, -2069, 669, -125], [-75087, 105948, -95580, 60237, -24354, -42, 10040, -9482, 5482, -1912, -74, 542, -433, 226, -82, 19], [-708588, 958635, -849285, 515565, -197316, 756, 85788, -81374, 46218, -16018, -84, 4292, -3624, 1975, -753, 193], [-347733, 502281, -474093, 312012, -142074, 9288, 45918, -48824, 29190, -10624, 510, 2648, -2301, 1207, -429, 100], [255879, -374220, 331290, -197181, 85716, -1116, -33490, 32084, -18728, 6358, 100, -1772, 1463, -784, 302, -61], [321489, -438615, 402489, -271836, 111618, -3870, -43500, 42964, -24816, 8450, 234, -2470, 2043, -1037, 351, -74], [165483, -231579, 214650, -134892, 53433, 1875, -22783, 20887, -11879, 3863, 205, -1225, 962, -482, 179, -41], [883548, -1358127, 1259955, -799713, 321678, -5346, -124578, 123088, -70872, 24788, 6, -7066, 5466, -2921, 1101, -275], [4617, -7452, 7371, -5247, 2499, -349, -661, 779, -497, 195, -17, -41, 38, -21, 8, -2]];
    Rf_basisdens := [1, 1, 34992, 69984, 34992, 23328, 11664, 69984, 7776, 69984, 34992, 23328, 23328, 11664, 69984, 432];
    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_336_bc();"
function MakeCharacter_336_bc()
    N := 336;
    order := 6;
    char_gens := [127, 85, 113, 241];
    v := [6, 6, 3, 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_336_bc_Hecke();"
function MakeCharacter_336_bc_Hecke( : Kf := false)
    N := 336;
    order := 6;
    char_gens := [127, 85, 113, 241];
    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, -1, 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], [0, 0, 0, 0, 0, -1, 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, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, -1], [0, 0, 0, 1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0], [0, 2, 0, -1, 0, -1, 0, 2, 1, -2, 0, -1, -1, 0, -2, 1], [0, 0, 1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, -1, 1, -1, 1, 0, -1, 1, 0, -1, 0, 1, 0, -1, 0, 0], [0, -1, 0, 2, 0, -2, 0, -1, 1, 4, 0, 1, 1, 0, 2, -1], [4, 0, 0, -1, -2, -1, -1, 0, 1, -1, 1, 0, 0, 1, 0, 0], [-1, 0, 2, -2, 1, -1, -2, 0, 2, -1, 1, 0, 0, -1, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 2, 1, 0, 2, 1, 0, -1, 2, -2, 0, 0, -1, 0, 0], [-1, -1, 1, -1, -1, -2, 1, 1, 0, -1, -2, -1, -2, 1, -2, -2], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -1, -2, 0, -1, 2], [2, 0, -2, 1, -1, 3, 1, -4, -3, 3, 1, 1, 2, 1, 3, 0], [1, 0, 1, 0, 1, 0, 1, 0, 0, 0, -2, 0, 0, 1, 0, 0], [0, 0, -1, 0, -2, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0], [1, 3, -1, 0, 0, -2, 0, 1, 1, 0, 1, 0, -1, 0, 0, 3], [2, 0, 1, 1, -1, 1, 1, 0, -1, 1, -2, 0, 0, -2, 0, 0], [-4, 0, 1, -2, 4, -1, -1, 0, 2, -1, 0, 0, 0, -1, 0, 0], [1, 2, -1, 2, -2, 0, 2, 0, 2, 0, -1, -3, 0, 2, 0, -2], [1, 0, -1, 2, 1, 4, -1, -2, 0, 2, 2, 2, 4, -1, 2, 0], [-2, 0, 1, 1, 4, 0, -3, 0, -1, 0, 1, 0, 0, -3, 0, 0], [-2, -4, 2, -2, 1, 2, -1, 0, -2, -2, -1, 4, 0, -1, 4, -2], [0, 0, -1, 0, 0, 2, -1, 0, 0, 2, 1, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 1, -1, -1, 2, 2, -1, 0, 4, 0, -1, 1, 0], [0, 0, 1, 1, -7, -1, 3, 0, -1, -1, -2, 0, 0, 2, 0, 0], [-1, -1, 1, 0, 0, -2, 0, 1, 1, 0, -1, 0, -1, 0, 0, -1], [6, 0, -3, 2, 0, 4, 0, 0, -2, 4, 3, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, -4, 0, 1, 0, 1, 0, -1, -2, 0, -1, 2], [0, 0, 0, -2, 1, -2, -1, 0, -2, 4, 1, 0, 2, -1, 2, 0], [-2, 0, -2, 3, 4, 0, 3, 0, -3, 0, -2, 0, 0, 3, 0, 0], [3, 6, -3, -4, -3, 2, 3, 0, -4, -2, 0, -4, 0, 3, -2, 0], [0, 0, 1, 0, -8, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, 0, -2, -4, -5, -4, -1, 0, 4, -4, 3, 0, 0, 3, 0, 0], [4, 0, -2, -2, -4, -1, 2, 0, 2, -1, 0, 0, 0, 2, 0, 0], [-1, -3, 1, 2, 2, 0, -2, -1, 1, 0, 1, 4, -1, -2, 0, 3], [2, 3, -2, 1, 2, 1, -2, -1, 0, 2, 4, 1, 2, -2, 0, 6], [0, 0, 0, 0, 1, 0, -1, 0, 0, -1, 1, 1, -1, -1, -2, 3], [-6, 0, 0, 3, 12, 0, 1, 0, -3, 0, 0, 0, 0, 1, 0, 0], [-1, -1, 1, 3, 1, -2, -1, 1, 4, 1, 0, -3, 0, -1, -2, 0], [0, 0, -4, -2, -5, 2, -4, 0, 2, 2, 0, 0, 0, 0, 0, 0], [-1, 1, 1, 0, 0, -6, 0, 3, 3, 0, -1, -2, -3, 0, -4, 1], [8, 0, 2, 1, -4, 1, 2, 0, -1, 1, -4, 0, 0, -4, 0, 0], [0, 0, -4, 0, 0, 0, -4, 0, 0, 0, 4, 0, 0, 4, 0, 0], [0, 0, -1, -1, 0, 0, -1, 0, 1, 0, -1, 0, 0, -1, 0, 0], [0, 6, 0, -6, 0, 0, 0, 6, 0, -9, 0, -2, -3, 0, -5, 3], [1, 0, -2, 0, 1, 1, -2, 0, 0, 1, -1, 0, 0, 3, 0, 0], [2, 5, -2, -3, -2, 1, 2, 1, -2, -2, 0, -3, 0, 2, -2, 0], [2, -2, -2, 4, 0, 4, 0, -6, -2, 8, 2, 6, 6, 0, 12, -2], [14, 0, -1, 3, -7, 3, 0, 0, -3, 3, 1, 0, 0, 1, 0, 0], [1, 0, -1, -4, -2, 0, 2, -2, -6, 0, -1, -3, -2, 2, 0, 0], [0, 3, 0, -3, 0, 1, 0, 3, 0, -10, 0, -3, -6, 0, -6, 6], [0, 0, 0, 3, 0, 3, 0, 0, 3, -3, 0, -2, 0, 0, 2, 0], [2, 2, -2, -1, -1, 3, 1, -2, -3, 0, 1, -4, 1, 1, -3, 1], [-6, 0, 2, 0, -6, -4, 2, 0, 0, -4, -5, 0, 0, 3, 0, 0], [0, 0, -3, 3, -1, -3, -5, 0, -3, -3, 2, 0, 0, -2, 0, 0], [1, 3, -1, -4, 0, 6, 0, 1, -3, -8, 1, -2, -1, 0, -4, 3], [0, 0, -3, 2, 0, 2, 2, 0, -2, 2, 1, 0, 0, 1, 0, 0], [2, 3, -2, -2, -4, 0, 4, -1, -3, 0, -2, -1, -1, 4, 0, -3], [-10, 0, 2, -3, 20, 0, -3, 0, 3, 0, 2, 0, 0, -3, 0, 0], [4, 4, -4, -3, -2, 7, 2, -4, -7, -1, 2, -4, 2, 2, -2, 2], [1, 0, 4, 0, 1, -4, 4, 0, 0, -4, -4, 0, 0, 0, 0, 0], [-1, -5, 1, 5, 1, -1, -1, -3, 2, 4, 0, -1, 0, -1, 1, 0], [6, 0, 3, 4, -6, 2, -3, 0, -4, 2, 1, 0, 0, -2, 0, 0], [-2, 0, -7, -2, 0, -4, -2, 0, 2, -4, 7, 0, 0, 2, 0, 0], [0, 0, 0, -4, -3, -4, 3, 0, -4, -3, -3, -1, -7, 3, -6, 1], [2, 0, 0, -4, -4, 0, -2, 0, 4, 0, 0, 0, 0, -2, 0, 0], [4, 2, -4, 7, -2, -1, 2, -6, 1, 10, 2, 9, 3, 2, 12, 1], [0, -4, 0, 2, 0, 1, 0, -4, -2, 3, 0, -4, 0, 0, 0, 0], [16, 0, -2, 3, -8, 3, 1, 0, -3, 3, 1, 0, 0, 1, 0, 0], [-5, 0, 0, 6, 5, 3, 0, 0, -6, 3, 1, 0, 0, 1, 0, 0], [8, 0, 4, -3, 0, -6, 3, 0, 3, -6, -4, 0, 0, -3, 0, 0], [1, 3, -1, -1, 1, 6, -1, 1, 0, -9, 2, -1, -2, -1, -4, 6], [0, 0, 0, 1, 0, 1, 0, 0, 1, -4, 0, -5, -3, 0, 2, 3], [1, 0, 2, 0, 1, 5, 2, 0, 0, 5, 3, 0, 0, -5, 0, 0], [-2, -1, 2, -1, 2, -1, -2, 3, 2, -2, 0, -5, 0, -2, -4, 0], [6, 0, -4, 10, -3, 10, 2, 0, -10, 10, 2, 0, 0, 2, 0, 0], [-2, 0, 2, 0, 4, 0, -4, 4, 4, 0, 2, 2, 4, -4, 0, 0], [2, 0, -2, 1, 0, 2, -1, 0, -1, 2, 2, 0, 0, 1, 0, 0], [0, 0, 0, 2, 3, 2, -3, 0, 2, 1, 3, 5, 3, -3, -2, 3], [-4, 0, 2, -3, 8, 0, 1, 0, 3, 0, 2, 0, 0, 1, 0, 0], [-4, 0, 4, 0, -4, 4, 4, 0, 0, 4, 1, 0, 0, -5, 0, 0], [1, 2, -1, 6, -1, -3, 1, 0, 6, 3, 0, -6, 0, 1, -3, 0], [1, -1, -1, -4, 0, 14, 0, -3, -7, -8, 1, 2, 3, 0, 4, -1], [5, 0, 4, 0, -5, 0, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0], [-2, -2, 2, -2, 4, 0, -4, 2, 0, 0, 2, 0, 2, -4, 0, 2], [-2, 0, -2, 1, 0, 2, -3, 0, -1, 2, 2, 0, 0, 3, 0, 0], [-1, -3, 1, 1, -1, -4, 1, -1, 0, 7, -2, 1, 2, 1, 0, -6], [-2, -6, 2, 12, 1, -10, -1, -2, 10, 13, -1, 9, 1, -1, 10, -3], [0, 0, 4, 3, -4, -3, -1, 0, -3, -3, 5, 0, 0, -5, 0, 0], [2, 9, -2, -4, 0, -2, 0, 5, 1, -8, 2, -3, -5, 0, -6, 9], [10, 0, 1, -4, -10, -2, -1, 0, 4, -2, 1, 0, 0, 0, 0, 0], [2, 4, -2, -4, -4, 0, 4, 0, -4, 0, -2, 2, 0, 4, 0, -4], [-2, -4, 2, 0, -2, -4, 2, 0, 0, 4, -4, 0, 0, 2, 5, -8], [2, 0, -2, 0, -1, 4, 1, -4, -4, 2, 1, -6, 2, 1, -4, 0], [0, 0, 7, 0, 0, -2, 7, 0, 0, -2, -3, 0, 0, -4, 0, 0], [-3, 0, 4, 6, 3, 3, -4, 0, -6, 3, 3, 0, 0, -1, 0, 0], [-14, 0, -4, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, -2, 0, 0], [0, 0, 0, 1, -5, 1, 5, 0, 1, -2, -5, -2, -1, 5, 1, -9], [2, -4, -2, 3, -1, 5, 1, -8, -5, 7, 1, -3, 4, 1, 1, -2], [-3, 0, 3, 2, 3, -4, -3, 6, 8, -2, 0, 2, 0, -3, -2, 0], [0, 0, -1, 2, 8, -2, 4, 0, -2, -2, -5, 0, 0, 5, 0, 0], [-6, 0, 0, 8, 3, 8, 2, 0, -8, 8, -2, 0, 0, -2, 0, 0], [-2, -7, 2, -2, 4, 0, -4, -3, -5, 0, 2, 3, -3, -4, 0, 7], [1, 2, -1, 0, 1, -4, -1, 0, 0, 4, 2, 0, 0, -1, -2, 4], [0, 0, 0, -3, -4, -3, 4, 0, -3, 5, -4, -2, 2, 4, 4, -10], [0, 0, 5, -6, 0, 0, 0, 0, 6, 0, 5, 0, 0, 0, 0, 0], [2, 0, -5, 0, 2, -9, -5, 0, 0, -9, 2, 0, 0, 3, 0, 0], [0, 0, 4, 2, -13, -2, -3, 0, -2, -2, 7, 0, 0, -7, 0, 0], [-2, -8, 2, 2, 0, 4, 0, -4, -2, 4, -2, 8, 4, 0, 16, -8], [-4, 0, 11, -2, 2, -2, -4, 0, 2, -2, -7, 0, 0, -7, 0, 0], [16, 0, 1, 1, 0, 2, 3, 0, -1, 2, -1, 0, 0, -3, 0, 0], [0, 0, 0, -3, -6, -3, 6, 0, -3, -4, -6, -11, -7, 6, 4, -5], [2, 0, -6, 1, -4, 0, 5, 0, -1, 0, -6, 0, 0, 5, 0, 0], [-8, -8, 8, 3, 4, -11, -4, 8, 11, -1, -4, 2, -4, -4, -2, -4], [1, 1, -1, 7, -1, -3, 1, -1, 6, 4, 0, 9, 0, 1, 5, 0], [3, 5, -3, -4, 0, 10, 0, -1, -5, -8, 3, 0, 1, 0, 0, 5], [-10, 0, 5, -1, 5, -1, -1, 0, 1, -1, -4, 0, 0, -4, 0, 0], [-18, 0, 7, -2, 0, -4, -2, 0, 2, -4, -7, 0, 0, 2, 0, 0], [-1, 1, 1, -3, -1, -3, 1, 3, 0, -6, -2, -3, -6, 1, -13, 2], [0, 0, 0, 7, 5, 7, -5, 0, 7, 7, 5, 3, 14, -5, 11, -4], [10, 0, 1, 0, 10, 6, 1, 0, 0, 6, -5, 0, 0, 4, 0, 0], [5, 6, -5, 2, 0, 4, 0, -4, -2, 4, 5, 3, 4, 0, 6, 6], [1, 0, -5, 0, -1, 0, 5, 0, 0, 0, -8, 0, 0, -3, 0, 0], [2, 8, -2, -4, 2, 5, -2, 4, 0, -17, 4, -4, -8, -2, -8, 16], [10, 0, -1, 8, -20, 0, 2, 0, -8, 0, -1, 0, 0, 2, 0, 0], [7, 0, -2, 0, 7, -3, -2, 0, 0, -3, 1, 0, 0, 1, 0, 0], [0, 0, -7, 0, -4, 0, -6, 0, 0, 0, -1, 0, 0, 1, 0, 0], [-6, -12, 6, -2, 0, 4, 0, 0, -2, -4, -6, 2, 0, 0, 4, -12], [-14, 0, -5, 6, 14, 3, 5, 0, -6, 3, 2, 0, 0, 7, 0, 0], [20, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -2, 0, 0], [-4, -5, 4, -3, -4, -11, 4, 3, 0, 2, -8, -3, -6, 4, 2, -10], [8, 0, -3, 0, -16, 0, -2, 0, 0, 0, -3, 0, 0, -2, 0, 0], [6, 4, -6, -6, -3, 14, 3, -8, -14, -2, 3, 0, 4, 3, 4, 2], [-32, 0, 0, -1, 16, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [3, 0, -3, -8, -6, 0, 6, -6, -14, 0, -3, -3, -6, 6, 0, 0], [0, 0, 0, -7, -2, -7, 2, 0, -7, 0, -2, -3, -7, 2, -4, 3], [-16, 0, 2, 2, 32, 0, -4, 0, -2, 0, 2, 0, 0, -4, 0, 0], [0, 10, 0, 2, 0, -6, 0, 10, 12, -4, 0, -8, 0, 0, -9, 0], [0, 0, -2, 3, 2, -3, -6, 0, -3, -3, 4, 0, 0, -4, 0, 0], [3, 4, -3, 4, 0, -4, 0, -2, 2, 8, 3, 7, 2, 0, 14, 4], [-6, 0, -7, 5, 3, 5, 7, 0, -5, 5, 0, 0, 0, 0, 0, 0], [0, 2, 0, -4, 0, 0, 0, 2, -2, 0, 0, -6, 2, 0, 0, -2], [8, 0, -8, 5, -16, 0, 5, 0, -5, 0, -8, 0, 0, 5, 0, 0], [2, 0, -2, 0, -1, 4, 1, -4, -4, 2, 1, -6, 2, 1, -4, 0], [2, 0, 2, 0, 2, -5, 2, 0, 0, -5, 0, 0, 0, -2, 0, 0], [4, 2, -4, 4, -4, 1, 4, -6, -2, 5, 0, -6, 0, 4, 0, 0], [11, 0, 5, -8, -11, -4, -5, 0, 8, -4, -2, 0, 0, -7, 0, 0], [-5, -3, 5, 8, 10, 0, -10, 7, 15, 0, 5, 4, 7, -10, 0, 3], [18, 0, 2, -1, 0, -2, -1, 0, 1, -2, -2, 0, 0, 1, 0, 0], [2, -2, -2, 6, 2, -3, -2, -6, 0, 21, 4, 6, 12, -2, 4, -4], [0, 0, 5, -6, 2, 6, 0, 0, 6, 6, 5, 0, 0, -5, 0, 0], [-3, -9, 3, 12, 0, -18, 0, -3, 9, 24, -3, 4, 3, 0, 8, -9], [-14, 0, 3, -20, 14, -10, -3, 0, 20, -10, 1, 0, 0, -2, 0, 0], [0, -5, 0, 5, 0, 7, 0, -5, 0, 8, 0, 5, 10, 0, 8, -10], [-14, 0, -3, 2, 28, 0, -2, 0, -2, 0, -3, 0, 0, -2, 0, 0], [-8, -20, 8, 8, 4, -4, -4, -4, 4, 10, -4, 7, 2, -4, 9, -10], [0, 2, 0, -16, 0, 7, 0, 2, -14, -9, 0, -10, 0, 0, -6, 0], [-1, -1, 1, 4, 0, -10, 0, 1, 5, 8, -1, 4, -1, 0, 8, -1], [-2, 0, 3, -2, 0, -4, 4, 0, 2, -4, -3, 0, 0, -4, 0, 0], [-1, 2, 1, -4, -1, 5, 1, 4, 0, -17, -2, -4, -8, 1, -17, 4], [0, 0, 0, 6, 7, 6, -7, 0, 6, 12, 7, 10, 18, -7, 8, -4], [-2, 6, 2, -12, 1, 2, -1, 10, -2, -17, -1, 7, -5, -1, 2, 3], [0, 0, 2, 1, -4, -1, 15, 0, -1, -1, -13, 0, 0, 13, 0, 0], [-2, 0, -11, -1, 1, -1, 9, 0, 1, -1, 2, 0, 0, 2, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_336_bc_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_336_2_bc_f();".
// 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_336_2_bc_f(:prec:=16)
    chi := MakeCharacter_336_bc();
    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_336_2_bc_f();".
// 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_336_2_bc_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_336_bc();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![256, 0, 12208, 0, 577961, 0, 199741, 0, 47026, 0, 6101, 0, 578, 0, 29, 0, 1]>,<13,R![4096, 0, 3584, 0, 836, 0, 55, 0, 1]>],Snew);
    return Vf;
end function;