
// Make newform 1368.2.g.b in Magma, downloaded from the LMFDB on 08 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1368_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1368_g_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_1368_2_g_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_1368_2_g_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 := [256, -256, 192, -128, 64, 32, -40, 48, -40, 24, -10, 4, 4, -4, 3, -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], [256, -192, 128, -64, -32, 40, -48, 40, -24, 10, -4, -4, 4, -3, 2, -1], [-64, 64, -64, 96, -8, 8, 8, -16, 14, -6, 8, 0, -1, 1, -1, 1], [320, 96, 96, 0, -32, 36, 0, 34, 0, 10, 0, -3, 4, 1, 2, 0], [128, 704, 192, 224, -16, -88, 48, 8, 76, 14, 4, -4, -2, 11, 4, 5], [-1152, -448, -832, 96, 48, -40, 16, -192, -4, -42, 20, 8, -10, -1, -12, -3], [-1152, -704, -448, -96, 176, -72, -48, -120, -68, -14, -4, 12, -10, -11, -8, -5], [-1152, 512, -448, 384, 16, -160, 64, -148, 92, -38, 28, 6, -18, 5, -8, 5], [-1920, -704, -960, -32, 112, -104, -80, -248, -52, -62, 12, 12, -18, -11, -16, -5], [-3712, 832, -1344, 928, 144, -456, 192, -496, 188, -98, 60, 16, -58, 11, -28, 9], [384, 640, 288, 224, -120, 0, 48, 44, 74, 6, 12, -10, 1, 7, 5, 5], [-2176, 2880, -320, 1504, -80, -456, 336, -288, 396, -66, 100, -8, -50, 35, -12, 25], [-5760, 2880, -1856, 2144, 16, -776, 400, -784, 500, -170, 148, 0, -94, 31, -40, 29], [2176, -1664, 704, -1056, 64, 368, -208, 300, -256, 66, -60, 6, 40, -19, 14, -15], [2816, -832, 960, -752, -72, 368, -144, 368, -162, 74, -52, -8, 43, -9, 19, -9], [3584, -2240, 1152, -1504, 32, 520, -304, 496, -368, 98, -100, 0, 64, -27, 22, -21]];
    Rf_basisdens := [1, 128, 64, 64, 256, 256, 256, 128, 256, 256, 64, 256, 256, 128, 64, 128];
    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_1368_g();"
function MakeCharacter_1368_g()
    N := 1368;
    order := 2;
    char_gens := [343, 685, 1217, 1009];
    v := [2, 1, 2, 2];
    // 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_1368_g_Hecke();"
function MakeCharacter_1368_g_Hecke( : Kf := false)
    N := 1368;
    order := 2;
    char_gens := [343, 685, 1217, 1009];
    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, 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, -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, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1], [0, 1, 0, 0, 0, 0, -1, 0, -1, 0, -1, -1, 1, 0, 0, 0], [0, 1, -1, -1, 0, 0, -1, 0, 1, 1, 0, 1, -1, 0, 1, -1], [1, 0, 0, 0, -1, 0, 0, -2, 2, -2, -1, 2, -1, -1, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, 0, -1, 0, 0, 0, -1, 1, 2, 1, 0, -1, -1, 0, 1], [0, -1, -1, 0, -1, -1, 3, 0, 1, 1, -1, 0, 0, 0, 1, -1], [2, 1, -1, -1, 1, 0, 0, 0, -1, 0, 0, 0, 2, 1, 0, 0], [0, -2, 0, 1, -1, -1, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0], [-2, 2, 0, 1, 1, -1, 0, 2, 0, 2, 1, -1, 1, 0, 0, 0], [0, -1, -2, 0, -1, 0, -1, 0, 1, -2, 0, 1, -2, -1, 0, -1], [-1, 2, -2, -1, 2, 1, 0, 2, -2, 2, 1, 0, 3, 1, 0, -2], [0, -1, 0, 1, 0, -1, -5, 0, -1, -1, -1, -1, 0, 0, -1, 0], [0, -1, 0, 0, -1, -2, -2, 0, 1, 2, 0, 1, -2, 1, 0, -1], [0, -3, -4, 0, 1, 2, 1, 0, 3, -2, 1, 2, 1, -3, 4, -1], [0, 4, 3, -1, 0, 0, 0, 0, -2, 2, 0, -1, 0, 2, -2, 1], [-4, 2, -3, 1, 1, 2, 0, 2, -2, 0, 0, 1, 2, 2, 0, 1], [-1, 4, -2, 0, 2, 2, 0, 4, -4, 0, 0, 0, 4, 2, 0, 0], [-2, -1, 2, -2, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 1], [0, 0, -2, 0, -1, 0, 4, 0, 0, -2, -1, 0, -1, -1, 0, -1], [4, 2, -2, -3, 1, 1, 0, -2, 0, -2, -1, 3, 3, 0, 0, -2], [0, 4, 1, 1, 1, 0, 0, 4, -2, 0, 0, -1, 2, 0, 0, -1], [0, 4, 0, -2, 2, 2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 2, 1, -3, 1, 0, 0, 0, -4, -4, -2, -1, 4, 2, 0, -3], [0, 3, -4, -2, 1, 0, 2, 0, 1, 2, 0, 3, -2, -1, 4, -3], [0, -3, 3, 2, -1, -3, -3, 0, -1, 1, -1, -2, 0, 2, -3, 1], [-2, 0, 4, 1, -1, -1, 0, 2, -2, -2, -1, -3, -1, 0, 0, -2], [0, -1, -3, -3, -1, 4, 0, -4, 1, -4, -2, 6, 0, -1, 0, -2], [0, 5, 0, -2, 1, 0, -1, 0, -1, 2, -2, -1, 0, 1, 0, -1], [-1, 0, -2, -2, 2, 0, 0, 0, -4, 0, 0, -2, 4, 4, 0, -2], [0, 3, -2, -2, 2, 0, -1, 0, 1, 4, 1, 3, -1, 0, 4, -2], [0, 1, 0, 0, 2, 0, -7, 0, -1, 2, 0, 0, 2, 0, 2, 0], [-6, 0, -2, 2, 2, 0, 0, 4, -4, 4, 2, -4, 2, 4, 0, 4], [0, -2, -2, 0, -1, -4, -2, 0, 2, 4, -3, 0, -1, 1, 2, -3], [0, 4, 0, 0, -2, 0, 4, 0, -4, -4, -2, -2, -2, 0, -4, 0], [-2, 1, 2, -2, -2, 2, 0, -2, 3, -4, -2, 5, -2, -5, 0, -1], [0, 2, 3, 1, 1, -2, 0, 0, -4, 2, -2, -3, 2, 2, -2, 1], [0, -3, -5, -3, 3, 4, 2, 0, 9, 0, 2, 4, 0, -3, 6, -2], [0, -6, 1, 1, -1, 0, 4, 0, 4, 2, 2, 1, 2, 0, 2, 1], [0, 2, 0, 0, 2, -1, 0, 3, -5, 0, 0, -4, 4, 4, 0, 3], [-4, 2, -1, 0, 0, -1, 0, -2, 6, 2, 1, 4, -1, -4, 0, -3], [0, 0, -2, 0, -5, 0, -4, 0, 0, -2, 1, 2, -3, -1, 0, -1], [-6, -2, 4, 2, -1, -1, 0, 3, -1, 2, 1, -4, -3, -1, 0, 4], [0, -3, -5, -1, -2, -2, -4, 0, 5, 4, -2, 2, 0, -1, 6, -4], [-2, -3, -2, 2, 0, 2, 0, 2, 3, 8, 4, 1, -4, -3, 0, 1], [0, -5, -4, 2, -3, 0, -4, 0, 1, -6, 2, 3, -4, -3, 0, -1], [0, 1, 2, 0, 5, 0, 9, 0, -1, 2, 1, 0, 1, 1, 0, 1], [1, 0, -4, 1, 0, 3, 0, 0, 4, 4, 2, 5, -2, -3, 0, 0], [-2, -2, 2, -4, -1, -1, 0, -5, 3, -2, -1, 2, -1, -3, 0, 0], [-2, -2, 2, 1, -1, 1, 0, 2, 0, 2, 1, -1, -3, -2, 0, 8], [0, 7, -2, -4, 3, 4, 7, 0, 1, 2, 2, 3, 0, -1, 4, -1], [0, 2, 5, -3, -1, -4, 0, -4, 4, -4, -2, 1, 0, -4, 0, -1], [3, 0, -4, 1, 0, 1, 0, -2, 0, -2, -1, 2, 1, 3, 0, -6], [0, 2, 2, -1, 1, -1, 10, 0, 0, 4, -1, -1, 1, 2, 0, 0], [9, -4, 0, 3, -2, 0, 0, -1, 3, 2, 1, 0, -5, -1, 0, -3], [0, -1, -2, -2, 1, 2, 11, 0, 5, 2, 5, 6, -3, -1, 4, -1], [-2, 2, 0, 2, -2, 2, 0, 0, 2, -4, -2, 4, -2, -2, 0, 0], [0, -1, -4, -2, 1, 4, -1, 0, 5, -6, 4, 5, -6, -3, 0, -1], [0, -1, 6, 2, 3, -4, -3, 0, -3, 7, 0, -2, 3, 4, -1, 2], [0, 1, 3, 1, -3, 0, -4, 0, -3, 0, -2, -4, 4, 1, -2, 2], [6, -2, 4, -2, -1, 1, 0, 1, 1, 2, 1, 0, -3, -5, 0, -2], [2, 0, -2, 1, -2, 5, 0, 0, 4, 0, 0, 7, -4, -5, 0, 4], [0, -1, 2, 0, 1, -2, 5, 0, 1, 1, 0, 0, -3, 2, -3, 0], [0, -4, 1, -1, 0, -2, 4, 0, 6, 6, -2, -1, 2, 2, 2, -1], [-4, 4, -1, 5, 1, -6, 0, 0, -2, -4, -2, -5, 4, 8, 0, -3], [0, 1, 8, 4, 0, -4, 5, 0, -9, 4, -3, -7, 7, 4, -4, 4], [0, -7, -2, 0, -6, 2, 5, 0, 7, 0, 4, 4, 0, -2, 4, 0], [8, -4, 2, -3, -1, 1, 0, -2, -2, -2, -1, -1, -1, 0, 0, 3], [-8, -4, 6, -2, -2, -1, 0, -1, -3, -4, -2, -4, -2, 0, 0, -3], [2, -4, 0, -2, 1, -4, 0, -4, 0, 2, 1, -4, 1, 3, 0, -3], [0, 3, 6, 2, 3, -4, -9, 0, -7, 1, -2, -4, -1, 4, -7, 2], [0, 1, 3, 3, -4, -6, -12, 0, -7, 4, -6, -6, 4, 3, -2, 0], [16, -5, 2, 0, -2, -4, 0, -6, 5, 0, 0, -1, -4, -1, 0, 1], [0, 1, 2, 2, -7, -2, -11, 0, -5, 0, -1, -2, 1, 1, -2, 1], [0, -1, 2, 2, 7, -2, 7, 0, -3, 6, -3, -4, 9, 1, 4, 1], [0, 4, 1, 2, 4, -5, 0, 6, -4, 6, 3, -8, 5, 4, 0, -1], [0, 0, -5, 0, 0, -1, 0, -6, 8, 2, 1, 6, -1, -2, 0, -3], [0, 12, 8, -2, 1, 0, -4, 0, -8, 6, 1, -2, 1, 5, -4, 3], [0, 7, -2, -5, 4, 7, 5, 0, 3, -3, 3, 3, -2, -2, 1, 0], [2, -6, -2, 0, -2, -4, 0, -10, 4, -4, -2, 0, -2, 4, 0, 2], [6, -2, 8, -3, -1, -5, 0, -2, 0, -2, -1, -5, -1, -2, 0, 6], [6, -5, 1, -1, -1, 4, 0, 2, -5, 0, 0, -2, -2, 1, 0, 4], [0, -5, -2, 2, 0, 0, 1, 0, 1, -4, -3, -3, 3, -2, 0, 0], [4, -2, 2, 2, -2, 4, 0, 4, -2, 0, 0, 0, -4, -2, 0, 4], [-9, 0, 2, 2, 0, -4, 0, 0, 0, 0, 0, -4, 0, 2, 0, -4], [0, -3, -12, -4, -3, 2, 5, 0, 11, 0, -1, 6, -3, -5, 10, -7], [3, 10, 0, 1, 2, -3, 0, 2, -2, -6, -3, 0, 7, 3, 0, -6], [0, -7, 4, 2, 1, -4, 1, 0, 3, 2, -2, -3, 0, 3, -4, 1], [-10, 4, -4, 6, 3, 4, 0, 12, -8, 6, 3, -4, 3, 5, 0, 3], [-2, -12, 1, -1, -5, 6, 0, -4, 2, 0, 0, 3, -10, -4, 0, 3], [0, 2, 2, -2, 6, 8, -8, 0, 2, -4, 6, 2, 2, -2, 0, 4], [0, -1, -4, -2, 3, 8, 1, 0, 5, -6, 4, 3, 2, -5, 4, 1], [-11, -4, 0, 7, -1, -2, 0, 3, 3, 8, 4, -4, -6, 0, 0, 2], [0, 2, 1, -1, -9, 2, 10, 0, 0, -2, 2, 1, -2, 0, -2, 1], [0, 2, 8, -1, -3, -7, 0, -6, 0, -14, -7, -3, 1, 2, 0, 0], [0, 5, 3, -1, 0, 2, 0, 0, -3, 0, 0, -2, 2, 1, -2, 2], [0, 3, -2, 0, -6, 6, -5, 0, -3, -10, 0, -2, 2, -4, -2, 2], [0, 4, 2, 0, 0, 2, 0, 0, -4, -8, 2, 0, -6, 0, -8, 2], [0, 1, 4, 2, -4, -2, 1, 0, -5, 0, 2, 0, -2, 2, -4, 2], [0, -9, 3, 3, -1, -2, 6, 0, 3, 0, -2, -4, 4, 1, -2, 2], [-8, 0, 1, 1, -1, 0, 0, 0, -2, -4, -2, -1, 0, 2, 0, -1], [0, 0, 4, 6, -9, -4, -16, 0, -12, -10, -7, -10, 1, 1, -12, 3], [7, 0, 0, 4, 1, 0, 0, 6, -2, 6, 3, -4, -1, 1, 0, 5], [0, 3, -2, -2, 4, 0, -1, 0, 1, 0, 3, 5, -7, 0, 0, -2], [10, 0, 4, -2, -4, -2, 0, -8, 4, -12, -6, 4, -2, -2, 0, 2], [4, 1, -5, -1, 5, 6, 0, 10, -7, 12, 6, -2, 4, 1, 0, 4], [-2, 0, 0, -8, -2, 8, 0, -4, -4, -12, -6, 8, 2, -2, 0, -2], [2, 0, -1, 1, 1, 4, 0, 6, -8, 0, 0, -3, 2, 4, 0, 7], [0, 1, 10, 2, -6, -4, -11, 0, -5, 6, -6, -10, 8, 6, -6, 4], [1, -4, -4, 3, -4, 5, 0, -4, 4, -4, -2, 7, -6, -1, 0, 2], [0, -4, -6, -2, 10, 8, 0, 0, 8, -4, 4, 4, 4, -6, 8, 0], [-3, 4, 4, 3, 0, -5, 0, 2, -4, -6, -3, -6, 3, 5, 0, 6], [4, 2, -11, 5, 3, 4, 0, 4, -4, 4, 2, 1, 4, 8, 0, 1], [0, -3, -2, -4, 5, 4, 11, 0, 11, 6, 2, 3, 4, -1, 8, -1], [-2, 6, 8, -2, 1, -7, 0, 1, -3, -6, -3, -6, 5, 1, 0, 4], [0, 1, 2, 0, 9, 0, 1, 0, -1, -2, -3, -4, 1, 1, -4, 1], [0, -21, 3, 7, 2, -6, 4, 0, 7, 4, 2, 0, 4, 1, 2, 2], [0, 15, 2, -4, -9, 2, -5, 0, -7, 1, -2, -2, -1, 2, -3, 0], [10, 4, -7, -5, 3, 6, 0, 0, -6, -4, -2, 5, 8, 4, 0, -9], [0, 9, -5, -6, -1, 9, 3, 0, 3, 1, 3, 4, 4, -4, 9, -1], [0, -2, -2, 2, -8, -4, -24, 0, -2, -4, -6, -4, -2, 0, -4, -2], [0, 1, 0, 0, -4, -4, -5, 0, -1, 4, -7, -5, 3, 2, 0, -2], [0, -8, 0, 6, -12, -2, 8, 0, -4, -8, -2, -4, 2, -2, -4, 2], [0, 2, 0, 0, -2, 0, -2, 0, -2, 6, 2, 2, 4, 0, 6, 0], [-4, -4, 0, 0, 2, -1, 0, 3, 1, 12, 6, -4, -2, -2, 0, 1], [-2, 0, 4, -2, 1, -1, 0, 3, 5, 10, 5, 0, -3, -9, 0, 0], [0, 6, 4, 0, 1, -2, 10, 0, -6, 0, -3, -4, -1, 3, -6, 1], [0, -5, -10, -2, -8, 4, 29, 0, 9, -4, 5, 9, -5, -6, 8, -4], [-2, -7, 5, -3, -1, -4, 0, -4, -1, 0, 0, -6, -2, 1, 0, -2], [4, 0, 5, 5, -3, -8, 0, -4, 10, 0, 0, -1, -6, -4, 0, 1], [0, 7, -4, -6, 3, 8, -1, 0, 5, -2, -3, -2, 5, -3, 4, -1], [-1, 4, -4, -4, 0, 12, 0, 4, -4, -4, -2, 10, 2, -4, 0, 0], [-3, -12, -2, -2, -2, 0, 0, -8, 12, 12, 6, 4, -10, -8, 0, 6], [0, -9, -6, -4, 1, 2, -1, 0, 17, 7, 6, 10, -3, -2, 11, -4], [0, 13, 0, -4, 7, 2, -8, 0, -5, 6, 2, 3, 0, 1, 4, -1], [0, 1, -3, -2, -5, -5, 5, 0, 3, 9, 1, 6, -6, 2, 5, -5], [-11, -8, 4, 4, -7, -4, 0, -10, 14, -2, -1, 4, -13, -7, 0, -7], [0, -21, 1, 3, 4, -4, -6, 0, 15, 8, 4, 4, 2, 1, 6, 0], [0, -13, -4, 4, -9, -6, -13, 0, 5, -2, -1, 2, -5, -1, 0, -3], [-4, 8, -4, 4, 4, 1, 0, 9, -1, 8, 4, 0, 4, 0, 0, 5], [0, 0, 6, 0, 1, 2, 14, 0, 0, 4, 8, 4, 0, 2, 0, 4], [0, -13, -8, 2, -13, -2, 13, 0, 9, -5, 6, 10, -11, -4, 3, -4], [-16, -2, -5, 7, -3, 2, 0, -2, 10, 4, 2, 7, -8, -4, 0, 9], [0, 12, 2, 2, -15, -6, -20, 0, -16, -2, -7, -6, -3, 3, -8, -1], [-2, -12, 2, -2, -2, 4, 0, 0, 0, 8, 4, -2, -8, -4, 0, 4], [0, 3, 2, -4, -1, -4, -1, 0, 5, 6, -8, -5, -2, 5, -4, -3], [20, -1, -7, -7, 1, 12, 0, 0, -3, 0, 0, 10, 2, -3, 0, -8], [0, 5, -1, -5, 6, 4, 7, 0, 5, 7, 4, 5, 1, 0, 7, -1], [9, 4, 0, 3, 0, -7, 0, -4, 4, -4, -2, -1, 2, 3, 0, -6], [0, -11, 6, 2, 4, -8, -11, 0, 7, 8, -3, -3, -1, 6, -4, 0], [12, 2, 2, -6, 0, 2, 0, -2, 0, -4, -2, 4, 2, -4, 0, 4], [0, 4, 6, 2, -7, -4, -10, 0, -8, -4, 0, -2, -8, 4, -12, 2], [12, -6, 5, -1, -5, 0, 0, -6, 6, -4, -2, 3, -8, -6, 0, 9], [-7, 4, 10, -3, 1, -10, 0, -1, -5, -8, -4, -10, 6, 4, 0, -12], [6, -12, 2, -7, -4, 1, 0, -12, 4, -4, -2, 3, -6, -3, 0, -6], [-9, -4, 0, 0, -3, 0, 0, -6, 6, -2, -1, 4, -5, -3, 0, 1], [0, 5, 9, -1, 4, 6, 11, 0, -3, 7, 8, 1, 9, 2, 3, 7], [0, 6, 2, -4, -3, -4, 0, 0, 2, 14, 1, 4, -3, 5, 4, -3], [0, 4, -4, 1, 3, -1, 0, 2, -6, -2, -1, -3, 7, 8, 0, -14], [18, 4, 2, -4, -1, 4, 0, 0, 4, -2, -1, 8, -1, -9, 0, -1], [0, 8, 6, 0, -2, 2, -8, 0, -8, -8, 0, -4, -4, 2, -12, 4], [12, 2, 5, 0, 2, -3, 0, 6, -6, 2, 1, -8, 3, 2, 0, 11], [-2, 2, -8, 8, 4, 2, 0, 10, -4, 12, 6, -4, 2, 6, 0, 6], [-4, 7, 2, -2, -2, -4, 0, -8, 3, -16, -8, 5, 4, 1, 0, -11], [0, -5, -2, 2, 5, -2, -25, 0, 1, -6, 1, 2, -7, -1, -4, -1], [-12, 4, -11, 2, 8, 7, 0, 14, -12, 14, 7, -4, 9, 8, 0, 1], [0, 7, 10, 0, 7, -4, 7, 0, -7, 14, 1, -2, 5, 7, 0, 3], [0, 6, 6, 2, -1, -4, 24, 0, -10, 4, -2, -4, 2, 4, -4, 2], [0, 19, 7, -4, 11, -1, -5, 0, -11, 11, -3, -4, 4, 6, -1, 1], [6, 8, 4, -6, 1, 4, 0, 4, -4, -6, -3, 4, 5, -5, 0, -7], [0, -4, 4, -1, -1, 1, 0, 2, 6, 10, 5, 1, -7, -10, 0, -7], [16, -4, 1, -1, -3, 2, 0, -4, -2, -8, -4, 1, -2, 2, 0, 7], [7, 4, 6, -5, -4, -1, 0, -8, 8, -12, -6, 9, -2, -9, 0, -10], [0, -8, 0, 4, -6, -4, 20, 0, 0, -4, 2, 2, -6, 0, -4, 0], [0, -9, 12, 6, -3, -4, -13, 0, -3, 0, -3, -10, 7, 5, -10, 7], [-10, 2, 4, 4, -8, 6, 0, -2, 8, -12, -6, 12, -10, -10, 0, 6], [0, -5, 6, 6, 8, 0, 19, 0, -7, -2, -4, -10, 14, 0, -2, 6], [6, 8, -8, -2, 7, -5, 0, -1, 1, 6, 3, 0, 11, 5, 0, -6], [0, -13, -7, -1, 4, 0, -2, 0, 15, 0, 0, 4, -2, -3, 6, -4], [0, 10, -7, -3, -3, 0, -4, 0, -4, 10, -8, -3, 12, -2, 14, -5], [2, 8, -5, -3, 5, 6, 0, 8, -6, 4, 2, 3, 8, 0, 0, 11], [6, 8, -8, 6, 1, -1, 0, -1, 5, -2, -1, 6, 3, 3, 0, 0], [0, 5, 2, 0, -16, 6, -15, 0, -5, -14, -2, -6, 0, -2, -10, 4], [0, 1, 0, -4, 13, -4, 15, 0, 7, 15, 0, 4, -1, 4, 7, -4], [0, -14, 6, 8, 4, -2, 22, 0, -2, 0, -4, -10, 16, 0, 0, 6], [22, -2, 7, -10, 0, -7, 0, -10, 2, -6, -3, -2, 3, -2, 0, -5], [-1, 4, 2, -9, 1, 2, 0, -3, -3, -8, -4, 4, 6, -2, 0, -2], [-2, -4, -5, 2, 0, 5, 0, 2, -4, 2, 1, 0, -1, 4, 0, 7], [0, -17, -4, 6, -7, -12, -1, 0, 5, 2, -4, 1, -6, 1, 0, -5], [0, 7, 1, 1, 5, 2, -12, 0, -9, -4, -4, -6, 6, -1, -2, 2], [0, 15, 0, -7, 0, 3, 9, 0, -1, 3, -1, 1, -4, 2, -1, -2], [0, 14, 8, 0, 8, 0, 4, 0, -14, -4, -4, -8, 0, 4, -12, 4], [-4, 4, -6, -2, 4, 4, 0, 4, -12, -4, -2, -2, 10, 10, 0, -2], [5, 0, -4, 4, -2, -4, 0, -8, 8, -4, -2, 4, -2, 2, 0, 6], [0, 12, 12, 0, -3, 2, 24, 0, -12, -2, -1, -8, 3, 5, -12, 7], [-14, -4, -4, -3, 0, 11, 0, 4, 0, 8, 4, 7, -4, -7, 0, -2], [26, -2, 0, 2, -1, -1, 0, -1, -1, -2, -1, -2, -1, 3, 0, -4], [0, 8, 17, 5, 3, -4, -6, 0, -18, 2, -2, -11, 6, 8, -14, 9], [16, 1, -2, 4, 0, -6, 0, -4, 9, 4, 2, 1, -2, -1, 0, -13], [0, 1, 4, -5, 10, 5, -3, 0, 9, 1, 7, 5, -6, 2, -3, 2], [-6, 10, -1, -2, 2, 1, 0, 2, 2, -2, -1, 6, 5, -4, 0, 7], [0, 6, -6, 0, -7, 4, -8, 0, -6, -18, -3, -2, -7, -5, -8, -1], [0, 5, -8, 0, -2, 0, 1, 0, -5, -14, -6, -2, -8, -4, -6, -4], [-16, -10, -4, 4, -8, 13, 0, -3, 5, -4, -2, 12, -14, -6, 0, 11], [0, 17, 8, -8, 15, 2, -8, 0, -1, 14, 0, -1, 2, 7, 0, 1], [13, 0, 2, -4, -1, 2, 0, -2, 2, -2, -1, 4, -1, -5, 0, 9], [0, -5, -10, 0, -5, 0, 2, 0, 5, -14, 8, 13, -22, -5, -4, -5], [-8, 4, 0, -1, -5, 5, 0, -6, 6, -14, -7, 13, -3, -6, 0, -7], [0, -10, -10, -2, -1, 10, -6, 0, 14, -8, 9, 10, -1, -9, 10, -1], [-11, -2, -2, 5, 4, -5, 0, 6, -14, 2, 1, -16, 7, 17, 0, 0], [0, -9, -10, -4, 3, 8, 16, 0, 17, -10, 10, 13, -12, -7, 4, -3], [-32, 15, 4, 6, 2, -2, 0, 12, -5, -4, -2, -3, 6, 1, 0, 7], [14, 8, 3, -8, 0, -1, 0, -6, -8, -22, -11, 2, 11, 6, 0, -11], [9, 8, 4, -5, 8, -8, 0, 7, -1, 14, 7, -8, 9, -3, 0, -5], [-12, -8, -6, -8, -4, 18, 0, -4, 0, -4, -2, 16, -6, -8, 0, 10], [-2, 1, 0, -6, 2, 2, 0, 0, -3, 0, 0, 1, 4, -1, 0, 3], [0, 13, -4, -4, 0, 12, 25, 0, -5, -10, 0, -2, 6, -6, 2, 2], [0, 8, -9, -5, 9, 0, 12, 0, 2, 10, 6, 13, -6, -2, 14, -7], [6, 5, 4, 6, -4, -8, 0, -6, 7, -12, -6, 1, -2, 1, 0, 3], [14, 14, 3, 2, 4, -15, 0, -2, 2, -6, -3, -6, 11, 6, 0, -13], [-20, -2, 4, 4, 0, -3, 0, 5, 1, 8, 4, -6, -4, -2, 0, 3], [0, -9, -3, 3, 6, -8, 4, 0, 3, 4, 4, 8, -10, 1, 2, -4], [0, 7, 10, 2, 0, 0, -9, 0, -11, -6, 2, -4, -4, 4, -14, 6], [14, -6, 8, -4, -2, -8, 0, -8, 10, 4, 2, -2, -6, -8, 0, -12], [-4, -5, -1, -9, 5, 6, 0, 6, -9, 12, 6, -4, 4, -1, 0, 6], [4, 4, -9, -1, 5, 10, 0, 10, -4, 12, 6, 5, 4, -2, 0, 3], [0, 1, 14, 6, -2, 0, -11, 0, -13, -8, 1, -9, 3, 4, -16, 10], [0, -13, -10, 4, -7, 0, 19, 0, 5, -6, 5, 8, -3, -7, 8, -3], [0, -10, -5, 5, -6, 6, 12, 0, 0, -18, 10, 7, -6, -8, -2, 3], [-28, 7, -12, 4, 8, 6, 0, 14, -7, 16, 8, -1, 8, 5, 0, -7], [14, -6, 5, -11, 3, 4, 0, 4, -12, 4, 2, -7, 4, 0, 0, 1], [0, -1, 2, -4, 9, -4, 7, 0, 9, 22, 0, 3, 6, 5, 12, -3], [4, 4, -10, 6, 0, -3, 0, -7, 11, 0, 0, 8, 0, 2, 0, -11], [5, 4, 6, -3, -6, 1, 0, -8, 8, -16, -8, 11, -4, -9, 0, -20], [0, -3, 2, -2, 1, -6, 5, 0, 7, 8, 1, 4, -9, 5, -2, -3], [0, 4, -12, -6, -3, 4, -20, 0, 8, -6, 3, 10, -13, -5, 4, -7], [-4, 3, -11, -5, 11, 0, 0, 6, -13, 12, 6, -8, 16, 13, 0, -14], [0, -8, 0, 4, 18, -4, -2, 0, 0, 8, 0, 0, 8, 0, 8, 0], [0, -3, 0, 4, -6, -4, -21, 0, -5, -4, 1, 1, -5, 0, -4, 0], [-23, -2, -4, -1, 1, 2, 0, -1, 1, 4, 2, 2, 0, 0, 0, -6], [0, 5, 10, -4, 7, 2, -7, 0, 3, 5, 10, 6, -9, 6, -7, 4], [-6, -2, 3, -6, -4, -7, 0, -18, 10, -14, -7, 6, -1, -2, 0, -7], [-11, -8, 4, -1, -4, -5, 0, -10, 8, -2, -1, 0, -7, -3, 0, -6], [-17, -8, 6, 11, -12, -3, 0, -10, 12, -14, -7, 4, -17, -3, 0, 6], [0, -1, 7, 1, 7, 4, 4, 0, -1, 12, 0, -6, 22, 1, 10, 6], [0, 24, 12, -4, 10, 4, -24, 0, -16, 0, -2, -8, 2, 6, -12, 6], [8, 14, -10, -6, 8, 7, 0, 7, -9, 0, 0, 6, 16, 4, 0, 9], [0, 0, -10, -8, -15, 4, -4, 0, 16, -2, 5, 12, -15, -3, 4, -7], [-3, 4, 10, 6, -3, -4, 0, 6, -6, -10, -5, -8, -1, 3, 0, -3], [5, -20, -10, 1, -3, 14, 0, -1, -1, 12, 6, 4, -12, 0, 0, 4], [0, -10, -4, 9, -5, 3, 0, -2, 16, 14, 7, 7, -17, -10, 0, 8], [0, 1, -10, -4, -9, 0, 5, 0, 7, 4, -5, 2, 1, -3, 10, -7], [0, 18, -10, -12, 1, 12, 24, 0, 6, 6, 9, 14, -3, -5, 16, -5], [0, -13, 2, 2, -10, -8, -5, 0, 9, 8, -10, -8, 6, 4, 0, -2], [0, 1, -1, 1, -5, -12, 18, 0, -3, 12, -6, 0, -4, 5, 2, -6], [0, -6, -8, -1, 3, 5, -8, 0, 8, -8, 11, 13, -11, -6, 4, -2], [-14, -4, 10, -4, 2, -10, 0, 0, 4, 12, 6, -10, -2, -6, 0, 4], [0, 3, -20, -2, 0, 22, 0, 0, 3, 0, 0, 25, 0, -5, 0, -3], [0, -6, 4, 2, 8, -2, 6, 0, 2, 0, 6, 4, -6, 2, -4, 2], [0, -9, 16, 10, -8, -10, -5, 0, -11, 10, -2, -10, 12, 8, -6, 8], [0, 21, -1, -9, 0, 12, 5, 0, -3, 1, 0, -1, 7, -2, 5, 1], [-5, 12, -6, 1, 4, -3, 0, 0, 4, 0, 0, 5, 8, 1, 0, 0], [0, -12, 2, 6, -3, 4, -6, 0, 0, -16, 8, 2, -4, -4, -8, 6], [0, -15, -4, 5, -6, -1, 35, 0, 5, -9, 1, 1, -2, -4, -1, 0], [7, 12, 2, 4, 8, -10, 0, 12, -12, 4, 2, -16, 14, 12, 0, -10], [0, -7, 4, 2, -9, -8, -10, 0, 3, 2, -4, -3, -6, 5, -8, -1], [0, -9, -4, 6, -8, 6, 11, 0, -3, -18, 6, 2, 0, -8, -2, 4], [0, -7, -2, -1, -12, 7, -1, 0, 9, -13, 9, 7, -10, -4, -5, 2], [0, -2, -8, 0, -17, 4, -8, 0, 2, -16, 0, 2, -8, -6, -4, -2], [27, -16, 6, -10, -7, -8, 0, -26, 14, -10, -5, 4, -9, -5, 0, -7], [10, -2, -14, -4, 5, 11, 0, 3, 3, 18, 9, 10, 1, -5, 0, 2], [-20, 10, 0, -5, 3, 5, 0, 6, -12, -10, -5, 1, 11, 4, 0, -4], [9, -10, 8, 11, -4, -20, 0, -9, 17, 10, 5, -12, -13, -1, 0, -3], [0, 9, 4, -4, 4, 4, 33, 0, -1, -4, -3, -5, -1, 2, -8, 2], [0, 5, -4, -3, 2, 7, 15, 0, 1, -5, 1, 1, 2, -4, 3, 0], [6, 4, 5, -3, -3, -8, 0, -12, 6, -16, -8, 3, 2, 0, 0, -15], [0, -7, 14, 6, 6, 4, 7, 0, -5, -8, 13, 1, -1, 2, -12, 12], [-8, 0, 3, 3, -1, -6, 0, -2, 4, 0, 0, -3, -2, 0, 0, -7], [-21, 4, 6, 3, 0, -4, 0, 5, -15, -14, -7, -12, 7, 13, 0, 5], [12, 8, 14, 10, 2, -15, 0, 13, -5, 4, 2, -20, 2, 4, 0, -1], [0, 15, 4, -6, -11, -6, -5, 0, -3, 7, -8, -4, -9, 8, -9, -4], [0, 15, 0, 0, 4, 4, 9, 0, -15, -12, -1, -3, -3, -2, -8, 2], [15, 2, -2, -13, 0, 1, 0, -14, -6, -22, -11, 6, 11, 7, 0, -16], [-7, -4, -2, 8, 3, 2, 0, 14, -6, 18, 9, -10, -3, 3, 0, 9], [0, -10, -9, -1, -1, -8, -18, 0, 12, 2, -12, -3, -4, 0, 2, -9], [10, 4, -6, 4, -3, 13, 0, 5, -1, -6, -3, 12, -3, -3, 0, -10], [-8, 6, -9, -3, 3, 22, 0, 16, -8, 8, 4, 13, 2, -8, 0, 3], [0, -6, 0, 4, 13, -2, 22, 0, -2, 4, -3, -4, 11, -1, 6, 1], [0, 13, -6, -6, -6, -4, -19, 0, -1, 4, -5, 3, -11, 2, 0, -8], [12, 2, -16, -4, 8, 0, 0, -4, 6, 16, 8, 6, 8, 2, 0, -10], [0, -4, -6, 2, -16, -8, -10, 0, 0, 0, -4, 2, -8, 0, 0, -6], [0, 4, 11, 7, -1, -4, -4, 0, -18, -10, -12, -19, 8, 4, -18, 7], [0, -14, -8, 8, -4, -4, -6, 0, -2, -10, -6, -4, 4, -6, 2, -2], [-2, 8, -17, 3, 15, 6, 0, 22, -24, 20, 10, -13, 20, 20, 0, -5], [0, 15, -8, -10, 10, 10, 3, 0, 5, 12, 10, 14, 2, -4, 20, -4], [0, 14, 10, 0, 7, 2, 2, 0, -14, 0, -12, -18, 16, 4, -8, 6], [0, -9, -2, 0, 15, 4, 5, 0, 9, -6, -2, -3, 4, -3, 0, 1], [7, 4, 8, -1, 0, 1, 0, 8, -8, -4, -2, -5, 2, -1, 0, -8], [0, -5, 0, -2, 15, 6, 37, 0, 9, 1, 8, 6, 1, -2, 5, 2], [38, -16, 8, -10, -5, 9, 0, -3, -9, -6, -3, -2, -7, -3, 0, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1368_g_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1368_2_g_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_1368_2_g_b(:prec:=16)
    chi := MakeCharacter_1368_g();
    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_1368_2_g_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_1368_2_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1368_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![82944, 0, 358912, 0, 425168, 0, 233080, 0, 68929, 0, 11532, 0, 1078, 0, 52, 0, 1]>],Snew);
    return Vf;
end function;