
// Make newform 216.3.b.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_216_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_216_b_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_216_3_b_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_216_3_b_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 := [65536, 0, -8192, 0, -512, 0, -896, 0, 400, 0, -56, 0, -2, 0, -2, 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], [0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8192, 0, 512, 0, 896, 0, -400, 0, 56, 0, 2, 0, 2, 0, -1], [-6144, 0, -512, 0, -896, 0, 400, 0, -56, 0, -2, 0, -2, 0, 1, 0], [0, -36864, 0, 15872, 0, -3584, 0, 720, 0, -168, 0, 78, 0, -10, 0, 1], [-28672, 0, 19968, 0, -6528, 0, 944, 0, -264, 0, 90, 0, -22, 0, 3, 0], [7168, 0, 3584, 0, -2048, 0, 336, 0, -40, 0, 14, 0, -10, 0, 1, 0], [0, -139264, 0, 46592, 0, -1664, 0, 464, 0, -568, 0, 166, 0, -10, 0, -3], [0, -38912, 0, 18944, 0, -1344, 0, -80, 0, -128, 0, 62, 0, -6, 0, -3], [0, 28672, 0, -19968, 0, 6528, 0, -944, 0, 264, 0, -90, 0, 22, 0, -3], [-73728, 0, 23040, 0, -1280, 0, 432, 0, -312, 0, 82, 0, -6, 0, -1, 0], [0, -15872, 0, 2048, 0, 752, 0, -200, 0, -30, 0, 4, 0, 5, 0, -1], [-102400, 0, 27136, 0, 1664, 0, -880, 0, -248, 0, 78, 0, 14, 0, -7, 0], [-106496, 0, 61952, 0, -16000, 0, 3280, 0, -568, 0, 278, 0, -58, 0, 5, 0]];
    Rf_basisdens := [1, 1, 1, 1, 4096, 2048, 8192, 4096, 1024, 8192, 4096, 4096, 2048, 1024, 4096, 4096];
    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_216_b();"
function MakeCharacter_216_b()
    N := 216;
    order := 2;
    char_gens := [55, 109, 137];
    v := [1, 1, 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_216_b_Hecke();"
function MakeCharacter_216_b_Hecke(Kf)
    N := 216;
    order := 2;
    char_gens := [55, 109, 137];
    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]];
    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 := 3;
    raw_aps := [[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, 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, -1, 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, 1, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0], [-4, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1], [0, 3, 0, 1, -1, 0, -1, 0, 0, 1, 0, -1, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0], [2, 0, -1, 0, 0, 0, 0, 1, -1, 0, 0, 0, -2, 0, 0, 1], [4, 0, -5, 0, 0, -2, 0, 1, -2, 0, 0, 0, -1, 0, -2, 1], [0, 6, 0, -4, 2, 0, -2, 0, 0, 0, -2, -2, 0, 0, 0, 0], [-4, 0, -7, 0, 0, 4, 0, 1, -1, 0, 0, 0, 0, 0, 0, -1], [0, -7, 0, 1, 1, 0, 1, 0, 0, 1, 2, -3, 0, 0, 0, 0], [0, 2, 0, -4, 0, 0, 0, 0, 0, -4, 2, 2, 0, 0, 0, 0], [0, -13, 0, -4, -4, 0, 0, 0, 0, 4, 0, 4, 0, 1, 0, 0], [-4, 0, 8, 0, 0, 4, 0, 0, 1, 0, 0, 0, -1, 0, -4, 0], [8, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 8, 0, 0, 0, 4, -4, 0, 0, 0, 0], [5, 0, 8, 0, 0, -4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 15, 0, 0, 8, 0, 1, -1, 0, 0, 0, -2, 0, 3, 1], [0, 18, 0, 0, 4, 0, 2, 0, 0, 4, 2, 6, 0, 4, 0, 0], [0, -8, 0, 3, 0, 0, 0, 0, 0, -3, 0, -3, 0, 5, 0, 0], [5, 0, -15, 0, 0, 6, 0, -7, 3, 0, 0, 0, 0, 0, 0, 3], [0, 32, 0, -4, -8, 0, -6, 0, 0, -4, 4, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, -4, 0, -2, -2, 0, 0, 0, 8, 0, -3, -2], [0, 5, 0, 4, 0, 0, 2, 0, 0, 0, 2, 2, 0, -3, 0, 0], [4, 0, -11, 0, 0, -6, 0, -1, 1, 0, 0, 0, 2, 0, 10, -1], [0, 38, 0, -3, 10, 0, -2, 0, 0, -1, -2, -3, 0, -1, 0, 0], [8, 0, -10, 0, 0, -4, 0, 2, -14, 0, 0, 0, 8, 0, 6, 2], [0, -44, 0, 8, -12, 0, 2, 0, 0, -4, 2, -2, 0, -6, 0, 0], [0, -30, 0, -1, -10, 0, -2, 0, 0, -3, -2, -5, 0, -15, 0, 0], [-28, 0, 7, 0, 0, -4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [0, -36, 0, -4, 8, 0, -8, 0, 0, -4, 8, -4, 0, 0, 0, 0], [-14, 0, 19, 0, 0, 8, 0, -3, 3, 0, 0, 0, 6, 0, 1, -3], [4, 0, -5, 0, 0, -2, 0, 1, -9, 0, 0, 0, 6, 0, 6, 1], [20, 0, 11, 0, 0, -8, 0, 4, 5, 0, 0, 0, 0, 0, 0, 5], [0, 67, 0, -3, -17, 0, -25, 0, 0, -3, 4, -1, 0, 0, 0, 0], [0, 34, 0, 4, -8, 0, 6, 0, 0, 4, -6, 2, 0, 0, 0, 0], [0, 16, 0, -20, 4, 0, -8, 0, 0, 4, -8, -4, 0, -4, 0, 0], [-8, 0, -2, 0, 0, -4, 0, -6, 13, 0, 0, 0, 5, 0, -20, -6], [0, -59, 0, -3, 17, 0, 9, 0, 0, -3, -6, 9, 0, 0, 0, 0], [17, 0, 9, 0, 0, -6, 0, -27, 3, 0, 0, 0, 0, 0, 0, 3], [0, 22, 0, 4, -8, 0, -7, 0, 0, 4, 6, -10, 0, 0, 0, 0], [8, 0, -10, 0, 0, -4, 0, 2, 2, 0, 0, 0, -8, 0, 5, 2], [68, 0, 9, 0, 0, -4, 0, -8, -1, 0, 0, 0, 0, 0, 0, -1], [6, 0, -15, 0, 0, -8, 0, -1, 1, 0, 0, 0, 2, 0, -8, -1], [0, 44, 0, -12, 12, 0, -8, 0, 0, -4, -8, -12, 0, -8, 0, 0], [4, 0, 10, 0, 0, 8, 0, 6, -6, 0, 0, 0, -12, 0, 0, 6], [0, -32, 0, -22, -8, 0, -4, 0, 0, 14, -4, 10, 0, 10, 0, 0], [0, 66, 0, -14, -14, 0, 42, 0, 0, -14, 4, 10, 0, 0, 0, 0], [-13, 0, 1, 0, 0, 2, 0, -35, -5, 0, 0, 0, 0, 0, 0, -5], [0, 10, 0, -12, 8, 0, -6, 0, 0, 0, -6, -6, 0, 16, 0, 0], [0, 54, 0, 10, 10, 0, 6, 0, 0, 2, 6, 8, 0, -6, 0, 0], [0, -58, 0, -14, 14, 0, -18, 0, 0, -14, 16, -2, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 7, 0, 0, 0, 14, -14, 0, 0, 0, 0], [-8, 0, 4, 0, 0, 0, 0, -4, 12, 0, 0, 0, 0, 0, -25, -4], [20, 0, -43, 0, 0, -22, 0, -1, 9, 0, 0, 0, -6, 0, 18, -1], [0, -30, 0, -2, -10, 0, 6, 0, 0, 14, 6, 20, 0, 10, 0, 0], [-148, 0, -19, 0, 0, 12, 0, -3, -5, 0, 0, 0, 0, 0, 0, -5], [0, -94, 0, 0, 24, 0, 37, 0, 0, 0, -2, 2, 0, 0, 0, 0], [-76, 0, 3, 0, 0, 4, 0, -21, -11, 0, 0, 0, 0, 0, 0, -11], [0, 23, 0, 1, -1, 0, -41, 0, 0, 1, -20, 19, 0, 0, 0, 0], [59, 0, 9, 0, 0, -6, 0, 53, 3, 0, 0, 0, 0, 0, 0, 3], [0, 82, 0, 0, -24, 0, -10, 0, 0, 0, 14, -14, 0, 0, 0, 0], [164, 0, 57, 0, 0, -28, 0, 2, -1, 0, 0, 0, 0, 0, 0, -1], [-19, 0, -43, 0, 0, 22, 0, -51, -1, 0, 0, 0, 0, 0, 0, -1], [0, -50, 0, 8, -16, 0, -4, 0, 0, -16, -4, -20, 0, -34, 0, 0], [8, 0, -13, 0, 0, -6, 0, 1, 22, 0, 0, 0, -25, 0, -22, 1], [0, -36, 0, -6, -12, 0, -4, 0, 0, -2, -4, -6, 0, -18, 0, 0], [0, 9, 0, -17, -3, 0, 53, 0, 0, -17, 20, -3, 0, 0, 0, 0], [10, 0, -35, 0, 0, -20, 0, -5, 17, 0, 0, 0, -2, 0, -1, -5], [-4, 0, -25, 0, 0, -18, 0, -11, 10, 0, 0, 0, 23, 0, -26, -11], [-40, 0, -28, 0, 0, 20, 0, -11, -12, 0, 0, 0, 0, 0, 0, -12], [0, 54, 0, 18, -14, 0, 34, 0, 0, 18, -16, -2, 0, 0, 0, 0], [0, -28, 0, 12, 8, 0, -23, 0, 0, 12, -16, 4, 0, 0, 0, 0], [-36, 0, 57, 0, 0, 26, 0, -5, 5, 0, 0, 0, 10, 0, 10, -5], [0, 44, 0, 6, 12, 0, 4, 0, 0, 2, 4, 6, 0, 10, 0, 0], [-73, 0, 17, 0, 0, -2, 0, 65, -13, 0, 0, 0, 0, 0, 0, -13], [0, 49, 0, -28, 12, 0, -4, 0, 0, 20, -4, 16, 0, 15, 0, 0], [20, 0, -19, 0, 0, -6, 0, 7, -30, 0, 0, 0, 9, 0, 42, 7], [0, 47, 0, -13, -13, 0, -53, 0, 0, -13, 18, -5, 0, 0, 0, 0], [32, 0, -19, 0, 0, 2, 0, -79, 15, 0, 0, 0, 0, 0, 0, 15], [34, 0, -65, 0, 0, -32, 0, 1, -1, 0, 0, 0, -2, 0, 8, 1], [0, -48, 0, -28, -16, 0, -2, 0, 0, 24, -2, 22, 0, 6, 0, 0], [0, -38, 0, -1, -2, 0, 2, 0, 0, 5, 2, 7, 0, 37, 0, 0], [-40, 0, 37, 0, 0, -22, 0, 73, 7, 0, 0, 0, 0, 0, 0, 7], [0, -4, 0, -16, 0, 0, 7, 0, 0, -16, 20, -4, 0, 0, 0, 0], [2, 0, -1, 0, 0, 0, 0, 1, -33, 0, 0, 0, 30, 0, 41, 1], [0, 21, 0, 52, 0, 0, 18, 0, 0, -16, 18, 2, 0, -19, 0, 0], [0, -52, 0, -2, 12, 0, 44, 0, 0, -2, 6, -4, 0, 0, 0, 0], [-18, 0, 27, 0, 0, 12, 0, -3, -25, 0, 0, 0, 34, 0, 25, -3], [0, -15, 0, -36, 0, 0, -10, 0, 0, 16, -10, 6, 0, 21, 0, 0], [296, 0, -14, 0, 0, 8, 0, -54, -2, 0, 0, 0, 0, 0, 0, -2], [0, -75, 0, 19, 13, 0, 85, 0, 0, 19, 4, -23, 0, 0, 0, 0], [0, 40, 0, -20, -8, 0, 19, 0, 0, -20, 12, 8, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0], [164, 0, 41, 0, 0, -32, 0, 2, 23, 0, 0, 0, 0, 0, 0, 23], [-24, 0, 21, 0, 0, 6, 0, -9, 10, 0, 0, 0, 17, 0, 14, -9], [-328, 0, 92, 0, 0, -44, 0, 45, -4, 0, 0, 0, 0, 0, 0, -4], [0, 68, 0, -4, -16, 0, -23, 0, 0, -4, 0, 4, 0, 0, 0, 0], [0, 14, 0, 36, 0, 0, 22, 0, 0, 8, 22, 30, 0, 16, 0, 0], [0, 20, 0, -37, 20, 0, -12, 0, 0, 13, -12, 1, 0, 61, 0, 0], [-388, 0, 43, 0, 0, -28, 0, -38, 13, 0, 0, 0, 0, 0, 0, 13], [5, 0, 48, 0, 0, -20, 0, -92, -8, 0, 0, 0, 0, 0, 0, -8], [0, -19, 0, 28, -8, 0, 14, 0, 0, 0, 14, 14, 0, 1, 0, 0], [0, 54, 0, -4, 26, 0, -14, 0, 0, -24, -14, -38, 0, 12, 0, 0], [0, -136, 0, 26, 36, 0, 20, 0, 0, 26, -34, 8, 0, 0, 0, 0], [176, 0, 47, 0, 0, -26, 0, 91, 5, 0, 0, 0, 0, 0, 0, 5], [-68, 0, 112, 0, 0, 52, 0, -8, 28, 0, 0, 0, -4, 0, -25, -8], [-64, 0, 89, 0, 0, 38, 0, -13, 38, 0, 0, 0, 1, 0, 62, -13], [0, -52, 0, 51, -12, 0, 16, 0, 0, -19, 16, -3, 0, 1, 0, 0], [248, 0, -92, 0, 0, 56, 0, 51, -20, 0, 0, 0, 0, 0, 0, -20], [46, 0, -95, 0, 0, -48, 0, -1, -31, 0, 0, 0, 34, 0, -51, -1], [0, 52, 0, 72, 12, 0, 24, 0, 0, -24, 24, 0, 0, -4, 0, 0], [476, 0, 61, 0, 0, -28, 0, 61, -5, 0, 0, 0, 0, 0, 0, -5], [0, 170, 0, -4, -46, 0, -14, 0, 0, -4, 18, -14, 0, 0, 0, 0], [0, 34, 0, -24, 0, 0, -30, 0, 0, -24, -10, 34, 0, 0, 0, 0], [0, -208, 0, -36, -56, 0, -14, 0, 0, 8, -14, -6, 0, -22, 0, 0], [0, 0, 39, 0, 0, 26, 0, 13, -36, 0, 0, 0, -3, 0, 10, 13], [-97, 0, -95, 0, 0, 46, 0, -87, 3, 0, 0, 0, 0, 0, 0, 3], [0, -66, 0, 8, 16, 0, -125, 0, 0, 8, -6, -2, 0, 0, 0, 0], [0, 70, 0, -4, 4, 0, 8, 0, 0, 20, 8, 28, 0, -26, 0, 0], [-484, 0, -73, 0, 0, 36, 0, -89, 1, 0, 0, 0, 0, 0, 0, 1], [0, -88, 0, 28, 16, 0, -39, 0, 0, 28, -4, -24, 0, 0, 0, 0], [28, 0, -20, 0, 0, -4, 0, 12, -11, 0, 0, 0, -25, 0, -84, 12], [0, 2, 0, 24, -2, 0, -50, 0, 0, 24, -18, -6, 0, 0, 0, 0], [-22, 0, 47, 0, 0, 24, 0, 1, 23, 0, 0, 0, -26, 0, -20, 1], [12, 0, -51, 0, 0, -30, 0, -9, -23, 0, 0, 0, 50, 0, -54, -9], [-148, 0, 63, 0, 0, -20, 0, 95, -23, 0, 0, 0, 0, 0, 0, -23], [0, -239, 0, -13, 57, 0, 25, 0, 0, -13, 24, -11, 0, 0, 0, 0], [56, 0, -70, 0, 0, -28, 0, 14, -26, 0, 0, 0, -16, 0, 85, 14], [52, 0, -98, 0, 0, -48, 0, 2, 7, 0, 0, 0, -13, 0, -8, 2], [0, -76, 0, 13, -28, 0, 8, 0, 0, 3, 8, 11, 0, -25, 0, 0], [-193, 0, -60, 0, 0, 56, 0, -60, -52, 0, 0, 0, 0, 0, 0, -52], [0, 126, 0, 8, -32, 0, -78, 0, 0, 8, -6, -2, 0, 0, 0, 0], [716, 0, -37, 0, 0, 24, 0, -62, -11, 0, 0, 0, 0, 0, 0, -11], [0, 296, 0, 20, -72, 0, 74, 0, 0, 20, -28, 8, 0, 0, 0, 0], [0, 232, 0, 0, 56, 0, 12, 0, 0, 24, 12, 36, 0, 28, 0, 0], [-4, 0, 49, 0, 0, -12, 0, 65, -25, 0, 0, 0, 0, 0, 0, -25], [0, -262, 0, -8, 56, 0, 123, 0, 0, -8, 46, -38, 0, 0, 0, 0], [-20, 0, 16, 0, 0, 4, 0, -8, -44, 0, 0, 0, 68, 0, -1, -8], [0, 55, 0, -28, 32, 0, -22, 0, 0, -16, -22, -38, 0, 35, 0, 0], [20, 0, 11, 0, 0, 14, 0, 17, 6, 0, 0, 0, -57, 0, -98, 17], [0, 128, 0, -18, -32, 0, 8, 0, 0, -18, 18, 0, 0, 0, 0, 0], [-60, 0, 156, 0, 0, 84, 0, 12, -5, 0, 0, 0, -31, 0, 4, 12], [0, 4, 0, -12, -20, 0, 0, 0, 0, 12, 0, 12, 0, -72, 0, 0], [-712, 0, 84, 0, 0, -36, 0, -29, -12, 0, 0, 0, 0, 0, 0, -12], [0, 129, 0, -3, -39, 0, 81, 0, 0, -3, 30, -27, 0, 0, 0, 0], [-88, 0, 122, 0, 0, 52, 0, -18, 65, 0, 0, 0, -11, 0, -36, -18], [0, -152, 0, -55, -40, 0, -4, 0, 0, 47, -4, 43, 0, 35, 0, 0], [-184, 0, 30, 0, 0, -20, 0, -153, 10, 0, 0, 0, 0, 0, 0, 10], [0, -46, 0, -24, 14, 0, -10, 0, 0, -24, 14, 10, 0, 0, 0, 0], [716, 0, -21, 0, 0, 0, 0, 12, 21, 0, 0, 0, 0, 0, 0, 21], [0, -46, 0, -44, 34, 0, 82, 0, 0, -44, -46, 90, 0, 0, 0, 0], [18, 0, -45, 0, 0, -24, 0, -3, 43, 0, 0, 0, -34, 0, -52, -3], [0, 256, 0, -38, 56, 0, -8, 0, 0, 22, -8, 14, 0, -18, 0, 0], [299, 0, -31, 0, 0, 10, 0, 69, 11, 0, 0, 0, 0, 0, 0, 11], [0, -4, 0, 40, -16, 0, -53, 0, 0, 40, 28, -68, 0, 0, 0, 0], [0, -93, 0, -92, -8, 0, -30, 0, 0, 32, -30, 2, 0, 63, 0, 0], [0, -136, 0, 37, -24, 0, 20, 0, 0, 3, 20, 23, 0, 63, 0, 0], [18, 0, 21, 0, 0, 20, 0, 19, -55, 0, 0, 0, -2, 0, 23, 19], [0, -147, 0, -52, -40, 0, -38, 0, 0, -24, -38, -62, 0, -75, 0, 0], [0, -42, 0, -8, -14, 0, 18, 0, 0, 44, 18, 62, 0, 48, 0, 0], [0, 229, 0, 17, -51, 0, 69, 0, 0, 17, -42, 25, 0, 0, 0, 0], [-74, 0, 79, 0, 0, 28, 0, -23, -29, 0, 0, 0, 98, 0, 115, -23], [-28, 0, 11, 0, 0, -2, 0, -15, -25, 0, 0, 0, 70, 0, -146, -15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_216_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_216_3_b_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_216_3_b_b(:prec:=16)
    chi := MakeCharacter_216_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_216_3_b_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_216_3_b_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_216_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![69696, 0, 42048, 0, 4032, 0, 120, 0, 1]>],Snew);
    return Vf;
end function;