
// Make newform 322.2.g.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_322_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_322_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_322_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_322_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 := [6561, 0, 8748, 0, 5913, 0, 2808, 0, 1045, 0, 312, 0, 73, 0, 12, 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, 6561, 0, 5913, 0, 2808, 0, 1045, 0, 312, 0, 73, 0, 12, 0, 1], [-661203, 0, -601668, 0, -323595, 0, -129853, 0, -39975, 0, -9244, 0, -1713, 0, -100, 0], [0, -769095, 0, -856251, 0, -452016, 0, -179573, 0, -59943, 0, -16250, 0, -3174, 0, -311], [0, -1237113, 0, -1285470, 0, -673740, 0, -286580, 0, -95700, 0, -23303, 0, -4650, 0, -452], [-1070901, 0, -985446, 0, -541512, 0, -227600, 0, -74748, 0, -19349, 0, -3660, 0, -386, 0], [0, -737991, 0, -668304, 0, -367725, 0, -149950, 0, -49464, 0, -12619, 0, -2346, 0, -199], [142155, 0, 194751, 0, 118908, 0, 55462, 0, 18906, 0, 5323, 0, 1155, 0, 148, 0], [0, -967383, 0, -985446, 0, -541512, 0, -227600, 0, -74748, 0, -19349, 0, -3660, 0, -386], [0, -433269, 0, -403254, 0, -221938, 0, -92585, 0, -30151, 0, -7784, 0, -1525, 0, -149], [-935793, 0, -907875, 0, -511680, 0, -211591, 0, -69813, 0, -18316, 0, -3498, 0, -361, 0], [3031182, 0, 3027618, 0, 1673847, 0, 703412, 0, 231384, 0, 60812, 0, 12012, 0, 1241, 0], [2703132, 0, 2756916, 0, 1557621, 0, 651076, 0, 216690, 0, 56644, 0, 11280, 0, 1249, 0], [2635335, 0, 2902716, 0, 1700397, 0, 730286, 0, 247134, 0, 65627, 0, 13236, 0, 1493, 0], [0, -4299642, 0, -4606875, 0, -2673774, 0, -1142809, 0, -382929, 0, -102067, 0, -20352, 0, -2365]];
    Rf_basisdens := [1, 1, 2187, 103518, 310554, 310554, 103518, 103518, 34506, 103518, 34506, 34506, 103518, 103518, 103518, 310554];
    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_322_g();"
function MakeCharacter_322_g()
    N := 322;
    order := 6;
    char_gens := [185, 281];
    v := [1, 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_322_g_Hecke();"
function MakeCharacter_322_g_Hecke(Kf)
    N := 322;
    order := 6;
    char_gens := [185, 281];
    char_values := [[0, 0, 0, 0, 0, 0, -1, 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, -1, 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, 1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, -1, -1, 0, -1, 0, 1, 1, 0, 0, 0, 0, -1], [0, 0, 1, 0, 2, 0, 0, -1, 0, 2, 1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0], [0, 0, 1, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0, -2], [0, 1, 2, 0, 1, 0, 0, -1, 0, 2, 0, 0, 0, 0, 0, 0], [0, -1, -1, 1, 0, 1, -1, 0, -1, 0, -1, 0, 0, 0, 2, 0], [-1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, 1, -1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0], [0, -1, 2, 0, 1, -1, 0, -1, 0, 3, 1, 0, 0, 0, 0, -2], [2, 0, 0, 1, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, -1, 0], [0, 2, -1, 0, -1, -1, 0, 3, 0, -1, 0, 0, 0, 0, 0, 1], [-2, 0, 0, 4, 0, 0, -1, 0, -2, 0, 0, 1, 2, -1, 1, 0], [0, -2, -3, 0, -1, 0, 0, -1, 0, -5, -1, 0, 0, 0, 0, 2], [2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 3, 0, 0], [0, -3, -1, 0, 1, -2, 0, -2, 0, 1, -2, 0, 0, 0, 0, 0], [0, 1, 5, 0, -2, 0, 0, -2, 0, 1, 3, 0, 0, 0, 0, -1], [-4, 0, 0, 1, 0, 0, 0, 0, -2, 0, 0, 0, 1, -1, 1, 0], [2, 0, 0, 2, 0, 0, -2, 0, -4, 0, 0, 3, 2, 1, 4, 0], [0, 1, -3, 0, -3, -2, 0, -2, 0, 3, 2, 0, 0, 0, 0, -4], [0, 3, -2, 0, -3, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, -1], [0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, -1, 0, 4, 4, 0, 4, 0, 2, -8, 0, 0, 0, 0, 6], [-6, 0, 0, 2, 0, 0, 6, 0, -4, 0, 0, 1, 2, -6, 4, 0], [0, 1, 2, 0, 6, 1, 0, -1, 0, 1, 1, 0, 0, 0, 0, 0], [0, 6, -3, 0, -5, -4, 0, -1, 0, -1, 4, 0, 0, 0, 0, -2], [0, 1, 6, 0, -1, 0, 0, -3, 0, 2, 4, 0, 0, 0, 0, -1], [0, -6, 4, 0, 7, 5, 0, -7, 0, 1, 0, 0, 0, 0, 0, -1], [2, 0, 0, 1, 0, 0, 0, 0, -3, 0, 0, 3, 3, -1, 3, 0], [-8, 0, 0, -2, 0, 0, -4, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, 2, -2, 0, -1, 0, 0, -2, 0, -5, 4, 0, 0, 0, 0, -2], [-3, 0, 0, 3, 0, 0, -6, 0, -3, 0, 0, 3, 3, 3, -3, 0], [0, 2, -2, 0, 0, 2, 0, 4, 0, -8, -2, 0, 0, 0, 0, 8], [-3, 0, 0, 6, 0, 0, -3, 0, -4, 0, 0, 2, -2, 4, 0, 0], [0, -1, -3, 0, -2, -2, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1], [0, 0, 0, 3, 0, 0, -4, 0, -4, 0, 0, 8, 0, 6, 0, 0], [-2, 0, 0, 2, 0, 0, -4, 0, -4, 0, 0, 4, -2, 2, 2, 0], [-2, 0, 0, 5, 0, 0, -1, 0, -8, 0, 0, 2, 4, -2, 2, 0], [-9, 0, 0, -2, 0, 0, -9, 0, -4, 0, 0, 2, 0, -1, 0, 0], [0, -5, -5, 0, 5, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 2], [0, 5, -5, 0, -6, 1, 0, -2, 0, -4, -1, 0, 0, 0, 0, -4], [9, 0, 0, -7, 0, 0, 9, 0, 2, 0, 0, -1, -5, -1, 0, 0], [-6, 0, 0, -3, 0, 0, 0, 0, -2, 0, 0, -2, 0, 3, 0, 0], [0, 5, -6, 0, -8, -8, 0, 5, 0, 1, 4, 0, 0, 0, 0, 1], [1, 0, 0, 3, 0, 0, 0, 0, -4, 0, 0, -2, 1, -3, 1, 0], [2, 0, 0, -3, 0, 0, 4, 0, 4, 0, 0, -4, 2, -3, -2, 0], [0, -6, -3, 0, -1, -2, 0, -1, 0, 5, 1, 0, 0, 0, 0, -2], [0, 2, -2, 0, -10, 1, 0, 4, 0, -3, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 5, 0, 0, -8, 0, 2, -2, 0], [7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -1, 0, -1, 0], [0, 1, 5, 0, 3, -6, 0, 0, 0, 4, 3, 0, 0, 0, 0, -3], [0, 4, 8, 0, -1, -3, 0, -3, 0, 2, 6, 0, 0, 0, 0, -4], [8, 0, 0, -7, 0, 0, 4, 0, 7, 0, 0, -1, -2, 1, -1, 0], [0, -2, -11, 0, -3, 0, 0, 1, 0, -13, -3, 0, 0, 0, 0, 2], [1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 5, 0, 9, 0, 0], [-4, 0, 0, -6, 0, 0, -2, 0, 9, 0, 0, -3, -6, 3, -3, 0], [0, 0, 0, -9, 0, 0, 0, 0, 8, 0, 0, -4, -5, -2, 0, 0], [0, -1, 6, 0, 7, 6, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0], [0, -7, 3, 0, 4, 4, 0, -7, 0, 4, -2, 0, 0, 0, 0, -5], [0, -1, 3, 0, -2, 3, 0, 3, 0, -6, -6, 0, 0, 0, 0, 2], [-3, 0, 0, -7, 0, 0, -6, 0, -5, 0, 0, 5, 0, -7, 0, 0], [-4, 0, 0, -1, 0, 0, 4, 0, 2, 0, 0, -1, -1, -2, -2, 0], [0, 2, -5, 0, -1, -2, 0, 7, 0, -3, -2, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 8, 0, -4, 0, 0, 6, 0, 4, 2, 0], [0, 0, 0, 0, 0, 0, -3, 0, 2, 0, 0, -5, 0, -1, 1, 0], [0, -1, 4, 0, 6, -7, 0, -3, 0, 5, 0, 0, 0, 0, 0, -5], [17, 0, 0, -4, 0, 0, 17, 0, 0, 0, 0, 0, 10, -7, 0, 0], [2, 0, 0, -2, 0, 0, 4, 0, -5, 0, 0, 5, 4, -2, -4, 0], [3, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -5, 0, 2, 0, 0], [0, -5, -1, 0, 2, -3, 0, 0, 0, 8, 3, 0, 0, 0, 0, 0], [0, -9, 9, 0, 10, 12, 0, -13, 0, 2, -6, 0, 0, 0, 0, -7], [0, -2, -4, 0, 1, 5, 0, 4, 0, -7, -5, 0, 0, 0, 0, 8], [0, 16, 6, 0, -4, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1], [0, -3, -7, 0, 7, -4, 0, 4, 0, -3, -4, 0, 0, 0, 0, 0], [0, 8, 5, 0, -8, 0, 0, 7, 0, 4, 1, 0, 0, 0, 0, -8], [12, 0, 0, -1, 0, 0, 6, 0, 5, 0, 0, 1, 2, -1, 1, 0], [0, 7, -7, 0, -7, -7, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, -1, 0, 2, 0, 0], [0, -6, 16, 0, 8, -2, 0, -2, 0, 9, 4, 0, 0, 0, 0, 7], [0, 8, -7, 0, -8, 6, 0, 14, 0, -10, 0, 0, 0, 0, 0, 10], [0, -5, -1, 0, 11, 0, 0, 1, 0, 11, -6, 0, 0, 0, 0, 5], [0, 3, 13, 0, 1, -4, 0, -4, 0, 12, 8, 0, 0, 0, 0, -10], [4, 0, 0, 2, 0, 0, 2, 0, -7, 0, 0, -2, -4, 2, -2, 0], [0, 0, 0, 2, 0, 0, -11, 0, -1, 0, 0, -5, 0, -3, 7, 0], [9, 0, 0, 1, 0, 0, 0, 0, 7, 0, 0, 1, -3, -1, -3, 0], [0, 12, -5, 0, -10, -6, 0, -3, 0, -3, 6, 0, 0, 0, 0, -6], [-4, 0, 0, -1, 0, 0, -8, 0, -1, 0, 0, 1, -5, -1, 5, 0], [5, 0, 0, -1, 0, 0, 0, 0, -3, 0, 0, -3, 0, 1, 0, 0], [0, 4, 2, 0, -2, -2, 0, 2, 0, 4, -2, 0, 0, 0, 0, 0], [0, -7, 1, 0, 8, 8, 0, 3, 0, -4, -4, 0, 0, 0, 0, 7], [4, 0, 0, 0, 0, 0, 4, 0, -12, 0, 0, 6, 6, -3, 0, 0], [4, 0, 0, -7, 0, 0, 0, 0, 20, 0, 0, 0, -10, 7, -10, 0], [0, 0, 0, 4, 0, 0, -3, 0, -3, 0, 0, 2, 0, 4, 4, 0], [0, 5, -1, 0, -4, -1, 0, -1, 0, -12, 2, 0, 0, 0, 0, 4], [6, 0, 0, 4, 0, 0, 3, 0, 4, 0, 0, 4, 8, -4, 4, 0], [0, 25, 0, 0, -13, -6, 0, 2, 0, -9, 3, 0, 0, 0, 0, -1], [0, 12, 2, 0, 1, 1, 0, 10, 0, 1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, -13, 0, -5, 0, 0, 10, 0, 6, 0, 0], [9, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, -1, -2, -1, 0], [0, 1, -2, 0, -10, 0, 0, 3, 0, -9, -2, 0, 0, 0, 0, -1], [0, -25, 1, 0, 12, 4, 0, -5, 0, 12, -2, 0, 0, 0, 0, -3], [0, -9, -10, 0, 2, 7, 0, 5, 0, -3, -7, 0, 0, 0, 0, 10], [0, 4, -5, 0, 3, 0, 0, -4, 0, -6, 8, 0, 0, 0, 0, -4], [1, 0, 0, 1, 0, 0, -1, 0, -2, 0, 0, 1, 1, 6, 2, 0], [9, 0, 0, 1, 0, 0, 18, 0, 4, 0, 0, -4, -6, 1, 6, 0], [-6, 0, 0, 1, 0, 0, -3, 0, -7, 0, 0, 2, 4, -2, 2, 0], [0, 0, 0, 3, 0, 0, 0, 0, 8, 0, 0, -4, 9, -3, 0, 0], [6, 0, 0, -7, 0, 0, 12, 0, -3, 0, 0, 3, 1, -7, -1, 0], [-24, 0, 0, 3, 0, 0, -12, 0, 0, 0, 0, 3, 6, -3, 3, 0], [0, -3, 8, 0, 9, 0, 0, -7, 0, 10, 4, 0, 0, 0, 0, 3], [0, -15, 10, 0, 19, 8, 0, -16, 0, 4, 0, 0, 0, 0, 0, -4], [0, 6, 5, 0, 0, 4, 0, -6, 0, -3, -2, 0, 0, 0, 0, -4], [0, -5, 5, 0, 6, 1, 0, -9, 0, 4, 0, 0, 0, 0, 0, -4], [0, -5, 3, 0, 7, 0, 0, 1, 0, 11, -6, 0, 0, 0, 0, 5], [0, -3, -8, 0, -2, 5, 0, 5, 0, -15, -10, 0, 0, 0, 0, 11], [5, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, -3, 0, -8, 0, 0], [0, 0, 0, -2, 0, 0, -7, 0, 7, 0, 0, -16, 0, -6, 2, 0], [0, -7, -6, 0, 3, -8, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 8, 0, 12, 8, 0, 0, 0, -4, -4, 0, 0, 0, 0, 4], [-26, 0, 0, -2, 0, 0, 0, 0, 11, 0, 0, -5, -8, 2, -8, 0], [0, 3, 2, 0, 20, 4, 0, 1, 0, -2, 4, 0, 0, 0, 0, 0], [21, 0, 0, -2, 0, 0, 21, 0, 0, 0, 0, 0, 6, -4, 0, 0], [4, 0, 0, 2, 0, 0, 2, 0, -13, 0, 0, 1, 2, -1, 1, 0], [0, 4, -3, 0, -6, 4, 0, 4, 0, -4, 0, 0, 0, 0, 0, 4], [0, 18, 6, 0, -9, -8, 0, -6, 0, -4, 8, 0, 0, 0, 0, -12], [10, 0, 0, -2, 0, 0, 5, 0, 7, 0, 0, 1, 2, -1, 1, 0], [0, 0, -10, 0, -1, 1, 0, 1, 0, 2, -2, 0, 0, 0, 0, -4], [0, 2, -11, 0, -9, -9, 0, 13, 0, -2, -9, 0, 0, 0, 0, 0], [-7, 0, 0, -6, 0, 0, -7, 0, -18, 0, 0, 9, 2, -4, 0, 0], [0, -20, 2, 0, 12, 8, 0, -12, 0, 2, 0, 0, 0, 0, 0, -2], [0, -1, 4, 0, 5, 9, 0, 5, 0, -13, -9, 0, 0, 0, 0, 10], [0, 9, 0, 0, -2, -1, 0, 7, 0, -3, 0, 0, 0, 0, 0, 3], [4, 0, 0, 5, 0, 0, 2, 0, -7, 0, 0, 0, 0, 0, 0, 0], [0, -11, -4, 0, 6, 5, 0, 5, 0, 10, -10, 0, 0, 0, 0, 2], [0, -19, -11, 0, -1, 2, 0, 0, 0, 8, -1, 0, 0, 0, 0, 1], [0, 3, -16, 0, -15, 10, 0, 0, 0, -9, -5, 0, 0, 0, 0, 5], [0, -2, 14, 0, 6, -4, 0, -4, 0, 9, 8, 0, 0, 0, 0, 3], [-1, 0, 0, -12, 0, 0, -1, 0, -2, 0, 0, 1, -2, -5, 0, 0], [-16, 0, 0, -7, 0, 0, -32, 0, -4, 0, 0, 4, -1, -7, 1, 0], [0, 0, 0, -1, 0, 0, 3, 0, -2, 0, 0, 10, 0, 4, -6, 0], [-14, 0, 0, 3, 0, 0, -14, 0, -2, 0, 0, 1, 3, 0, 0, 0], [0, 9, -9, 0, -16, -1, 0, 11, 0, -5, 0, 0, 0, 0, 0, 5], [15, 0, 0, 4, 0, 0, -15, 0, -8, 0, 0, 13, 4, 0, 8, 0], [0, -8, -2, 0, 6, 2, 0, 2, 0, -7, -4, 0, 0, 0, 0, 19], [-5, 0, 0, -6, 0, 0, -10, 0, -3, 0, 0, 3, 6, -6, -6, 0], [-6, 0, 0, 13, 0, 0, 6, 0, -26, 0, 0, -2, 13, -11, 26, 0], [-26, 0, 0, 6, 0, 0, -13, 0, 4, 0, 0, -2, -4, 2, -2, 0], [0, 0, 0, 5, 0, 0, -6, 0, -15, 0, 0, 21, 0, 1, 9, 0], [0, 0, 0, -1, 0, 0, 2, 0, 2, 0, 0, -13, 0, -11, 9, 0], [0, -3, 23, 0, 8, -5, 0, -5, 0, 28, 10, 0, 0, 0, 0, -12], [21, 0, 0, -2, 0, 0, 0, 0, 7, 0, 0, -7, -7, 2, -7, 0], [8, 0, 0, 5, 0, 0, 4, 0, 6, 0, 0, -2, -4, 2, -2, 0], [0, -4, 6, 0, 7, 0, 0, -9, 0, 4, 5, 0, 0, 0, 0, 4], [0, -6, -5, 0, -11, 5, 0, -7, 0, 1, 0, 0, 0, 0, 0, -1], [0, -20, -17, 0, 5, 13, 0, 7, 0, 0, -13, 0, 0, 0, 0, 14], [-14, 0, 0, -2, 0, 0, -7, 0, 10, 0, 0, -2, -4, 2, -2, 0], [0, -10, 0, 0, 7, 3, 0, 3, 0, -5, -6, 0, 0, 0, 0, 19], [0, 14, -6, 0, -11, -12, 0, 10, 0, -3, 6, 0, 0, 0, 0, 4], [0, 0, 0, -8, 0, 0, 0, 0, 12, 0, 0, -12, 0, -4, -12, 0], [0, 13, 10, 0, 13, -4, 0, 6, 0, -1, 0, 0, 0, 0, 0, 1], [-5, 0, 0, -2, 0, 0, 0, 0, -9, 0, 0, -1, 4, 2, 4, 0], [0, 6, 13, 0, 14, 7, 0, -7, 0, 6, 7, 0, 0, 0, 0, 0], [0, 3, 12, 0, 5, 0, 0, -5, 0, 12, 8, 0, 0, 0, 0, -3], [0, 29, -3, 0, -20, -18, 0, 4, 0, -3, 9, 0, 0, 0, 0, -5], [4, 0, 0, 6, 0, 0, 4, 0, 8, 0, 0, -4, 0, 3, 0, 0], [10, 0, 0, 3, 0, 0, -10, 0, -6, 0, 0, -6, 3, 3, 6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_322_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_322_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_322_2_g_b(:prec:=16)
    chi := MakeCharacter_322_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(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_322_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_322_2_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_322_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -15, 68, 105, 33, -21, -4, 3, 1]>],Snew);
    return Vf;
end function;