
// Make newform 1323.2.h.f in Magma, downloaded from the LMFDB on 15 November 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1323_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1323_h_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_1323_2_h_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_1323_2_h_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 := [9, -9, 36, -3, 90, -36, 40, -8, 9, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-117, -873, -387, -204, -282, -72, -61, -3, -9, -1], [-1557, 36, -108, 303, -240, 204, -23, 48, -12, 7], [63, 270, 204, 123, 30, 68, 8, 12, -1, 2], [-180, 684, -351, 1680, -780, 663, -176, 156, -39, 16], [-63, 369, 1620, 1020, -57, 624, -4, 141, -24, 20], [-1368, -1071, -4401, -2757, -69, -1668, -11, -375, 60, -53], [720, -2898, -432, -7260, 2850, -3264, 632, -732, 165, -82], [801, -3222, 648, -8070, 3174, -3633, 704, -813, 174, -91]];
    Rf_basisdens := [1, 1, 189, 567, 63, 567, 567, 567, 567, 567];
    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_1323_h();"
function MakeCharacter_1323_h()
    N := 1323;
    order := 3;
    char_gens := [785, 1081];
    v := [1, 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_1323_h_Hecke();"
function MakeCharacter_1323_h_Hecke(Kf)
    N := 1323;
    order := 3;
    char_gens := [785, 1081];
    char_values := [[-1, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 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]];
    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, 1, 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, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, -1, 1, 0, 0], [1, 1, 1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 1, -3, -1, 0, 0], [-1, 2, 1, 0, -1, 0, -1, -1, 0, 0], [0, 0, 0, -2, 0, 0, 0, 2, -2, 1], [0, 0, 0, 2, 0, 1, 1, -2, 1, 0], [-1, 0, 1, 1, 1, 0, 0, 0, 1, -1], [0, 2, 0, 0, -2, 0, 0, 0, 0, 0], [0, 1, 2, 0, 1, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, -2, 3, -1, 1, -3], [-4, -1, -1, 2, -1, 1, 0, 0, -1, 1], [0, 0, 0, 0, 0, 1, -5, 0, 0, 2], [-6, -1, -1, -1, 1, 1, 0, 0, 1, 1], [-2, -1, 1, 1, -1, 1, 0, 0, -1, -1], [-1, 4, -1, 2, 1, -4, 0, 0, 1, 1], [-2, 2, 2, -1, 3, -2, 0, 0, 3, -2], [0, 0, 0, 1, 0, 0, 4, -1, 3, 0], [3, -4, -1, 2, 0, 4, 0, 0, 0, 1], [0, 0, 0, 2, 0, 0, -1, -2, 4, 2], [7, -2, 1, 0, -2, 0, 7, 0, 0, 0], [0, 0, 0, 4, 0, -1, -2, -4, 1, -2], [1, 4, -1, 0, -3, 0, 1, -2, 0, 0], [0, 0, 0, 1, 0, -1, 2, -1, 3, 2], [2, 2, -4, 0, -1, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 4, 3, 0, 1, 0], [-3, -3, 0, 0, -3, 0, -3, 1, 0, 0], [5, -2, 2, 1, 1, 2, 0, 0, 1, -2], [0, 0, 0, -2, 0, 5, -12, 2, 0, 0], [-1, 1, 2, 0, -1, 0, -1, -3, 0, 0], [-5, 2, 0, 0, 1, 0, -5, -4, 0, 0], [0, 0, 0, 2, 0, 2, -8, -2, 4, 3], [-3, -2, 1, 0, -3, 0, -3, -5, 0, 0], [0, 5, 0, -1, 4, -5, 0, 0, 4, 0], [-3, 3, 2, 0, 2, 0, -3, 2, 0, 0], [2, 4, -2, 0, 0, 0, 2, -3, 0, 0], [-10, 6, 1, 1, 3, -6, 0, 0, 3, -1], [0, 0, 0, 0, 0, -1, 8, 0, 1, -1], [4, 2, 1, 7, 2, -2, 0, 0, 2, -1], [-1, -5, -7, -1, 2, 5, 0, 0, 2, 7], [-4, 3, 3, -6, 0, -3, 0, 0, 0, -3], [8, -7, -3, 0, 1, 7, 0, 0, 1, 3], [0, 0, 0, 0, 0, 5, -6, 0, -1, -6], [2, -7, -1, 0, 3, 0, 2, 3, 0, 0], [0, 0, 0, 0, 0, 1, 5, 0, 1, 3], [5, -5, 0, 0, -2, 0, 5, -7, 0, 0], [0, 0, 0, 2, 0, 4, -12, -2, -6, 1], [8, 0, 4, 0, 0, 0, 8, 5, 0, 0], [-4, 2, -3, 0, 4, 0, -4, -2, 0, 0], [3, -1, 2, 0, 3, 0, 3, -4, 0, 0], [-5, -2, 0, 3, 2, 2, 0, 0, 2, 0], [0, 0, 0, -5, 0, -3, 1, 5, 1, 0], [-6, 8, 2, 0, 4, 0, -6, -6, 0, 0], [0, 0, 0, 0, 0, -4, 2, 0, 6, 1], [7, -7, -3, 0, -2, 0, 7, 6, 0, 0], [2, 1, -3, 0, 5, 0, 2, 0, 0, 0], [0, 0, 0, -2, 0, -8, -1, 2, 3, -2], [0, 0, -1, -4, -7, 0, 0, 0, -7, 1], [2, -4, 0, 0, 1, 0, 2, 2, 0, 0], [-9, 7, 1, -4, 0, -7, 0, 0, 0, -1], [-17, 7, 7, 1, 7, -7, 0, 0, 7, -7], [-1, 4, -3, -7, -1, -4, 0, 0, -1, 3], [-5, 1, 1, 4, -5, -1, 0, 0, -5, -1], [4, -4, -9, 4, 1, 4, 0, 0, 1, 9], [-3, 0, 2, 0, -1, 0, -3, 5, 0, 0], [7, 2, 7, 1, 1, -2, 0, 0, 1, -7], [0, 0, 0, -6, 0, 11, -4, 6, -6, 2], [-6, 4, 7, 0, 5, 0, -6, 3, 0, 0], [5, -8, 3, 0, -2, 0, 5, 4, 0, 0], [-5, -8, -5, 0, 3, 0, -5, 5, 0, 0], [0, 0, 0, -2, 0, 3, -9, 2, -7, 4], [2, 7, -4, -5, 1, -7, 0, 0, 1, 4], [0, 0, 0, -1, 0, -12, -6, 1, -1, -4], [4, -10, 0, 0, -7, 0, 4, 1, 0, 0], [-11, 0, 2, 0, -1, 0, -11, -7, 0, 0], [0, 0, 0, -3, 0, 4, -10, 3, 1, 2], [-8, 12, 6, 0, -3, -12, 0, 0, -3, -6], [18, -1, 7, 0, -4, 0, 18, 3, 0, 0], [0, 0, 0, 2, 0, 4, 1, -2, -6, 4], [0, 0, 0, 4, 0, -3, 2, -4, 3, 7], [-3, 7, -5, -4, -3, -7, 0, 0, -3, 5], [10, 6, -3, 2, 0, -6, 0, 0, 0, 3], [15, -1, 2, 5, 1, 1, 0, 0, 1, -2], [18, -12, -3, 6, -6, 12, 0, 0, -6, 3], [0, 9, 7, -2, 4, -9, 0, 0, 4, -7], [0, 0, 0, 3, 0, 4, -19, -3, 3, 7], [10, -1, -5, 0, -10, 0, 10, -8, 0, 0], [-6, -11, 0, 0, 0, 0, -6, -1, 0, 0], [20, 0, -2, 0, 6, 0, 20, 9, 0, 0], [0, 0, 0, -2, 0, 2, 2, 2, 4, 1], [12, -7, 7, 0, 2, 0, 12, -4, 0, 0], [0, 0, 0, 3, 0, -7, 4, -3, 2, -3], [-9, 0, 3, -6, -6, 0, 0, 0, -6, -3], [0, 0, 0, 11, 0, 14, -7, -11, -5, 5], [0, 0, 0, 10, 0, 10, -3, -10, 3, 6], [-6, 0, 13, 0, 7, 0, -6, 7, 0, 0], [-2, -4, -5, 0, 8, 0, -2, 3, 0, 0], [0, 0, 0, -4, 0, -4, 6, 4, -3, -4], [0, 0, 0, -6, 0, 9, 6, 6, -4, -1], [-4, 11, -5, -11, 0, -11, 0, 0, 0, 5], [-10, 1, 5, 5, 5, -1, 0, 0, 5, -5], [-15, -3, 1, 2, 1, 3, 0, 0, 1, -1], [0, 0, 0, -3, 0, 4, -16, 3, -5, 9], [0, 0, 0, 4, 0, -1, -1, -4, 5, -9], [-7, 11, 4, 0, 0, 0, -7, 6, 0, 0], [4, -6, -12, -6, -1, 6, 0, 0, -1, 12], [0, 0, 0, 2, 0, 4, 5, -2, 7, 3], [0, 0, 0, 7, 0, 1, -13, -7, -2, -6], [0, 0, 0, 9, 0, 2, -8, -9, 8, 9], [8, 8, 0, 0, 4, 0, 8, 10, 0, 0], [24, -12, -5, 0, -5, 0, 24, -2, 0, 0], [-14, 6, 11, -8, -7, -6, 0, 0, -7, -11], [-25, 10, -1, 0, -1, 0, -25, 3, 0, 0], [3, -2, -12, 0, -7, 0, 3, -1, 0, 0], [0, 0, 0, -8, 0, -15, -1, 8, -1, -9], [0, 0, 0, 7, 0, -12, 3, -7, 4, -5], [0, 0, 0, -2, 0, -18, -5, 2, 2, 1], [-22, -3, 4, 9, 7, 3, 0, 0, 7, -4], [0, 0, 0, 0, 0, 5, 4, 0, 11, 9], [-6, -5, 7, 10, 11, 5, 0, 0, 11, -7], [0, 0, 0, -3, 0, 4, -2, 3, 3, 5], [2, -7, -4, 5, 3, 7, 0, 0, 3, 4], [-13, 10, 4, 10, 6, -10, 0, 0, 6, -4], [-3, -8, 12, -3, -4, 8, 0, 0, -4, -12], [0, 4, 0, 0, 6, 0, 0, 11, 0, 0], [0, 0, 0, -9, 0, -14, -2, 9, 2, -4], [0, 0, 0, 0, 0, -7, 9, 0, -4, -3], [0, 0, 0, -1, 0, 14, -10, 1, 1, 7], [-3, 7, 5, 0, -5, 0, -3, -15, 0, 0], [0, 0, 0, -7, 0, 10, -27, 7, 1, 3], [0, -5, 9, 5, -1, 5, 0, 0, -1, -9], [0, 0, 0, 3, 0, -19, 10, -3, 11, -2], [6, 2, -1, 0, 6, 0, 6, 2, 0, 0], [0, 0, 0, -4, 0, -1, 9, 4, -13, 8], [-2, -7, -5, 1, 8, 7, 0, 0, 8, 5], [1, -1, 2, 0, 3, 0, 1, -8, 0, 0], [0, 0, 0, 1, 0, -15, 14, -1, -6, -2], [4, -9, 0, 0, -15, 9, 0, 0, -15, 0], [-17, 6, 2, -13, 0, -6, 0, 0, 0, -2], [-6, -2, 0, 6, -1, 2, 0, 0, -1, 0], [-29, 5, 8, -1, 3, -5, 0, 0, 3, -8], [0, 0, 0, 8, 0, 15, -6, -8, -5, 5], [0, 0, 0, -6, 0, -12, -2, 6, -10, -8], [0, 0, 0, 3, 0, -12, 10, -3, -3, -3], [-24, -5, -11, 0, 2, 0, -24, -6, 0, 0], [0, 0, 0, -5, 0, -7, 20, 5, -3, 2], [-10, 3, 1, 0, 3, 0, -10, 5, 0, 0], [0, 0, 0, 3, 0, -16, 25, -3, 5, -9], [5, -14, -5, 4, 9, 14, 0, 0, 9, 5], [9, -7, -1, -10, -9, 7, 0, 0, -9, 1], [0, 0, 0, -4, 0, 14, 5, 4, 4, 2], [-4, -4, -4, 0, 12, 0, -4, 8, 0, 0], [0, 0, 0, 4, 0, 6, -8, -4, 2, -11], [5, 11, -4, 0, -3, 0, 5, -4, 0, 0], [1, 13, 5, -13, -6, -13, 0, 0, -6, -5], [30, -21, -7, -6, -10, 21, 0, 0, -10, 7], [22, -4, -9, 9, 11, 4, 0, 0, 11, 9], [-1, 8, 10, -8, -9, -8, 0, 0, -9, -10], [14, -9, -16, 5, -4, 9, 0, 0, -4, 16], [-1, -1, -4, 0, 6, 0, -1, 3, 0, 0], [12, 1, -9, 0, 0, 0, 12, -1, 0, 0], [13, 3, 15, -3, 2, -3, 0, 0, 2, -15], [17, -5, -6, 0, -11, 0, 17, -7, 0, 0], [0, 0, 0, 3, 0, -3, 14, -3, 0, 3], [27, 0, 7, 0, 7, 0, 27, -5, 0, 0], [2, -15, -9, 14, 0, 15, 0, 0, 0, 9], [0, 0, 0, 7, 0, 9, -8, -7, -11, -3], [14, 1, 2, 0, -1, 0, 14, 3, 0, 0], [-31, 14, 7, 0, 2, 0, -31, 7, 0, 0], [0, 0, 0, -14, 0, -18, 6, 14, -2, -2], [15, -24, -10, 0, -1, 0, 15, 4, 0, 0], [-6, -2, -11, 7, 9, 2, 0, 0, 9, 11], [8, -19, -6, 0, 1, 0, 8, 5, 0, 0], [0, 0, 0, -3, 0, -8, -20, 3, -6, -14], [0, 0, 0, 0, 0, 6, 27, 0, 8, 2], [-8, -13, 7, 8, 2, 13, 0, 0, 2, -7], [-10, 7, 6, -9, -4, -7, 0, 0, -4, -6], [14, -5, 2, -3, -14, 5, 0, 0, -14, -2], [0, 0, 0, -9, 0, -16, -15, 9, 4, -2], [-10, 3, 7, 0, -5, 0, -10, -7, 0, 0], [0, 9, -2, 0, -7, 0, 0, -21, 0, 0], [-22, 2, -1, -10, -4, -2, 0, 0, -4, 1], [-18, 17, 0, 0, -3, 0, -18, 2, 0, 0], [0, 0, 0, -10, 0, 3, -15, 10, -19, 1], [3, 7, -12, 0, -4, 0, 3, 4, 0, 0], [0, 0, 0, -1, 0, -6, 3, 1, 5, -11], [0, 0, 0, 1, 0, 2, -14, -1, 7, -9], [25, 0, -7, 0, 11, 0, 25, 12, 0, 0], [0, 0, 0, 2, 0, 5, -22, -2, -7, -3], [-17, -9, -3, 0, 0, 0, -17, 12, 0, 0], [-8, 6, -18, -6, -7, -6, 0, 0, -7, 18], [0, 0, 0, 1, 0, -7, 10, -1, -1, -17], [-9, 5, 11, -4, 1, -5, 0, 0, 1, -11], [4, -7, 3, -8, -14, 7, 0, 0, -14, -3], [6, 16, 3, -12, 5, -16, 0, 0, 5, -3], [0, 0, 0, -5, 0, -8, 7, 5, -2, -9], [30, -20, 0, 0, -6, 0, 30, 2, 0, 0], [13, 8, -11, -11, -8, -8, 0, 0, -8, 11], [0, 0, 0, -9, 0, -14, 19, 9, 11, 8], [0, 0, 0, 11, 0, -3, 17, -11, 8, -5], [-20, -4, -12, 0, -2, 0, -20, 3, 0, 0], [-24, -11, -1, -4, 1, 11, 0, 0, 1, 1], [0, 0, 0, 6, 0, 10, 28, -6, 11, 4], [-16, 5, -11, 0, -13, 0, -16, -11, 0, 0], [0, 0, 0, -1, 0, -13, 37, 1, 7, -12], [0, 0, 0, -12, 0, 7, -6, 12, 2, -7], [12, -6, 8, -13, -1, 6, 0, 0, -1, -8], [1, 4, 2, 0, 1, 0, 1, -16, 0, 0], [-14, 16, 3, 0, 8, 0, -14, 8, 0, 0], [0, 0, 0, -6, 0, -26, 10, 6, 6, -2], [11, -2, -3, 12, 16, 2, 0, 0, 16, 3], [-14, -4, -6, -15, -7, 4, 0, 0, -7, 6], [11, -17, -3, -12, -4, 17, 0, 0, -4, 3], [27, 3, -2, 5, -2, -3, 0, 0, -2, 2], [9, -14, 7, 0, -7, 0, 9, 9, 0, 0], [-5, 12, 3, 0, 2, 0, -5, -14, 0, 0], [-18, 1, -4, 0, -2, 0, -18, -6, 0, 0], [0, 0, 0, 8, 0, 7, 1, -8, 12, -5], [14, -15, 3, 0, -12, 0, 14, 5, 0, 0], [0, 0, 0, 17, 0, 9, -22, -17, 8, 3], [27, -24, -10, 0, -4, 0, 27, 1, 0, 0], [39, -14, -10, 0, -14, 0, 39, -11, 0, 0], [0, 0, 0, -3, 0, 5, 2, 3, 4, 7], [9, -3, -1, -13, -23, 3, 0, 0, -23, 1], [0, 0, 0, -3, 0, 2, -28, 3, -14, -1], [4, -1, 13, 9, -1, 1, 0, 0, -1, -13], [39, -1, -13, 2, 1, 1, 0, 0, 1, 13], [24, -26, -12, 4, -7, 26, 0, 0, -7, 12], [0, 0, 0, -5, 0, 11, 0, 5, -10, 12], [-6, 5, -9, 0, 10, 0, -6, -5, 0, 0], [9, -10, 6, -9, -9, 10, 0, 0, -9, -6], [0, 0, 0, 0, 0, -6, -27, 0, -5, -7], [25, 2, 4, 0, 1, 0, 25, -6, 0, 0], [0, 0, 0, -7, 0, -14, 19, 7, 8, -16], [-23, -4, -2, 0, -2, 0, -23, -13, 0, 0], [15, 1, 2, 0, 1, 0, 15, 6, 0, 0], [-26, 6, -2, -5, -8, -6, 0, 0, -8, 2], [20, -11, -1, 0, -16, 0, 20, 0, 0, 0], [-6, 11, -7, 0, 0, 0, -6, 3, 0, 0], [-5, -27, -12, 2, 9, 27, 0, 0, 9, 12], [-29, 2, 1, 0, 2, 0, -29, -2, 0, 0], [-14, 4, -6, 0, -16, 0, -14, 8, 0, 0], [-30, 6, 2, -1, 7, -6, 0, 0, 7, -2], [18, 1, -2, -4, 9, -1, 0, 0, 9, 2], [-26, -10, 3, -3, -4, 10, 0, 0, -4, -3], [-32, 6, -10, -15, -1, -6, 0, 0, -1, 10], [15, -9, 9, 0, -9, 9, 0, 0, -9, -9], [4, 7, 5, 0, 4, 0, 4, -6, 0, 0], [-3, -5, -9, 0, 24, 0, -3, 5, 0, 0], [-9, 2, -3, 3, 15, -2, 0, 0, 15, 3], [0, 0, 0, -9, 0, 17, 8, 9, -6, 2], [6, 7, 1, 0, -1, 0, 6, 6, 0, 0], [-12, -12, 8, 0, -8, 0, -12, -9, 0, 0], [0, 0, 0, 13, 0, 17, -3, -13, 3, 11], [-23, 0, -1, 0, 14, 0, -23, 11, 0, 0], [10, -7, 11, 14, 6, 7, 0, 0, 6, -11], [8, -20, -7, 0, 7, 0, 8, 6, 0, 0], [16, 8, 1, 0, 2, 0, 16, 12, 0, 0], [0, 0, 0, 7, 0, -2, 8, -7, 18, -11], [31, 6, -6, 3, 12, -6, 0, 0, 12, 6], [-11, 16, 8, -21, -5, -16, 0, 0, -5, -8], [4, 26, 0, 6, 8, -26, 0, 0, 8, 0], [-29, 15, -3, 12, -3, -15, 0, 0, -3, 3], [17, 4, -8, 4, 3, -4, 0, 0, 3, 8], [0, 0, 0, -15, 0, 3, -2, 15, -16, 4], [-17, 20, 7, 0, 14, 0, -17, -11, 0, 0], [-5, -11, -10, 20, 0, 11, 0, 0, 0, 10], [0, 0, 0, 7, 0, 5, -27, -7, -1, 15], [0, 0, 0, -12, 0, -15, 5, 12, -12, 12], [25, 17, 0, 0, 1, 0, 25, -6, 0, 0], [0, 0, 0, 11, 0, -8, 1, -11, -3, 13], [-6, -3, -14, 0, 4, 0, -6, 7, 0, 0], [6, -4, -4, 0, -6, 0, 6, -19, 0, 0], [0, 0, 0, -4, 0, -5, 21, 4, 6, 6], [-39, 17, 12, 0, 7, 0, -39, 3, 0, 0], [27, -9, -1, 0, -7, 0, 27, 11, 0, 0], [-18, 2, 15, 3, 0, -2, 0, 0, 0, -15], [-15, -18, -5, 4, -7, 18, 0, 0, -7, 5], [2, -13, 10, 13, 2, 13, 0, 0, 2, -10], [0, 0, 0, 12, 0, 13, -28, -12, 3, 1], [0, 0, 0, 2, 0, -6, -26, -2, -9, 0], [0, 0, 0, 20, 0, 22, 0, -20, 9, 7], [28, -11, -7, 0, 12, 0, 28, 12, 0, 0], [0, 0, 0, 3, 0, -13, -2, -3, 11, -9], [-34, 14, 18, 0, -5, 0, -34, -5, 0, 0], [-14, -2, 16, 0, 0, 0, -14, 7, 0, 0], [8, 10, 10, 13, -3, -10, 0, 0, -3, -10], [-10, -9, 12, 0, 13, 0, -10, 28, 0, 0], [0, 0, 0, -14, 0, -25, 13, 14, 10, -3], [0, 0, 0, -4, 0, 14, -38, 4, -14, 3], [10, -5, 3, 0, 5, 0, 10, 5, 0, 0], [0, 0, 0, 3, 0, -26, 31, -3, 9, -11], [17, -1, 1, 13, 6, 1, 0, 0, 6, -1], [30, -15, 8, 2, -13, 15, 0, 0, -13, -8], [0, 0, 0, 5, 0, -1, 31, -5, -9, -1], [2, 10, 17, 0, -16, 0, 2, -5, 0, 0], [0, 0, 0, -3, 0, 4, 46, 3, -4, -7], [0, 0, 0, -8, 0, 4, 34, 8, -1, 8], [45, -1, 12, 0, 0, 0, 45, 19, 0, 0], [0, 0, 0, 5, 0, -2, -18, -5, 10, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1323_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1323_2_h_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_1323_2_h_f(:prec:=10)
    chi := MakeCharacter_1323_h();
    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_1323_2_h_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_1323_2_h_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1323_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 3, 9, -5, -2, 1]>,<5,R![9, -54, 378, 294, 402, 51, 79, -16, 21, -4, 1]>],Snew);
    return Vf;
end function;