
// Make newform 1323.2.c.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1323_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1323_c_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_c_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_c_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 := [1, 0, 200, 0, 996, 0, 1928, 0, 1815, 0, 872, 0, 212, 0, 24, 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, 102636, 0, -400750, 0, -2653934, 0, -4029525, 0, -2451006, 0, -677054, 0, -82183, 0, -3566], [-9938, 0, 1280801, 0, 4485192, 0, 5537952, 0, 3069810, 0, 809596, 0, 95961, 0, 4109, 0], [-8680, 0, -1459060, 0, -4816716, 0, -5829935, 0, -3196410, 0, -836669, 0, -98695, 0, -4213, 0], [0, 206322, 0, 3918733, 0, 12631032, 0, 15693672, 0, 8786075, 0, 2330454, 0, 277026, 0, 11880], [0, 89795, 0, 3821459, 0, 13061987, 0, 16395526, 0, 9191538, 0, 2437027, 0, 289535, 0, 12411], [0, 184528, 0, -2969225, 0, -12258167, 0, -16190613, 0, -9248819, 0, -2472050, 0, -294796, 0, -12661], [16978, 0, -3904397, 0, -14787914, 0, -19531408, 0, -11245322, 0, -3026016, 0, -362454, 0, -15611, 0], [-5170, 0, -736177, 0, -2573576, 0, -3232355, 0, -1808300, 0, -478539, 0, -56782, 0, -2432, 0], [18, 0, 5980, 0, 21096, 0, 26564, 0, 14872, 0, 3936, 0, 467, 0, 20, 0], [0, -109968, 0, -6487208, 0, -22235265, 0, -27854760, 0, -15583888, 0, -4125450, 0, -489639, 0, -20975], [62386, 0, 7015238, 0, 24752870, 0, 31483261, 0, 17764742, 0, 4726427, 0, 562578, 0, 24141, 0], [0, 246543, 0, 8361354, 0, 27549033, 0, 33876484, 0, 18772057, 0, 4943611, 0, 585089, 0, 25023], [-19311, 0, -7440426, 0, -26500974, 0, -33715265, 0, -19017892, 0, -5058697, 0, -602052, 0, -25833, 0], [0, -437333, 0, 6225156, 0, 25519016, 0, 33405881, 0, 18998083, 0, 5068077, 0, 603881, 0, 25926], [0, -173250, 0, 8990368, 0, 33528527, 0, 43160577, 0, 24444196, 0, 6512858, 0, 775702, 0, 33297]];
    Rf_basisdens := [1, 7231, 7231, 7231, 7231, 7231, 7231, 7231, 1033, 7, 7231, 7231, 7231, 7231, 7231, 7231];
    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_c();"
function MakeCharacter_1323_c()
    N := 1323;
    order := 2;
    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_c_Hecke();"
function MakeCharacter_1323_c_Hecke(Kf)
    N := 1323;
    order := 2;
    char_gens := [785, 1081];
    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]];
    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, 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, -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, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1], [0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -2], [0, 0, 0, 0, 3, 0, 1, 0, 0, 0, -2, 0, 0, 0, 2, 0], [2, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, -1, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, -2, 0, 3, 0, 0, 0, 0], [0, -2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 1, 0, 0, -2], [0, 0, -3, -4, 0, 0, 0, 0, 0, 2, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, -1, 0, 0, 0, -4, 0, 0, 0, 1, 0], [3, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 5, 0, 0, -1], [0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 1, 0, 0, 0, -1, 0], [-4, 0, 0, 0, 0, 0, 0, -1, 3, 0, 0, 0, 0, 3, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, -3, 0, 0, 0, 0, 0, -5, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 5, 0, 0, 0, -6, 0, 0, 0, 2, 0], [0, 0, 3, -3, 0, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, -3, 0], [0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2], [8, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, -1, 0, 0], [0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, -3], [5, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, -2, 0, 0], [0, 0, 1, 5, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 1], [0, 0, 0, 0, -8, 0, -1, 0, 0, 0, 6, 0, 0, 0, -2, 0], [0, -2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 9, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, 0, 0, -4, 0, 0], [0, 0, 0, 0, 6, 0, -3, 0, 0, 0, 1, 0, 0, 0, -3, 0], [0, 0, 0, 0, 0, 0, 0, -2, -4, 0, 0, 0, 0, -1, 0, 0], [0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 5, 7, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0], [0, 1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 5], [0, 0, 0, 0, 3, 0, -2, 0, 0, 0, 7, 0, 0, 0, -2, 0], [0, -3, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 5], [6, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, -1, 0, 0], [0, -3, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0], [8, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 1, 0, 0, 0, -12, 0, 0, 0, 1, 0], [0, 0, 1, 7, 0, 0, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 6, 0, 0, 0, -2, 0], [0, 3, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 2, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, -6, 0, 0, 6], [0, 0, 0, 0, -6, 0, -7, 0, 0, 0, -5, 0, 0, 0, 2, 0], [0, 0, 7, 7, 0, 0, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0, 0, 0, 0, 5, 0, -1, 0, 0, 0, 0], [0, -5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 9, 0, 0, 2], [0, 0, -9, -2, 0, 0, 0, 0, 0, -4, 0, -9, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, -2, 0, 0, 0, -3, 0, 0, 0, 1, 0], [-7, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -2, 0, 0], [0, 8, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, -5, 0, 3, 0, 0, 0, -6, 0, 0, 0, -3, 0], [0, 0, -5, -8, 0, 0, 0, 0, 0, 2, 0, -7, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, -8, 0, 0, 0, 6, 0, 0, 0, 1, 0], [0, 0, -6, -2, 0, 0, 0, 0, 0, 9, 0, -7, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 4, 0], [0, 6, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 3, 0, 0, -5], [4, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, -1, 0, 0], [7, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 2, 0, 0], [0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 14, 0, 0, -1], [0, 0, 0, 0, -9, 0, -4, 0, 0, 0, 1, 0, 0, 0, -2, 0], [0, 0, -1, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3], [0, 0, 0, 0, 3, 0, -2, 0, 0, 0, -6, 0, 0, 0, -2, 0], [15, 0, 0, 0, 0, 0, 0, -1, -7, 0, 0, 0, 0, -2, 0, 0], [-3, 0, 0, 0, 0, 0, 0, -4, -9, 0, 0, 0, 0, 4, 0, 0], [0, 0, 2, -3, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -5, 0, 0, 8], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 3, 0, 0, 0, -6, 0], [0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, -1, 0, 0, 5], [0, 0, 0, 0, 3, 0, 4, 0, 0, 0, -5, 0, 0, 0, 4, 0], [0, 0, 7, -4, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0], [0, 4, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0], [0, 0, 0, 0, 2, 0, 7, 0, 0, 0, 1, 0, 0, 0, 2, 0], [0, 0, 0, 0, -4, 0, -2, 0, 0, 0, 14, 0, 0, 0, -2, 0], [0, -5, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0], [0, 5, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, -7, 0, 0, -9], [-15, 0, 0, 0, 0, 0, 0, -3, 1, 0, 0, 0, 0, 3, 0, 0], [0, 0, 1, -2, 0, 0, 0, 0, 0, 14, 0, -1, 0, 0, 0, 0], [-18, 0, 0, 0, 0, 0, 0, -2, -11, 0, 0, 0, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 5, 0, 0, 0, 0], [0, 0, 2, 11, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 0, -4, -16, 0, 0, 0, 0, 2, 0, 0], [0, -5, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 15, 0, 0, -2], [-6, 0, 0, 0, 0, 0, 0, -5, -2, 0, 0, 0, 0, -3, 0, 0], [0, 0, 9, 11, 0, 0, 0, 0, 0, -3, 0, 2, 0, 0, 0, 0], [0, 0, 5, 4, 0, 0, 0, 0, 0, -6, 0, 5, 0, 0, 0, 0], [0, 0, 4, -4, 0, 0, 0, 0, 0, 1, 0, 10, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, 1, 0, 0, 0, -2, 0, 0, 0, 4, 0], [5, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, -2, 0, 0], [2, 0, 0, 0, 0, 0, 0, -4, -2, 0, 0, 0, 0, 5, 0, 0], [0, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -4, 0, 0, -6], [0, 0, 6, 9, 0, 0, 0, 0, 0, 5, 0, 10, 0, 0, 0, 0], [0, 7, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, -8, 0, 0, -2], [-3, 0, 0, 0, 0, 0, 0, -5, 12, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -17, 0, 4, 0, 0, 0, 7, 0, 0, 0, 0, 0], [0, 0, -2, -2, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0], [0, 0, 2, -11, 0, 0, 0, 0, 0, -12, 0, 5, 0, 0, 0, 0], [0, 5, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 4, 0, 0, -7], [0, 0, 0, 0, -9, 0, -3, 0, 0, 0, -1, 0, 0, 0, -3, 0], [0, 0, 0, 0, -7, 0, -1, 0, 0, 0, -14, 0, 0, 0, 4, 0], [-11, 0, 0, 0, 0, 0, 0, 3, -9, 0, 0, 0, 0, -1, 0, 0], [0, 11, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, -10, 0, 0, -9], [0, 0, 0, 0, 3, 0, 9, 0, 0, 0, -8, 0, 0, 0, -1, 0], [-15, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, -8, 0, 0, -3], [0, 0, 0, 0, -5, 0, 4, 0, 0, 0, -9, 0, 0, 0, -9, 0], [0, 0, 6, -3, 0, 0, 0, 0, 0, -1, 0, -7, 0, 0, 0, 0], [0, 1, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, -3, 0, 0, -4], [0, 6, 0, 0, 0, -12, 0, 0, 0, 0, 0, 0, -10, 0, 0, 8], [0, 0, 0, 0, 11, 0, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0], [-10, 0, 0, 0, 0, 0, 0, 1, -16, 0, 0, 0, 0, -4, 0, 0], [0, 0, 6, -12, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 9, 0, 0, -11], [0, 0, 0, 0, -2, 0, -9, 0, 0, 0, 16, 0, 0, 0, -5, 0], [0, 4, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, 3, -12, 0, 0, 0, 0, -9, 0, 0], [0, 0, -9, -4, 0, 0, 0, 0, 0, 4, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, -17, 0, -1, 0, 0, 0, 18, 0, 0, 0, -1, 0], [0, 0, 0, 0, 1, 0, 8, 0, 0, 0, -1, 0, 0, 0, 0, 0], [8, 0, 0, 0, 0, 0, 0, 2, -4, 0, 0, 0, 0, -6, 0, 0], [0, -4, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 7, 0, 0, 5], [-4, 0, 0, 0, 0, 0, 0, 5, 7, 0, 0, 0, 0, 4, 0, 0], [-5, 0, 0, 0, 0, 0, 0, 6, -1, 0, 0, 0, 0, -4, 0, 0], [0, 0, -14, -3, 0, 0, 0, 0, 0, -12, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 7, 0, 6, 0, 0, 0, -11, 0, 0, 0, 4, 0], [0, 0, -7, 8, 0, 0, 0, 0, 0, -8, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, -17, 0, -4, 0, 0, 0, -14, 0, 0, 0, 0, 0], [0, 0, -3, -3, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0], [0, 3, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 4, 0, 0, -12], [0, 0, 0, 0, 20, 0, -1, 0, 0, 0, 1, 0, 0, 0, 2, 0], [0, 8, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2], [1, 0, 0, 0, 0, 0, 0, 10, 7, 0, 0, 0, 0, -2, 0, 0], [0, -2, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, -4, 0, 0, 7], [0, 0, 0, 0, 1, 0, -13, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 11, 11, 0, 0, 0, 0, 0, -1, 0, 21, 0, 0, 0, 0], [0, 0, 0, 0, -16, 0, -10, 0, 0, 0, 6, 0, 0, 0, -2, 0], [0, 0, -18, 1, 0, 0, 0, 0, 0, -8, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, -8, 0, -12, 0, 0, 0, -2, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, -11, 0, 0, 4], [-9, 0, 0, 0, 0, 0, 0, 5, -2, 0, 0, 0, 0, 8, 0, 0], [0, 0, 6, -8, 0, 0, 0, 0, 0, -10, 0, -2, 0, 0, 0, 0], [21, 0, 0, 0, 0, 0, 0, -6, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, -5, 0, 0, 0, 0, 0, -5, 0, 4, 0, 0, 0, 0], [4, 0, 0, 0, 0, 0, 0, -1, -4, 0, 0, 0, 0, 3, 0, 0], [0, -6, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 5, 0, 0, 4], [32, 0, 0, 0, 0, 0, 0, -3, 11, 0, 0, 0, 0, 1, 0, 0], [0, 0, -3, 10, 0, 0, 0, 0, 0, -10, 0, 5, 0, 0, 0, 0], [0, 0, 0, 0, 24, 0, -3, 0, 0, 0, 15, 0, 0, 0, 3, 0], [0, 0, 16, 5, 0, 0, 0, 0, 0, 2, 0, 8, 0, 0, 0, 0], [0, -2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 6, 0, 0, -10], [0, -7, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, -9], [-9, 0, 0, 0, 0, 0, 0, -9, 13, 0, 0, 0, 0, 9, 0, 0], [0, 0, 0, -10, 0, 0, 0, 0, 0, 9, 0, -10, 0, 0, 0, 0], [0, 13, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, -11], [0, 0, 5, 2, 0, 0, 0, 0, 0, -8, 0, -7, 0, 0, 0, 0], [21, 0, 0, 0, 0, 0, 0, 1, -20, 0, 0, 0, 0, -8, 0, 0], [0, 0, 0, 0, -2, 0, -3, 0, 0, 0, -22, 0, 0, 0, 10, 0], [-12, 0, 0, 0, 0, 0, 0, -2, -3, 0, 0, 0, 0, -11, 0, 0], [0, 0, 6, 7, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 0], [0, 9, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, -7, 0, 0, -1], [0, 0, 0, 0, 17, 0, 7, 0, 0, 0, -23, 0, 0, 0, 2, 0], [0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 13], [0, 0, 0, 0, 0, 0, 0, 3, 12, 0, 0, 0, 0, -5, 0, 0], [0, 0, 0, 0, -19, 0, -5, 0, 0, 0, -8, 0, 0, 0, -5, 0], [0, 0, -7, -21, 0, 0, 0, 0, 0, 3, 0, -10, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 8, 9, 0, 0, 0, 0, -4, 0, 0], [0, -18, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 10, 0, 0, 14], [0, 0, 0, 0, 3, 0, 4, 0, 0, 0, -11, 0, 0, 0, 2, 0], [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 13, 0, 0, 0, -9, 0], [7, 0, 0, 0, 0, 0, 0, 5, -7, 0, 0, 0, 0, 8, 0, 0], [0, 0, 15, 3, 0, 0, 0, 0, 0, 5, 0, -5, 0, 0, 0, 0], [23, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -9, 0, 0], [0, 0, -1, -7, 0, 0, 0, 0, 0, -3, 0, 3, 0, 0, 0, 0], [0, 13, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 10, 0, 0, -1], [0, 0, 9, 9, 0, 0, 0, 0, 0, -15, 0, 2, 0, 0, 0, 0], [16, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 12, 0, 0], [0, 0, 0, 0, 13, 0, -9, 0, 0, 0, -5, 0, 0, 0, -1, 0], [20, 0, 0, 0, 0, 0, 0, 1, -13, 0, 0, 0, 0, -2, 0, 0], [0, 0, 10, 9, 0, 0, 0, 0, 0, -2, 0, 10, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, -11, 0, 0, 0, 7, 0, 0, 0, 8, 0], [0, -11, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, -7, 0, 0, 17], [3, 0, 0, 0, 0, 0, 0, 5, 8, 0, 0, 0, 0, 10, 0, 0], [0, 0, -4, 7, 0, 0, 0, 0, 0, 8, 0, -13, 0, 0, 0, 0], [0, 10, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0], [0, 0, 5, 1, 0, 0, 0, 0, 0, -9, 0, -9, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, -5, 1, 0, 0, 0, 0, 1, 0, 0], [-19, 0, 0, 0, 0, 0, 0, 3, 14, 0, 0, 0, 0, -6, 0, 0], [0, 0, -25, -13, 0, 0, 0, 0, 0, -12, 0, -8, 0, 0, 0, 0], [0, 0, -16, -6, 0, 0, 0, 0, 0, 5, 0, -15, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, -2, 0, 0, 10], [0, 0, 0, 0, 2, 0, -2, 0, 0, 0, 16, 0, 0, 0, -8, 0], [0, 0, 0, 0, -26, 0, -4, 0, 0, 0, 12, 0, 0, 0, -5, 0], [0, 0, 0, 0, 7, 0, -6, 0, 0, 0, 9, 0, 0, 0, 1, 0], [0, 0, -14, -5, 0, 0, 0, 0, 0, -14, 0, -4, 0, 0, 0, 0], [0, 0, -22, -2, 0, 0, 0, 0, 0, -19, 0, -10, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 7, 0, 0, 0, -14, 0, 0, 0, 0, 0], [0, 16, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, -6, 0, 0, 4], [0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 4, 0, 0, -10], [0, 0, 0, 0, 16, 0, 3, 0, 0, 0, -20, 0, 0, 0, 5, 0], [-32, 0, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 7, 0, 0], [0, 0, 11, 5, 0, 0, 0, 0, 0, 20, 0, 14, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0, 0, 5, 11, 0, 0, 0, 0, 10, 0, 0], [0, 0, 13, 2, 0, 0, 0, 0, 0, 8, 0, 14, 0, 0, 0, 0], [0, 0, 7, 12, 0, 0, 0, 0, 0, 16, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 12, 0, 7, 0, 0, 0, -20, 0, 0, 0, 4, 0], [8, 0, 0, 0, 0, 0, 0, -7, 17, 0, 0, 0, 0, 7, 0, 0], [0, 0, -14, -12, 0, 0, 0, 0, 0, -2, 0, -13, 0, 0, 0, 0], [0, -14, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 14, 0, 0, -17], [0, -3, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, -11, 0, 0, 13], [-3, 0, 0, 0, 0, 0, 0, -3, -18, 0, 0, 0, 0, -14, 0, 0], [0, 0, 0, 0, 2, 0, -9, 0, 0, 0, -8, 0, 0, 0, 5, 0], [0, 5, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 1, 0, 0, -18], [6, 0, 0, 0, 0, 0, 0, 3, -33, 0, 0, 0, 0, -5, 0, 0], [0, 0, -7, -10, 0, 0, 0, 0, 0, 4, 0, -8, 0, 0, 0, 0], [-23, 0, 0, 0, 0, 0, 0, 7, 10, 0, 0, 0, 0, 4, 0, 0], [0, 0, 1, -9, 0, 0, 0, 0, 0, 7, 0, 9, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, -7], [0, 0, 0, 0, 6, 0, 4, 0, 0, 0, 3, 0, 0, 0, 8, 0], [0, -10, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, -2, 0, 0, 11], [12, 0, 0, 0, 0, 0, 0, -2, -32, 0, 0, 0, 0, -6, 0, 0], [0, 0, 0, 0, 21, 0, -1, 0, 0, 0, 22, 0, 0, 0, 6, 0], [-7, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, -6, 0, 0], [0, 7, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, -25, 0, 0, 9], [0, 0, 0, 0, 9, 0, 10, 0, 0, 0, -22, 0, 0, 0, 12, 0], [0, 0, 3, 9, 0, 0, 0, 0, 0, -2, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -2, 0, 0, 0, -5, 0, 0, 0, 2, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0], [0, -12, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 1, 0, 0, -12], [0, 0, 12, 17, 0, 0, 0, 0, 0, -10, 0, -1, 0, 0, 0, 0], [0, 3, 0, 0, 0, -17, 0, 0, 0, 0, 0, 0, 13, 0, 0, -9], [0, 0, 0, 0, 2, 0, -20, 0, 0, 0, 22, 0, 0, 0, -2, 0], [0, 0, 0, 0, -19, 0, 9, 0, 0, 0, -7, 0, 0, 0, 11, 0], [-3, 0, 0, 0, 0, 0, 0, -5, 15, 0, 0, 0, 0, 5, 0, 0], [0, 0, 20, 13, 0, 0, 0, 0, 0, -15, 0, 20, 0, 0, 0, 0], [0, 0, -1, -9, 0, 0, 0, 0, 0, 5, 0, -11, 0, 0, 0, 0], [0, 0, 0, 0, 10, 0, -12, 0, 0, 0, 5, 0, 0, 0, -8, 0], [0, 0, 11, 11, 0, 0, 0, 0, 0, -11, 0, 18, 0, 0, 0, 0], [22, 0, 0, 0, 0, 0, 0, -1, 30, 0, 0, 0, 0, -3, 0, 0], [0, -7, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, -8, 0, 0, 5], [24, 0, 0, 0, 0, 0, 0, -1, -4, 0, 0, 0, 0, -5, 0, 0], [0, 0, -10, 0, 0, 0, 0, 0, 0, -4, 0, 2, 0, 0, 0, 0], [0, 3, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, -4, 0, 0, -7], [0, 0, 0, 0, 10, 0, 12, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 1, -5, 0, 0, 0, 0, 0, 21, 0, 3, 0, 0, 0, 0], [0, -12, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 5, 0, 0, 4], [3, 0, 0, 0, 0, 0, 0, 5, 8, 0, 0, 0, 0, -4, 0, 0], [0, 0, 0, 0, -20, 0, -14, 0, 0, 0, 29, 0, 0, 0, -6, 0], [0, 0, 20, 2, 0, 0, 0, 0, 0, 2, 0, 21, 0, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, 0, 0, 0, -7, 0, 0, 0, -12, 0], [-1, 0, 0, 0, 0, 0, 0, 8, 12, 0, 0, 0, 0, 7, 0, 0], [0, -4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 11, 0, 0, 2], [0, 0, 0, 0, 23, 0, 0, 0, 0, 0, 15, 0, 0, 0, -4, 0], [0, 0, 0, 0, -12, 0, 7, 0, 0, 0, -7, 0, 0, 0, 9, 0], [0, 0, -7, 6, 0, 0, 0, 0, 0, 11, 0, 12, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, 4, 0, 0, 0, -21, 0, 0, 0, 8, 0], [0, -8, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, -6, 0, 0, -6], [0, 0, -13, -3, 0, 0, 0, 0, 0, 8, 0, -24, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, 5, -16, 0, 0, 0, 0, -1, 0, 0], [0, 10, 0, 0, 0, -21, 0, 0, 0, 0, 0, 0, -7, 0, 0, 4], [0, 0, 0, 0, -12, 0, -1, 0, 0, 0, 34, 0, 0, 0, -5, 0], [6, 0, 0, 0, 0, 0, 0, -1, -17, 0, 0, 0, 0, 10, 0, 0], [0, 0, 0, 0, -19, 0, 17, 0, 0, 0, -18, 0, 0, 0, 6, 0], [-21, 0, 0, 0, 0, 0, 0, 2, 30, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 34, 0, 14, 0, 0, 0, -22, 0, 0, 0, 13, 0], [0, 0, 0, 0, -14, 0, -5, 0, 0, 0, 1, 0, 0, 0, 3, 0], [0, 9, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, -5, 0, 0, -11], [0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, -1, 0, 0], [19, 0, 0, 0, 0, 0, 0, 6, -9, 0, 0, 0, 0, 2, 0, 0], [0, 0, -24, 2, 0, 0, 0, 0, 0, -15, 0, -9, 0, 0, 0, 0], [0, 0, 24, 6, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0], [-22, 0, 0, 0, 0, 0, 0, -2, 16, 0, 0, 0, 0, 3, 0, 0], [0, 0, -5, 3, 0, 0, 0, 0, 0, -15, 0, -6, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, -8, 0, 0, 0, -8, 0, 0, 0, 4, 0], [0, 0, 0, 0, 21, 0, 2, 0, 0, 0, 10, 0, 0, 0, -4, 0], [0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 8], [9, 0, 0, 0, 0, 0, 0, -2, 27, 0, 0, 0, 0, 4, 0, 0], [0, -9, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0, 4, 0, 0, -7], [0, 0, 0, 0, 21, 0, 2, 0, 0, 0, 20, 0, 0, 0, 2, 0], [0, 0, -13, 1, 0, 0, 0, 0, 0, -5, 0, -18, 0, 0, 0, 0], [0, -3, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 10, 0, 0, 5], [0, 0, 1, -16, 0, 0, 0, 0, 0, 4, 0, -16, 0, 0, 0, 0], [0, -9, 0, 0, 0, 26, 0, 0, 0, 0, 0, 0, -24, 0, 0, 8], [0, 0, 24, 17, 0, 0, 0, 0, 0, 12, 0, 3, 0, 0, 0, 0], [-17, 0, 0, 0, 0, 0, 0, -8, 1, 0, 0, 0, 0, 6, 0, 0], [0, 0, 7, -3, 0, 0, 0, 0, 0, -23, 0, 8, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 18, 0, 0, 0, 8, 0, 0, 0, 1, 0], [0, 0, -8, -20, 0, 0, 0, 0, 0, 15, 0, -17, 0, 0, 0, 0], [0, 0, 24, -1, 0, 0, 0, 0, 0, -6, 0, 31, 0, 0, 0, 0], [0, 0, 0, 0, 12, 0, -9, 0, 0, 0, -17, 0, 0, 0, -1, 0], [0, 0, 0, 0, 7, 0, -27, 0, 0, 0, 18, 0, 0, 0, -8, 0], [0, -6, 0, 0, 0, -26, 0, 0, 0, 0, 0, 0, -14, 0, 0, 8], [-2, 0, 0, 0, 0, 0, 0, 8, 4, 0, 0, 0, 0, 12, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1323_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1323_2_c_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_c_f(:prec:=16)
    chi := MakeCharacter_1323_c();
    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_c_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_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1323_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 0, 24, 0, 38, 0, 12, 0, 1]>],Snew);
    return Vf;
end function;