
// Make newform 243.2.e.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_243_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_243_e_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_243_2_e_c();".
// 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_243_2_e_c();".
// 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 := [3, -27, 108, -258, 420, -504, 463, -330, 186, -80, 27, -6, 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], [9, -58, 145, -224, 249, -199, 127, -58, 22, -5, 1, 0], [-25, 155, -445, 803, -1030, 1006, -747, 445, -197, 71, -16, 3], [25, -195, 649, -1285, 1762, -1805, 1390, -849, 384, -140, 32, -6], [49, -348, 1118, -2170, 2918, -2942, 2232, -1344, 601, -216, 49, -9], [-62, 421, -1286, 2411, -3178, 3144, -2360, 1402, -623, 221, -50, 9], [-61, 442, -1417, 2724, -3643, 3653, -2761, 1657, -739, 265, -60, 11], [85, -610, 1958, -3802, 5122, -5176, 3936, -2375, 1064, -383, 87, -16], [110, -742, 2272, -4267, 5631, -5577, 4191, -2491, 1108, -393, 89, -16], [-209, 1486, -4708, 9024, -12058, 12076, -9134, 5472, -2443, 873, -198, 36], [217, -1519, 4772, -9096, 12120, -12110, 9150, -5476, 2444, -873, 198, -36], [257, -1785, 5589, -10648, 14172, -14157, 10689, -6396, 2853, -1019, 231, -42]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1];
    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_243_e();"
function MakeCharacter_243_e()
    N := 243;
    order := 9;
    char_gens := [2];
    v := [4];
    // 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_243_e_Hecke();"
function MakeCharacter_243_e_Hecke(Kf)
    N := 243;
    order := 9;
    char_gens := [2];
    char_values := [[0, 0, 0, 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+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 := [[1, 0, 0, 0, 0, -1, 0, 0, -1, 1, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, -1, -1, 1, 0, 1, 0, 0, -1, 0, 0], [1, 0, 1, -1, 0, -1, -1, -1, 1, 1, 0, -1], [-1, 0, -1, -1, 1, 0, 1, 0, 0, 0, -2, 0], [1, 1, 1, 0, 1, -1, 0, 0, -1, 1, 0, 0], [-1, -1, -1, 0, 1, 0, 1, -2, -2, 0, -2, 1], [0, 0, 1, 2, -1, 1, 0, 1, 2, -1, -1, 2], [2, 0, 0, 0, 0, -1, 1, 0, 2, 0, 2, 0], [1, 0, 0, 0, 0, 2, 0, 3, -1, -2, 3, 2], [-1, -1, 1, 2, 0, 2, -2, -1, 1, 0, -2, 1], [5, 3, 1, 2, -1, -2, -3, 1, -1, 2, -1, -2], [-6, -3, 0, 0, 1, 3, 3, -1, 0, 0, 2, 2], [-4, -1, -1, -4, 3, 0, 4, 0, 2, -1, 2, 1], [3, 2, 0, 3, -2, 0, 1, -1, 0, -2, 0, 0], [-3, 0, 0, 0, 3, 3, 0, -3, 3, 0, 0, 0], [1, 2, -1, 2, -2, 3, 1, 3, 0, -7, 0, 3], [-1, 1, -5, 1, -1, 5, 0, 2, -3, 0, 0, 3], [-1, -3, -4, 0, 1, 3, -1, 1, -1, 1, -1, -1], [-1, -1, 2, 0, -2, 0, 1, 4, 4, 0, 4, 7], [-2, 0, -3, -3, 0, 0, 3, 6, 0, -6, 6, -1], [6, 1, 0, -1, 0, -5, 0, -2, -6, 6, -2, -6], [-5, -3, 0, 3, -3, -1, 3, 3, 5, -5, 3, 5], [1, 0, 0, -4, 4, -4, 1, -2, -5, 2, 3, -5], [-1, -1, 5, 2, -3, 0, -2, 0, -1, 2, 2, 1], [-6, -7, -3, -6, 7, 3, 1, -7, 3, 4, 0, -3], [-11, -1, -4, -2, 2, 3, 4, 8, 0, -4, 7, 1], [3, 3, -3, 3, -6, -3, 0, 3, -3, -6, -6, 0], [-5, -6, -2, 2, 2, 4, 0, 1, -1, -1, -1, 0], [4, -1, 2, -4, 4, -3, -2, -9, 0, 8, -9, 0], [0, 0, 1, 5, -4, 4, 0, -2, -4, 2, 2, 5], [-7, 0, 0, -3, 0, -1, 1, 3, 5, 0, -4, -3], [1, 3, 0, -3, 6, -4, -6, -12, -1, 4, -12, -4], [8, 5, -5, 2, 0, 2, -2, 5, -5, 0, 1, -5], [-9, -3, -6, 0, 4, 3, -3, -7, 0, 0, -1, -1], [12, 3, -4, 4, -1, 0, -4, 0, -2, -5, -7, -7], [0, -3, -2, 1, -1, -1, 2, -2, 0, -1, -7, 5], [4, 3, 4, -4, 2, -2, 0, 1, -1, 8, 8, 0], [10, -1, 2, 8, -8, -3, -2, -3, 0, -4, 0, -3], [-1, 3, 2, 5, -2, 0, -5, 0, -12, 9, 1, -3], [14, 8, 5, 6, -5, -6, -8, 7, 1, 6, 1, -2], [0, 0, -2, -7, 5, -5, -3, -8, 5, 8, -13, -7], [-2, 6, 0, -6, -3, -1, 3, 12, 2, 1, 12, -1], [-6, -2, 2, -10, 0, -3, 3, 3, 7, 0, 6, -3], [1, 0, 3, 8, -5, 5, -8, 1, -2, -1, 3, 7], [-1, 3, 1, -4, -1, 4, -3, 1, 8, -7, 8, 1], [15, 3, 4, 10, -10, -1, -4, -2, 0, 2, 2, -4], [4, -3, 4, 2, 3, -4, 5, 2, -5, 10, 0, 5], [17, 3, 8, 8, -5, 0, -8, 0, 6, -6, -5, 0], [1, -5, 1, 0, -2, 5, 6, 9, 11, -11, 3, 3], [5, 5, 5, 0, -5, 0, -5, -2, -8, 6, -8, -2], [2, 0, 0, 6, 0, 2, -2, 0, 8, 0, -4, 0], [-3, -5, 0, 5, 3, 13, -3, 1, 3, -3, 1, 3], [7, 0, -3, -1, -2, 2, -2, -8, -5, 8, -3, -8], [9, 9, 6, 0, 1, -9, -3, -4, -9, 9, -4, -4], [6, 8, 3, 0, -8, -3, -8, -1, 0, -2, 0, 0], [-6, -6, -6, 6, 0, 6, 0, 0, 0, 3, 3, 0], [-4, 0, 3, -3, 3, 0, 0, 3, -3, -3, -3, 0], [4, 9, 7, 8, -9, -7, -1, 5, -8, -5, 0, 8], [-16, -6, -1, -10, 4, 0, 10, 0, 3, 0, 4, 3], [-5, -2, -5, 0, 4, 2, -3, -12, 2, -2, -9, -9], [11, -3, 3, 3, 0, 2, -2, 3, -1, 0, 11, -3], [-3, -3, 9, 0, -9, 0, 3, 12, 12, 0, 12, 16], [6, -3, -3, 0, -2, 3, 0, 5, -9, 9, -4, -4], [11, -4, 5, 2, -6, 0, -2, 0, 11, -13, 8, -2], [0, 2, 0, 3, -2, 0, 1, 8, 3, -8, 0, -3], [10, 3, -2, 2, -3, 2, -1, 8, 10, -5, 0, -10], [4, 7, -2, 0, -2, -7, -9, 0, -7, 7, 3, 3], [-4, 9, -9, 0, 0, -1, 1, -3, -7, 0, -13, 3], [9, 13, 0, -13, 6, -5, -6, -2, -9, 12, -2, -12], [19, 3, 0, -3, 0, -7, 0, -12, -19, 4, -12, -4], [7, 0, 0, -1, 1, -1, 1, 1, 13, -1, -12, -8], [-15, -9, -1, -5, -4, 0, 5, 0, 10, 1, 5, 11], [6, 3, -2, 1, -1, 5, 2, -2, 0, -4, 2, -4], [1, -6, -5, 5, -1, 4, 0, -11, 11, -7, -7, 0], [-5, 2, 2, -1, 1, 0, -2, 6, 0, -7, -3, 9], [-4, 6, -13, 2, 4, 0, -2, 0, -12, 12, -5, 0], [6, 0, 4, 8, -4, 4, -9, 10, 2, -10, 8, 2], [-7, -3, 3, -12, 0, -7, 7, -9, 5, 0, 8, 9], [2, 8, -8, 5, 0, -1, 1, 2, -20, 0, -11, -2], [0, -6, 0, 0, 1, 6, 6, 11, 6, -6, 5, 5], [-16, -1, -10, -7, 6, 0, 7, 0, -16, 8, 8, -8], [-6, 6, 3, -3, 0, -3, 0, 18, -18, 9, 9, 0], [13, 15, -5, 5, -4, 1, 0, 13, -13, -7, -7, 0], [-13, -2, -2, -2, 2, 2, 0, -7, 0, -9, 0, 0], [-1, 0, -1, -7, 7, 0, 7, 0, 6, -6, 13, 0], [-1, -7, 2, 6, -2, -6, 7, -5, -5, 0, -5, -2], [-3, -5, 0, 5, -9, -5, 9, 4, 3, -3, 4, 3], [-11, -9, 0, 9, -3, 17, 3, 6, 11, -11, 6, 11], [-4, -7, 7, 5, 0, 8, -8, -7, 4, 0, -2, 7], [4, 0, 0, -1, 1, -1, 4, -5, -11, 5, 6, -5], [-24, -7, -12, 3, 7, 12, 10, -7, -6, -14, 0, 6], [-8, 3, -5, 5, -1, 4, 0, 13, -13, 5, 5, 0], [4, 2, -4, -4, 4, 6, 4, -6, 0, -7, 9, -15], [-7, -6, 2, 0, 4, 6, 8, 7, -1, 1, 8, 8], [-10, -7, 8, -3, -8, 3, 7, 1, 13, -12, 13, 19], [11, 9, -9, 9, 0, 14, -14, 3, -10, 0, -7, -3], [-5, 0, 6, 5, 1, -1, 13, 1, 4, -1, -3, 10], [15, 9, -3, 6, 3, -6, -9, -12, -15, 3, -15, -14], [7, 9, 13, -13, -7, -20, 0, 1, -1, 11, 11, 0], [-7, -5, -5, -5, 5, 5, 0, -13, -9, 12, 0, 9], [5, 8, -4, -3, 4, 3, -8, -5, 4, -9, 4, -20], [-13, -15, 15, 6, 0, 5, -5, -15, 5, 0, -4, 15], [10, 6, 0, -6, -6, -13, 6, -3, -10, 4, -3, -4], [-7, -4, 4, -13, 0, -1, 1, -1, 10, 0, 10, 1], [0, 0, 3, 0, -3, 0, 0, 3, 6, -3, 6, 13], [-6, -1, -3, 3, 1, 3, 4, 8, -21, 13, 0, 21], [21, 6, 6, -6, -6, -12, 0, -15, 15, 0, 0, 0], [-14, -4, -4, -10, 10, 0, 4, 0, 0, -4, 15, -15], [-14, -15, 1, -10, 15, -1, 5, -4, 10, 1, 0, -10], [10, 10, 7, 0, -11, -10, -3, 6, -10, 10, 6, 6], [13, 0, 6, 9, -3, 3, -18, -18, -9, 18, -9, -4], [2, 0, 0, -3, 0, -16, 16, 9, -19, 0, 5, -9], [7, 5, 0, -5, 2, -2, -2, 10, -7, 11, 10, -11], [-1, -3, -8, 2, 8, -2, 3, 1, 2, -1, 2, -8], [6, 5, 18, 6, -5, -18, 1, 14, -9, 4, 0, 9], [13, 8, 11, 7, -7, -3, -11, 2, 0, 2, -14, 16], [-18, -21, -3, 3, 6, 9, 0, -3, 3, -6, -6, 0], [-14, 2, -7, 8, -8, 9, 7, 24, 0, 11, 6, 18], [-4, -12, -10, -7, -5, 0, 7, 0, 3, -15, 7, -12], [2, -3, -1, 0, 10, 3, 2, 7, 5, -5, 2, 2], [9, -14, 0, 14, 6, 1, -6, -26, -9, 15, -26, -15], [-14, 6, 0, -6, 3, -1, -3, 0, 14, -11, 0, 11], [-23, 0, 6, -4, 10, -10, -5, -23, -8, 23, -15, 19], [20, 8, 17, 8, 0, 0, -8, 0, 8, -4, 5, 4], [0, -6, -6, 6, -6, 0, 0, -3, 3, -6, -6, 0], [1, 9, 1, 2, -9, -1, -7, 14, -5, -5, 0, 5], [-7, 5, -25, -12, 25, 12, -5, -14, -20, 6, -20, -14], [-11, -1, 0, 1, 11, 7, -11, -2, 11, 2, -2, -2], [-4, -4, 4, 5, 0, -1, 1, 5, 19, 0, -5, -5], [-19, -12, -11, -7, 11, 7, 12, 7, -4, 11, -4, -11], [-9, -3, 3, 0, 13, 3, 6, -16, 0, 0, -10, -10], [-12, -3, -5, -5, 5, 2, 5, 4, 0, 8, 11, -7], [-22, -15, 12, -12, 6, -6, 0, -30, 30, 0, 0, 0], [22, -1, 11, 11, -11, -12, -11, -12, 0, 2, -12, 0], [-8, 4, -2, 0, -5, -4, -6, -3, 11, -11, 9, 9], [20, 8, 11, 12, -11, -12, -8, -2, -14, 12, -14, -14], [-18, -8, 8, 2, 0, -12, 12, -12, 7, 0, -12, 12], [3, 3, 0, 0, -11, -3, -3, 14, 0, 0, 14, 14], [15, 18, 3, -3, -12, -15, 0, 12, -12, 6, 6, 0], [1, 6, -5, 5, 2, 7, 0, 16, -16, -1, -1, 0], [17, 0, 5, 17, -17, 0, -17, 0, 12, 0, 4, 12], [31, 13, 7, 0, -20, -13, -6, 15, 5, -5, -3, -3], [-10, -4, 8, -6, -8, 6, 4, 19, 22, -3, 22, 10], [27, 0, 10, 8, 2, -2, -9, -8, 2, 8, -10, -19], [13, -9, 0, 9, 9, 2, -9, 3, -13, 22, 3, -22], [-6, -3, -19, -17, 14, 0, 17, 0, 4, -14, -10, -10], [21, 5, 18, -6, -5, -18, -11, -7, 6, 16, 0, -6], [0, -3, -8, -17, 17, 5, 8, -20, 0, 2, -7, -13], [3, 6, 3, -3, 0, -3, 0, 6, -6, 21, 21, 0], [-11, -11, -5, 0, -11, 11, 6, 6, 14, -14, 6, 6], [5, -1, 11, 6, -11, -6, 1, -2, 16, -18, 16, 10], [-9, 0, -14, -16, 2, -2, 3, -8, -10, 8, 2, -7], [-13, -12, 12, -9, 0, 5, -5, -9, 32, 0, 8, 9], [8, 2, -4, 8, -6, 0, -8, 0, 8, 2, 11, 10], [12, 11, 12, 0, -11, -12, -11, -25, -9, 10, 0, 9], [-4, -12, 12, -12, 0, -12, 0, -18, 18, -6, -6, 0], [-10, 3, 2, -1, 4, 0, 1, 0, -15, 12, -11, -3], [-24, 0, -11, -19, 8, -8, 9, -17, -16, 17, -1, 5], [14, 6, -6, 18, 0, 8, -8, -3, -1, 0, -10, 3], [-14, -3, 0, 3, 3, 17, -3, -9, 14, 7, -9, -7], [10, 0, 6, 5, 1, -1, -2, 13, 7, -13, 6, -5], [19, 5, 8, 16, -16, -3, -8, 2, 0, -13, -5, 7], [-6, 15, 6, -6, 3, -3, 0, 3, -3, -9, -9, 0], [10, 8, 14, -4, 4, -6, -14, -6, 0, 14, 3, -9], [-25, -6, 14, -4, -2, 0, 4, 0, 18, 15, 13, 33], [1, 0, 15, 15, 0, 0, -21, -12, -3, 12, -9, 14], [19, -1, 0, 1, 2, -5, -2, 10, -19, 2, 10, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_243_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_243_2_e_c();".
// 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_243_2_e_c(:prec:=12)
    chi := MakeCharacter_243_e();
    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_243_2_e_c();".
// 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_243_2_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_243_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, -54, 135, -207, 279, -279, 195, -81, 27, -6, 3, -3, 1]>],Snew);
    return Vf;
end function;