
// Make newform 81.2.e.a in Magma, downloaded from the LMFDB on 18 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_81_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_81_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_81_2_e_a();".
// 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_81_2_e_a();".
// 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_81_e();"
function MakeCharacter_81_e()
    N := 81;
    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_81_e_Hecke();"
function MakeCharacter_81_e_Hecke( : Kf := false)
    N := 81;
    order := 9;
    char_gens := [2];
    char_values := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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 := [[1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, -1, 0, -1, 0, -1, -1, 1, -1, -1], [-1, -1, 0, -1, 1, -1, 0, -1, 0, 1, 0, -1], [-1, -1, -1, 0, 1, 1, 0, -1, 2, -2, -1, -1], [0, 1, -1, 0, -1, 1, -1, 1, 0, -1, 0, 0], [-1, 0, 0, 1, -1, 1, -1, -1, -1, 1, 0, 2], [1, 0, 2, 1, -2, -1, 0, -1, 1, -2, 1, 1], [-1, 0, 0, -1, 1, 0, 1, 0, 2, 0, 3, 2], [1, 0, 0, -2, 0, 0, 0, 3, -1, 0, 3, -3], [-4, -1, -2, -3, 2, 0, 3, 0, 1, 0, 1, 1], [2, 0, 2, 1, 1, -1, -3, -4, -2, 4, -2, -1], [3, 3, 1, 3, -3, -1, 0, 5, 0, -3, 0, 0], [4, 4, 3, 0, -1, -4, -1, 1, -2, 2, 1, 1], [3, -1, 2, 0, 0, -3, -2, -4, 0, 6, -3, -1], [0, 0, 0, 0, 3, 3, 0, 3, -3, 0, 0, 0], [-2, -2, 0, 2, -3, 2, 3, -1, 2, -5, -1, 5], [1, 0, -1, 5, -5, 1, 1, 4, 0, -5, -2, 6], [-4, -3, -1, 1, 3, 1, 4, -2, 1, -3, 0, -1], [-4, 0, 0, -2, 2, -2, -1, -1, 2, 1, -3, 2], [-6, -3, -3, -3, 3, 3, 3, 0, -3, 3, -3, 1], [-6, 0, 0, -5, 0, -1, 1, 0, 2, 0, -1, 0], [-5, -3, 3, 1, 0, -3, 3, -3, -1, 0, -3, 3], [1, 1, 4, 0, -4, 0, -1, -1, 5, -6, 5, 2], [1, -2, -3, 0, 5, 2, -1, -5, -2, 2, -8, -8], [-6, -1, -7, -3, 3, 6, 7, 2, 0, -3, 3, -1], [0, -1, 0, 1, 3, 2, -3, -1, 0, -4, -1, 4], [6, 3, -3, 3, -6, -3, 0, 3, -3, 3, 3, 0], [3, 6, 2, -2, -2, -4, 0, 5, -5, 1, 1, 0], [7, 1, 0, -1, 3, -4, -3, -1, -7, 1, -1, -1], [1, 0, 5, 1, -5, -1, 0, 8, 10, -2, 10, 4], [-4, 0, -3, -1, 1, 0, 1, 0, -10, 3, 3, -7], [1, 6, -6, 4, 0, 3, -3, 3, -1, 0, -9, -3], [8, 5, -2, 3, 2, 0, -3, 0, -8, 0, -5, -8], [9, 3, 4, -3, -3, -4, -6, -4, 9, 0, 0, -9], [-12, -4, -1, 0, -4, 4, 3, 0, 4, -4, 8, 8], [-5, -3, 0, 3, -1, -1, 1, -1, 5, 1, -1, -1], [3, -3, -4, 4, -2, 2, 0, 2, -2, 4, 4, 0], [-8, 1, 0, -1, 3, 8, -3, -1, 8, -5, -1, 5], [-1, 5, 2, 0, -2, -5, -3, 2, -7, 7, 8, 8], [-7, 0, -6, -5, -1, 1, 8, 11, 8, -11, 3, 2], [1, 3, -7, -2, 7, 2, -3, -10, -5, -5, -5, -8], [-2, -3, 3, 1, 0, 6, -6, -6, -1, 0, 0, 6], [4, -2, 10, 1, -3, 0, -1, 0, 11, -5, 4, 6], [-11, -8, -8, -3, 8, 3, 8, -7, -1, -6, -1, 2], [-1, 0, -4, 1, -5, 5, -3, -1, 4, 1, -5, 2], [13, 3, 0, -3, -1, -10, 1, -7, -13, 10, -7, -10], [-4, 5, 3, -4, 4, 2, -3, 5, 0, 7, -3, 8], [17, 8, 5, 0, -8, -8, -3, 5, 2, -2, -4, -4], [0, -5, 2, -6, 5, -2, -1, -2, 12, -1, 0, -12], [2, 0, 0, -5, 5, -5, 5, -13, -7, 13, -6, -7], [2, 0, 6, 2, -2, 0, -2, 0, 2, 0, 12, 2], [3, -3, 3, 13, 0, 5, -5, -3, -1, 0, -7, 3], [1, -2, 1, 3, -1, -3, 2, -7, -10, 3, -10, -7], [-9, -9, 1, -3, 9, -1, 6, -4, 3, 0, 0, -3], [6, 8, 8, 3, -3, 0, -8, 5, 0, 6, 0, 5], [0, -6, -6, 6, 0, 6, 0, 0, 0, -9, -9, 0], [-1, 0, -3, 3, -3, 0, 0, 6, -6, -6, -6, 0], [-4, -1, -9, -7, 7, 8, 9, -4, 0, -5, 0, -4], [-16, -10, -4, 0, 1, 10, 6, 2, 11, -11, 8, 8], [0, -2, -4, 3, 2, 4, 5, 7, -6, 5, 0, 6], [2, 3, 3, 5, -2, 0, -5, 0, 5, 6, -6, 11], [4, 0, 0, 9, -9, 9, 3, 9, -3, -9, 12, 5], [-3, 3, -2, 0, -3, 2, -3, 14, -9, 3, 0, 9], [-11, -2, -6, 0, 5, 2, -4, -11, 1, -1, -2, -2], [0, -1, 2, 0, 0, -3, -2, -1, 0, -3, -6, 5], [-10, -1, -3, 2, -2, 2, 3, 11, 0, 1, 12, -1], [-3, 7, 2, 9, -7, -2, 2, 7, -12, -7, 0, 12], [-7, -9, 0, -10, 1, 0, 10, 0, 8, -12, 12, -4], [-9, -6, 6, -5, 0, -13, 13, -3, 2, 0, 2, 3], [19, 0, 0, 7, 0, 3, -3, 15, 8, 0, 12, -15], [1, 1, 1, 0, -1, 0, -1, 14, 2, 12, 2, -7], [15, 5, -4, 0, -1, -5, -9, 3, -8, 8, -7, -7], [10, 3, 0, -3, 5, -1, -5, -4, -10, 13, -4, -13], [6, 6, 5, -5, 1, -4, 0, -10, 10, -2, -2, 0], [10, -2, 0, 2, 0, -1, 0, -10, -10, 7, -10, -7], [-4, 2, -4, 0, 13, -2, -6, -19, -16, 16, -13, -13], [13, 9, 8, 4, -8, -4, -9, 2, -11, 13, -11, 1], [5, 3, -12, -4, 7, 0, 4, 0, -1, -6, -6, -7], [11, 8, -5, 9, -1, 0, -9, 0, -8, 6, -2, -2], [6, 6, 1, 6, -6, -1, 0, 5, -12, 6, 0, 12], [16, 7, 6, 0, -10, -7, -1, 1, 10, -10, -8, -8], [12, 6, 3, -3, 0, -3, 0, -15, 15, -6, -6, 0], [-6, -15, 5, -5, 4, -1, 0, -7, 7, -2, -2, 0], [13, 0, 2, 2, -2, -2, -2, -11, 0, -5, -2, -9], [-1, -7, -7, 0, 1, 7, 0, -1, -7, 7, -7, -7], [5, 0, -6, -2, -4, 4, -7, -4, -1, 4, -3, -7], [3, 9, -9, -5, 0, 5, -5, 9, -4, 0, -1, -9], [-11, -3, 3, -17, 0, -9, 9, -3, 8, 0, 9, 3], [-22, -7, -5, -15, 8, 0, 15, 0, 4, 0, 13, 4], [4, 4, 1, 0, -1, 0, -4, -10, -7, -3, -7, -7], [-24, -10, -7, -12, 12, -3, 7, 2, 0, 15, 3, -1], [-12, -3, 5, -5, 1, -4, 0, 5, -5, 10, 10, 0], [-17, -2, 0, 2, -6, -4, 6, -1, 17, -14, -1, 14], [-10, -6, -4, -8, 6, 4, -2, -11, 7, -3, 0, -7], [-1, 0, 3, -8, 11, -11, -7, -16, -10, 16, -6, -7], [2, -9, 9, 5, -14, 0, -5, 0, 17, -6, 3, 11], [7, 13, -5, -6, 5, 6, -13, 5, -10, 15, -10, -7], [4, 0, 6, -3, 9, -9, -9, -12, 3, 12, -15, -7], [-9, -9, -13, 13, 7, 20, 0, 5, -5, -2, -2, 0], [7, 0, 5, 5, -5, -5, -5, -2, 0, -8, -14, 12], [11, 0, 3, 4, -1, 1, 8, 14, -4, -14, 18, -7], [35, 15, 6, 20, -5, 0, -20, 0, -13, 0, -24, -13], [10, -6, 6, 13, 0, 6, -6, 0, 8, 0, 3, 0], [-4, -4, 13, -3, -1, 0, 3, 0, 10, -3, 10, 7], [10, 0, 0, 3, -3, 3, 0, 0, 0, 0, 0, -7], [-6, -4, -1, -3, 3, -3, 1, -1, 0, -3, 18, -19], [-9, 6, 6, -6, -6, -12, 0, 15, -15, 6, 6, 0], [-5, 4, 0, -4, 0, -10, 0, -10, 5, 10, -10, -10], [14, 5, 15, -1, 1, -10, -15, -10, 0, 22, 0, -10], [12, 10, 11, 3, -10, -11, -7, -8, -3, 2, 0, 3], [24, 18, 9, 6, -9, -6, -18, 12, 6, 6, 6, -2], [-25, 0, -3, -16, 16, 0, 16, 0, -7, 9, -6, 2], [-7, -2, 2, -2, 0, -5, 5, 2, 11, 0, -3, -2], [-1, 0, 2, -8, 10, -10, 3, -19, -11, 19, -8, -7], [6, -1, 5, 18, -18, -6, -5, 11, 0, -21, 3, 8], [-12, 8, 0, -8, -3, -7, 3, 5, 12, -1, 5, 1], [-9, -21, -3, 3, 6, 9, 0, -15, 15, 3, 3, 0], [1, -2, 0, 2, -9, 8, 9, 35, -1, -2, 35, 2], [-4, -7, 5, 0, 10, 7, 12, 14, 2, -2, 11, 11], [-13, -3, -10, -2, 3, 10, 1, 10, 7, -15, 0, -7], [-9, -6, 6, 1, 0, 14, -14, 0, -1, 0, -4, 0], [-14, 3, -3, 1, 0, 6, -6, 0, -1, 0, -18, 0], [-11, -5, 4, -6, -4, 6, 5, 2, 2, 0, 2, 38], [-20, -8, 0, 0, 17, 8, 8, -5, 4, -4, 7, 7], [0, -6, -6, 6, -6, 0, 0, 3, -3, 0, 0, 0], [-1, -7, -9, -1, 1, 2, 9, -7, 0, 4, -3, -4], [-4, 0, 12, 25, -13, 13, 5, 11, -1, -11, 12, 29], [11, -11, 11, 7, 0, 1, -1, 2, 14, 0, 15, -2], [2, -4, -5, -3, -1, 0, 3, 0, 13, -9, -17, 4], [-19, 0, -7, -11, 4, -4, 12, 8, -11, -8, 19, 8], [0, 3, 13, 6, -3, -13, 3, -4, 0, -9, 0, 0], [4, -3, 0, 3, 2, 5, -2, -10, -4, -5, -10, 5], [-1, 15, -12, 12, -6, 6, 0, -3, 3, -12, -12, 0], [1, 1, 0, -1, 12, 11, -12, -10, -1, 1, -10, -1], [-6, 4, 5, 6, -4, -5, 2, 16, -18, 2, 0, 18], [5, 0, -12, -11, -1, 1, 8, -4, -1, 4, -3, -16], [-8, -8, -2, 4, -12, 0, -4, 0, 14, 4, -5, 18], [3, -3, -11, -3, 3, 11, 0, 5, 3, 6, 0, -3], [15, 18, 3, -3, -12, -15, 0, 3, -3, -3, -3, 0], [-3, -6, 5, -5, -2, -7, 0, 5, -5, 4, 4, 0], [17, 17, 17, 0, -5, -17, 0, -7, -16, 16, -7, -7], [30, 13, 20, 6, -13, -20, -7, 4, 12, -1, 0, -12], [8, 0, 6, -8, 14, -14, -4, -10, 5, 10, -15, -16], [19, 9, 8, 10, -8, -10, -9, -4, -5, 1, -5, 16], [13, 9, -9, -2, 0, -9, 9, 0, -28, 0, 6, 0], [6, 17, 14, 0, -19, -17, -3, 6, -11, 11, 17, 17], [21, 11, 5, 18, -18, 6, -5, 8, 0, -12, 0, 8], [1, -3, 0, 3, 5, 17, -5, 2, -1, 13, 2, -13], [3, 6, 3, -3, 0, -3, 0, -3, 3, -18, -18, 0], [0, -11, 11, -6, 11, -11, 5, -14, 6, 11, 0, -6], [-4, 0, -6, -11, 5, -5, -1, 5, -7, -5, 12, -7], [-17, -3, -16, -14, 16, 14, 3, 8, 7, 1, 7, -8], [26, 12, -9, 17, -5, 0, -17, 0, -16, 3, -6, -13], [-8, -8, -6, 0, -4, 8, 2, 16, 25, -25, 16, 16], [12, 11, 11, 12, -12, 0, -11, 11, 0, 24, 9, 2], [8, 12, -12, 12, 0, 12, 0, 6, -6, -18, -18, 0], [-10, -1, -4, 0, -2, 1, -3, 2, 11, -11, 11, 11], [-20, -9, -19, -11, 19, 11, 9, 8, 10, -2, 10, -23], [5, -6, 18, 2, -8, 0, -2, 0, 23, -9, 21, 14], [-14, 3, -3, -17, 0, -3, 3, -18, 17, 0, 0, 18], [-8, -2, -5, -6, 5, 6, 2, 8, 17, -9, 17, -7], [6, 5, 0, -5, -3, -16, 3, 8, -6, 8, 8, -8], [18, 15, 6, -6, 3, -3, 0, 6, -6, 15, 15, 0], [1, -8, 0, 8, 6, -4, -6, -10, -1, 10, -10, -10], [-25, -4, 2, 0, -14, 4, 6, -13, 11, -11, 8, 8], [36, 21, 15, 15, -15, -15, -21, 3, -9, 12, -9, -20], [-19, -2, 2, -5, 0, 1, -1, -19, 20, 0, -12, 19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_81_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_81_2_e_a();".
// 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_81_2_e_a(:prec:=12)
    chi := MakeCharacter_81_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_81_2_e_a();".
// 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_81_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_81_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;