
// Make newform 2299.2.a.n in Magma, downloaded from the LMFDB on 27 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2299_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2299_2_a_n();".
// 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_2299_2_a_n();".
// 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 := [-2, 7, 6, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-2, -1, 1, 0, 0], [3, -3, -2, 1, 0], [1, 7, -2, -3, 1]];
    Rf_basisdens := [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_2299_a();"
function MakeCharacter_2299_a()
    N := 2299;
    order := 1;
    char_gens := [970, 1332];
    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;

function MakeCharacter_2299_a_Hecke( : Kf := Kf)
    return MakeCharacter_2299_a();
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, 0, -1, 0, 0], [0, 0, 0, 1, 0], [-1, -1, 1, 0, 0], [-2, 1, 1, 0, 0], [0, 0, 0, 0, 0], [-1, 1, 0, 0, -1], [2, -2, 0, -2, 0], [1, 0, 0, 0, 0], [-1, 1, 1, 1, 3], [-1, -3, 1, -2, 1], [2, 2, -1, -1, 0], [1, 1, -1, -1, -3], [-3, 3, 2, 4, -1], [-4, 0, -2, 3, 1], [-2, 0, -2, -2, -4], [-4, -2, 2, 2, 4], [3, -5, -1, -1, 1], [4, -2, -2, -2, 0], [4, -4, 1, -5, 0], [4, 2, 2, -3, -2], [-2, 6, -2, 0, 2], [-8, -2, 0, 0, 0], [6, -2, -4, -3, -1], [-5, 1, -1, -1, -1], [5, 1, -5, 1, -3], [-4, 4, 2, 4, 4], [9, 1, -1, 2, 1], [2, 4, 2, 0, 0], [-10, 4, 0, 2, 2], [-5, 1, 1, -1, 1], [-6, 2, 2, -2, 2], [0, -5, -1, 2, 2], [-1, 2, -4, -3, -1], [-10, -2, 2, 1, 3], [-4, -2, 0, 0, -6], [-8, 0, 0, -2, -4], [-5, -1, 3, 0, 0], [-4, -2, -4, -4, -8], [2, 0, 0, -2, 2], [1, -3, 5, 2, 5], [-6, 2, 1, 7, 2], [1, 3, -1, 1, 3], [3, -1, -7, -1, -5], [1, 7, -1, 4, -5], [2, 6, 2, 2, 0], [0, -2, 2, 0, 4], [8, -2, -6, -6, -4], [5, 1, 5, -3, -5], [6, -2, -2, 6, 0], [-9, 0, -6, -1, -1], [4, -10, 2, -2, -4], [-8, 8, 0, 3, -3], [1, -3, -7, 2, -3], [-1, 1, -9, -7, -7], [-4, -6, 0, -2, 4], [8, -8, 0, 3, 1], [-2, 2, -6, -4, -4], [0, 3, -9, -4, 2], [2, 2, 2, 8, 0], [-7, -1, -5, 2, 7], [6, -9, -5, -6, -8], [-17, -3, 8, -2, 5], [-2, -4, 0, -2, -6], [6, -6, 0, 2, -4], [5, -3, -9, 2, -6], [3, -3, -3, -7, -5], [2, -8, 4, 3, -4], [-11, 3, -10, -4, -1], [-6, 6, 2, 6, -2], [8, -2, -4, -6, -10], [3, 7, 1, 1, 3], [-16, 5, -3, -4, -6], [-5, 7, 7, 1, -1], [-5, -1, 11, 0, 11], [14, -2, -1, -1, 0], [-12, 8, -1, 5, 4], [-13, 7, 3, 6, 2], [-8, 3, 3, -6, 2], [8, -6, 4, 4, -2], [15, -7, -11, -8, -7], [-2, 2, -6, -12, -6], [-6, 10, 4, 8, -4], [-14, -10, 10, 0, 8], [-5, 7, -1, -1, 1], [-4, 8, -6, 4, 2], [9, -7, 1, -3, -1], [-13, 3, 11, 9, 9], [6, -2, 6, 2, -2], [-12, -6, -4, 2, -10], [-3, -1, -3, 11, -3], [-5, -7, 11, -3, 1], [-16, 7, 3, 6, -6], [12, 0, -4, -3, 0], [0, -5, 7, -14, 2], [-8, 8, -4, 8, 0], [8, 0, 6, -7, 7], [11, -11, 3, -1, -5], [-7, -1, 15, 3, 7], [6, 4, 10, 8, 6], [2, 12, -6, 10, -2], [10, -10, 4, 2, -2], [-6, 12, 4, 14, 2], [-4, 4, 6, 8, -6], [-15, -5, -4, -8, 1], [6, -9, 3, 2, -4], [13, 12, -10, 5, -3], [-4, -2, -6, -2, -10], [26, -10, -10, -8, -2], [-11, 7, 5, -4, -5], [-7, 3, -10, 6, -3], [-8, 0, 6, -2, 10], [4, -6, 12, 10, 8], [2, -7, -3, -6, 4], [3, -1, -3, 3, 5], [21, -3, -3, -5, 7], [-9, 9, -1, 3, 7], [17, -11, -9, -7, 7], [-21, 11, -9, 5, 5], [23, -16, -6, -5, -5], [8, -4, 10, -6, 0], [7, -11, 1, 5, 7], [-13, 1, -4, -4, -1], [-11, -7, 9, 4, 13], [-4, -6, -4, -4, 8], [-5, -9, 7, -1, 11], [-8, 10, 0, 10, 4], [-11, 8, 6, -5, 5], [-25, 13, 9, 5, -5], [-5, -7, 1, 5, 3], [-8, 4, 6, 14, 4], [-6, 2, 4, -1, 9], [-4, -2, 0, 6, 12], [23, -5, -1, 3, 5], [-16, 14, 10, 17, 3], [-6, -8, 14, 4, 6], [-24, 8, 10, 14, 10], [-16, 14, -6, 2, 4], [22, -6, 0, -2, -4], [3, 3, 3, 9, 3], [6, 4, 2, 0, 6], [2, 4, -6, -10, 0], [2, -12, 8, -4, -10], [-3, 1, 11, -3, 5], [12, -4, 4, -8, 6], [13, 5, -19, -7, -7], [20, -8, 0, -7, 4], [16, 2, -16, -2, -4], [1, -5, 0, -10, -1], [1, 7, 1, -5, 1], [-15, 3, 2, -6, 1], [13, 9, -15, -2, -3], [-5, 2, 8, 9, -5], [10, 12, -12, 10, -8], [12, -8, 8, 2, 4], [8, 6, -4, 0, 4], [4, 2, -12, 4, 4], [-10, -6, 2, 5, -5], [0, -13, -5, 6, 6], [28, 10, -8, -4, -14], [-12, 6, 10, 4, -8], [-7, -11, 5, 7, 17], [8, 2, -2, -12, 0], [-2, 2, 6, -2, -2], [32, -10, -2, 7, 2], [3, -11, 19, -3, -1], [10, 6, 7, -7, 10], [31, -11, -6, 6, -1], [14, -20, 0, 0, 2], [9, -9, 0, -6, -9], [-37, -4, 18, 3, 11], [4, -6, -6, 6, 4], [-14, 8, 8, -4, 0], [-26, 6, 6, -3, 3], [-8, 12, -6, -8, -12], [-5, 15, -1, 3, -11], [3, 7, -1, 4, -12], [2, 1, 5, -10, 0], [-29, 11, 5, 13, 7], [-20, 6, 12, 4, 2], [2, -14, -8, -2, 6], [-2, -22, 12, -8, 2], [-20, 8, 6, 10, 8], [7, -15, 1, -19, -5], [1, 5, -7, 0, -7], [17, -5, 7, 9, 15], [18, 0, 12, -16, 0], [-10, 4, -4, 18, 2], [9, -13, 1, -16, -9], [-13, 7, -1, -14, -5], [24, 0, 0, 1, -5], [7, 13, 9, 1, 7], [12, -1, -21, -8, -10], [24, 0, -4, 3, -8], [-2, 2, -2, 2, 10], [-16, 21, 5, 0, -2], [-9, -11, -7, -11, 1], [18, 4, 2, 2, -2], [-7, 14, -4, 3, 13], [-34, 10, -6, 14, 12], [-6, 14, -6, 2, 2], [-19, 7, 10, 2, 1], [-20, -4, 10, -6, 4], [-44, 8, -8, 12, 10], [-9, 11, 19, 10, 19], [-15, -7, 19, -1, 11], [13, -11, -1, -1, -3], [11, 15, 3, 9, 1], [12, -4, 20, -4, -4], [-20, 16, 12, -4, 6], [-15, -5, 17, 8, 11], [-14, 18, 4, 16, 8], [-11, -23, 5, -14, -2], [-11, 11, 19, 3, -5], [-11, 15, 2, 12, -5], [0, -2, 16, -2, -2], [7, 19, -11, -3, -11], [-26, 6, 10, 14, 14], [-6, 16, 4, 10, 14], [9, -5, -2, -12, 7], [9, -10, -4, 3, 5], [7, -3, -18, -16, -7], [-4, -4, -14, -10, -10], [3, -5, -1, 1, -1], [35, -15, -11, -7, 5], [8, 0, -8, 14, -2], [10, -12, -2, 2, -10], [2, -2, -2, 0, 0], [21, -1, -23, 0, -9], [20, 14, -6, 6, -2], [4, -12, -14, -17, -11], [3, 3, 15, 10, 2], [12, 16, -4, 10, 6], [-14, 12, 2, 2, -6], [25, -13, 8, -6, -1], [-19, 5, -11, -5, -11], [-22, 9, 17, -2, 8], [-7, 6, -12, 1, -3], [4, -8, -4, -6, -6], [-19, 21, 7, 17, 1], [-12, 12, 17, -9, 8], [24, 16, -20, -3, -16], [0, -7, 1, -24, 2], [4, 26, 2, 6, 6], [7, -27, -1, 11, 1], [-12, 6, 14, 4, 16], [-22, -3, -11, 0, -8], [-2, 0, 6, -10, 8], [3, -23, 15, -4, -3], [4, -4, 6, 0, 8], [-20, 0, 12, 8, 12], [10, 4, -20, -4, -14], [8, -12, 12, -8, 2], [7, -3, 5, 13, -7], [-8, 4, -22, -4, -6], [-52, 4, 12, 0, 2], [-8, 8, -16, 9, 3], [-28, -14, 10, -1, -3], [28, -10, -32, -8, -12], [-63, 11, 15, 5, 5], [24, -2, 14, 16, 16], [-16, -10, 2, 4, -20], [-6, 6, -14, 2, -6], [6, 10, 16, -10, 4], [19, -11, -3, 10, 13], [-20, 6, -10, 1, 7], [-8, 2, -24, -12, 4], [17, 11, 1, -11, -13], [-21, -13, 19, 0, 16], [0, 16, -8, -8, -16], [-34, 10, 28, 16, 8], [-5, -5, -1, -5, -15], [-8, 6, 2, -5, -14], [-10, 17, 5, 10, 8], [-22, -6, -4, 11, 9], [15, -25, -13, -24, -17], [-33, -9, 7, 5, 15], [7, 3, 15, -1, -7], [-1, -21, 11, -12, 3], [23, -3, 21, 12, 13], [-32, 0, 30, 5, 15], [22, -2, -6, 6, 0], [11, -23, 7, 9, 13], [-10, 10, 10, 3, -15], [-9, 25, -3, -2, 1], [-4, 4, -4, 21, -5], [-1, -9, 5, 19, -1], [-6, -25, 7, -2, -12], [39, -19, -12, -24, -9], [-3, 25, 1, 17, -17], [-3, -3, -3, 2, 13], [-5, 4, -14, 5, -5], [6, -14, 4, 4, 2], [-15, -25, 26, 2, 5], [-8, 4, -12, -12, -26], [-11, -19, 13, 0, -7], [-12, -12, 18, -16, 16], [28, 0, -3, 11, -4], [39, -9, 1, -3, 7], [-14, 0, -4, 8, -14], [24, -20, 0, -21, -7], [8, -10, 14, -4, 18], [17, 3, -28, -10, -21], [-30, 14, -14, 2, 10], [21, 9, -9, -10, -11], [13, 9, -15, 13, -3], [-43, 33, 11, 23, 17], [30, -18, -9, -21, -12], [13, 23, -11, -11, -11], [-14, -20, 10, 4, 22], [4, 20, 0, -6, -16], [6, 11, 11, -2, -4], [-8, -22, 4, -9, 1], [-21, 17, 8, 22, 5], [46, 8, -16, -3, 0], [-44, 1, 33, 0, 14], [-6, -8, -12, 10, -10], [-17, 5, 9, 17, 23], [-26, 24, 12, 16, 10], [0, 2, -24, -7, -17], [-22, 2, -2, -12, 6], [-20, 10, 4, -2, -4], [9, -15, 11, 11, 13], [-16, 10, 6, 0, 4], [7, -25, -5, -25, 13], [-20, 2, 32, 8, 4], [13, -15, 7, -5, -3], [-22, 10, -5, -1, -16], [-24, 6, 6, 19, -18], [28, 4, -18, 8, -2], [6, 2, -4, 4, -14], [50, -4, -8, -16, -10], [-36, 6, 18, 18, 8], [-20, 0, -14, 27, 1], [6, -6, 18, -2, 10], [32, -24, -4, -27, -5], [51, -15, 3, 5, -3], [-5, -25, 3, 9, 15], [-9, 13, -14, 2, 19], [3, 27, -7, 5, -13], [-8, 4, -2, -3, 11], [5, -13, -25, -7, -15], [13, -11, -7, -6, 14], [-3, 15, -10, 2, 5], [46, 4, 8, 3, 4], [3, 11, -1, 3, -1], [-16, -6, -20, -2, -10], [-41, 10, -4, 15, 15], [12, 32, -13, -1, -2], [-2, 10, -2, -8, 2], [27, -5, 29, -11, -13], [-24, 18, 6, 28, 6], [1, 29, -1, -7, -29], [-22, -13, -13, -6, 0], [26, 6, -30, -2, -12], [-1, 1, -2, -22, -9], [19, -3, -3, 0, -19], [7, -1, 3, -19, -11], [-16, -8, -24, 7, -11], [-18, 2, -22, 8, -12], [-8, -24, 13, 5, -2], [-26, 12, -4, 6, 6], [-6, -8, 2, 14, 20], [-14, 16, -8, 23, 12], [0, 14, 10, -16, 10], [-5, 29, 9, 11, 17], [23, -11, 3, -15, -13], [-30, -16, 0, 2, -10], [-16, 2, 8, 0, 4], [15, -9, -25, -10, 3], [-15, -9, 9, 1, -5], [-3, -13, -2, -8, -9], [40, 0, -12, 8, 0], [-9, 7, -9, -10, 11], [-44, -4, 24, 4, 12], [37, -23, -9, -1, 5], [-5, 5, -14, -22, -5], [-12, 27, -5, 14, 18], [28, -4, -20, 14, -6], [-2, -10, 16, -22, 8], [22, -16, 12, -14, 14], [-20, 16, 6, 18, 4], [-23, -16, 10, -13, 5], [-32, 7, -1, 16, 14], [4, 2, -42, -22, -10], [-10, 2, 14, 10, 4], [5, 1, 17, -15, 3], [18, 2, -14, -12, -22], [-21, 19, 13, 26, 18], [-10, -9, 11, -30, 8], [-26, 30, -6, 13, 10], [-3, 13, 7, 23, -17], [-3, 9, 7, -9, -3], [11, 13, 5, 19, -5], [-9, -25, 11, 5, -9], [-24, -16, 27, 15, 34], [-11, -21, 3, -24, -5], [-30, 16, 16, 12, -8], [-12, 6, 32, 10, 0], [9, -13, -2, 8, -5], [10, -6, 10, -26, -8], [5, -13, -12, 14, -13], [-49, 11, 5, -2, -10], [-24, -18, -10, -12, -12], [-22, -14, 16, -8, 12], [-39, -3, -7, 4, 13], [-38, 26, 18, 22, 22], [-12, 3, 31, 22, -6], [-42, 20, -8, -6, 2], [-8, 2, 0, 20, -20], [-37, -5, 3, 5, 7], [-6, 12, 16, 10, -2], [10, 18, 6, 14, 0], [-28, 6, -8, 16, 4], [-40, 20, 6, 17, -17], [-8, 28, -10, 2, 6], [25, -3, -17, -9, -7], [10, -8, -20, -8, 20], [-16, -12, -5, -27, -4], [16, -5, -5, -18, -34], [22, 0, -10, -20, 6], [-17, -21, -5, -31, -7], [-16, -6, 2, -8, 0], [4, 18, 2, -6, -34], [28, -2, 8, 18, 12], [22, 14, 2, 16, 12], [-15, -27, 17, -19, 15], [16, 12, -29, 5, -28], [-36, 10, -12, 18, 20], [-9, -31, 9, -11, 11], [-10, -14, -14, 17, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2299_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2299_2_a_n();".
// 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_2299_2_a_n(:prec:=5)
    chi := MakeCharacter_2299_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_2299_2_a_n();".
// 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_2299_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2299_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 5, -10, -6, 2, 1]>,<3,R![-1, 7, 11, -9, -1, 1]>,<7,R![-64, -119, -62, -1, 6, 1]>],Snew);
    return Vf;
end function;