
// Make newform 1449.2.a.r in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1449_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_1449_2_a_r();".
// 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_1449_2_a_r();".
// 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 := [-27, 16, 17, -9, -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], [-4, 0, 1, 0, 0], [11, 5, -8, -1, 1], [19, -5, -10, 1, 1]];
    Rf_basisdens := [1, 1, 1, 2, 2];
    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_1449_a();"
function MakeCharacter_1449_a()
    N := 1449;
    order := 1;
    char_gens := [1289, 829, 442];
    v := [1, 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_1449_a_Hecke(Kf)
    return MakeCharacter_1449_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+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, -1, 0, 0, 0], [0, 0, 0, 0, 0], [1, 0, 0, 0, 1], [1, 0, 0, 0, 0], [1, -1, 1, 1, 1], [-1, 0, 0, 1, 1], [3, 0, -2, 0, -1], [2, -2, 0, 0, 0], [1, 0, 0, 0, 0], [0, -1, 3, 0, 0], [6, 0, 0, -1, 0], [0, 2, 0, 2, 0], [-1, 0, 0, -1, 1], [-2, 0, 0, 0, 2], [-2, 2, -2, 3, 0], [-4, 0, 2, 0, 0], [-5, 2, 0, 0, 1], [-3, -2, 0, -2, -1], [-3, 5, 1, -1, -1], [-1, 0, 0, -3, -1], [1, 0, -4, -1, -1], [7, -1, -1, 1, 1], [-2, 0, 0, 0, -2], [5, -2, 0, 2, 1], [-3, 2, -2, 0, -3], [2, 2, -2, 2, 0], [6, 0, -2, 0, 0], [2, 4, -2, 2, 0], [2, 0, 4, -2, 0], [0, 0, 0, 2, 2], [3, -2, -2, 3, 3], [6, 2, 2, -1, 0], [2, 2, 4, 0, -2], [-2, 2, 2, 5, 2], [0, 4, -2, 0, -4], [9, -2, -2, -3, -3], [-9, 0, 2, 0, -1], [1, 2, -2, -1, -3], [-3, 0, -6, 4, 1], [-4, 2, 2, 2, 2], [1, 6, -2, 1, 1], [-3, -2, 0, 0, 1], [-6, 0, -2, -2, 0], [5, 2, -4, -3, 1], [-5, 4, -6, -3, -5], [10, 4, 0, 2, -2], [-4, -2, -2, -6, -4], [1, -4, 6, -2, 3], [-8, 4, 4, 0, 0], [7, 0, -4, 4, 1], [-6, 3, 3, 2, 2], [-1, -2, 2, -1, -1], [-1, 0, 2, 2, 3], [-8, -2, 4, -2, -2], [-5, 4, -8, 3, -3], [5, -5, 1, -3, 1], [-1, 2, 2, 3, -3], [3, 4, 6, 2, -3], [-6, -1, 7, -2, -2], [4, -4, 2, -6, -4], [2, 0, 4, -2, 6], [-3, 2, -2, 4, -3], [7, -4, 6, 4, 3], [-8, -4, -4, -1, 4], [-15, -2, -2, 2, -1], [-7, -3, 3, -3, -1], [5, 0, 0, 5, 5], [0, -4, 2, 2, 4], [-4, -2, -2, -6, 4], [-1, -8, 0, -5, 1], [-7, 2, 2, 3, 3], [-5, -5, -1, 1, -3], [-12, 6, -6, -2, -4], [6, -4, 4, 4, 4], [15, 9, -3, -3, 1], [10, -6, 4, -6, 0], [14, 4, 0, -2, 0], [7, 0, 0, -3, 5], [-12, 4, 0, -8, -2], [-5, 2, -6, 1, -3], [14, -6, 2, -2, -2], [4, 0, 0, -2, -6], [-2, 4, -2, 0, 0], [-9, -8, 2, -4, 1], [16, -2, 2, -7, -4], [5, -2, 2, 7, 1], [-3, -7, -5, -3, -1], [-2, -6, -2, 4, -4], [3, 6, 2, -1, -3], [18, -4, 0, 4, 2], [-8, -2, -6, -4, 0], [-4, 6, 0, -10, -2], [15, -8, -8, 5, -1], [13, 0, 4, 3, -3], [-9, -4, 4, -5, 7], [-6, -6, 4, 4, 0], [-3, -4, 4, 1, 3], [5, -6, -4, -8, -5], [-6, 4, -6, 4, -4], [-6, 11, -9, 2, -6], [-1, -2, -2, -5, 3], [-6, 4, 2, 10, 2], [26, 0, 2, 0, 0], [4, 2, 0, 10, 8], [-4, 4, -2, 0, 2], [-1, 2, 6, 9, 9], [-4, 6, 6, -11, 0], [-8, -6, 2, -2, 6], [4, 2, 6, 2, 0], [-1, 2, -6, 9, -3], [11, -6, -4, -6, -5], [-20, 4, -2, -2, -4], [8, -2, -8, -6, -8], [-12, 0, 2, 0, 2], [7, -1, -3, -1, -5], [8, -6, 6, -4, 10], [-4, -4, 6, -2, -2], [0, 2, 2, -1, -6], [13, -8, -6, -5, -7], [-2, 8, -10, 4, 0], [9, -4, -4, 0, 5], [6, 5, 1, 2, -10], [1, -6, -2, 2, 5], [-5, -2, 6, 9, 7], [15, -8, 6, 0, 7], [-30, -2, 4, 0, 2], [-2, 0, 4, -10, 0], [-21, 10, 8, -2, -1], [-4, 4, -2, -6, 2], [-1, -12, 6, -2, 1], [-5, 12, 0, 1, 3], [13, -7, -5, 1, 1], [5, 1, -13, -3, 1], [6, 6, -2, -10, -4], [-1, -12, 8, 5, 5], [-1, -10, 10, -4, 1], [9, -2, 2, -10, 3], [-16, 6, -6, -6, -12], [13, 2, -6, -4, 3], [6, -2, -2, 10, 0], [-2, -2, -6, -7, -8], [7, -5, -11, 7, 1], [7, 6, -6, 1, -9], [0, 12, 0, 0, 0], [2, 4, -4, 0, 4], [-14, 10, -4, 6, -4], [-12, -8, -4, -8, 6], [7, -12, -4, 11, 1], [-6, 0, -8, 9, -6], [-3, 4, 4, 3, -7], [9, -7, 11, -11, 3], [-7, -2, 2, 6, -9], [28, -6, -6, 2, 4], [14, -14, -10, 5, 2], [8, 4, 2, 2, -6], [27, -7, 3, 3, 3], [-17, 13, 5, -7, -7], [-1, 8, -8, -5, -7], [-1, 0, 4, -2, -3], [13, -12, 12, 0, 7], [-7, 4, 4, -7, -3], [-16, -2, 2, -2, -2], [-3, -10, -2, -5, 1], [8, -2, 0, -4, -2], [-10, 12, 2, 8, 2], [0, 10, -4, -12, -2], [0, -4, 4, 0, 0], [0, 6, -2, -6, -6], [6, -10, -4, -10, -2], [15, -14, 2, 1, 5], [-2, -4, 6, -10, -12], [19, 0, -8, 3, 1], [-12, 0, -4, -12, -8], [-16, 0, 6, -10, 8], [10, -12, -4, -3, 0], [7, 10, -6, -6, 1], [3, 4, 12, -7, 3], [-11, -4, -2, -5, 1], [10, 2, 4, -12, 4], [-3, 6, -4, 4, -9], [-1, -2, 6, 11, -1], [8, -4, -12, 8, -8], [-3, 2, 4, -15, -3], [-8, -8, 16, 0, 6], [-16, -4, 12, 10, 0], [-13, -2, -8, -12, 1], [5, 9, -9, -3, -9], [2, -2, 2, -12, -2], [2, -13, 3, -6, 6], [-7, -4, -6, -2, -9], [-17, -10, 14, 7, 1], [32, -8, -8, 0, -8], [0, -8, -10, 6, 2], [-32, 8, -4, -8, -6], [25, 9, 1, -1, -1], [13, 8, 10, -2, -5], [-2, 0, -2, 16, 2], [-14, 14, -8, 0, -2], [13, -12, 0, 2, 9], [-16, -8, 12, 13, 0], [2, 12, 6, -2, -6], [-3, -2, 8, -2, 9], [-15, 6, 6, -7, 3], [-27, -10, 8, 2, -5], [26, -16, -2, 0, 4], [-53, 2, -2, 3, -3], [16, -2, -12, 8, -14], [-29, 6, -2, 7, -5], [-8, 1, -7, 8, -8], [-10, 12, -12, -11, -8], [38, 1, 5, -6, -2], [3, -6, -6, 5, 5], [-7, 15, -9, -5, -5], [14, 6, 4, 4, -4], [-29, -2, 20, 2, 7], [-13, 8, -8, 4, -3], [17, -2, 2, -11, 5], [-1, -4, -4, -11, -11], [-14, -12, 2, -2, 0], [25, -8, -18, -5, -7], [-21, -10, 16, 3, 3], [6, 2, -12, -4, 0], [15, 2, -4, 3, 7], [8, -24, 2, -8, 2], [16, 2, -2, 1, -8], [-20, 11, -13, 0, -12], [3, -4, -2, -12, -7], [17, 2, 6, -19, 1], [-10, 4, -18, 2, -4], [12, -8, 0, -12, 0], [-34, 12, 0, -4, -4], [-26, 4, 18, 4, 12], [27, -3, -1, -1, -5], [-1, -9, -3, -5, -3], [-2, 12, 2, 16, 4], [2, 2, -12, 14, 4], [11, -10, 12, -2, 7], [-4, 18, 6, 10, 6], [17, -4, -4, -15, -3], [-16, 0, -8, 3, -2], [20, 4, -10, 0, -6], [-30, 14, 2, -3, -4], [-8, 0, -4, -5, -14], [19, 12, -2, 5, 3], [6, -20, 8, -4, 12], [12, 8, 4, -11, -6], [-4, 10, 2, 7, -4], [-36, 8, -8, 6, -4], [0, -4, -2, 2, -10], [6, -4, 12, -6, 6], [-16, -2, 12, 8, -4], [-1, 16, -2, 0, 1], [5, 5, -3, -1, -5], [-1, -18, 6, -6, -3], [-18, 6, -2, -10, -8], [-15, 14, 2, 5, -3], [20, -6, -8, -10, -8], [-8, 2, 2, 4, 4], [9, 8, 4, -3, 11], [1, -12, -4, 15, 3], [21, -5, -5, 15, 3], [14, 0, 2, 12, 12], [-14, 0, 4, 8, 0], [11, -2, 8, 16, 5], [-10, 8, 4, 8, 0], [32, 10, -2, 4, 0], [32, 10, 6, 2, -2], [-25, 0, 4, -6, 7], [-1, 3, 15, -11, 1], [-9, -8, 6, -7, -1], [-19, 2, 14, 5, 15], [17, -9, -7, -3, 1], [11, 12, -4, 0, -11], [-15, -1, -5, 19, 7], [-5, -14, 6, -3, -7], [-6, 2, 14, 5, -8], [8, -8, -8, -16, 0], [1, 17, 3, -3, -1], [-18, 8, 4, -4, 8], [4, -8, 8, -4, -8], [-36, 18, 2, -13, -12], [22, 12, -2, 6, -8], [36, 10, 0, -4, 2], [17, 0, -10, -2, 5], [-3, 0, 2, 14, -5], [14, -8, -12, 12, 2], [8, -8, 22, -12, 8], [14, 4, -8, 0, 4], [13, 18, 0, -4, 7], [-24, 12, 12, -4, -10], [14, -12, -4, 16, 4], [6, 4, -2, 8, 4], [-3, -3, -1, 17, 3], [-18, -6, -6, -16, -10], [-23, 8, 18, -1, 1], [-11, 2, -8, -8, 1], [30, -2, 2, 14, -2], [33, 12, -4, 3, 7], [-18, -6, -2, -13, -2], [-24, 6, -2, 9, 12], [37, -10, 8, -2, 13], [2, -14, 2, -18, 0], [-2, -12, -2, 0, -8], [-29, -16, -4, -17, 3], [22, -8, -6, -12, -4], [-1, 14, -16, -5, -13], [-10, -6, 14, -2, -2], [7, 12, -12, 6, -13], [-22, 4, 4, 2, -6], [14, 22, -10, 6, -8], [26, -14, 6, 19, 12], [20, -2, -16, -18, -12], [38, -12, 8, -6, 0], [-21, -4, 0, 9, 7], [15, -11, -5, -5, 3], [-9, 16, -10, 6, -7], [16, 4, 8, -3, -4], [-21, 20, 8, -7, 3], [-7, 10, -8, -8, -9], [5, -18, 20, -7, 17], [0, -12, 2, -12, 2], [22, -18, 8, 8, 14], [-13, 10, -10, 9, 9], [30, -8, 8, 8, 10], [-22, -12, 0, 20, -8], [21, 6, -12, 12, -5], [8, -4, 12, -18, 12], [8, 4, 12, -19, -10], [27, 1, -5, 7, -9], [-18, 4, 16, -2, -12], [22, -13, -5, -6, -2], [13, 2, -4, 1, -15], [16, -12, 0, -11, 6], [-56, 2, 10, 3, 12], [-51, -1, -3, -7, -3], [-6, -4, 12, 9, 18], [6, -12, -10, 0, 2], [25, 10, 0, -6, 1], [-5, 6, -10, 3, -3], [-4, -6, 18, 22, 4], [38, -3, -11, 6, 2], [28, 10, -14, 4, -2], [30, -6, 6, -2, 16], [38, 4, -20, -10, -2], [14, 12, -14, 8, 6], [-4, 12, 0, -12, -4], [41, 10, -10, -13, 3], [-34, -12, 0, -4, 2], [44, -10, 6, 16, 12], [-9, 2, -4, 0, -17], [-12, 6, -6, 5, -6], [5, 22, -4, 9, -11], [5, 9, 3, 5, -17], [20, -8, -16, 2, -12], [-16, -12, 4, 6, -10], [-18, -2, -18, 2, -4], [12, 6, 10, 10, 12], [-18, -6, 4, 8, -8], [-44, 7, -1, -8, 0], [12, 12, -8, -4, 4], [14, -6, 12, -10, 14], [21, 14, 2, 3, -5], [33, 0, -12, 7, -11], [-25, 1, 19, 3, 7], [16, -10, 22, -1, 6], [21, -12, 18, 7, 5], [21, 4, 12, -9, 3], [-29, -3, 31, 3, 7], [16, 0, 2, 0, 24], [27, 10, 2, -11, -1], [11, -16, 10, -6, -7], [27, 14, 6, 15, 3], [10, -6, -14, 0, -12], [6, 2, -4, -2, -12], [-5, -10, -8, -5, -13], [52, -8, -4, 3, 2], [38, -4, 6, 18, 8], [-17, 12, -18, 0, -7], [11, 6, 14, 8, -1], [-29, 26, 2, -1, -7], [3, -2, -10, 18, -17], [-8, -2, 14, -2, -6], [-10, 2, 10, -6, -10], [5, 0, -6, -5, 21], [4, 14, -4, -10, -10], [20, 2, -14, -1, -12], [23, 12, 4, -7, 7], [0, 26, -10, -14, -6], [17, 1, 17, -1, -9], [20, 4, -2, -8, 2], [-18, 16, 12, 14, 12], [-14, 8, 4, 8, 8], [-41, -6, 2, -5, 3], [21, 18, -4, 8, 7], [-15, 5, -9, 1, 1], [56, -4, -8, -8, -6], [12, 22, 2, -6, 8], [13, -14, 2, -22, -17], [12, -12, 6, -16, -6], [17, -11, 7, 5, 3], [-37, 24, 12, 9, 9], [8, 15, -9, -12, -4], [-1, 23, 1, 15, -1], [15, 28, -8, 12, -7], [3, -26, 14, -7, -3], [0, -10, -10, 7, 0], [40, -2, 6, -14, -6], [11, 8, 2, -3, 3], [26, -10, 4, 2, 4], [-21, 8, 22, 4, 3], [7, 6, -10, -9, -15], [45, -22, 4, -7, 1], [4, -4, -2, 6, 18], [-10, -4, 4, 10, -6], [-8, -22, 32, -2, 16], [-29, 2, -18, 11, -3], [8, -14, -10, 2, -4], [-5, -10, -4, 2, -1], [-53, 10, -10, 16, 3], [-18, 8, 6, -16, -16], [8, -8, -2, 18, -8], [-13, 8, 20, 16, 5], [4, -6, -6, -18, -8], [-29, 0, 2, 22, 9], [47, -18, -10, 9, 1], [5, 8, -8, -9, 7], [-17, 25, -25, 7, -9], [-6, 6, 6, 14, -4], [40, 6, -14, -9, -8], [8, 6, 2, -3, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1449_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1449_2_a_r();".
// 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_1449_2_a_r(:prec:=5)
    chi := MakeCharacter_1449_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_1449_2_a_r();".
// 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_1449_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1449_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![27, 16, -17, -9, 2, 1]>,<5,R![-168, 52, 54, -14, -4, 1]>],Snew);
    return Vf;
end function;