
// Make newform 4002.2.a.be in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4002_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_4002_2_a_be();".
// 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_4002_2_a_be();".
// 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 := [-18, 26, 7, -11, -1, 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, 8, -5, -2, 1], [4, 8, -7, -2, 1], [8, -6, -2, 1, 0]];
    Rf_basisdens := [1, 1, 2, 2, 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_4002_a();"
function MakeCharacter_4002_a()
    N := 4002;
    order := 1;
    char_gens := [2669, 3133, 553];
    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_4002_a_Hecke(Kf)
    return MakeCharacter_4002_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 := [[-1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [0, 0, 1, 0, 0], [-1, 0, 0, 0, 1], [1, 1, -1, 0, 0], [-2, -1, 0, 0, 0], [1, -2, 0, 0, 1], [-1, 2, 0, 0, -1], [-1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [2, 1, 0, 2, -2], [2, -3, 0, -1, 0], [1, 1, 1, 1, 0], [-3, 0, 1, 1, 0], [3, -2, -2, 0, -3], [-2, 0, -1, -3, 1], [5, -2, -1, -1, -1], [-1, 0, 1, -4, 3], [3, 0, -2, 3, -2], [-4, 1, -2, 0, 0], [-2, 0, -3, -3, -1], [-2, -1, 1, -1, 0], [-2, 4, -4, 4, 0], [6, 4, -1, 1, 1], [-2, 0, 1, 3, -1], [2, -4, 0, 0, -1], [-10, 2, 0, -1, 1], [3, -3, -2, 0, 2], [-4, 4, 0, 4, 2], [-5, 0, 3, -3, 0], [-6, -2, 0, -4, 1], [2, 1, -1, 3, 0], [-2, 4, 1, 1, 1], [-6, -4, 1, -3, 1], [0, 0, 4, 1, 3], [-7, 1, 1, -1, 1], [3, -1, 2, -6, -2], [-3, 7, 1, 3, 4], [-2, 4, 1, -3, 0], [5, 1, 2, 4, 0], [8, 0, 2, -4, -6], [-8, -5, 0, -4, 3], [-12, -1, -1, 0, -1], [-10, -2, 3, 1, -1], [-5, 1, -1, 7, -1], [3, -2, -1, 6, -3], [-5, 1, 1, 3, 6], [-8, 4, -3, 1, -3], [-3, -1, -1, -3, 3], [0, -3, -1, 2, 3], [4, -2, 0, 0, -6], [-2, -4, -1, 3, 4], [-5, 1, -4, 0, 0], [1, 0, 6, -3, -2], [0, 7, -1, -5, -2], [-5, 3, 2, 0, -4], [8, 1, -5, 5, 1], [-6, -8, 2, -4, -1], [-18, 1, -2, 0, -4], [-12, 2, 3, -3, 5], [-11, 1, -5, -2, -2], [12, -4, 0, -2, -4], [-6, 0, -2, 4, -1], [-11, 0, 6, 0, -3], [-3, -1, -2, 0, 0], [-4, 1, 2, -2, 2], [1, -3, 1, 1, -3], [5, 2, 3, 0, 9], [1, 2, 1, 1, 0], [2, -2, -3, 5, -8], [-6, 2, 5, -7, -1], [-5, 3, -6, 2, -4], [-6, -1, 6, -8, 7], [-8, 4, 3, -5, -1], [2, -2, -7, -3, 1], [-2, 1, -3, 1, 2], [6, 8, -1, 1, -5], [8, 3, -2, 2, 1], [13, -2, -3, 6, -9], [-8, 5, -1, 9, 4], [-3, -7, 3, -1, -3], [-1, 0, -2, 5, 0], [-4, 5, -1, 3, -2], [-4, 5, -1, 1, 2], [-7, 5, -3, -1, -5], [-14, -2, 1, 3, -1], [7, -7, 7, -9, 0], [-19, -3, -1, 3, -1], [-20, -3, 5, -11, 5], [-6, 4, -1, 3, -3], [-13, 1, -1, 4, 6], [-11, 2, 3, -2, 9], [3, -1, 4, 0, -4], [16, -7, -5, 7, 0], [-8, -5, -4, -6, -1], [-9, 3, -2, 8, -4], [1, -7, 4, 4, 2], [4, 7, 1, -3, 10], [3, 4, -2, -7, 6], [-7, 4, -3, 11, 4], [-6, -2, 8, -4, -4], [-14, 0, 7, 0, 2], [-3, -4, -4, 1, 2], [27, 2, 2, 6, 1], [7, -7, 6, -1, 7], [4, -3, -7, 3, 4], [-1, -2, 1, -7, -6], [4, 2, -4, 4, -10], [-16, 0, 3, 1, 3], [10, 0, 5, 3, 5], [-24, -3, -2, 2, -3], [6, 8, 0, 0, -8], [-5, 4, -5, -5, 0], [13, 10, 0, 2, -1], [-12, -12, 5, 0, -2], [11, 0, -1, 1, -10], [8, 3, 1, -1, -4], [12, -3, 7, -5, 1], [-4, -6, 3, 1, -4], [2, -16, 1, -5, -7], [-6, 8, 6, -4, -4], [-9, 0, -1, -13, 4], [6, 4, 2, 6, 8], [-3, -14, 5, -3, 5], [0, -1, 0, 4, 3], [-20, 2, -8, -3, -3], [4, 3, 5, 7, 4], [0, -3, 2, 0, -10], [20, 6, -4, 18, -8], [1, 8, -2, -7, 0], [8, 2, 5, -3, -6], [12, -5, -1, 4, -9], [-4, 6, 1, 7, 1], [30, -7, 2, -3, -2], [8, -11, 6, -2, 7], [13, 1, -2, -13, -1], [20, -5, 2, -10, -1], [-29, 7, 4, -1, 7], [-4, 3, 5, 3, 10], [4, 12, -3, 11, -2], [-8, -4, 1, -3, 1], [4, 10, -9, 9, 1], [6, -5, -1, 11, -9], [17, 14, -4, 3, 4], [-11, 0, 2, 12, 3], [20, 1, -5, -4, 7], [17, 6, -10, 10, -3], [-2, 9, -2, -2, 5], [3, -13, -7, -3, -3], [6, 1, -4, 8, -2], [-4, 3, -2, -8, 1], [-8, -2, 1, 1, 15], [-16, 11, -3, -7, -6], [8, 4, -4, -14, 4], [32, 6, -2, 12, -10], [25, -2, -3, 0, -3], [-2, 10, 3, -2, 14], [16, 4, 1, -1, 3], [27, -5, -6, 12, -2], [-17, -10, 0, -3, 4], [-4, 9, -3, -1, -8], [6, -10, 5, -1, 9], [-4, -1, 0, -8, -3], [-11, 9, -2, 7, -1], [-3, 3, 4, 13, 3], [-3, -10, 12, -7, 6], [5, -7, 7, -9, 7], [-3, 2, -4, -8, -3], [-7, 5, -3, 13, -7], [-10, 3, 0, -14, 9], [0, 0, -3, 1, -13], [10, 0, -4, 10, -8], [-34, 2, 3, -5, 3], [8, 1, 1, 1, 12], [-7, -5, 7, 1, 1], [-23, 9, 6, 3, 7], [2, -2, -1, 1, -5], [0, -12, 0, 4, 2], [-24, -1, 2, -2, -7], [0, 0, 9, 15, 5], [8, -7, 2, -4, -4], [29, -4, 7, -6, 3], [16, 1, 1, 1, 7], [8, -4, -7, -3, -1], [-16, 3, 4, 0, 1], [7, 8, 9, 6, -3], [11, 0, -4, -2, -11], [19, 0, 5, -5, -2], [8, 1, -2, 0, 13], [38, -8, -12, 8, -6], [15, -4, 3, -3, 7], [8, 11, -11, 9, 0], [-8, 0, 0, -10, 20], [4, 13, 6, 2, 3], [-11, 3, 8, -1, 13], [-24, -5, -3, -5, -6], [-23, -14, 2, -13, 6], [0, 7, 4, 6, 5], [6, -2, 3, -1, 15], [-22, 0, -7, -9, -2], [-22, -12, 5, -3, 3], [-7, -1, 6, 0, -2], [-19, -2, 1, 1, 0], [-4, -4, -5, 3, 11], [22, 15, -7, 11, -2], [-29, 1, 5, -3, 7], [8, -3, -5, 3, 10], [-7, -6, 15, -17, 3], [-2, -2, 11, 1, 3], [-16, 0, -8, 8, 5], [10, -8, 8, -4, 0], [-3, 7, -12, 4, 2], [-6, 5, 2, -20, 3], [-22, -1, 1, -13, -2], [27, -8, -4, -2, -5], [-12, 2, -6, 6, -2], [-15, -5, 7, -19, 15], [4, 0, -14, 4, -7], [-20, -3, 6, 0, 5], [-4, 7, 0, -20, 4], [22, 6, -16, 18, -6], [-3, 2, 6, -1, -6], [-11, 5, 1, -5, -2], [-36, 16, 2, 0, 0], [-4, 0, -10, 8, -2], [5, 0, 2, -2, -3], [7, -4, 3, -3, -6], [12, -4, 10, 0, 4], [10, 5, -4, 10, 1], [6, -10, -2, -6, 1], [-4, 0, 3, -3, -5], [6, 9, -7, 11, -12], [0, 0, 5, 3, 9], [-29, 5, 3, -5, -9], [21, 7, 3, 10, -18], [-9, 11, 8, 4, 18], [23, -8, 6, -11, -16], [4, -11, -3, 3, 2], [-7, -12, -3, 3, 5], [27, -4, 7, 1, 2], [10, 1, -5, -15, 2], [16, 6, 0, 10, -10], [0, -4, 2, 12, -4], [27, 14, -11, 3, 4], [10, 10, -1, 7, 5], [-12, 17, 4, 6, 3], [22, 8, -7, 11, -13], [-10, -2, 10, 4, -8], [-11, -6, 4, 3, 8], [15, -16, 0, 13, 2], [-48, -10, 1, -1, -1], [-9, -16, 9, -2, -1], [13, -15, 4, -10, -10], [-9, 6, 4, -9, 12], [-10, 4, -4, 0, -12], [15, -12, 4, -6, -14], [3, 1, -2, -10, -8], [3, -10, 10, -8, 9], [10, -4, -11, 5, -11], [0, 10, -14, 6, 1], [24, -8, 1, -7, 7], [-18, -8, 13, -7, -3], [20, 8, 3, 5, -6], [23, -8, -5, 14, -1], [-23, 1, 7, -19, 17], [-5, 10, 10, 0, -5], [4, 8, 6, 6, -8], [1, 10, 11, 3, 10], [19, -10, 7, -1, 20], [21, -9, -2, 14, 4], [10, -5, 0, 2, -10], [-12, 5, 13, 7, -2], [-19, 9, 5, -1, 3], [-4, -20, 3, -7, -13], [-14, -5, 13, -3, -8], [-20, -1, 6, -8, 11], [-32, 3, 17, -7, -2], [4, -11, 4, -4, 9], [-7, 17, -5, -8, -2], [-11, 1, -2, 12, -16], [0, 4, 4, -6, 13], [-57, 6, 8, -2, 3], [-14, -2, -11, 1, -13], [32, -9, 2, -4, -1], [-45, -3, 1, -7, 7], [15, -12, 12, -8, 17], [-40, 3, 5, -3, 2], [-16, 10, -3, 5, -15], [3, -3, -11, 9, 7], [16, 1, -10, 2, 7], [-14, 4, 11, -15, -7], [24, -4, 17, 1, 9], [18, -7, 2, -8, -17], [26, -9, 7, 1, -4], [-2, -10, 9, -5, -7], [6, 10, -3, 2, -6], [12, 5, -11, -5, -8], [-14, 10, 0, -8, 6], [21, 14, 0, -6, 0], [-11, -17, 13, -13, 3], [-5, 7, 8, 5, 19], [20, -16, -4, -13, -13], [2, 0, -17, 5, -11], [-42, -2, 2, 4, -8], [0, 2, -11, 7, 13], [21, 6, 9, 7, 10], [8, 12, 0, -2, 8], [28, -3, -5, 11, 0], [-12, -3, 3, 1, 14], [-2, 10, -12, 14, -18], [10, -8, -5, 7, 4], [44, 4, -3, 3, 1], [-17, -14, 13, -4, -3], [14, -15, -2, 2, -19], [16, 6, 5, -3, 15], [3, -27, 0, -6, -10], [2, 21, -6, 14, 3], [-34, -8, -4, 8, 3], [35, -21, 1, 7, 13], [49, -7, 6, -6, -1], [41, -8, -3, 1, 2], [-3, 17, -3, 3, 17], [-9, 3, -4, 2, 12], [-4, -18, 8, -4, 10], [6, 4, -15, 15, -9], [-19, 13, -4, 13, 1], [-11, 2, 5, 10, 1], [32, 8, -5, 3, -7], [-7, -4, -1, -6, 19], [-12, 2, 18, -7, -9], [-1, 2, -2, 16, 3], [-19, 3, -9, 11, 7], [-31, 15, 9, -3, 1], [-24, 1, -12, -8, -7], [-40, 14, -5, 1, 1], [41, 4, -5, 1, -4], [-22, 1, 4, -10, 13], [-22, -2, 6, 0, -4], [-6, -3, 17, -3, 12], [-40, 4, 3, -4, -4], [5, -22, 5, -7, 12], [-9, -7, -19, -8, 4], [34, 6, -6, -6, -1], [0, 14, 1, 0, 8], [-1, -10, 2, 8, -1], [8, 5, -7, -3, 16], [-5, 18, 1, 5, 2], [-11, 9, -12, 2, 7], [-45, 8, 19, -4, 3], [-26, 8, 2, 20, 12], [-4, -1, -14, 4, 1], [-26, 4, -17, 3, -18], [-2, -8, 8, 6, -7], [-7, -4, 10, 7, 10], [7, 5, -2, -12, -4], [-13, -7, 3, 1, 4], [3, 4, 9, -6, 15], [-5, 5, 0, -2, -18], [36, -2, -10, 14, -19], [24, 3, 14, 2, 10], [-10, 12, -3, -11, 9], [10, -3, -15, 3, 16], [-23, 4, -1, 5, -14], [24, 14, 1, 5, 1], [-19, -5, -17, -9, 4], [68, -5, -7, 1, -4], [36, -5, -2, 6, 9], [6, 4, -3, -6, 0], [18, -3, 1, -7, -6], [30, -8, 0, -6, 0], [-4, 4, 0, 4, 6], [2, 12, 6, 2, 0], [-30, -20, 15, -23, 9], [14, 8, 6, -5, -9], [28, 0, 6, 0, 9], [8, 3, 3, -13, -2], [-8, -13, -14, -5, -10], [-17, 3, 7, 4, 8], [5, -20, 9, -18, 3], [5, -6, -7, 1, 14], [-14, -10, 3, -11, 3], [-39, 20, 8, -5, 10], [-2, 6, -5, 17, 4], [23, 3, -18, 2, -9], [14, 20, -7, 9, -7], [-16, 2, 8, 4, 7], [20, 2, -12, 12, 6], [-26, -19, 4, 8, 3], [7, -4, 4, 4, 3], [13, -18, 6, -3, 16], [49, -1, 0, 9, -15], [1, 11, -1, -13, 7], [-6, -6, 18, -14, -6], [13, -25, -8, -10, -11], [6, 3, -12, 13, -12], [15, -11, -9, -11, -19], [-25, -2, 10, -22, 13], [21, 6, -9, 19, -2], [-17, 2, -7, -2, 3], [28, 15, 0, 6, -5], [0, -6, 6, -22, -3], [-38, -2, 2, 8, 13], [-10, 15, 9, -5, 20], [-15, -8, 0, -17, 12], [27, -13, -6, -14, -6], [-2, -12, 9, -23, 5], [4, 6, 18, -2, 0], [-39, 9, -12, -2, -5], [-26, -22, 6, -2, 4], [37, -8, -10, 0, 8], [-22, -10, 2, 4, 9], [35, 13, 3, -3, -13], [29, 9, 7, 9, 3], [20, -7, -1, -13, -12], [-3, -15, 3, 14, 0], [-44, -20, 4, -20, 18], [37, -12, -3, 13, 4], [-25, -9, 6, 8, -8], [8, -10, 2, 18, -4], [8, -13, 9, 2, 15], [18, -10, 7, 2, -12], [-8, -8, 13, -13, 17], [-1, 2, -7, 17, -10], [14, 14, -3, 13, 3], [18, -6, 10, 12, 9], [14, 4, -17, 23, -23], [-15, -1, -13, 7, -3], [35, 11, 17, 15, -5], [-12, 24, -8, -8, 0], [-25, -6, 16, -20, 17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4002_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4002_2_a_be();".
// 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_4002_2_a_be(:prec:=5)
    chi := MakeCharacter_4002_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_4002_2_a_be();".
// 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_4002_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![49, 85, 4, -18, -1, 1]>,<7,R![20, 20, -26, -7, 4, 1]>,<11,R![42, -122, 99, -17, -5, 1]>],Snew);
    return Vf;
end function;