
// Make newform 4008.2.a.f in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4008_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_4008_2_a_f();".
// 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_4008_2_a_f();".
// 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, 10, 5, -6, -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], [3, 8, -3, -3, 1], [6, 9, -4, -3, 1], [11, 14, -9, -5, 2]];
    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_4008_a();"
function MakeCharacter_4008_a()
    N := 4008;
    order := 1;
    char_gens := [3007, 2005, 1337, 673];
    v := [1, 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_4008_a_Hecke(Kf)
    return MakeCharacter_4008_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, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, -1, 1], [-1, 0, -1, 1, 0], [0, 0, 0, -1, -1], [-2, -1, 1, 0, -1], [-3, 0, 0, 2, 0], [0, -1, 1, 0, 0], [-2, 2, 1, 1, 0], [4, -1, 2, -1, -2], [0, 2, 1, 0, -1], [-1, 0, -3, 0, 0], [-4, 2, -2, 2, -3], [-4, -1, 2, -2, 2], [-1, 1, 1, 4, 0], [-2, 1, -3, 3, 1], [2, -5, 3, -3, 1], [-4, -5, 2, -2, -1], [-4, 1, -1, -4, 3], [-2, -1, -1, -1, 4], [-5, 2, -3, 3, 3], [-2, -1, 0, -4, -2], [-2, -2, -5, 1, -1], [-8, -3, 1, -2, 0], [-6, 2, 0, 1, 2], [-1, 5, 2, -1, -3], [4, 0, -5, 1, 2], [-4, 2, -1, -4, 2], [3, -3, 0, 1, -3], [-1, 1, -5, 6, -1], [3, -3, 0, 0, 1], [5, -2, 0, 1, -2], [-1, 2, -1, -1, 3], [1, -2, 1, 3, -1], [10, -2, -2, 3, -2], [1, 8, 1, 1, -2], [-8, 7, 1, 1, 1], [1, 1, 3, 2, -1], [1, 0, 0, 0, 0], [2, 1, 3, -3, 3], [-5, 9, -4, 8, 5], [-13, 3, -2, -4, 4], [1, -3, -6, 1, 4], [-1, 0, 0, 3, 1], [2, 1, -1, -2, 5], [7, -6, -3, -5, 5], [-5, -1, 0, -7, 1], [-9, 0, 0, 6, 1], [-6, 3, 3, -6, 7], [0, 4, -1, 1, 4], [-7, -4, -1, -4, 2], [5, -2, 2, 5, -7], [-2, -10, 3, -4, -6], [0, 4, -6, 3, -3], [3, -11, 4, -2, 1], [-5, -4, -7, -2, -1], [5, -5, 6, 4, -4], [15, 1, 9, -5, -8], [-6, 4, 8, -2, -3], [7, -1, 6, -8, 3], [-2, -3, -5, 3, -1], [4, -1, -2, -8, 1], [1, 0, 9, 4, -2], [17, -4, 5, -10, 0], [-2, 3, 2, -1, -1], [11, -8, 9, -10, -2], [-8, 12, -1, 9, -3], [-12, 14, -5, 7, 8], [-1, -1, 6, -2, -6], [-13, 5, 5, -4, 1], [-3, 10, 1, -1, -1], [0, -3, 7, -3, 4], [-4, 4, -7, 7, -6], [-7, 7, -2, -2, 0], [10, 3, 5, 3, -5], [-3, -12, 8, -3, -6], [-7, -6, 2, -7, 2], [-10, -4, -2, -2, -5], [7, 8, -7, 4, -10], [-13, -4, -6, -1, -2], [7, -10, 9, -9, 1], [7, -4, -2, 0, -1], [14, -3, 3, 2, -1], [-1, -6, 5, 5, -8], [-16, 2, -13, 8, 6], [6, -5, -8, -2, 6], [-4, -8, 3, 3, 5], [-1, 3, -9, 8, -6], [-2, 3, -9, 14, 3], [10, -2, 6, -2, 7], [2, -10, -2, 2, -1], [12, 7, 0, 4, 6], [9, 8, -10, 5, -1], [18, 2, -4, -6, -2], [9, 10, 3, -2, -7], [-2, -5, -3, -1, 6], [-8, 5, 4, -1, -5], [-2, -2, 7, -1, 1], [3, -2, 8, 9, -3], [-3, 1, 9, -13, 4], [14, -14, 2, -7, -2], [8, 7, -3, 10, -1], [-14, 13, -7, 11, -4], [7, -3, 4, -16, -4], [0, -6, -4, 1, -6], [-9, -1, 1, -5, 8], [-9, 5, -2, 9, -12], [1, -18, 7, -7, -2], [1, 1, -14, 5, -2], [-16, 0, -7, 8, 8], [9, -7, 0, -5, 9], [13, -8, 7, 2, -1], [-11, 8, -2, 14, -7], [-12, 5, 2, -4, -4], [5, 4, -7, -2, 2], [4, 8, 8, 0, 5], [-1, 4, -6, -1, 3], [-8, 16, -2, -4, 1], [17, -2, 2, -5, -10], [-13, 3, -1, 1, 15], [7, 2, 1, -1, -12], [-11, 16, -4, 8, -4], [-1, -4, -10, -5, 10], [12, -3, 4, 7, 0], [15, -9, 2, -17, -2], [4, 3, -12, -5, 4], [3, -8, 13, -14, -2], [4, 0, 7, 6, 7], [2, -9, 17, -18, 1], [-14, -2, -10, -1, 3], [14, 1, 11, 6, -5], [-4, 4, -11, 7, 5], [13, -9, -5, 9, -9], [-9, 5, 1, 2, -5], [17, -1, -4, 12, -6], [-20, 6, 3, 2, 0], [7, -16, 3, 1, -2], [3, -6, 2, -11, -7], [25, -10, -3, -1, 5], [1, 13, -5, -7, -4], [-9, -13, 17, -17, -2], [8, 0, 5, -1, 13], [6, -1, 18, 3, -5], [15, 2, 4, -3, 1], [-8, 12, -8, 3, -7], [20, -12, -5, -8, 2], [-10, 2, 7, 0, 11], [20, -7, 5, -9, -10], [-12, 13, -10, 16, 11], [15, -5, -2, 8, 5], [-9, 7, -7, 24, -1], [11, -5, -14, -10, 11], [-13, 12, -8, 4, -8], [4, 12, 11, -1, -2], [-10, -10, 3, -2, -8], [-6, 5, -9, -2, 7], [10, 3, 0, 0, 2], [-14, 3, 2, 7, 5], [7, -8, 10, -14, 10], [-6, 0, -10, 6, 5], [-10, -10, -12, -1, -1], [-5, 5, -3, 7, -3], [1, 11, -4, -8, 3], [18, -3, -7, 12, 0], [23, 9, -11, -3, 6], [11, -20, 4, 0, 5], [5, -9, -7, 2, -12], [-4, 3, -3, -7, 18], [-13, -10, 21, -7, -4], [-2, -5, 5, -2, -2], [1, -4, 6, -1, -15], [2, -1, 15, 8, -6], [-13, -1, 8, 10, -7], [-3, -7, -4, 6, -1], [28, -5, 3, -14, 5], [28, 3, 17, -2, -8], [5, -12, -1, -11, 5], [10, 4, 7, 10, -8], [-16, -2, 6, -7, 17], [-8, 6, -6, 14, 3], [-18, -4, 6, 7, 0], [-3, 8, -4, 8, 3], [13, 5, -19, 9, 1], [16, -14, 7, -15, -8], [-8, 10, -12, 2, -4], [-23, -18, 21, -10, 0], [14, -10, 7, -10, 2], [-9, 6, -9, -12, 15], [-18, -4, -2, 1, 14], [14, 9, -2, -9, -4], [2, -4, 5, 1, 6], [3, 12, 13, 8, -1], [11, 11, 0, 7, -16], [-5, 12, 2, 13, -15], [-13, 11, -6, 15, -7], [-29, 0, -1, 8, -7], [19, -15, 6, 1, -5], [15, 8, -4, -3, 4], [-20, -6, 0, -12, -1], [14, -13, 10, -11, -9], [-16, 16, 7, 2, 3], [-30, 2, -2, 11, 3], [9, 8, -2, -3, -15], [-1, 0, 19, -10, -5], [-9, -10, -2, -9, -9], [-13, -4, -23, 8, 11], [30, -5, 4, -3, -16], [-10, 7, 15, -10, -4], [0, -6, 0, -11, -9], [-9, 7, 7, -16, 4], [29, 3, 11, 10, 0], [15, 3, 18, -4, -2], [14, 4, -21, 2, 6], [-32, 11, -8, 9, 12], [22, -10, 12, -3, -11], [27, -14, 10, -8, -7], [7, 0, -8, 16, -18], [4, -2, -5, 6, 13], [-8, -13, 3, -6, 10], [-15, -1, -6, 0, 21], [-2, -21, -2, -14, -4], [-2, -8, 6, 5, 14], [-35, 15, -2, 18, -1], [-9, 23, -10, 9, 6], [-11, -2, 12, 7, -5], [-39, 6, -11, 8, 15], [-19, 18, -1, 12, 5], [14, 2, -2, -13, -17], [19, -4, -3, -15, 5], [-34, 10, -12, 15, -1], [5, -26, 2, -2, -8], [8, 0, -8, 3, 3], [4, 7, -8, -8, 1], [50, -12, 5, -8, 0], [-7, 13, -13, 12, -22], [-6, 22, -3, 8, -9], [-23, 4, -11, 20, 10], [-11, 0, -3, 2, 3], [-13, -12, 4, 9, -6], [-6, 10, 13, -6, -7], [10, -24, 12, -1, -20], [-17, 16, -13, 11, -7], [-27, 13, -7, 17, 3], [4, -5, -24, -11, 12], [-11, -1, 15, -3, -1], [-29, 18, -5, 1, -2], [28, 0, 1, -7, 5], [-10, 19, -12, 2, 11], [-29, 18, 6, 10, -7], [11, -10, 4, -14, -7], [13, -31, 18, -25, -10], [2, 11, -2, 10, 3], [33, -9, -1, -19, 13], [-23, -4, -1, 15, 6], [13, 15, -10, 16, -1], [-13, -7, 6, -3, -5], [-6, -4, 0, -6, 10], [-28, 1, 2, -3, 13], [-4, 3, 14, -8, -3], [20, -13, -11, -3, 7], [12, -2, -3, -3, 10], [-23, 18, 1, 21, -14], [-31, -3, -15, 6, 16], [3, 1, -3, 15, 8], [-13, -15, -21, -3, 17], [3, -25, 5, -2, -11], [5, 17, -16, 11, 1], [-20, -2, -1, 6, 18], [9, -21, 4, -1, 10], [7, 16, 0, 9, 0], [2, 25, -1, 6, -4], [-9, -2, -20, 16, -8], [-37, -11, 5, 4, 8], [-17, 2, 0, -11, -3], [-21, 1, -9, -1, 11], [18, -9, 14, -21, -13], [-1, -10, 15, -26, 5], [-19, 6, -18, 12, -1], [16, 17, -12, 23, 5], [9, 10, -20, 0, -3], [2, -13, -11, 0, -13], [3, 3, -10, 28, 2], [-47, 5, 3, 1, 5], [-55, 16, -3, 24, -5], [12, 1, 0, -16, 20], [-6, -15, 19, -2, -12], [2, -27, 18, -8, -14], [-8, 4, -23, 31, 7], [-1, 4, 11, 2, 8], [-13, 4, 10, 6, -2], [36, -17, -2, 4, -2], [-25, 12, 1, -1, 6], [-39, 1, -8, -13, 1], [-5, -4, 8, -6, -13], [15, 2, -5, -20, 18], [5, -18, -5, -6, -10], [16, -11, -12, 13, -12], [0, -13, 14, -25, -9], [-7, -10, 18, 14, 3], [-4, 10, -22, 20, -14], [22, 0, 21, -18, -6], [5, 20, -5, 13, 2], [-3, -9, 0, -4, -13], [-27, 13, 0, 4, 13], [-17, 18, -3, 17, 5], [-37, 14, -5, 28, -19], [-9, -4, -7, -11, 23], [-4, 6, 2, 13, -24], [-1, 14, 0, -2, 13], [-37, 16, -2, 6, 7], [-42, 14, -3, 30, -1], [-7, 16, 21, -5, -6], [-30, 14, 1, 26, -9], [-16, -3, 6, -7, -6], [-13, 24, 8, -2, -7], [5, -3, 0, 3, -11], [22, -11, 4, -20, -5], [-33, 24, -4, 12, 14], [28, 9, -22, -2, 5], [7, 19, -13, -4, 4], [22, 0, 12, 10, -9], [32, -6, 4, -4, -5], [-11, 7, -14, -25, 19], [2, 2, 4, -15, 9], [-12, -12, -21, 13, 8], [3, 16, 1, 21, 11], [12, -29, 9, -6, 2], [-6, 15, -10, -11, -9], [-15, 9, -8, 31, -15], [-11, -11, 25, -9, 9], [25, -13, 8, 21, -9], [41, -8, -3, -10, -1], [-5, 6, -2, 4, 21], [-6, -1, 16, -30, -3], [-37, -11, -6, 12, 28], [-53, 15, -10, 17, 13], [-37, 3, -1, 3, 9], [-14, 27, -17, 18, 5], [-12, 0, -34, 17, 2], [11, -12, -3, 5, -5], [-38, 3, -6, 10, -7], [39, -1, 5, -6, -20], [-32, -3, -11, -1, 16], [-27, 4, -10, 11, 14], [-1, 21, 15, 14, -3], [13, -8, -4, -23, 13], [6, -24, 3, -15, -1], [-16, 14, -1, 8, 4], [-24, -3, 6, 3, 12], [3, 28, -9, 1, -14], [-30, -1, -12, -1, 2], [-9, 7, 2, 2, 2], [-3, 18, -8, 20, -8], [-18, -3, 21, -14, 4], [13, -26, 15, -15, 9], [-11, 1, -19, -18, -6], [12, 19, 9, -5, -20], [20, 4, -3, -6, 4], [-19, 0, 16, 12, -9], [33, -2, 1, 4, 9], [-23, 11, 20, 2, -12], [12, 9, -15, 15, -18], [3, -2, -8, -8, -1], [23, -7, 4, 9, -2], [17, 6, -8, 23, -8], [34, -3, 18, -13, -18], [-25, 10, -21, 0, 14], [-9, -5, 5, 22, -19], [23, 5, -13, 10, -12], [-17, 13, -18, 3, 1], [29, -14, 13, -23, 12], [8, -20, -4, 10, -25], [-26, 41, 5, 4, 1], [27, 23, -7, 13, 4], [12, 7, 2, 9, -21], [1, 10, -17, -8, -13], [-8, -13, -12, 1, -20], [27, 11, 14, 4, -16], [4, -10, 28, -36, 10], [-14, 10, -20, 9, 3], [-10, -31, 7, -1, 11], [-29, -29, 14, -2, 13], [53, 0, 4, 4, -19], [-23, 2, -8, -15, 0], [-8, 4, 24, -15, -2], [0, 5, -11, 11, 21], [18, -7, 21, -21, 1], [22, -10, 18, 10, -1], [-10, 37, -22, 33, 13], [-28, 24, 0, 8, 19], [-11, 7, -24, -7, 16], [1, -15, 18, -2, -8], [10, 14, 23, 13, -14], [-9, -9, -9, -22, 19], [2, -15, 12, 8, -14], [7, -1, 12, -1, 10], [-4, 12, -18, 27, -12], [-20, -15, -1, 0, 20], [11, -18, 12, 9, -32], [8, 1, -12, 28, 8], [-25, 5, 4, -1, -5], [-43, -6, 4, -2, 15], [51, -19, 2, -12, 16], [-5, 4, 13, 5, -12], [-31, 23, -10, 27, 14], [18, -18, 8, -28, -19], [-4, -7, 9, -14, 28], [-28, 11, 4, -25, 17], [57, -2, 7, -10, -2], [7, -9, -22, -2, 14], [-2, 3, 1, -27, 4], [-17, -13, 28, 2, -7], [2, -6, -19, -15, 16], [27, -6, -2, 29, -5], [34, -7, 13, 0, -17], [8, 16, 17, 12, -13], [11, 2, 25, -6, -19], [29, -15, 5, -4, 4], [-1, -2, 2, -15, 24], [12, 28, -9, 16, 2], [-19, -35, 32, -18, -13], [-6, 5, -20, -16, 32], [-3, -5, 14, 7, -13], [-8, 3, 17, -11, -7], [9, 10, 2, -2, 2], [37, 4, 13, -11, 3], [-32, 8, 9, -12, 13], [-21, -3, 31, -26, 9], [32, 7, 5, -7, -21], [25, -23, -1, 17, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4008_2_a_f();".
// 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_4008_2_a_f(:prec:=5)
    chi := MakeCharacter_4008_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_4008_2_a_f();".
// 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_4008_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![5, 19, 12, -13, -1, 1]>,<7,R![-5, 17, -13, -5, 4, 1]>],Snew);
    return Vf;
end function;