
// Make newform 8008.2.a.j in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8008_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_8008_2_a_j();".
// 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_8008_2_a_j();".
// 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, -1, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [2, -3, -8, -1, 1], [5, -10, -10, 0, 1], [-6, 2, 9, 1, -1], [15, -19, -29, -1, 3]];
    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_8008_a();"
function MakeCharacter_8008_a()
    N := 8008;
    order := 1;
    char_gens := [6007, 4005, 3433, 4369, 4929];
    v := [1, 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_8008_a_Hecke(Kf)
    return MakeCharacter_8008_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], [0, -1, 0, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [4, 0, 0, 1, 0], [-1, -1, -1, 1, 0], [1, -1, 1, 1, 0], [-4, -2, -1, 0, -1], [1, 2, 0, 0, -1], [2, -1, 1, 2, -1], [2, 1, 0, 1, -2], [-5, -2, 1, 2, 0], [1, 1, -1, -1, 2], [4, 0, 1, -1, -1], [3, -4, 1, -2, 0], [5, -1, 1, 0, 0], [5, 0, 0, -2, -1], [6, 0, 0, 0, 2], [-1, -1, 3, 0, 0], [-3, 0, 0, -1, -3], [-1, -3, 3, 0, -2], [-7, 1, -1, 1, 0], [3, -3, 2, 1, -1], [-3, -3, -3, 2, 0], [-9, -4, 0, -1, -1], [-4, 1, 0, -1, 2], [-1, -6, 0, 4, -1], [6, 2, 0, 3, -2], [4, -1, -6, 0, 2], [-6, 1, -2, 1, -2], [-2, 3, -3, 1, 5], [0, 3, 3, 0, 1], [7, 3, -3, -3, 0], [-4, -5, -2, 2, -2], [-1, 3, 2, 0, -3], [-3, 1, 2, 0, -3], [9, 1, -2, 0, 1], [-2, 0, -1, 1, -1], [0, 2, -3, -1, 5], [6, 1, 3, -1, -1], [3, -1, 5, -2, 0], [0, -6, -5, 4, -1], [-3, -2, 2, 1, -3], [6, 2, -1, -2, 3], [5, -1, 3, 3, -2], [13, -6, -1, -1, 2], [-3, -6, -3, -3, 0], [3, 2, 3, 0, 4], [2, -2, 1, -5, -3], [-7, -1, -1, 1, 2], [3, 2, -4, 2, -1], [6, 2, -4, -2, -2], [15, 4, 4, 1, 1], [-10, 2, 2, -2, 0], [2, 1, 1, -2, -1], [9, 7, 2, 1, 3], [6, 2, -4, 1, 2], [-4, 1, 3, -6, 3], [-5, 7, 0, -3, -1], [5, 8, -2, 0, 3], [4, 4, -7, 1, 1], [1, -4, -4, 0, -1], [0, 8, -4, -3, 4], [8, 0, 0, -5, 0], [1, -3, -1, 5, -2], [-19, -4, -2, -1, -3], [6, 1, 2, 4, -4], [-2, 1, 2, 0, 6], [2, 2, 0, -2, 0], [2, -2, -2, -7, -2], [-3, -3, 2, -1, -5], [-1, 5, -1, 0, 4], [2, 5, -4, -3, 0], [5, -5, 6, -2, -3], [4, -3, -4, 5, 0], [3, 6, -1, -2, -4], [14, -4, -3, 4, 3], [2, -1, 3, -7, -1], [-2, 5, 6, -6, 0], [1, -1, 6, 3, 1], [-4, 2, -5, 0, -3], [20, 8, -5, -4, -1], [-10, 9, -4, -2, 6], [-4, -5, -1, 1, -1], [3, -5, 0, 2, -5], [1, -3, 1, -3, -2], [26, 7, -5, -2, 1], [-3, 4, 0, 0, -1], [14, 5, -2, 3, -2], [9, -2, 1, -1, 0], [-2, -8, 3, 2, -1], [-3, 6, 5, 3, 0], [-5, 10, -5, -1, 8], [11, -3, 6, -1, -7], [-4, -2, 8, 1, -4], [1, -3, 0, 4, 1], [-10, -4, 3, 5, 5], [10, -2, 4, 1, 2], [-3, -1, 2, 9, -1], [-8, 0, -1, -3, 5], [11, 5, 2, -1, 1], [-5, -3, -8, 1, -3], [6, 5, -2, 5, 6], [-9, -1, -4, -3, -7], [-16, 8, 4, -7, 2], [-3, 1, -5, 2, -2], [-5, -1, 4, 8, 1], [1, 4, 5, 3, 2], [3, -4, -7, 3, -2], [-16, 3, 9, -3, -1], [18, -1, -2, 2, 2], [4, -8, 5, -5, -3], [10, 13, -7, 1, 3], [6, -5, 3, 6, -3], [-4, 11, -1, -4, -3], [-23, -8, -1, 0, 0], [20, 7, -5, -3, -3], [0, 9, -6, 4, 2], [-12, -12, 0, 3, 0], [-6, 3, 9, 2, -5], [-8, -9, 3, 1, -11], [-12, 3, 2, -4, -6], [20, 5, -8, 0, 0], [2, 11, 6, -5, 4], [8, -7, 0, -1, -4], [-7, 2, 12, 3, -7], [-12, 1, 4, -5, 6], [-13, -8, -6, 5, -1], [-19, 6, 5, 2, -2], [10, 8, 1, 2, -3], [13, 4, 3, 9, -2], [12, 9, 3, -5, 1], [-7, -2, 3, -2, 6], [5, -3, 4, -5, -1], [21, 0, 7, -8, -2], [-16, 8, 6, 1, -4], [3, -4, 6, 8, -7], [2, -11, 4, -4, 0], [-23, 11, 1, 1, -4], [-5, -1, -5, 0, 8], [0, 7, -8, -7, 2], [-28, 8, 2, -1, 0], [0, 6, -13, -5, 9], [10, 8, -1, -4, 7], [-17, -17, -1, 6, -4], [22, 16, -4, -3, 2], [-9, 2, -4, 4, 5], [21, -4, 5, 4, -2], [-4, -7, -1, -3, -3], [-18, -5, -5, 5, -1], [-30, 0, -1, 0, -1], [30, 1, 0, 1, -4], [3, 1, -6, 0, 7], [0, 3, -8, -1, -2], [-33, 4, 1, -7, 2], [9, -1, 4, 3, 3], [17, -2, 2, 0, -1], [17, 6, 6, -9, 1], [-4, -13, -2, 6, -6], [-13, 10, 11, -4, -4], [7, 5, 7, 0, -2], [29, 10, 2, 1, 7], [10, -1, 1, 1, -3], [7, -6, 4, 3, -1], [28, 8, -6, 2, -2], [-33, 7, -2, 1, 3], [-4, 2, 11, -6, -1], [24, 4, -2, 7, 2], [1, 5, 5, -8, -6], [-14, 10, 7, 2, -7], [-26, -6, 2, 9, 2], [-6, -4, -4, 6, 4], [-13, -2, -7, -2, 0], [2, -5, -4, 2, 4], [-23, 9, 5, -2, -2], [-24, -7, 9, -1, -9], [-23, -2, -3, 5, 0], [-3, -6, -10, -8, 3], [29, -2, -2, 3, 3], [1, 14, -5, -3, -2], [1, 0, 6, 6, -1], [-12, 4, -2, 4, -4], [14, 1, 7, -10, -1], [0, 0, -2, -9, 2], [4, 6, 10, -11, -2], [-1, 1, -10, 3, 5], [12, -12, -1, 11, 1], [-13, -5, 6, -8, -1], [-9, 0, -1, 15, 6], [-30, 0, 5, 0, -7], [-10, -18, 3, 2, -5], [2, 2, -7, -9, 5], [-16, 4, -7, 0, 3], [31, 0, 6, 8, 1], [32, -12, 1, 7, -5], [-9, -7, 13, -4, -2], [-7, -2, 6, -2, -9], [15, -9, -3, -6, 4], [-12, -13, 4, -9, 0], [-5, 16, -7, 1, 10], [5, -2, -3, 9, -2], [-12, -4, -3, -10, 5], [-8, -2, -10, 6, -10], [-13, -2, 10, 1, -5], [-13, 6, -1, -4, -6], [11, 2, -2, 3, -11], [-16, -12, 3, -2, -5], [-30, -6, -1, -5, 1], [28, 9, 1, -4, 5], [6, -14, 0, 12, -8], [9, -1, -3, 4, 14], [1, 8, -4, -4, 7], [8, -4, 10, 0, -2], [-8, -2, -7, -9, -5], [1, 5, 0, 0, -5], [-19, 6, 12, 2, -11], [-1, -8, -1, 0, 2], [2, -5, -11, 7, -7], [-17, -4, 5, -8, -6], [-31, 2, 0, -5, 5], [12, -3, 4, 3, 8], [13, 4, 6, -1, -5], [-9, -3, 6, -7, -3], [13, 0, -13, 3, 0], [12, -6, -6, -2, 8], [-12, 10, -4, -1, -6], [-7, -10, 12, 4, -9], [-14, 9, 10, 0, -4], [17, -11, -5, 6, 2], [24, -2, 11, -8, -3], [-9, -1, -2, -3, 1], [15, 18, 2, -14, 1], [5, 12, 7, -11, -2], [30, 16, -9, -1, 1], [-6, 1, -1, 1, -13], [-9, 15, 0, -12, 11], [-19, -8, 2, 7, -5], [1, -12, 3, -5, -2], [16, -10, -2, -9, 0], [-19, 1, 5, 5, -4], [-13, -11, -5, -1, 0], [14, 2, 5, -10, 3], [23, -6, -2, 6, 9], [10, -11, -1, 5, -3], [1, 4, -4, -3, 9], [-13, 1, -16, -2, 9], [-5, 2, 7, 5, 0], [-6, 4, -15, 5, 3], [-18, -18, 7, 0, -7], [-23, 10, 3, -2, -2], [0, 20, -3, -4, 7], [11, -10, -7, 13, 4], [-10, -7, 0, -2, 4], [-8, 14, 11, -7, -5], [4, -10, -4, 14, -6], [-3, -6, 12, -5, -15], [-4, 14, 1, -4, 5], [25, 5, 7, -5, -4], [21, 12, -4, -10, 3], [4, 2, -4, -17, 2], [-17, -10, 4, -3, 1], [-15, 6, 0, -5, 7], [13, 6, 1, 13, 6], [-28, -5, -2, 4, 6], [25, 8, -11, -1, 10], [35, -9, -4, 8, -1], [-16, 6, 3, -3, -13], [17, -18, -1, 17, -4], [0, 13, 11, -3, 1], [-24, 4, -13, -1, 7], [-37, 7, -3, -7, 8], [-10, -1, 4, 15, 0], [-29, -11, 1, -7, -4], [27, -11, 6, 3, -3], [-26, -7, 8, -1, -4], [-6, 8, -7, -10, -1], [5, 19, -7, -7, 10], [-5, 0, -10, -10, 3], [37, -8, -4, -7, 1], [7, -1, 20, -8, -11], [-37, 5, 3, 4, -2], [11, 2, -1, -6, 12], [0, -5, 2, 1, 4], [-27, 2, 2, -2, -7], [-32, 6, 1, -12, -3], [-8, -5, -21, 0, 7], [36, -13, -1, 8, 1], [-8, 10, 6, 5, -2], [-19, 8, -10, -4, 15], [-1, 21, 8, -1, -1], [-25, 1, 9, -2, -2], [20, 16, -5, 3, 19], [9, 22, -14, 3, 9], [-6, -9, 10, -6, -18], [9, 11, -7, 3, 12], [-2, -3, -8, -1, 6], [-5, 0, -10, -2, -5], [23, 5, 12, -5, -5], [-6, 11, -2, -8, 2], [17, 8, -9, 2, 14], [-21, 22, 10, -14, -1], [-21, 2, -1, -1, -4], [13, 8, -15, -1, 8], [-27, -6, -3, 11, 2], [22, -15, 11, 3, -11], [17, -15, 5, 0, -4], [19, 1, 3, 10, -6], [14, -2, 4, 9, -6], [5, 12, 4, 1, -7], [-30, -14, 3, 17, 3], [-20, -7, -7, 11, -1], [20, -5, 7, 2, -3], [-17, 9, -14, 10, 3], [-4, 7, -12, 1, 8], [4, -16, -5, 4, -3], [-15, 18, 9, -2, 2], [20, -4, -11, 3, -5], [17, 6, -18, -2, 3], [-7, -5, 10, -3, 7], [45, -4, -1, 1, 12], [6, 1, 18, 2, 2], [19, 8, 6, 3, -3], [0, 14, 0, -17, 0], [0, -15, 3, 1, -3], [5, -1, 5, 2, 10], [14, -7, -2, 7, -14], [35, 13, -3, 1, 14], [-20, 1, -6, -2, -16], [-4, 1, 13, 0, -7], [18, 2, 14, -7, 12], [6, 10, -17, -1, 9], [33, -14, -4, 14, -1], [-29, 5, -8, -2, 13], [21, 28, -8, 0, 7], [-8, 4, 7, -10, -9], [32, -1, -3, 3, -5], [11, -20, 8, -11, 5], [9, 4, -4, -7, -3], [18, 23, 3, -8, 9], [-25, 0, -7, -2, -2], [1, 8, 11, -14, -6], [1, 20, -9, 8, 0], [-1, -5, 9, 19, -4], [7, 1, 11, 17, 0], [-26, -6, -6, -6, 2], [16, 16, -18, -2, 10], [29, 13, 0, 10, 7], [-8, -11, 1, 9, -5], [-29, 10, -5, -11, 4], [-12, -15, 4, -7, 4], [-8, 10, 14, -9, 2], [15, -20, 10, -10, -1], [42, -21, 0, -5, 2], [14, -13, 9, -5, 1], [-16, -13, -4, -2, 8], [-4, 14, -2, 11, 4], [-38, -6, -13, -6, -3], [46, 9, 14, -6, 4], [-4, -5, 16, -14, 6], [-15, -17, 11, -13, 8], [-16, 10, -3, 8, -9], [15, 15, -4, 5, 15], [-22, -12, 6, -2, -8], [40, 13, -7, -10, 5], [12, 10, -1, 7, 5], [-5, -15, -5, 13, -4], [12, 4, -3, 2, 9], [-23, 9, 7, 2, -4], [-9, -6, 14, 4, 3], [2, -4, 17, 5, -9], [-18, 2, -4, 1, 2], [3, 4, 0, 18, 5], [-15, 23, 11, -12, -2], [-8, 3, -3, 11, 5], [14, 2, -5, 3, 9], [23, 13, 5, 6, 2], [-5, -26, -6, 3, -1], [-21, 8, -9, -9, -2], [-1, 3, -7, 2, 10], [20, 2, -4, 13, 16], [12, 10, 15, 3, -5], [-10, -6, -7, 6, -1], [12, 4, 7, -15, -5], [15, 4, 2, -1, 1], [-8, 5, 17, -12, -1], [-46, 1, -3, 2, -5], [-24, -13, -10, -7, -8], [-63, 11, 4, 1, -7], [19, 4, -4, 4, -9], [10, 14, -3, 1, 5], [10, -12, -11, 1, -1], [10, -13, 8, 3, -16], [-55, 2, 3, -10, 0], [9, 0, 6, 1, -19], [-2, -5, -4, 3, 4], [-12, -13, -2, 9, -6], [-43, -2, -9, -9, -6], [-1, 2, -6, -3, 3], [30, -15, 1, 9, 1], [-34, 0, -8, -9, -2], [16, -13, -16, 11, 6], [23, 13, -1, 5, 2], [11, -14, 1, 9, -12], [14, -10, 5, 2, -3], [18, 6, -2, -12, -2], [-27, -5, 1, 1, -6], [-28, -2, 1, 4, -1], [13, -12, 7, -10, 8], [-25, -26, -1, -2, -12], [11, 2, 10, -14, 1], [-5, -11, 12, 9, -5], [-51, -6, 1, 1, -8], [-43, -14, 13, 13, -2], [-19, 6, 7, -17, 4], [-11, -1, 9, -1, -16], [-17, 8, 4, -20, 9], [41, -1, 2, -12, -5], [44, -3, -4, -9, -2], [-15, -13, 6, -6, -13], [-6, -3, 7, -9, 3], [-15, -20, 4, -10, -5], [11, 1, 3, 9, -8], [13, 12, -8, -18, 1], [41, 17, -6, -3, -3], [6, 2, 2, -3, 2], [-22, 1, -9, -3, 11], [-17, -17, 1, 19, 6], [20, -25, 4, -16, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8008_2_a_j();".
// 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_8008_2_a_j(:prec:=5)
    chi := MakeCharacter_8008_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_8008_2_a_j();".
// 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_8008_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 11, 12, -10, -1, 1]>,<5,R![10, 39, -16, -18, 1, 1]>,<17,R![-18, 327, -344, 126, -19, 1]>],Snew);
    return Vf;
end function;