
// Make newform 8016.2.a.r in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8016_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_8016_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_8016_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 := [-11, 31, 20, -16, -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], [-7, 0, 1, 0, 0], [-3, -11, 1, 1, 0], [17, 8, -14, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2, 4];
    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_8016_a();"
function MakeCharacter_8016_a()
    N := 8016;
    order := 1;
    char_gens := [3007, 2005, 5345, 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_8016_a_Hecke(Kf)
    return MakeCharacter_8016_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, 1, 0, 0], [-2, 0, 0, 0, 1], [0, 0, 0, -1, 0], [1, 1, 0, 0, 0], [1, -1, 0, 0, 0], [-2, 0, 0, 1, 0], [0, 0, 0, 1, 0], [-2, 2, 0, -1, 0], [-2, 0, 0, -1, -1], [1, 1, -1, 1, 0], [1, -1, -2, 1, 0], [-4, 2, 0, -1, -2], [2, -2, 0, 0, -1], [0, 0, -3, -1, -2], [-3, 3, 1, 1, 2], [4, -2, 0, 0, -2], [-2, 0, 1, -2, -2], [2, -2, -2, 1, 2], [2, 0, 2, 1, 2], [-1, -1, -2, 1, 0], [-1, 1, 1, 0, 4], [-2, 2, 0, 0, 3], [-2, 4, 2, 0, 1], [0, 0, -1, 3, 0], [1, -3, 0, 1, 0], [0, 0, -2, 0, -4], [3, -1, 0, -2, 0], [1, -1, -2, -1, -2], [-4, 2, -2, -1, -3], [1, -1, 2, 2, -2], [0, 0, -2, -2, 1], [0, -2, -1, 0, -4], [-2, 2, -1, 3, 0], [-1, -1, 0, -1, 2], [6, -4, -2, -2, -2], [-6, 4, -1, -1, 0], [1, 0, 0, 0, 0], [-2, -4, -2, -2, 0], [-2, 2, 4, -1, 0], [-6, -4, 2, -1, 2], [6, 0, 0, -3, -1], [-10, 0, -2, 1, 4], [-2, 2, 2, -1, 0], [2, 2, 2, 4, -2], [0, -2, 0, 2, 4], [10, 0, 2, 0, 5], [1, -1, -3, -1, -4], [-10, 0, 0, 2, 2], [0, 0, 2, 1, 1], [6, 0, 2, 0, 4], [-8, -2, 4, -1, 2], [-4, 4, 2, 1, 0], [3, -3, -2, 2, 0], [4, 2, 0, 0, -7], [-6, 6, 3, 2, -2], [9, 1, 4, 0, -4], [1, 1, 3, -2, -4], [-8, -4, 2, -1, 5], [0, -2, 2, -2, 0], [-8, -4, 0, -2, 4], [0, -2, 3, -1, 4], [0, 0, -2, -1, 1], [-10, 0, -2, 1, 2], [-8, 2, -4, -3, 2], [0, -2, 4, 3, -2], [12, -4, 0, 1, -3], [-13, 1, 1, 3, -2], [-19, -3, 1, 1, 0], [4, -4, 4, 7, -1], [-2, -4, -2, 4, 0], [8, -4, -2, -2, -6], [-5, 7, -1, -2, -2], [0, -2, -3, 3, 2], [-8, -4, 2, 0, -2], [-8, -4, -8, -1, -2], [8, 6, 2, 3, -4], [5, -5, 0, 3, 0], [2, 0, -4, 0, 0], [0, 0, 4, -4, -4], [-4, -6, -6, -1, 4], [-12, 0, -8, -1, 3], [-16, -6, -2, 1, 1], [-1, -1, 4, 3, 6], [-11, -1, 3, 5, 2], [-4, 4, 0, 8, -1], [-2, 4, 0, -2, 0], [-4, -2, -2, -3, -4], [-3, 1, -2, 3, 2], [-20, -4, -4, 2, -2], [10, 2, -2, -2, -2], [-5, 3, -2, 2, 0], [-6, -6, -4, 0, -4], [-4, 2, -4, 5, -2], [-8, -4, 6, -4, -2], [-10, 4, 2, 0, 4], [-3, 3, -2, 0, 4], [-18, 4, 2, 3, 0], [1, 1, 5, -4, 0], [0, -2, -9, -6, -2], [2, -2, 2, -5, 2], [4, -4, 0, 5, 2], [-13, 1, 4, -2, 4], [-6, -8, -3, -3, 2], [6, 2, 4, 2, -3], [9, 3, -3, 2, -10], [23, 1, 2, 0, 4], [12, 6, -6, 0, -5], [-18, -4, -4, 0, -1], [-27, 1, 0, -3, -4], [0, 2, 10, 7, 6], [8, -2, -2, 2, -6], [-6, 4, -2, 1, 8], [4, -6, -10, 3, 3], [-3, 3, -2, -1, 8], [-18, 4, -4, -1, 6], [-4, 4, 10, 2, 2], [-14, -4, 8, 4, 2], [3, -3, 5, -2, -4], [-15, 5, 0, -2, 8], [6, -4, 4, 1, -2], [26, -2, -2, -1, -2], [17, -5, -4, 2, 0], [2, -4, 0, -5, -6], [2, 4, 4, -2, 2], [-7, -3, -3, 2, -2], [12, 0, 0, 8, 4], [-1, -1, 0, -4, 2], [16, -2, 2, -3, -4], [-4, 2, 3, -1, -2], [0, -6, 8, 2, 3], [-9, 7, 2, 2, 0], [8, -6, -6, -3, 0], [14, 10, -4, -1, -1], [-18, -4, -12, -3, 4], [8, 4, -5, 1, -6], [-16, 2, 0, 5, 2], [-4, 4, 6, 1, 0], [8, 4, -2, 2, -7], [0, -2, 11, 0, -2], [2, -2, 6, 5, 0], [-3, -11, 2, -4, 2], [9, 3, 2, -2, -10], [-9, -1, 7, -6, -2], [26, 4, 0, 1, -1], [-18, -4, -12, 2, 2], [6, 6, 0, 9, -1], [-10, -4, 0, -5, 6], [-6, 12, 4, -2, 2], [22, -8, 0, -1, 6], [9, 3, 0, 1, -8], [-14, 0, -2, -1, 8], [-18, 6, -6, -6, -8], [-2, 0, -12, 3, 2], [0, 0, -4, -5, 5], [-2, 0, -2, -2, -11], [-14, 2, 6, -7, -7], [20, -6, -2, 1, -6], [-4, 4, -2, -7, -4], [-2, 2, -8, 10, 6], [9, 3, 0, -2, -6], [-6, -2, 4, -2, -4], [-33, -3, -1, -2, -2], [9, 3, 14, 0, 6], [2, 10, 6, -2, -6], [-5, 3, 0, 0, -6], [2, 0, -4, -1, -4], [6, 2, 10, 0, -3], [22, -4, -4, -2, -4], [21, 3, 3, -6, 6], [0, 2, 10, -8, -6], [4, 2, -10, 2, 5], [8, -6, 0, 1, 9], [-11, 9, 6, -5, -2], [-20, 2, 6, -2, -2], [-20, 0, -2, 2, -6], [20, -8, 3, 4, 12], [2, -10, -8, -6, 0], [9, 5, 5, -2, -8], [-10, 8, -4, -5, 5], [-16, 4, 2, 0, 4], [21, -7, -1, -2, -2], [25, -1, 6, 2, 2], [26, -2, -2, 1, -4], [-10, -2, -4, 3, 4], [-8, -2, -6, -7, -8], [-14, 6, 6, 7, 4], [-20, 10, -2, -5, 5], [-2, -10, -4, -3, -4], [40, -2, 2, 2, 8], [-3, 3, -13, -10, -2], [-12, -8, -6, -6, -4], [-8, 2, -8, 7, 8], [24, 0, 8, -1, 12], [-8, -4, 6, -10, -5], [-2, 10, 2, 0, -3], [-12, 14, 4, 6, 4], [-2, 2, -2, 10, 1], [-2, 8, 6, 1, -1], [-16, -8, 6, -5, -4], [22, 12, -6, 5, 3], [23, 3, 3, -6, 0], [20, -6, -14, -4, -8], [3, -3, 0, 0, 8], [2, -14, -10, 3, 2], [-1, -1, 2, 10, -4], [4, -4, 0, 3, 6], [7, -7, 0, 4, -8], [4, 12, 2, -4, -2], [-16, 6, 8, 7, 11], [-8, 2, 14, 3, 2], [13, -15, -4, 0, -4], [7, -7, -2, -2, -2], [30, 0, 4, 4, 2], [-28, -6, 8, -3, -1], [-13, 11, 0, 6, 12], [12, 6, -2, 2, -2], [-12, 0, -4, 9, 1], [-4, 4, -5, 4, 2], [13, 1, 2, 2, -2], [-24, -2, 4, -3, -7], [17, 7, 4, 3, -8], [10, 6, 16, 0, 0], [-5, 5, -2, -8, -2], [20, 6, 0, 2, 0], [16, 8, 16, 0, 1], [2, 10, 4, 2, 0], [1, 9, 16, 0, 4], [-4, 4, -6, 2, 4], [9, 5, 1, 0, -6], [-8, -6, 2, -1, 0], [3, -5, 12, 6, 12], [-3, -9, 2, -10, -16], [-12, -8, -12, 8, 2], [-22, -2, 6, 3, 6], [-2, 4, 2, 16, 0], [22, 2, 4, 5, -10], [-1, 1, 1, -6, 2], [2, 16, 4, 2, -6], [19, 5, 6, -4, -6], [-16, 8, 2, 0, -2], [-13, -13, -8, -1, -6], [7, -5, 3, 0, -2], [4, 2, 8, 10, 12], [-20, 8, 6, 2, 18], [30, 4, 6, 2, 5], [11, 1, 5, 3, 4], [-20, 0, 0, -4, -2], [-8, -16, 4, 2, 2], [20, -6, -2, 8, 10], [-4, 10, 10, 4, 6], [24, 0, 2, 4, 8], [14, 0, 14, 7, 2], [-6, 6, 2, -11, -8], [20, 4, -4, -3, -2], [-7, 9, 5, -1, 14], [14, -10, 2, 1, -10], [12, 12, 3, -5, 10], [-4, 0, 10, 1, 7], [29, -7, 6, 6, 0], [-29, -7, -1, 2, 4], [-9, 11, -1, 1, 10], [1, 1, 7, 3, -4], [8, -2, 4, 6, -4], [-18, 10, 12, 1, -2], [-20, 8, -10, -9, 4], [-3, -9, -4, -5, 0], [-30, 4, -7, -10, 6], [20, 4, 10, -4, 0], [25, 1, 2, -8, 2], [-4, 2, 2, 2, -6], [-10, -12, -12, 5, 4], [-4, -10, 6, -10, -13], [26, 6, -8, 0, -6], [19, 3, 6, 4, 12], [25, -1, -5, 6, -2], [11, 3, -7, 11, 6], [-28, -6, -4, 1, 6], [-32, -4, -10, -2, -6], [16, -4, -2, 6, 6], [15, -5, 2, -9, 0], [-4, 4, 12, 5, 10], [8, 6, 2, 2, 2], [4, -6, -11, -5, -2], [26, -2, -4, -3, 4], [24, 2, 8, -1, -15], [28, -16, -10, -1, 4], [-12, -14, 2, 5, -3], [21, -9, 0, 4, 0], [-10, -8, -4, -10, 0], [17, -17, -3, 5, -2], [-18, 6, 8, -4, 7], [24, -12, -4, -3, 2], [46, -8, -2, 12, 14], [-20, 8, 2, 2, 2], [-14, 0, -4, 4, 19], [4, 8, 3, 1, -6], [-5, 5, -15, -8, -4], [8, -10, 10, 8, 4], [24, 2, -14, -2, -14], [0, 0, 21, 10, 10], [-12, 10, -4, 4, 9], [15, 9, -6, 14, 6], [-20, 0, -4, -4, 16], [-10, 0, 12, 10, 4], [-23, 23, 11, 4, 4], [-2, -8, 8, -5, -12], [14, -14, -4, -16, 2], [12, 2, -8, -1, 4], [-38, -10, -6, -4, 12], [-16, -2, -6, 3, -18], [-2, 8, 14, 6, 10], [24, -2, 2, 7, 8], [16, -4, -2, 3, 6], [-10, 6, 8, 4, 10], [-13, -11, 3, -7, -6], [8, 10, 6, 0, -1], [-4, 6, 12, -6, 4], [11, -11, -8, 3, 0], [32, 2, 0, 6, 14], [-10, 12, -4, 4, 1], [8, 16, -4, -6, 6], [5, 5, 8, 5, 10], [7, -7, -6, -1, -2], [-18, 8, 16, 2, 9], [4, 6, -8, 0, -20], [-14, 6, 12, 7, 6], [-18, -6, -16, 6, -3], [-2, 2, 4, 5, -10], [-24, -10, -8, 1, -6], [-30, -6, -10, 3, 0], [-13, -1, 0, -2, -10], [18, 6, 0, -3, 4], [-10, 20, 0, -3, -10], [8, -8, -14, -7, -8], [-20, -2, -2, -12, -16], [23, 1, 0, -9, 8], [-34, 0, 2, 11, 2], [18, 4, 6, 1, 19], [-32, 10, -2, -5, -6], [6, 0, -14, 0, -2], [20, -8, -5, 3, -8], [15, 7, -4, -10, 6], [-16, -2, 6, -3, 4], [18, 6, -2, 6, -4], [30, 8, 6, 2, -10], [20, 2, 16, 6, -2], [0, 0, 6, 7, -8], [-12, 12, 12, 6, 14], [-6, -2, -8, 11, -2], [-10, -12, -8, -12, 6], [2, 10, -5, 2, -6], [1, 9, -2, -9, 10], [13, 1, -3, 6, -8], [-3, 3, 4, -1, 18], [-4, 16, 6, 3, -5], [-55, -5, 3, -7, -4], [-5, 5, -8, -6, 2], [-10, 4, -6, -11, 5], [-28, -6, -8, 15, -10], [9, 15, -10, -1, 2], [0, -12, -2, -6, 8], [-8, -4, 2, -8, 0], [8, -22, -3, -5, 0], [-38, 4, -4, 0, 12], [22, 2, 11, -4, 2], [-19, -7, 0, -5, -6], [0, 8, 2, -3, 5], [-43, -5, -5, -4, 2], [-14, -12, -4, -15, 2], [38, -8, 0, 10, 18], [-22, 4, -8, -2, 2], [-15, 13, 10, 7, 18], [19, 7, 5, 4, 12], [-21, 9, 2, -8, -10], [2, -2, -14, 1, 8], [0, -10, 4, 1, 17], [19, 5, 16, 1, 2], [2, -18, -4, -8, 2], [-12, 0, 24, 4, 18], [-4, 4, -8, -5, -11], [-1, -1, 2, 1, -4], [-28, 16, -6, -11, -7], [-26, 6, -4, -7, -4], [-26, -10, 6, -9, 8], [-1, -1, 6, 17, -4], [11, 3, 17, -3, 12], [36, -8, -2, -4, -14], [-42, -6, -10, -3, 14], [-34, 12, -4, -3, 12], [0, 12, -12, -8, 6], [12, -24, -14, 1, 0], [32, 2, 7, 3, -4], [-22, -2, -4, 8, -6], [-32, -14, 2, -3, 12], [-8, 6, 6, -2, -7], [-36, 0, -8, -10, 3], [-58, 8, -3, -5, -12], [32, 4, 19, 2, 8], [14, -12, -14, -4, 4], [0, -6, 14, -4, -10], [-19, 17, 10, -8, -2], [-5, -7, -14, -8, -28], [-20, -4, 9, -1, 16], [-43, 5, -6, 6, -4], [-19, 9, -1, 7, 20], [9, 15, 0, -3, 16], [26, 8, 7, -3, 12], [-8, -16, 6, -9, -20], [-38, -8, 6, -12, -4], [36, -12, 0, 11, 18], [44, -8, -16, -5, -8], [-26, -6, 14, -1, 8], [-38, -8, -6, 0, -18], [52, -4, 2, -8, -2], [-2, -22, -10, 4, -12], [8, -10, 10, 13, 1], [58, 2, 6, -3, -7], [-2, 2, -6, -12, 3], [-4, -8, 10, 1, 16], [-11, 13, -3, -3, 12], [-70, -2, 0, -2, 3], [-52, 4, -2, 3, 2], [-16, -6, -2, -15, -3], [42, -6, 2, -1, -12], [-14, 2, 2, -10, -18], [-38, -10, 12, -8, 3], [0, -14, -2, 0, -4], [-18, -6, -6, 7, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8016_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_8016_2_a_r(:prec:=5)
    chi := MakeCharacter_8016_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_8016_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_8016_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![24, 60, -21, -19, 1, 1]>,<7,R![-72, -132, -49, 15, 9, 1]>,<11,R![-288, 144, 60, -28, -2, 1]>,<13,R![-8, -48, 52, -2, -6, 1]>],Snew);
    return Vf;
end function;