
// Make newform 4020.2.a.e in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4020_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_4020_2_a_e();".
// 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_4020_2_a_e();".
// 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, 17, 9, -9, -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, 0, 1, 0, 0], [11, 0, -8, 0, 1], [16, -6, -9, 1, 1]];
    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_4020_a();"
function MakeCharacter_4020_a()
    N := 4020;
    order := 1;
    char_gens := [2011, 2681, 3217, 1141];
    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_4020_a_Hecke(Kf)
    return MakeCharacter_4020_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], [1, 0, 0, 0, 0], [-1, 1, 1, -1, 0], [1, -1, 0, 1, -1], [-2, 0, 1, 0, 0], [-2, 0, -1, 1, 0], [1, 0, -2, 0, 1], [1, 0, 0, 0, 2], [-2, 1, -1, -3, 0], [0, -2, -2, 0, -1], [-1, -4, 0, 2, 0], [-4, 2, 0, -2, -1], [-2, 1, -3, 2, -1], [-1, -1, 1, 2, 1], [-4, -1, 0, -2, 2], [3, -1, 3, 2, 1], [0, 1, 0, 2, 0], [-1, 0, 0, 0, 0], [2, -2, -1, 2, -1], [-4, 2, 1, 1, 0], [0, -5, 1, -2, -1], [-4, 6, 3, -4, -1], [-5, -1, -1, 2, 0], [-3, -2, 2, -1, -4], [-2, 5, 4, -4, -1], [-8, -1, -1, -2, -3], [2, 4, 0, 1, -1], [-4, -2, 1, -2, 1], [-7, 1, -1, -5, -2], [-2, 7, 2, -7, 2], [5, -1, 1, -1, 1], [-5, 2, -2, 1, 0], [-3, 3, -1, 3, 0], [5, 0, 0, 8, 1], [9, 0, -1, 4, 3], [-6, 1, 1, 1, 2], [-2, 5, 2, -3, 2], [3, -7, -6, 5, 2], [1, 4, 0, 4, 3], [2, -5, -1, 0, 0], [-8, 5, 5, -1, 2], [5, -3, 1, 3, 1], [-12, 1, -2, 3, 1], [-9, -1, 2, -3, -1], [-4, 2, 1, 3, 1], [-2, 2, -3, -2, 1], [-9, -7, 2, 2, -2], [-4, 0, -3, -7, -3], [-11, 0, -4, -1, 0], [-4, -7, -3, 8, -1], [-2, 7, 6, -4, -2], [-1, 7, 4, 0, -2], [1, -1, 6, -1, -2], [3, -10, -4, 10, -2], [0, 8, 6, -4, -1], [-14, -2, -2, -8, 1], [0, 1, 2, -2, 3], [-3, -3, -2, 5, -2], [6, -3, 7, 0, 2], [-2, -6, 4, 1, 3], [-8, 2, 0, -10, -1], [-1, -4, -1, 0, 0], [-2, 3, -9, 2, -1], [-4, -5, -6, 8, -2], [-6, 3, 3, -6, 5], [-4, -7, -2, 0, -7], [-9, 8, 3, 1, -1], [20, 5, 0, -2, 1], [-14, 4, -1, -8, 3], [-1, -5, -4, -1, -4], [-5, 0, -3, -6, -5], [4, 2, 7, 2, 5], [-9, -1, 2, 1, -2], [9, 3, 7, -5, -1], [0, 10, -2, -8, 1], [7, -5, 6, -1, 1], [3, 6, -1, 8, 4], [16, 1, 1, 0, 5], [6, 1, -2, 2, 7], [7, -7, -5, 4, 0], [0, 1, -4, -5, -4], [10, 3, -4, -1, 3], [-14, 1, -3, -4, 4], [-8, -4, 7, -1, -5], [-8, -3, 7, 0, -4], [-6, -3, 8, -3, 2], [6, -11, -8, 5, -6], [-4, 12, 0, -5, 2], [-8, 7, -4, -8, -2], [5, -5, -7, -1, 0], [-6, 8, 4, -3, 10], [6, 1, -9, 4, 0], [16, -2, 4, 3, 3], [5, 0, 5, 13, 3], [-11, 0, 3, 5, 0], [5, -4, -7, -4, -3], [5, 4, -3, 7, 2], [-6, 3, -13, -1, -1], [-5, -3, -7, 11, 3], [-5, 13, 4, -11, 1], [6, 5, 0, -2, -5], [-4, 1, 1, 0, -6], [-20, 10, 6, -5, -5], [-12, 13, 3, -5, -5], [7, 5, 2, 3, -5], [-1, 7, 10, -7, -3], [-12, -10, 5, -2, 0], [-4, 2, 3, 4, 2], [8, 7, 7, -5, 5], [-8, -4, 3, 0, -2], [12, -1, 1, -3, 3], [-2, 0, -6, 5, -2], [11, 4, 5, 3, 2], [5, -8, 6, -1, -5], [15, 4, 9, -11, -2], [-8, -14, -3, 8, 1], [16, -4, 4, 0, 0], [14, -2, 2, 8, -7], [6, -6, 4, 5, -5], [3, -11, -7, 3, 6], [3, -14, 3, 7, 3], [7, -5, 2, 5, -9], [17, 6, 1, 7, 9], [8, 9, 4, -3, 3], [15, -5, -5, 3, -6], [19, -2, 4, 13, -5], [0, 7, 4, -1, 11], [-19, -1, -4, 1, -5], [-22, 9, -1, -2, 0], [11, 1, 8, -11, -5], [3, -1, 4, -7, -9], [3, -7, 2, 8, 4], [13, 4, -12, 1, -2], [17, -4, -2, 2, -2], [-5, -2, 1, -1, -9], [14, 1, 4, 5, -3], [-10, 5, 3, -6, -4], [29, -2, 4, 7, 9], [-10, -1, -9, -4, -6], [10, -4, 6, -4, -4], [13, -3, -8, -8, -3], [10, 8, 7, -9, 1], [11, 8, -7, 6, -6], [8, -6, 2, 3, -4], [-10, 1, -10, -7, 2], [-23, 7, -2, -12, -1], [-22, 3, 2, -2, -6], [12, -14, -4, -4, -6], [8, -16, -8, 3, 4], [-6, -8, -12, 3, 6], [9, -13, -5, 8, 1], [8, -4, -4, 10, 7], [10, -4, 1, -3, -4], [-3, -13, -10, 6, -3], [19, 4, -2, 0, 4], [-7, 14, 4, -7, -4], [-18, 4, -4, 0, 9], [8, 5, -2, -4, -4], [3, -1, -5, -15, -3], [19, 10, -9, -2, 3], [12, 0, 6, 1, -8], [11, 3, 13, -7, 6], [14, -6, -1, 15, -4], [11, -16, 7, 6, 3], [-8, -6, 5, 0, -4], [4, -5, -3, 8, 3], [-3, -3, -9, 5, 5], [8, -6, 11, 5, 3], [17, -10, -1, -3, -10], [3, -2, -5, 2, 6], [-5, 10, 16, -6, 10], [10, 14, -2, -7, -1], [-19, -1, 9, 1, 3], [-13, -13, 0, 3, -5], [-8, 5, 5, -4, 1], [4, -2, 9, 4, -7], [28, 6, -1, 9, 10], [-18, 16, 7, -10, 0], [-5, 4, -14, 4, 3], [-13, 16, -2, -5, 1], [11, -6, 6, 15, -2], [23, -6, -5, 14, 8], [-20, 8, 9, -2, 7], [2, 0, -5, 20, 5], [6, -9, 1, 2, 4], [6, -11, -3, -6, -9], [-22, -9, -8, 2, 2], [1, 17, -5, -9, 2], [11, -17, -3, 9, -8], [-21, 15, 2, -17, 1], [-2, -1, -7, 1, -10], [-19, -13, 5, -1, 0], [-28, 11, 8, -4, -1], [11, -8, -14, -1, 5], [-9, 21, 7, -9, -5], [-6, 12, 2, -15, 10], [-4, 1, 9, -16, -1], [-17, -1, -11, 9, 3], [6, -7, -12, 0, 3], [-12, -9, 2, -16, 1], [-34, -1, 0, 1, -6], [10, 2, -9, 8, 15], [-19, -11, 8, 1, 1], [-4, -6, -9, 8, -3], [-1, 11, 8, -18, -1], [-8, 6, -14, 0, 0], [-7, -8, 9, 2, 6], [-20, 8, 2, -5, 2], [1, -11, -6, -5, 3], [4, -11, 15, -4, 3], [17, -6, 12, 3, -1], [-14, -14, -8, 10, -3], [2, 0, 21, -6, 8], [-14, 3, -14, 18, 5], [11, -7, -4, 9, 7], [5, -14, 1, 12, -1], [-21, 18, 5, -11, -2], [13, -2, -13, 9, -1], [15, -14, -9, 16, -5], [-31, 22, 5, -21, -2], [10, -12, 1, 19, -1], [22, 6, 2, 2, 2], [-28, 0, -8, -15, 8], [13, -8, -4, 9, 0], [7, -2, -3, -12, 0], [-8, 6, -2, -10, -1], [-16, 13, -10, -10, -7], [-8, -13, -7, -11, -3], [-7, 11, -5, 7, 4], [21, -2, 6, 11, -9], [-20, 23, -1, -8, -8], [-2, 20, 0, 1, 2], [11, -2, 2, 1, -11], [-14, 6, -2, -9, -10], [-4, 1, -1, -14, -8], [3, 8, 16, -10, -7], [18, -13, 0, -10, -12], [13, -16, -5, 0, -1], [-8, -3, -16, -6, 1], [-13, 7, -3, -12, 10], [27, 8, -7, 5, 9], [20, -3, 5, 8, -1], [11, -4, -6, 7, -10], [1, -22, -9, 13, 0], [2, -7, -2, 12, 11], [17, 17, 4, -12, 1], [-15, 15, 10, -13, 11], [28, -6, 11, 14, 7], [-14, 9, -13, -12, -2], [2, -2, 1, -11, -11], [2, -17, -7, 10, -14], [22, -13, 0, 2, -5], [9, 2, -11, 8, 5], [-5, -9, -5, 13, -7], [-29, -6, 0, -11, 0], [10, 0, -8, -8, 0], [27, -6, -4, 3, 3], [-19, 17, 8, 5, 8], [-3, 3, 3, -10, -7], [-18, 16, 8, -3, 1], [40, -8, -11, 0, -3], [13, -6, 20, -11, 7], [-13, -20, 6, 19, -5], [-17, 3, -6, 3, 2], [0, 19, -8, -7, -2], [8, 11, -12, 4, 10], [-7, -24, -3, 5, -1], [4, -8, 7, 2, -8], [19, -8, 1, 3, 8], [7, -1, 11, -11, -8], [-1, -3, 12, -11, -14], [13, 2, 18, -15, -5], [-18, 2, 13, -6, -10], [-16, 13, 20, -5, 11], [2, 2, 8, 2, 11], [25, 16, -1, -5, 8], [-5, 15, 20, -9, -4], [-6, -6, -15, -8, 2], [29, 8, -2, -1, 13], [8, 0, 10, 11, 6], [-3, 13, 14, -5, -4], [0, 12, 2, 0, 7], [-18, -2, 9, -6, 3], [-1, -15, -3, 3, -21], [-7, 10, -6, 9, -8], [0, -5, -17, 11, -11], [-33, 9, -20, -8, -9], [-14, 4, -2, -12, 13], [8, -7, -3, 2, 7], [-10, 13, 4, 1, 6], [-1, -14, 9, -13, -3], [23, 16, -4, -7, 14], [0, -13, 9, 2, -9], [5, 8, 5, 1, 9], [12, -20, -18, 18, -1], [19, 15, 8, -19, 3], [5, -2, -4, -7, 4], [-17, 5, 2, -16, 5], [15, -13, -9, 13, -9], [-11, 4, -15, -11, 2], [-8, 9, 8, -7, 10], [-6, 6, 16, -2, 1], [40, 5, 7, -2, -5], [-27, 22, 4, -18, 0], [8, 13, 18, 2, -4], [15, -20, -1, 24, 2], [51, -3, -2, -2, 5], [-25, -1, 5, -4, 0], [-24, -10, 1, 2, -9], [3, 7, 7, 1, 2], [10, -3, 7, 23, -2], [31, -27, -19, 8, 0], [0, -22, 6, 23, -4], [3, -10, -12, 22, 4], [3, 10, 6, 11, -7], [-29, 2, -9, 6, -7], [-18, -16, 4, -2, -3], [-3, 18, 0, -10, 5], [-37, 8, -15, 3, -5], [-5, -12, -11, 2, -4], [13, -13, -15, 10, 12], [-17, -15, 7, -12, 1], [10, -1, 0, 24, 7], [3, -22, 14, -2, 3], [3, -1, -1, -5, 13], [-48, 15, 15, 1, 5], [-36, -2, -11, -12, 11], [41, -8, 18, 3, 13], [53, 9, -1, 5, 5], [9, -7, -11, 10, -10], [14, -6, 24, 1, -4], [-34, -3, -2, -20, 1], [-26, -2, -1, 2, -2], [36, 9, 17, 2, -11], [-13, 15, -23, 10, -7], [-3, 8, -9, -1, -3], [0, 26, 21, -25, -6], [-20, -1, 4, -2, -8], [-13, -1, 9, -12, 10], [-17, 10, -2, -15, -4], [-25, 7, -2, 7, -6], [-1, 21, -3, 5, -9], [-24, 14, -6, 3, 1], [21, -13, -8, 12, 4], [28, -10, 0, 26, -6], [10, -10, -11, 14, -4], [-18, 17, -6, -4, -6], [25, -13, 9, 19, -4], [-17, 6, 17, -5, 2], [63, 2, 7, -5, 4], [32, 5, -9, 8, 8], [15, -9, -3, 14, 8], [-34, 9, 0, -8, -5], [-17, -5, -15, 1, -21], [12, 8, 5, -8, 3], [-44, -16, 8, 7, -5], [2, 16, 9, -2, 3], [-18, -8, -1, 8, -14], [-2, 13, 22, -4, -4], [-17, 5, -6, -9, 21], [-24, -7, 1, 0, -17], [-21, 12, -19, -2, -3], [-13, -13, 0, 5, 3], [47, -9, 2, 22, -7], [32, -3, -5, -2, -5], [4, -5, 9, 0, 9], [-4, -8, -11, 3, 10], [23, -5, -5, 17, 6], [-27, 5, -8, -11, 19], [37, -17, -26, 3, -4], [14, -1, 11, 7, -12], [-39, -2, -20, 7, 2], [-44, 6, -9, -12, -3], [25, 1, -11, -9, 3], [-30, 26, 11, -29, 11], [5, 1, 5, 9, -12], [19, -13, 25, -5, 5], [-53, -5, -4, 13, -1], [-1, 6, -4, 28, 3], [21, -26, -7, 27, -6], [7, -14, -9, -7, 5], [-31, -1, -13, 7, 14], [-51, 1, -1, -7, -2], [6, 15, 15, -22, 8], [10, -27, -6, 10, 12], [30, -11, -6, 28, -4], [-1, -2, 8, 18, 11], [-40, 6, -13, 12, -5], [-38, 3, 3, -20, -9], [0, -1, 13, -9, 6], [4, -14, -18, 0, 5], [22, -36, -7, 26, -1], [3, -6, -1, -11, -12], [-56, 14, 6, -10, -1], [21, 7, 8, 6, 12], [5, -11, -4, 31, 5], [-16, -5, 9, 21, 1], [-3, 18, -5, -29, 1], [4, 31, 9, -12, -9], [38, 12, -6, -4, 8], [-7, 16, 12, -17, 0], [-13, -12, -22, -1, 14], [17, 10, -12, 7, 18], [-40, -11, -8, 6, -7], [-30, -3, 10, -12, -6], [5, -27, 4, 29, 0], [1, -11, 7, 0, -7], [-15, 17, 0, -2, 8], [-19, 5, -20, -1, -2], [20, 7, 12, -11, 4], [-25, 6, -1, 12, -5], [4, 16, -22, 5, 10], [51, -30, -11, 22, 5], [-16, -9, 4, -4, 5], [23, -4, -5, -4, 6], [-18, -24, 15, 15, -10], [-21, 5, 15, -8, 0], [-23, -3, 11, 0, 5], [-18, -3, -14, 21, -6], [-19, 9, -1, 3, 4], [-42, 3, -9, 19, 3], [-16, -30, -7, 12, 3], [-14, 11, 9, -2, -7], [8, -16, -5, 22, -10], [16, 1, 18, 3, 11], [21, 22, 0, -4, 0], [-9, -16, 19, 14, 0], [-25, 0, 10, -9, -10], [-27, 15, 9, -13, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4020_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4020_2_a_e();".
// 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_4020_2_a_e(:prec:=5)
    chi := MakeCharacter_4020_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_4020_2_a_e();".
// 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_4020_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4020_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 3, -16, -12, 3, 1]>],Snew);
    return Vf;
end function;