
// Make newform 3640.2.a.y in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3640_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_3640_2_a_y();".
// 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_3640_2_a_y();".
// 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 := [4, 10, -3, -8, 0, 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, -1, 1, 0, 0], [1, -5, -1, 1, 0], [7, 3, -7, -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_3640_a();"
function MakeCharacter_3640_a()
    N := 3640;
    order := 1;
    char_gens := [911, 1821, 1457, 521, 561];
    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_3640_a_Hecke(Kf)
    return MakeCharacter_3640_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], [-1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [-1, 0, 0, 1, 0], [-1, 0, 0, 0, 0], [0, 0, -1, 0, 1], [0, 1, 1, -1, 0], [-1, 0, -2, -1, 0], [-1, -1, -2, -1, -2], [2, 1, 0, 0, -2], [-2, 0, 0, -1, -1], [-1, 1, 1, 2, 0], [0, 3, 2, 1, 3], [-3, 3, 0, 0, 1], [-4, -2, 0, 0, 2], [-1, 0, 2, 0, 1], [-3, -1, 2, 0, 1], [-5, -3, -1, 0, -2], [-2, -1, -2, -1, -1], [5, 3, 0, 2, 1], [-1, 5, 3, 0, 2], [0, 4, 2, 2, 0], [-5, 5, 2, 1, 0], [-1, -1, 2, 0, -1], [-3, -3, -4, 0, -1], [4, 1, -3, -1, -4], [-2, 1, 0, -1, -3], [-2, 4, 4, 0, 0], [-3, -3, 2, 0, 1], [-5, -6, 2, 1, 2], [2, -2, 2, -2, -2], [-5, -3, 2, -3, 0], [0, -2, -6, -2, -4], [-7, 1, 2, 2, -1], [0, 3, -2, 1, 3], [-2, -6, -4, -3, -3], [-5, 1, 0, 1, 4], [-6, 2, 0, 2, 4], [-5, -3, -2, -3, 0], [-2, -4, -1, 4, 1], [-11, -1, 2, 2, 3], [-1, 3, 0, -1, 2], [-8, 4, 1, 0, -1], [-8, 0, 0, 3, 1], [12, -4, -4, -2, -2], [4, -6, 1, 0, -3], [-3, -5, 0, -2, -1], [-4, -6, 2, 0, 2], [-8, 4, -1, 0, 1], [-1, 3, 4, 0, -3], [-4, -5, 0, -3, -3], [4, -8, -3, -4, -1], [11, 3, 2, 2, -1], [4, -2, 4, 4, 2], [-2, 3, 0, -5, 1], [-9, 3, 4, 0, 1], [7, 8, 0, 1, 0], [2, 0, -6, -2, -8], [-2, -8, 2, 0, -2], [5, -8, -3, 1, 3], [-16, 4, -2, -2, -2], [2, 2, -6, -2, -2], [4, -2, 2, -2, 4], [-3, -5, 2, -1, 0], [-1, -3, 2, 4, 3], [15, 2, -4, -1, 2], [-5, -5, -2, 2, -3], [-10, 1, 8, 1, 7], [-17, 1, -6, -3, 0], [7, -3, 2, 4, -1], [-10, -1, 0, 1, -5], [12, -1, 6, 3, -3], [-8, -2, -4, 2, 4], [-2, -3, 0, 1, 3], [-3, -5, 0, -6, 3], [-12, 8, 9, 0, 3], [-2, 2, -6, 2, 4], [0, 6, -2, 0, -4], [10, 0, -4, 0, -4], [5, -3, 0, 0, 5], [7, 3, -2, -2, -3], [8, 1, 4, -3, -3], [-4, 8, 3, 0, 1], [14, -6, -8, 0, -2], [12, -3, -8, -1, -5], [-6, 8, 0, 6, 6], [11, 5, 2, -3, 8], [-1, -7, -6, -5, -2], [2, 6, 1, 6, -3], [-14, -3, -5, -3, -2], [13, -2, -2, 2, -3], [3, -3, 4, 5, 0], [0, -6, 0, 6, 2], [5, 6, 8, 5, 2], [4, 9, 2, 5, 3], [-2, 10, -1, -4, -5], [8, -4, 2, 4, 0], [7, -10, -4, 3, 2], [14, 0, 6, 4, -2], [-6, 1, 8, 1, 3], [-23, 5, 7, 0, 2], [5, 0, 3, 1, 1], [0, -10, 0, -5, -3], [1, -5, -4, -5, 0], [4, -8, -6, 4, 0], [6, -12, -2, -6, -6], [16, -2, -6, -2, 2], [1, -15, -2, 0, 3], [6, -8, 0, 6, 2], [-3, -4, 2, 4, -1], [-23, -3, -2, 2, -5], [-2, 2, -5, -4, 3], [19, 0, -2, 0, 1], [2, -15, -4, 7, 1], [6, 6, 4, 1, 1], [4, 12, 2, 2, -4], [-13, -12, 2, -2, -1], [6, 8, 6, 4, -2], [-14, 2, 1, 2, 5], [-10, 10, 9, 4, 5], [3, -5, 0, 0, -3], [9, 9, 6, 6, 7], [-9, -3, -1, 0, -6], [1, -10, 2, -2, -7], [-6, -2, -1, -4, -5], [1, 1, 0, 4, -1], [-8, 0, -6, 4, 6], [-7, -2, -2, 8, 3], [-12, -10, -4, 4, -6], [0, -7, 4, -3, 5], [-12, 2, 1, 0, 1], [-4, 8, -2, -3, -1], [-8, 2, -4, 0, 2], [14, -10, 0, -1, 3], [-3, 9, 10, 6, 7], [-2, 14, 0, -2, 6], [-2, 6, 4, 4, 2], [4, 0, 8, 1, 3], [-8, -6, -5, 2, 3], [4, 13, 10, 5, 7], [14, 0, -2, -4, -2], [5, -14, -4, -7, -10], [-40, 8, 4, 0, 0], [4, 12, 10, 4, 8], [3, -2, 4, 3, -6], [-6, -10, -2, -4, -2], [-9, 1, -10, -1, 6], [-1, 3, 4, 0, -5], [-12, 10, 11, 0, 3], [23, 1, -11, 2, -4], [4, 0, -4, -4, 6], [2, 9, 2, -5, -1], [-1, -2, -6, 0, -3], [0, -3, 6, -3, 3], [2, -8, 3, 4, 5], [24, -4, -10, -1, -3], [6, 10, 8, -5, 7], [-10, 6, -1, 0, -9], [16, -12, -9, -4, 5], [-6, 0, -1, 0, 9], [20, 2, -6, 2, -2], [-7, -1, -4, -9, 0], [3, 1, 2, -2, -5], [0, -6, -3, 2, -7], [2, 4, 6, -4, 4], [-17, -1, 3, -2, 0], [5, 7, 5, -2, 6], [-2, -12, 4, -2, -2], [0, 4, 13, 0, 7], [6, -7, -6, 5, -7], [-9, -15, -6, -3, -8], [19, 11, 2, 8, 1], [18, 14, 4, 10, 6], [8, -8, 2, 2, -4], [-14, -12, -2, 4, 2], [15, 0, -8, -5, 2], [-10, 8, -2, 0, -6], [27, -2, -6, 0, -7], [-9, 7, 0, -2, -5], [-11, 5, 4, 1, 6], [-23, -7, 0, 6, -1], [-12, -2, -14, -8, -12], [3, -5, -2, 0, -9], [-26, 4, 0, -2, 4], [4, 4, 2, 4, -8], [-22, -8, 2, 2, 4], [4, -4, -6, -8, -6], [-4, -14, -10, -10, -2], [4, -7, 8, 0, 0], [2, -6, -7, 8, -3], [-8, 10, 2, 1, -9], [21, -11, -12, -3, -2], [-4, -8, 7, 0, 5], [-14, -1, 0, 1, 11], [12, -10, -12, 0, -10], [-10, 20, 6, -4, -2], [-8, 14, -2, -2, 2], [20, 12, 2, 0, 4], [9, -22, -10, -2, -7], [-33, 3, 0, -10, -1], [8, -4, -2, -6, 8], [-8, 14, 4, -6, 4], [-8, -16, 0, -6, -4], [4, 15, 9, 13, 2], [12, 12, 3, 12, 5], [-10, -5, -3, 3, 6], [24, 5, -8, -1, -1], [4, -14, -12, -8, -10], [17, -19, -18, 0, -1], [37, 9, -2, -4, -7], [-7, -1, -12, -4, 11], [13, 20, -2, 5, 4], [-12, 8, 0, 2, -6], [11, -4, 4, -3, 4], [0, -16, -8, 1, -1], [-9, -8, 0, -5, -2], [0, -6, 2, -2, -2], [1, 11, -4, 7, 8], [15, -15, 5, 6, 8], [-4, -18, -1, -6, -9], [-5, 16, 21, 7, 19], [-4, -2, 8, 0, 2], [-8, -6, -1, 0, -5], [-13, 20, 9, 7, 3], [-9, 15, 2, 1, 4], [37, 7, 2, -5, -12], [24, -16, -14, -1, -7], [30, 9, -5, 5, -6], [-9, -4, -4, -9, 2], [-26, -4, -6, -2, -4], [24, 2, 6, -2, 0], [-16, 17, 2, -3, 3], [8, 0, -4, -5, -11], [7, 12, 1, 3, 3], [-23, 5, 6, -6, -3], [-13, 23, 2, -6, 1], [-30, 8, -8, -5, -1], [-12, 22, -1, -4, -1], [18, -14, -2, -6, -14], [-22, 8, 10, 5, 13], [1, 1, 6, -6, 1], [41, 2, 6, 4, 7], [4, -6, 6, -2, 6], [-7, 9, 0, -4, -5], [-40, -4, 10, -2, 8], [-11, -1, -8, -8, 9], [-2, 15, -3, -5, -2], [9, 14, 12, 3, 0], [-27, -10, 2, -1, 10], [-33, 7, 7, 4, 4], [-24, 8, -1, 0, 9], [-42, -14, 4, 0, -6], [-20, 10, 12, -2, 4], [17, 21, -2, 2, 11], [-18, -9, -2, -9, 7], [-26, -4, -9, -2, 5], [-2, 4, 4, 10, -8], [-3, -9, -1, -10, -8], [29, -9, 0, -8, -1], [-2, 13, 0, 5, 11], [-38, 21, 8, 7, 1], [1, -5, -18, 1, 2], [-20, -4, 12, 3, 1], [11, 9, 2, 0, -1], [32, 6, -14, 2, -8], [17, -3, -16, -4, -9], [-21, 29, 5, -2, 4], [-24, 0, 2, 5, 11], [-2, 16, 16, 0, 4], [-5, -9, 0, 12, -3], [-27, -5, -15, -8, -14], [5, 21, 16, 4, 15], [-18, -8, 13, 4, 11], [-4, -4, 5, 4, 15], [-18, 6, 4, 6, -8], [6, 2, -1, 10, 7], [-3, -7, -10, -2, -5], [0, -2, -2, 10, -10], [2, -20, -8, 4, 2], [-14, -7, -23, -7, 0], [4, 3, -14, 1, -5], [13, 4, 10, 2, -3], [4, -6, -8, -8, -14], [-16, -4, 0, -4, -8], [-43, 23, 5, 0, 2], [-23, 9, 8, -12, -1], [-18, -12, -20, -12, -12], [9, -10, -25, -1, 3], [-8, -12, -18, -4, -20], [-18, -6, 8, 2, 14], [-5, 28, -4, -7, 4], [-7, 15, -2, 1, -10], [20, -15, 1, -9, -12], [-4, -2, -2, -8, 0], [-20, -14, -2, 6, 2], [28, -16, -10, -2, 0], [-17, 1, -5, -2, -10], [17, 13, -12, -2, -13], [36, -12, 8, 6, -2], [-12, 0, 5, 4, 15], [-15, -13, -1, 10, 8], [41, -7, -8, -3, 2], [-36, 1, 6, 5, 7], [8, 10, 9, 0, -7], [-18, 12, -3, 6, 3], [-16, 17, 8, 5, 5], [-26, 20, 2, -4, 6], [-20, -7, 2, 5, -1], [-28, 30, 10, 0, -4], [-6, -23, 9, -3, -4], [-8, 6, 24, 6, 20], [-45, -9, -3, 0, -6], [-2, 3, -8, 1, -17], [-43, 5, -4, 1, 6], [26, -6, 10, 10, 0], [-8, 11, -18, -1, 9], [-6, 3, -6, 3, 3], [-20, 16, -6, -12, -8], [3, 17, -9, -2, -2], [-7, 21, 12, 6, 3], [18, 0, 17, 6, 3], [-31, 7, 8, 4, 7], [15, 5, 12, 7, 6], [19, -9, -10, -3, -8], [-22, -2, -4, 0, 16], [-5, -10, -2, 3, -4], [12, 4, -18, 4, 2], [6, 3, 6, 7, -13], [18, 16, 6, -6, 0], [-13, 11, 8, 6, 11], [8, 14, 4, -16, -6], [40, -2, 2, -6, -12], [-6, -4, 4, -4, -10], [-11, 4, 16, 7, 12], [12, 6, 8, -2, 14], [8, 10, -6, -2, -10], [-3, -7, -4, 6, -3], [5, 15, 2, 7, 8], [-5, 0, 6, -10, 3], [-23, 23, 6, -1, 8], [3, -25, -12, -2, -9], [-3, -5, -2, -1, 12], [-2, -14, -2, 9, -3], [-23, -3, -2, -10, -9], [-13, 27, 0, -9, -6], [-10, -24, 4, 0, -12], [36, 16, -4, 8, -12], [1, 20, 7, 5, 9], [11, -17, -10, -10, -7], [-35, -2, 14, 2, 5], [-18, 8, 4, 0, 12], [23, 19, 2, 10, -3], [-5, -30, -6, 10, -5], [23, -11, -2, 3, -2], [1, 19, 14, 5, -10], [-19, -18, 11, -5, -1], [13, 2, 14, -4, -9], [15, -5, -10, -8, -9], [15, 15, 8, 15, -6], [17, 21, 6, -7, 6], [-4, 4, -16, 2, 0], [-9, 4, 8, 3, 14], [-23, -6, -10, -13, -6], [-2, 9, -20, -6, 4], [-13, 15, -14, -4, 11], [-4, 14, 16, 16, 18], [-28, -20, 1, -6, -5], [-34, 10, 0, 3, 15], [0, -18, -1, 8, 7], [25, 15, 22, 4, 11], [-37, -19, -8, 5, -4], [6, 6, 12, 2, 18], [-26, 18, 2, 0, 0], [-6, -12, -6, -10, -16], [20, 13, 5, -5, -4], [11, 5, -9, -4, 6], [-7, -3, -10, 4, 1], [-36, 16, 2, 2, 2], [27, -35, -6, 1, 0], [-13, -15, -2, -11, 8], [2, 10, 4, 6, 2], [14, -32, 1, 0, -5], [-26, 0, -5, 6, 1], [-13, 8, -10, -6, 3], [-2, -12, -4, 16, 2], [18, -5, 0, -5, -7], [11, 9, 9, 6, -16], [-8, 16, 4, 0, -8], [13, -4, 10, 2, -3], [36, -6, -4, 10, 0], [-12, 16, 2, 4, 20], [5, 12, 14, 7, 10], [7, -10, -10, -10, 7], [6, -31, -6, -15, -11], [21, 1, 4, -2, 17], [22, 11, -4, 0, -18], [-14, 14, 0, 4, -8], [21, -27, 0, -3, -6], [-42, 0, 0, 8, -4], [-4, -28, -3, 0, -5], [2, 20, 2, 4, 14], [-21, -25, -4, -3, -8], [33, -33, 0, 8, 3], [-10, -6, -19, -8, -3], [2, 6, 0, -6, 12], [-11, -23, -3, 4, -8], [-4, -18, -10, 2, -14], [16, -10, 6, 2, -4], [-16, 9, -18, -7, 3], [50, 12, 0, 4, -4], [-12, 18, 4, 16, 14], [18, -5, -20, -5, -7], [34, -2, 25, 10, -3], [-35, -23, 2, 2, 3], [-12, -22, -10, -14, -18], [-32, -1, 6, -1, -7], [-6, 17, 6, -13, 3], [27, 1, 14, 4, 15], [-37, 14, 2, 0, -7], [4, -4, -8, 10, 4], [14, 12, 8, 0, 12], [24, 0, 9, 10, -1], [0, 22, -12, -10, -2], [-18, -12, -10, 0, 6], [-22, -6, 12, 8, 8], [-14, 25, -10, -1, 7], [8, -10, 4, -6, -4], [22, 5, 6, 11, 3], [42, 12, 12, -12, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3640_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3640_2_a_y();".
// 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_3640_2_a_y(:prec:=5)
    chi := MakeCharacter_3640_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_3640_2_a_y();".
// 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_3640_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3640_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 10, 3, -8, 0, 1]>,<11,R![64, 96, -84, -10, 7, 1]>],Snew);
    return Vf;
end function;