
// Make newform 3640.2.a.z 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_z();".
// 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_z();".
// 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, -1, -10, 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], [5, 8, -9, -1, 1], [9, 8, -10, -1, 1], [13, 8, -19, -1, 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_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, -1, 0, 0, 1], [-1, 0, 0, 0, 0], [1, 0, 0, -1, 0], [1, 1, -1, -1, 1], [1, -1, 0, 0, -1], [1, 2, 0, 0, 1], [2, -1, 0, 0, 0], [2, 0, 1, 0, -1], [0, 2, -1, 1, 0], [0, 1, -1, 2, 1], [-1, -2, -1, 0, 2], [6, 0, 0, 0, -2], [1, 1, 1, 0, -2], [3, -2, -1, 0, 0], [4, 0, 1, -3, -2], [0, -1, 1, 2, 1], [-1, 2, 1, 2, 0], [-2, -2, -1, 1, 2], [0, 0, -2, 0, -2], [1, 2, -2, 2, -1], [-1, -2, -1, 0, -4], [5, 4, -1, -2, 0], [-1, 1, -3, 1, 1], [-4, 1, -1, -2, 1], [6, 0, 2, 0, 0], [3, 2, -3, 0, 2], [-1, 1, 2, -2, 1], [0, 0, 0, 4, 0], [5, 0, 2, 2, 1], [-4, 0, 0, 2, -2], [-1, 2, 1, -2, 2], [4, 1, 1, -4, -1], [6, 0, 1, 0, 3], [3, 0, 2, 0, 3], [-4, -6, 2, -2, 0], [1, -2, 0, 2, -1], [1, 0, 6, 1, -4], [7, 2, -3, 2, 0], [3, 0, 2, -2, -3], [3, 2, -4, 3, 0], [4, 4, -3, 0, -3], [2, 2, 0, -4, -2], [5, 2, -2, 3, 2], [-9, 4, -1, 2, 4], [-8, 0, -2, 0, 6], [-1, 4, 6, 1, -4], [9, 2, 3, -2, -4], [8, 5, -3, 0, 3], [5, -4, 0, -3, 2], [7, 0, -1, -2, 0], [12, -4, 0, -6, 2], [-2, 7, 1, 2, -5], [-3, 4, 1, 2, -2], [3, 3, 0, -2, -3], [4, -8, 0, 4, 0], [6, 8, 0, 2, -2], [-4, 7, 4, -1, -3], [8, -2, 0, -6, 4], [-6, -6, -2, 2, 2], [6, 4, -4, 2, 4], [-5, -2, -2, 0, -1], [1, -2, -3, 2, -2], [7, 1, 2, -8, -3], [1, 2, 1, 2, 2], [-4, 1, 5, -2, -1], [13, -2, -4, 0, 1], [-3, -2, 5, 2, -6], [-10, -9, -1, 2, 5], [-18, 1, 1, 4, -1], [-10, 4, -2, 2, 6], [2, 9, 1, 2, 1], [-3, -4, 7, 4, -4], [9, -4, -2, -1, 0], [8, 2, -2, -4, 4], [4, -8, 2, 2, 2], [-14, 0, -2, -2, 2], [-5, 6, -3, 2, 2], [1, -2, -1, 4, -4], [10, 7, 1, 4, 1], [-7, -8, 4, -1, -4], [-10, 10, 0, 2, 4], [-6, 7, -3, 4, 3], [14, 4, 6, -2, -4], [-15, 4, 2, 2, 1], [5, -6, 2, 4, -1], [5, 4, -2, -9, -2], [-1, 3, -5, 1, -3], [-1, -3, -1, 2, -2], [-17, 2, 0, 0, 1], [8, 4, 0, 0, -4], [5, -5, -2, 4, -5], [-2, -1, 3, -4, 5], [-7, -4, 4, -3, 0], [4, 0, 0, -4, -6], [-13, 1, 0, 0, -3], [2, 4, -2, -8, 0], [-6, 5, 5, -2, 1], [14, -10, -1, 1, -4], [-10, 3, 4, -3, 5], [-4, -2, -3, 2, 7], [1, 2, 4, 2, -3], [-16, -8, 0, 0, 2], [-8, 8, 2, -4, -2], [8, -8, 0, -6, 0], [3, -2, -3, -2, 2], [4, -4, 6, -2, 2], [-21, -1, 1, -4, -2], [3, -2, -5, 2, 2], [-17, 2, 2, 5, 0], [-23, 1, 5, 0, -2], [2, 5, -3, 6, -5], [6, -8, 1, -6, -3], [2, -4, 6, -6, 2], [-15, 9, 1, -8, 4], [-6, 12, 2, 0, -4], [7, -6, -4, 3, 2], [3, -12, 0, 7, 4], [-7, -2, 3, 6, -8], [11, 2, 7, 2, -2], [-8, 6, -1, 5, -8], [-5, 9, 1, 4, -8], [25, -4, 0, -3, 4], [-7, -2, 7, 6, -6], [0, 0, 0, -4, -4], [-9, 1, 1, 8, -10], [-8, 0, 2, -10, -2], [10, -9, 3, -4, 5], [7, -2, 6, -1, 4], [-2, 4, -1, 4, 7], [-10, 0, -4, -2, -8], [-6, -8, -7, 4, 5], [7, 6, 5, -8, -4], [-8, 6, -6, -6, -2], [-14, 14, 4, 0, -2], [0, 4, -5, 0, 7], [3, 2, 4, 1, -6], [-2, -9, -3, 2, -1], [2, 0, -6, 6, -2], [-1, 7, 4, -6, -5], [-8, 0, 0, 0, -4], [2, 4, 0, 2, -6], [3, -7, 2, -2, -1], [-6, 2, -4, 6, -2], [-11, 10, 0, 4, 7], [-19, 8, 5, -2, -4], [13, 4, -4, -3, 2], [6, 6, 1, -5, 4], [-4, 0, -8, 8, 2], [18, -3, -5, 10, -1], [5, 7, 3, 0, 2], [-32, -1, -1, 0, 3], [13, 4, -6, 1, 8], [-14, -10, -1, 0, 7], [-30, 4, 5, 0, -3], [15, 4, 6, -9, -8], [7, 4, 2, -7, -4], [7, 0, 6, -3, 2], [0, 4, -4, -2, 12], [-3, -6, -4, -4, -5], [-9, 0, 1, -4, -8], [11, 4, 2, -5, 6], [14, -2, 2, 0, 4], [-2, 2, -1, -7, 10], [8, 2, 3, -5, -8], [-24, 8, -2, 6, 2], [-17, -4, 2, 7, 2], [20, -3, 5, -2, 3], [37, -6, 0, 4, -7], [21, 4, -3, 2, 4], [-14, -8, 8, -8, 0], [28, -4, 2, -6, 8], [-12, 8, 4, -4, 2], [5, -9, -6, 2, 5], [10, -4, 6, 2, 2], [1, 7, -1, -4, 6], [-3, 12, 3, -2, -4], [-7, 2, -8, 4, -5], [15, -4, -3, 8, -8], [4, -8, 4, 6, 2], [-5, -2, 3, 2, 2], [-10, 4, -6, 12, -4], [-16, -12, 8, -4, 6], [10, 12, -8, 2, 4], [-18, -8, 8, -2, 0], [-12, -8, -4, 0, 2], [12, 5, 2, 4, -12], [-15, -8, -2, 5, 4], [22, -8, -3, -2, -5], [5, -8, 2, 2, -1], [27, 12, -2, 1, -8], [-8, 19, 5, 0, -3], [-4, 16, 0, 6, -6], [-4, 8, 4, 2, -12], [-36, -8, -2, 0, 2], [10, 8, 0, 4, 0], [3, 9, -1, -4, -8], [19, 2, -1, -14, -4], [-20, -6, 8, -4, -4], [24, 0, 4, 2, -12], [-16, -4, -6, 0, 12], [-15, -1, 1, 3, -5], [-27, 4, -2, -3, 14], [7, -3, 7, 5, -9], [30, -1, 5, -8, -9], [-2, -4, 2, -6, 0], [-21, -2, 5, -2, -2], [3, 2, 5, -2, 12], [1, -6, -1, 4, -6], [-7, -11, 4, 8, -1], [16, 14, -4, 8, 0], [19, -5, 8, 2, -7], [-32, -4, -3, 8, 11], [-15, -5, 12, -10, 1], [-18, -16, 8, 6, -2], [9, -12, -2, 8, 1], [26, -10, 5, -1, -8], [-17, -4, -2, 5, -4], [-8, -3, -2, -7, -5], [-12, -4, -6, 0, 0], [-19, 6, -2, 13, 0], [-18, 1, -4, 11, -1], [29, -10, 2, -4, -5], [19, 10, -6, -8, 7], [18, -14, -9, 4, 3], [-21, 11, 1, -9, 7], [31, -5, 6, -6, -5], [-6, 8, 8, -2, 4], [4, 6, -2, -12, -2], [20, -1, -3, -8, -5], [0, 4, 1, -2, 7], [-6, -7, 0, 3, -1], [15, -10, 7, 0, 2], [-11, 2, -1, 8, 4], [22, -4, 5, -8, 3], [27, -10, -6, 5, -2], [-6, -6, 6, -4, -8], [12, 18, -3, 2, -3], [-9, -4, 1, -4, 0], [-1, -15, -7, 4, 6], [4, -12, -2, 0, 10], [-9, -14, 5, 2, 8], [24, 14, -4, 4, 4], [13, 14, -1, 8, 4], [15, 5, 5, -5, -7], [3, -1, -12, -8, 7], [15, 7, -6, 6, -7], [16, -2, 1, 1, 8], [-13, -8, 4, -1, -10], [-10, 2, -12, 8, -2], [6, 0, -12, 8, 8], [19, -6, 3, -10, 10], [20, 7, -5, 12, 3], [1, 0, -4, 13, 4], [-24, 12, -2, -8, 6], [34, -2, 3, 1, 0], [-33, 2, -1, 6, -10], [18, 9, 3, -4, -1], [12, -7, 3, 0, -1], [15, -18, -2, -8, 9], [0, 20, 1, 0, -3], [1, 2, -7, 4, -8], [-26, 4, -4, 8, 2], [7, -14, -5, 4, -6], [12, -2, -1, -11, -4], [18, 4, -5, -2, 9], [-6, 12, -2, -12, 4], [13, -20, -7, 6, 2], [28, -6, 1, -5, -4], [7, 14, 1, -8, 2], [-25, -6, 8, 7, -6], [23, -12, -4, 9, 4], [-2, -6, 6, -14, 6], [11, -16, 6, 1, 2], [13, -6, 5, -10, 4], [-8, -16, 4, 0, -10], [20, -16, 0, -6, 6], [1, -13, -7, 5, -1], [-24, 13, 1, 14, 1], [-33, -17, -5, 0, 8], [28, 8, 6, 4, 0], [-30, 0, -4, 8, 6], [30, 16, -1, 3, 0], [25, -22, -3, -4, -4], [-18, -20, 8, 0, 8], [40, 5, -8, 3, 3], [2, -4, -4, 4, -8], [6, -4, 0, 4, 4], [-17, -9, 6, 4, 3], [7, -4, 8, -10, 1], [21, -19, -1, -5, 13], [10, -8, 0, 4, -6], [-16, 2, -10, 14, -6], [-14, 6, 8, 2, -8], [24, -10, 5, -11, 0], [25, 14, 11, 2, -2], [-6, -10, -4, -8, 6], [-25, 4, 4, 5, -4], [-2, 22, -1, 13, 8], [5, -14, -4, 6, 1], [22, -9, 7, 0, -7], [-19, -12, 12, -1, -4], [-19, -8, -8, -1, 10], [-4, 3, -5, 2, 5], [34, -6, -6, 2, 4], [18, 11, -3, -6, -9], [-38, 4, 8, -6, -4], [-11, -5, 1, 13, 7], [18, -4, 2, 0, -4], [-12, -28, -5, -1, 12], [10, 7, 5, 2, -3], [5, 6, 12, -12, -5], [18, -4, 8, -2, -2], [-6, 13, -1, 10, -3], [26, -5, 1, -12, 1], [-16, 0, -16, -2, 12], [18, 10, -3, -3, 2], [1, -22, -1, 4, 0], [-15, 0, -16, 11, 6], [5, 6, -1, 2, -4], [3, 0, -12, 0, 9], [53, 2, 6, -2, -11], [4, -24, 2, -14, -6], [-3, 13, -2, 0, 7], [-38, 4, -8, 0, -6], [14, 7, 1, 4, -11], [-14, -16, 4, 2, -4], [23, -6, 7, -4, -2], [-6, 0, -12, 0, 2], [-8, -2, 14, -10, 8], [-10, -4, 8, -12, 18], [15, -3, 0, 10, -11], [18, -10, -6, 6, -6], [36, 12, 2, 6, -4], [41, 6, 7, -2, -12], [27, -2, -2, -14, 9], [17, -7, 5, 0, -20], [-17, -6, -8, 2, 9], [7, 2, 1, -10, -12], [-33, -6, -4, 0, 7], [-20, 0, 5, -4, 3], [-13, 2, 19, -2, -10], [51, -10, 4, 6, -9], [28, -10, 2, 2, 10], [40, 14, 6, 0, -6], [4, 15, 6, -9, -11], [-3, 14, -7, 10, 6], [-5, -7, 9, -4, 8], [-2, -2, -2, 16, -14], [-11, -10, -5, 2, -2], [-35, 3, -7, 12, -4], [29, -10, -2, 0, -5], [-5, 2, -4, 20, 5], [28, 17, -8, 3, 7], [-9, -19, -1, 8, -2], [3, -30, -9, -6, 2], [-5, 2, -4, 16, 1], [-25, 10, -12, -2, 13], [-18, -4, 10, 2, -12], [9, -5, 4, -2, -3], [-9, -5, -4, 0, -13], [-22, -21, 8, 0, -10], [-21, -6, -1, -4, 0], [30, 12, -6, 10, 0], [-5, 24, 0, -1, -2], [6, -8, 11, 6, -1], [-5, -12, 0, -3, 12], [-5, 0, 9, 0, 0], [43, 8, -6, 4, -1], [30, -12, -4, -12, 12], [-46, 0, -6, 4, -8], [20, -16, -4, -2, 6], [5, -7, -13, 7, 1], [14, 8, 1, 5, -4], [-17, 14, -3, 14, -4], [-12, 4, 6, -8, -4], [21, 26, -2, 8, 1], [-31, -14, 4, -4, 9], [18, 0, 0, 4, -12], [-39, -20, 14, -1, -6], [15, -8, -16, 5, 10], [5, -11, 11, -12, -12], [14, 0, -8, -8, 10], [-32, 19, 5, -4, -9], [38, -14, 1, -1, -8], [12, 16, 0, 4, -12], [-1, -1, 7, -16, 4], [20, 12, 12, 0, -10], [10, 14, -10, -18, 8], [41, -7, 0, -4, 9], [-11, 19, -5, 2, -6], [40, 1, 7, -6, 7], [29, 10, 7, -2, -8], [46, 13, -4, 4, -12], [-4, 4, 14, -12, -4], [-3, -4, -18, -2, 7], [40, 0, -2, 6, 0], [-15, 20, -2, -5, -4], [-14, 20, -8, -12, 16], [19, 0, 4, -4, -19], [11, 2, 3, -14, -10], [11, -8, 4, 15, -4], [-74, 0, 0, 4, -2], [24, 2, 3, 11, 0], [-34, 24, 4, 4, 0], [12, 14, -2, -6, -16], [18, 7, -3, -2, 15], [22, 0, -2, 6, 6], [-12, -24, 4, 0, 16], [14, 3, -1, -8, 21], [-5, 4, -18, -5, 16], [13, 2, -1, -20, 6], [-12, -22, -6, -6, -10], [-14, 21, 3, 4, 3], [-32, -15, 3, -4, -13], [37, -26, -7, 4, 4], [-23, -17, 11, -8, -2], [-56, 4, 2, -4, 0], [10, 20, -4, -14, -2], [-17, -22, 6, -1, -4], [-42, 12, 0, -6, 4], [-32, 0, 0, 0, 6], [-14, 10, 12, 0, -4], [26, -7, -5, 12, 5], [14, 28, 0, 8, -8], [-8, -7, 9, -10, -3], [4, -16, 4, 10, -10]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : 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, 1, -10, 0, 1]>,<11,R![-16, 80, 68, -8, -7, 1]>],Snew);
    return Vf;
end function;