
// Make newform 539.2.a.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_539_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_539_2_a_h();".
// 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_539_2_a_h();".
// 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 := [3, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 0, 1]];
    Rf_basisdens := [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_539_a();"
function MakeCharacter_539_a()
    N := 539;
    order := 1;
    char_gens := [199, 442];
    v := [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_539_a_Hecke(Kf)
    return MakeCharacter_539_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, 1], [0, -1, 0], [-1, 1, -1], [0, 0, 0], [-1, 0, 0], [-4, 1, 0], [-1, 0, -1], [-3, -2, 1], [4, 0, -1], [-2, -3, -1], [-2, 3, 1], [-2, 2, 2], [-2, 1, 4], [1, -1, -2], [-1, 0, -1], [6, -1, 2], [3, -1, 6], [-8, 0, 2], [-6, 2, 0], [1, 4, -3], [-5, -5, 0], [0, 3, -1], [-3, -2, -1], [0, 1, -1], [4, -3, 0], [0, -2, -3], [0, 2, 1], [-1, 3, 3], [-11, 1, -3], [-1, -6, 3], [-1, 5, 6], [-6, -1, -3], [-3, 3, 2], [-7, 2, 0], [-1, 0, 0], [12, -3, -2], [-5, 2, -4], [-5, 0, -6], [-8, 5, -1], [10, -5, 0], [-5, -5, -2], [-7, -1, -8], [5, -4, -4], [8, -7, 6], [-6, 8, -5], [-2, -5, -3], [-2, -6, 4], [13, 0, 4], [1, 7, -5], [-7, 2, 8], [-9, 4, -4], [-8, 5, -7], [-9, 5, -2], [15, -2, 5], [6, -9, 9], [-6, 6, 3], [-6, -3, 3], [-5, 4, -12], [-3, 11, -4], [17, 1, 1], [0, 6, -7], [9, -4, -2], [-1, 5, -9], [23, -1, -5], [7, -8, -9], [9, 1, -4], [1, -6, 10], [15, -8, -5], [-10, 5, -1], [-7, 4, 4], [0, 7, 4], [2, -12, -1], [-2, -5, 5], [-1, 14, -8], [-11, 4, -5], [14, -9, 8], [12, -7, -9], [-18, -6, -2], [-15, 7, 6], [-14, -3, 9], [13, 4, 6], [-14, 10, -9], [-5, -2, -4], [2, -6, -8], [4, -2, -5], [15, 1, 1], [15, 0, 10], [5, 4, -1], [-15, 10, 3], [-2, -1, 9], [6, 8, 5], [8, -1, 11], [5, 1, 4], [9, -12, -12], [2, -7, -7], [12, -7, 4], [-4, 2, 9], [10, 2, -7], [-7, 0, -3], [8, -7, -5], [-3, -10, 10], [23, -7, 3], [-19, 15, 9], [-24, -6, 1], [-5, -14, -3], [23, -4, -3], [-15, -7, 4], [18, 3, 3], [-3, 13, 2], [-25, -5, 5], [8, 1, 12], [16, 8, 6], [-24, 4, 8], [8, 2, 16], [-9, 9, -6], [-24, -1, 9], [4, 4, 2], [6, -4, 2], [-26, -3, -4], [-8, -2, 3], [-25, -11, 5], [16, 12, -6], [-1, -22, 7], [6, -3, -8], [-12, -12, 0], [-3, 2, 9], [12, -9, 23], [19, 13, 1], [26, -15, -7], [-31, 6, -7], [9, -13, 20], [0, 1, -5], [31, -3, -5], [-1, -17, 13], [6, 4, -9], [25, -18, -12], [3, 10, 1], [2, -7, 10], [14, -16, -3], [3, -6, -16], [-25, 7, -13], [3, 27, -3], [2, -5, -17], [-16, 3, 2], [-29, -3, 0], [-15, -3, -11], [8, 14, 6], [20, 10, -9], [-8, 21, -12], [-29, 7, 9], [7, 3, -12], [-12, 11, 16], [-44, 1, -8], [1, -2, -17], [-7, 2, 6], [-9, 13, -11], [-9, 0, -4], [21, -3, -3], [6, 7, 13], [7, -3, -4], [5, 13, -7], [14, 12, -11], [-19, -5, 12], [2, 2, -11], [8, 6, -5], [3, -15, -8], [-5, -4, -5], [-6, -5, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_539_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_539_2_a_h();".
// 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_539_2_a_h(:prec:=3)
    chi := MakeCharacter_539_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(997) - 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_539_2_a_h();".
// 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_539_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_539_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, -5, 0, 1]>,<3,R![-3, -4, 1, 1]>],Snew);
    return Vf;
end function;