
// Make newform 209.2.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_209_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_209_2_a_b();".
// 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_209_2_a_b();".
// 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 := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_209_a();"
function MakeCharacter_209_a()
    N := 209;
    order := 1;
    char_gens := [134, 78];
    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_209_a_Hecke(Kf)
    return MakeCharacter_209_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, 1], [-1, -1], [-1, 0], [-2, -1], [-1, 0], [-2, 3], [2, 1], [-1, 0], [-3, 0], [-2, -3], [-5, -1], [3, 5], [4, -4], [6, -4], [6, 2], [4, -6], [-3, 1], [-4, -5], [-9, -1], [-11, -1], [4, -6], [-16, 1], [2, 1], [-5, 7], [1, 1], [2, 3], [8, 4], [10, 4], [-6, 4], [-5, 1], [-18, -1], [6, 6], [3, -4], [2, 13], [10, 8], [-2, -6], [5, 0], [-6, -10], [-6, -9], [-8, 7], [9, 5], [-1, -13], [-3, 8], [4, -2], [6, 7], [-2, 10], [-14, 1], [-15, -1], [12, -5], [-3, -8], [4, -3], [6, 0], [0, -12], [-17, 4], [-22, 2], [-18, 6], [-2, 0], [-8, 10], [6, -6], [18, 4], [-4, -9], [-2, -21], [6, 1], [-6, -4], [3, -12], [-27, -1], [13, 15], [4, 9], [4, 21], [2, 15], [-13, -8], [8, 6], [-7, -2], [-14, 8], [-1, -21], [1, 23], [-17, -8], [26, 0], [-18, -12], [-12, 9], [-2, -18], [-14, -10], [22, 6], [-7, 13], [-10, -3], [5, -12], [5, -13], [24, -5], [-4, 12], [-13, 18], [21, -8], [-8, -23], [-3, 11], [32, -3], [-10, -16], [-14, -10], [13, -15], [-15, 11], [8, -4], [-12, 15], [-2, -9], [6, -24], [-2, -15], [-22, -3], [4, 21], [13, 12], [2, 4], [20, 12], [2, 8], [-4, 11], [-12, 7], [-12, -1], [20, 2], [15, 6], [-25, 12], [-9, 7], [19, 4], [3, -24], [-17, 6], [2, 14], [-23, -19], [-18, 10], [10, 8], [-8, 22], [9, 12], [10, -8], [3, -20], [27, -12], [13, -22], [12, 1], [-26, -1], [-14, 5], [19, -11], [8, 24], [20, -10], [16, -10], [14, 4], [36, 0], [-9, -3], [44, -6], [48, -3], [-6, -23], [-23, 0], [14, 7], [-13, -17], [25, -11], [-14, 0], [18, -21], [-1, 20], [8, -28], [-24, -10], [-15, -26], [42, 6], [-6, 32], [-36, 10], [-4, 30], [-38, -13], [-28, 20], [8, -11], [24, 3], [-11, -12], [16, -7], [-12, 11], [13, -9], [-1, 31], [23, 7], [-20, -8], [-22, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_209_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_209_2_a_b();".
// 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_209_2_a_b(:prec:=2)
    chi := MakeCharacter_209_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_209_2_a_b();".
// 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_209_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_209_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 0, 1]>],Snew);
    return Vf;
end function;