
// Make newform 2009.2.a.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2009_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_2009_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_2009_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 := [-1, -1, 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_2009_a();"
function MakeCharacter_2009_a()
    N := 2009;
    order := 1;
    char_gens := [493, 785];
    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_2009_a_Hecke(Kf)
    return MakeCharacter_2009_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, -1], [0, 0], [-1, 2], [3, 0], [1, 4], [3, -3], [1, 1], [6, -3], [9, -1], [4, -6], [-1, 0], [3, -8], [2, -2], [-11, 5], [-7, 1], [11, 0], [5, 1], [-1, -2], [15, 0], [-4, 6], [-7, 8], [9, -9], [7, 3], [-3, -2], [2, 9], [6, -6], [-2, 8], [-5, 8], [-3, 14], [9, 1], [-9, 1], [11, -12], [10, -16], [9, -14], [0, -12], [8, 4], [-3, 6], [-3, -6], [9, 7], [-2, -12], [-8, -8], [-7, 12], [-1, 5], [3, 6], [0, -3], [7, 3], [-3, -4], [-12, 6], [0, -3], [18, -9], [-10, 6], [7, -20], [-3, 0], [3, 12], [-10, 17], [-21, 14], [1, -6], [27, 0], [0, -16], [5, 8], [10, -16], [-10, 13], [-3, 18], [-9, -12], [0, -16], [-10, -8], [24, -18], [1, 6], [2, -16], [-8, -2], [-18, 25], [1, 17], [-18, 2], [7, 4], [-2, 7], [19, -19], [-32, 10], [-13, 22], [12, -9], [14, 4], [-19, -8], [27, -17], [-21, -2], [-33, 1], [-18, 16], [-17, 6], [15, 10], [-22, 27], [5, -22], [-7, -13], [24, 6], [20, -31], [0, 2], [-2, -1], [-34, 5], [-21, -9], [-1, -24], [-15, -11], [9, -10], [-36, 9], [23, -10], [9, -15], [22, -24], [1, -16], [11, 4], [-36, 0], [6, 18], [18, -9], [-19, -12], [23, -33], [21, 2], [-6, 5], [19, -6], [-13, -8], [-4, 25], [15, -40], [-15, 27], [-33, 0], [32, -14], [-35, 6], [27, -7], [15, 18], [-36, 17], [19, 11], [-4, 30], [1, 20], [24, -21], [21, -12], [4, 3], [30, 9], [29, -8], [0, -6], [36, -4], [16, 14], [-17, 8], [-24, 6], [-9, -4], [-21, -13], [39, -2], [-5, -8], [-6, -17], [-39, 18], [5, 28], [-14, 22], [-26, 0], [8, -5], [23, 1], [-21, -18], [20, -21], [-18, 0], [15, -9], [24, -28], [-18, -25], [-4, 28], [-2, 9], [0, -27], [-3, 22], [-13, -17], [16, -5], [2, -7], [24, -36], [-39, 8], [4, -7], [17, -28], [9, 12], [4, -7], [-8, -27], [-47, 25], [-15, 0], [-22, -14], [35, -41], [-36, -6], [34, -6], [26, -49], [18, 9], [21, -18], [-15, 24], [-13, 18], [0, 27], [25, -22], [-24, 32], [-24, 41], [33, 13], [0, 0], [9, -20], [-4, 15], [-37, -2], [51, -19], [-37, 37], [-9, -30], [29, 22], [15, -36], [-29, -8], [-47, -5], [-7, 28], [33, -43], [-31, -1], [12, 23], [-1, 32], [1, 0], [60, -12], [30, -23], [-6, 21], [-36, 32], [29, 10], [-23, -4], [2, -34], [-29, 45], [-22, 24], [-24, 38], [-35, 0], [3, 30], [-30, 0], [-20, 21], [-35, 24], [12, 14], [-41, 41], [-27, 9], [-9, -3], [-6, 0], [-5, 34], [-18, 28], [54, -33], [24, -11], [42, -24], [-31, -7], [-17, -15], [6, 17], [-36, 24], [48, -34], [-22, 12], [-59, 7], [35, -8], [-38, 10], [-6, -18], [19, -1], [-16, 46], [-49, 17], [39, -21], [19, 0], [10, 5], [-9, -39], [20, 10], [18, -11], [32, -15], [27, 12], [21, 18], [-66, 14], [-34, 0], [-32, 10], [42, -27], [9, -21], [-20, 55], [-37, -7], [35, -38], [-34, 30], [-6, -20], [-15, 0], [47, -42], [-30, -13], [33, 17], [-18, 42], [-12, 30], [59, -14], [12, 17], [63, 12], [-31, 34], [3, -9], [-56, 11], [-26, -10], [34, -13], [24, 10], [4, 45], [-49, 25], [-34, 47], [-36, 55], [23, 4], [-18, 31], [24, -8], [36, -8], [27, -19], [-47, 33], [-59, 21], [-15, 15], [-35, -9], [38, 8], [-9, -32], [27, -60], [-6, -2], [-43, 8], [32, -26], [42, -39], [-39, 3], [-15, -9], [-47, 50], [-2, -38], [45, -21], [60, -7], [-18, 11], [-51, 52], [-15, 52], [-15, 20], [-59, 32], [-9, 38], [60, -34], [12, -44], [-37, 52], [10, 8], [-5, -23], [-23, 16], [-31, 23], [27, 1], [6, -39], [33, -10], [-10, -1], [36, -3], [-36, -5], [3, -14], [2, -14], [-40, 28], [23, -38], [-30, 35], [27, -52], [36, 10], [18, 5], [-84, 4], [57, -26], [-37, 7], [-41, 23], [-41, -32], [-37, 34], [-5, 10], [-29, 18], [-8, 28], [-57, -18], [-3, 33], [-36, 44], [38, -3], [5, -9], [-33, 15], [-18, 39], [37, 33], [-31, 5], [-44, 5], [-17, -22], [78, -27], [29, -14], [-63, 6], [59, -54], [-29, 0], [45, -44], [-21, 5], [9, -30], [-24, 63], [-63, 42], [-20, 7], [-26, -38], [-18, -24], [15, 9], [-23, 67], [-12, -22], [16, -64], [28, -31], [-15, -18], [-55, -25], [-63, 23], [13, -60], [24, -12], [-51, 50], [-3, 36], [39, -38], [-64, 3], [-39, 36], [-54, 33], [-40, 28], [-37, -20], [61, -5], [58, -30], [33, 24], [27, -24], [-3, 21], [-39, 24], [-4, 49], [-64, -10], [17, 15], [-6, 30], [19, 3], [-11, -10], [66, -9], [33, -8], [78, 4], [49, 0], [64, -56], [-24, -25], [-55, 16], [-21, -6], [44, -33], [-19, 41], [-58, 3], [1, 40], [11, -6], [-57, 28], [62, -55], [44, -39], [53, -60], [42, -78], [80, -8], [24, 30], [35, -38], [-39, 68], [-49, -4], [-41, 27], [-23, -42], [7, 7], [3, -28], [20, -45], [-69, 0], [65, 11], [68, 17], [22, 35], [-18, 41], [-12, 10], [-6, -46], [-7, -2], [18, 2], [53, -46], [34, 18], [1, -40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2009_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2009_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_2009_2_a_b(:prec:=2)
    chi := MakeCharacter_2009_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_2009_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_2009_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2009_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1, 1]>,<3,R![1, -3, 1]>],Snew);
    return Vf;
end function;