
// Make newform 2009.2.a.k in Magma, downloaded from the LMFDB on 25 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_k();".
// 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_k();".
// 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, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 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_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 := [[1, 1, 0], [-2, 1, 0], [0, 0, 2], [0, 0, 0], [0, -2, 2], [-1, -3, 1], [-2, 4, 1], [2, 2, -3], [-6, 3, -2], [-2, 4, -4], [4, -6, 6], [3, -3, 3], [-1, 0, 0], [4, 2, 3], [-5, 1, -1], [-8, 2, -2], [0, -6, -2], [-6, 4, -6], [-10, -4, 2], [-2, 0, -4], [2, -4, 0], [8, -4, 0], [6, -4, 8], [-5, 5, -11], [-4, -1, -2], [4, -5, 10], [-4, 2, 4], [0, -2, 3], [0, 0, -6], [4, 1, 4], [9, -1, -5], [2, 2, 4], [-8, 2, -2], [-6, 2, 8], [6, -4, 10], [8, -4, -4], [-14, 12, -15], [-2, 1, -6], [14, 2, 3], [0, 6, 2], [0, -2, 8], [-4, 3, -6], [-2, 2, -6], [18, -4, 2], [12, -12, 11], [5, 3, -7], [-2, -4, -10], [18, -6, 4], [8, 0, -4], [-8, 7, 10], [12, -4, 8], [-2, 2, 8], [2, -14, -6], [12, -2, 4], [4, -13, -2], [8, -2, 8], [-4, 10, 6], [26, -4, 6], [0, 1, -4], [12, -20, 14], [-8, -8, 0], [10, -4, 8], [16, -6, 6], [10, 6, 1], [6, -8, 11], [-18, -4, -6], [8, -16, 16], [32, -3, 4], [-20, 4, -2], [8, 14, -12], [-18, 14, -4], [-1, 5, -11], [-2, -4, -12], [-4, -4, -7], [-18, 19, -10], [6, -7, 8], [7, -7, 7], [-18, 4, -16], [-3, -1, 1], [4, 0, -14], [-4, -6, 0], [-10, 26, -12], [20, 0, -8], [-16, 4, 12], [2, 13, 0], [15, -15, 25], [-12, 21, -8], [14, -14, 24], [6, -16, 12], [12, 6, 4], [-22, 10, -10], [18, 1, 8], [8, 10, -13], [6, -25, 14], [-8, 22, -18], [2, 7, -8], [1, -9, -13], [7, -27, 1], [-2, -10, -6], [4, 8, 5], [24, 6, 2], [-8, 12, -10], [13, -5, -7], [-28, 12, -5], [4, 10, 6], [-8, -5, 6], [2, 30, -13], [-11, 11, -25], [-3, 7, 3], [-14, 8, -1], [0, -4, 12], [32, -7, 4], [-8, 4, -11], [24, -8, 8], [15, -15, 5], [2, -20, 2], [13, -1, -3], [-12, -2, -8], [22, 2, 10], [6, 2, -20], [-16, 0, -6], [18, -2, -8], [38, -16, 24], [12, 8, 6], [-3, 11, 5], [-13, 9, -17], [6, 2, 12], [0, -4, -24], [10, -2, 17], [-14, 24, -4], [-7, 19, -5], [8, -10, -21], [14, -2, -10], [-32, -6, 2], [24, -6, 2], [4, 4, 14], [-30, 16, -31], [36, -28, 40], [8, -14, -4], [-2, 18, 2], [8, 12, 4], [4, -32, 3], [16, -6, 26], [0, 22, 6], [22, -10, 10], [30, -23, 36], [-2, 0, 0], [6, -18, 0], [-30, 14, -32], [16, 16, -4], [-4, -19, -4], [-6, 20, -16], [-16, -8, 18], [-25, -11, -1], [11, -19, -15], [-8, 6, 19], [-22, 6, -10], [0, -21, 10], [-13, 13, 5], [22, -8, -6], [22, -11, -10], [9, 7, 17], [-30, 28, -40], [-10, 6, 15], [8, -6, 6], [24, 10, 10], [4, -10, -4], [-16, 15, 22], [36, -16, 29], [-26, -8, 3], [-6, 10, 2], [6, -20, 18], [-8, 28, -8], [11, 13, -21], [2, -6, -11], [26, 16, -8], [-2, -16, 8], [-4, -33, 24], [-17, -31, 7], [30, -16, 28], [-31, 7, -5], [-18, -2, -16], [-14, 20, -12], [-30, 16, -20], [-1, 1, -3], [-12, -2, 10], [19, -27, 11], [-8, 12, -18], [18, -24, 10], [29, 3, 9], [-40, 0, 4], [-54, -2, 4], [8, -10, -7], [16, 14, -8], [20, 8, 0], [-26, 6, -38], [0, -10, -16], [4, 6, 24], [0, -13, -2], [-24, 20, -20], [-14, -24, 20], [-20, 34, -30], [-26, 4, 3], [-6, 8, 13], [39, -31, 43], [14, -16, 28], [2, 12, -10], [-8, 12, 10], [-16, 11, -32], [-26, 8, -30], [36, -14, 2], [13, -33, 27], [47, -11, 33], [16, -36, 6], [6, -13, 16], [24, 2, 8], [4, -8, 0], [46, 4, -10], [20, 20, -16], [-18, 16, -8], [-4, 16, 30], [-28, 24, -42], [-16, 18, 10], [32, -30, -2], [-6, 6, -12], [-44, 20, -38], [4, -26, 14], [12, -10, -15], [-2, -10, 21], [51, -15, 1], [24, -21, -28], [25, 7, -15], [-30, 19, -42], [44, -25, 36], [16, 8, 20], [-51, -1, 9], [53, -21, 31], [-24, 48, -42], [-9, 21, -23], [-34, 22, -13], [-60, 14, -12], [14, -21, -8], [38, -15, 16], [-24, 17, -40], [20, 18, -16], [2, -44, 8], [-18, 40, -30], [10, -17, 28], [-16, 2, -9], [-16, 20, -28], [8, -20, -9], [0, -36, 30], [20, -30, -3], [-34, 4, -12], [1, -33, 3], [-52, 22, -19], [-22, 12, 24], [14, -25, 20], [-40, 21, -2], [-30, 32, -11], [22, -27, 22], [8, -16, -2], [2, 24, -5], [10, -28, -1], [-20, -16, 16], [32, -28, 34], [-38, 26, -16], [-18, 10, 6], [-20, -26, 9], [-38, 36, -48], [18, -16, 44], [15, -7, 9], [-6, 2, -34], [13, -33, 15], [-7, -29, 19], [56, 12, 2], [-50, -12, 0], [-12, 14, 0], [8, 28, -22], [-2, -3, 16], [-14, 10, 1], [-58, 14, -14], [-52, 30, -42], [22, 26, -18], [21, 7, -19], [58, -30, 52], [-2, -14, -24], [-26, 12, -32], [30, 17, 4], [-3, -33, 19], [-36, 18, -42], [10, 10, -6], [-10, 6, 30], [-38, 36, -52], [8, 20, 6], [6, -32, 23], [17, -5, 25], [26, -14, 12], [-47, 27, -11], [-30, 15, 12], [-16, 26, -12], [-10, -30, 24], [0, 42, -5], [0, -46, 24], [-23, 27, 19], [-52, 15, 20], [-20, 62, -27], [-8, 0, -32], [24, -6, -22], [12, 24, -20], [-30, -3, 16], [34, -16, 34], [-12, -12, 35], [9, -33, -21], [28, -22, 51], [-8, -40, 34], [34, -20, 0], [14, -5, -18], [-42, 20, -5], [44, 0, -16], [-2, 8, -18], [44, -8, 29], [2, 14, -36], [6, -14, 50], [-3, -25, 29], [-18, -20, 16], [22, -6, 16], [54, -2, 1], [-10, -2, -50], [32, -28, 45], [28, 22, 2], [-54, 8, -28], [44, -24, 44], [34, -6, 7], [-50, 11, 2], [-18, -3, -28], [-52, 4, 0], [12, -8, -36], [33, 3, -9], [32, -20, 20], [24, 24, 12], [8, -39, 28], [78, -4, -8], [20, 40, -30], [-12, 69, -36], [9, 11, -29], [60, -20, 52], [-22, 19, -14], [1, -53, 1], [2, -16, 14], [2, 10, -11], [0, -4, 3], [22, -44, 24], [72, -12, -2], [4, -34, -18], [-4, -7, -34], [-10, -16, 0], [16, 38, 6], [12, 20, 17], [32, -30, 55], [-17, 9, 11], [-42, 36, -55], [-68, -8, -8], [-10, 11, -38], [-34, 5, 4], [4, -6, -16], [24, -35, 10], [-20, 22, -5], [45, -21, -15], [4, 14, 12], [-46, 37, -56], [14, -11, 6], [8, 22, 0], [30, 36, -14], [-46, 2, -14], [-18, 34, -22], [-24, 16, -20], [-6, 26, 34], [10, 40, 15], [-8, -39, 10], [6, -20, -11], [-32, -12, -9], [40, 8, 16], [56, 4, -3], [48, 20, 0], [-8, 18, -2], [-24, 16, -30], [-32, -31, 2], [-52, 12, 4], [16, 20, -20], [-2, 26, -8], [75, 1, 17], [2, -2, -32], [44, -50, 20], [-10, 37, -50], [24, 46, -20], [-21, -23, -25], [-30, 72, -46], [-6, -17, -10], [42, -2, 8], [18, 40, -10], [28, -34, -44], [66, -29, 10], [72, 11, -10], [-10, -10, -8], [-26, 3, -16], [-28, 4, -35], [-4, -24, -28], [39, -23, -17], [-46, 12, -26], [-24, -34, 4], [4, -15, -24], [-8, -52, 30], [42, 20, 20], [30, -46, 42], [-50, 16, -32], [23, 25, -3], [-26, 30, 33], [42, -32, 13], [10, 32, -20], [7, 1, -37], [24, -31, -6], [7, 9, -15], [62, -15, 12], [-62, 12, 22], [-88, -2, -4], [-38, -18, 2], [-58, 6, -26], [-27, 31, -55], [14, -10, 15]];
    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_k();".
// 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_k(:prec:=3)
    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_k();".
// 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_k( : 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, 3, -4, 1]>,<3,R![1, 6, 5, 1]>],Snew);
    return Vf;
end function;