
// Make newform 99.2.f.b in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_99_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_99_f_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_99_2_f_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_99_2_f_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, -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_99_f();"
function MakeCharacter_99_f()
    N := 99;
    order := 5;
    char_gens := [56, 46];
    v := [5, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_99_f_Hecke();"
function MakeCharacter_99_f_Hecke(Kf)
    N := 99;
    order := 5;
    char_gens := [56, 46];
    char_values := [[1, 0, 0, 0], [0, 0, 0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0, -1], [0, 0, 0, 0], [1, -2, 1, 0], [0, 0, 0, -3], [-4, 3, -2, 2], [-5, 3, -3, 5], [-1, 1, -1, 0], [0, -3, 4, -3], [3, 0, 4, -4], [4, -4, 0, -2], [2, 1, -1, -2], [-2, 2, 0, 3], [0, -8, 7, -8], [3, 0, -2, 2], [0, 1, -1, 1], [-8, 9, -9, 8], [7, -7, 0, -1], [3, -6, 3, 0], [5, 0, 9, -9], [9, 0, 9, 0], [2, -2, 0, 0], [-3, 7, -7, 3], [-6, -3, -6, 0], [-7, 0, -4, 4], [-7, -6, 6, 7], [-3, 3, -3, 3], [0, 0, 0, 6], [0, -2, 3, -2], [0, 0, 0, 0], [0, 6, 3, 6], [-6, 0, -6, 0], [12, 0, 3, -3], [-4, 5, -4, 0], [3, -3, 0, -4], [0, 15, 0, 0], [0, 0, 2, 0], [0, 6, 0, 6], [6, 9, -9, -6], [-11, -2, 2, 11], [0, -1, -16, -1], [0, -2, 1, -2], [-4, 15, -4, 0], [-4, 4, 0, -1], [-9, 13, -9, 0], [-26, 0, -1, 1], [-7, 0, 6, -6], [-2, -13, 13, 2], [0, -6, -3, -6], [-16, 16, 0, 1], [-10, 10, -10, 10], [13, -20, 20, -13], [0, 9, 3, 9], [6, 0, 18, -18], [10, 1, -1, -10], [7, -7, 0, -23], [5, 0, -7, 7], [4, 18, 4, 0], [15, -15, 0, 3], [-16, 25, -25, 16], [-2, -2, -2, 0], [0, -10, -6, -10], [-8, 8, 0, 13], [-14, 0, -9, 9], [0, -4, -5, -4], [24, -11, 24, 0], [13, 7, -7, -13], [-9, 0, -22, 22], [-6, 6, 0, 10], [8, -16, 8, 0], [0, 18, 1, 18], [4, 0, -4, 4], [2, -2, 0, -16], [-9, 9, 0, 9], [15, 0, 12, -12], [-12, -9, -12, 0], [1, -17, 17, -1], [15, -15, 0, 0], [-29, 0, 6, -6], [5, -23, 5, 0], [24, -16, 16, -24], [11, 0, 17, -17], [0, 15, -28, 15], [-12, 29, -29, 12], [0, 0, 0, 6], [13, 0, 6, -6], [0, -18, 30, -18], [-18, 22, -22, 18], [-11, -15, -11, 0], [13, 0, -5, 5], [2, 0, 11, -11], [-2, -11, -2, 0], [5, -25, 5, 0], [0, 10, 23, 10], [23, -23, 0, -12], [9, -9, 0, 3], [0, 26, -12, 26], [0, 1, -23, 1], [-24, 24, 0, 24], [-21, -1, -21, 0], [-5, 14, -14, 5], [0, -1, 21, -1], [-5, 5, 0, 23], [17, -1, 1, -17], [0, -26, 8, -26], [-9, 0, -27, 27], [-2, -12, -2, 0], [-28, 28, 0, 7], [15, 0, 23, -23], [-6, -12, -6, 0], [-24, 24, 0, 5], [-2, 9, -9, 2], [0, -14, 26, -14], [-39, 0, -12, 12], [0, -10, -15, -10], [-15, 15, 0, -22], [0, 19, -24, 19], [-9, 33, -9, 0], [10, -21, 21, -10], [13, -13, 0, 30], [-12, 0, -14, 14], [-18, 0, -5, 5], [7, 5, -5, -7], [4, 16, 4, 0], [29, 0, 16, -16], [36, -30, 30, -36], [0, -27, -6, -27], [19, -37, 19, 0], [12, -12, 0, -21], [34, 0, -3, 3], [6, -6, 0, 28], [25, -25, 25, -25], [-10, 29, -10, 0], [0, -29, -6, -29], [11, -3, 3, -11], [-5, 21, -21, 5], [-31, 0, -27, 27], [0, -36, 9, -36], [6, 14, 6, 0], [2, 12, 2, 0], [7, 14, 7, 0], [0, 19, 15, 19], [12, -12, 0, 21], [29, -33, 33, -29], [24, -42, 42, -24], [-27, 27, 0, 26], [0, -6, 33, -6], [-8, 5, -8, 0], [17, 0, 40, -40], [33, 0, 16, -16], [-16, 2, -16, 0], [0, -12, -1, -12], [-22, 0, -5, 5], [0, 12, 18, 12], [0, 0, 0, 3], [-21, 30, -21, 0], [31, -23, 23, -31], [42, -21, 42, 0], [16, -23, 16, 0], [-19, -3, 3, 19], [35, -23, 35, 0], [-7, 0, 7, -7], [-10, 10, 0, 54], [9, 0, 27, -27], [0, 32, -18, 32], [-37, 17, -17, 37], [-28, 28, 0, 18], [-3, 0, -15, 15], [21, -21, 0, -21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_99_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_99_2_f_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_99_2_f_b(:prec:=4)
    chi := MakeCharacter_99_f();
    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_99_2_f_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_99_2_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_99_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -2, 4, -3, 1]>],Snew);
    return Vf;
end function;