
// Make newform 99.4.d.b in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_99_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_99_d_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_4_d_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_4_d_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_99_d();"
function MakeCharacter_99_d()
    N := 99;
    order := 2;
    char_gens := [56, 46];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_99_d_Hecke();"
function MakeCharacter_99_d_Hecke(Kf)
    N := 99;
    order := 2;
    char_gens := [56, 46];
    char_values := [[-1, 0], [-1, 0]];
    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 := 4;
    raw_aps := [[3, 0], [0, 0], [0, 14], [0, 12], [33, -11], [0, -21], [126, 0], [0, 63], [0, -85], [-24, 0], [-70, 0], [182, 0], [294, 0], [0, 3], [0, 77], [0, 104], [0, -364], [0, -231], [-880, 0], [0, 239], [0, -126], [0, -546], [1218, 0], [0, 1085], [-196, 0], [-546, 0], [-826, 0], [108, 0], [0, 837], [0, -1147], [0, -126], [-588, 0], [0, 239], [0, -2037], [-2082, 0], [0, -1158], [-2338, 0], [128, 0], [1176, 0], [1092, 0], [0, -274], [-1330, 0], [0, -2407], [0, 2418], [-3798, 0], [560, 0], [0, 1347], [-2968, 0], [-3444, 0], [-574, 0], [5310, 0], [-1236, 0], [0, -2436], [0, -4018], [0, -1057], [-4056, 0], [0, 4424], [0, -1134], [0, 3549], [2874, 0], [0, -1533], [9240, 0], [0, -2709], [0, 203], [-5992, 0], [0, 4460], [-1708, 0], [0, -4854], [2922, 0], [0, 5691], [0, -5719], [5436, 0], [-11536, 0], [0, 7287], [-3724, 0], [0, 3605], [0, 50], [8498, 0], [0, -1849], [0, -336], [0, -9058], [6554, 0], [-10692, 0], [7616, 0], [0, -9492], [0, 12956], [0, -463], [0, 4260], [-4452, 0], [14978, 0], [0, -6538], [5124, 0], [5096, 0], [13164, 0], [-4228, 0], [10332, 0], [0, -3892], [0, 455], [0, 7875], [0, -11529], [0, 3549], [4932, 0], [-17598, 0], [-18546, 0], [0, 14217], [-3220, 0], [0, 12740], [-6426, 0], [0, -14827], [0, 6384], [0, 11886], [0, -4095], [0, 41], [15680, 0], [-24334, 0], [0, -15709], [5096, 0], [0, 9779], [0, -1282], [-16566, 0], [-6118, 0], [0, -14802], [12558, 0], [0, 1562], [2828, 0], [-8424, 0], [-8890, 0], [0, -16933], [-10024, 0], [0, 4389], [0, 26397], [-14340, 0], [-33334, 0], [-4354, 0], [-22638, 0], [0, -19110], [0, 14504], [0, 26691], [0, 13622], [-27486, 0], [0, -28917], [25770, 0], [-42208, 0], [-8082, 0], [-11914, 0], [0, 19229], [0, 3675], [24402, 0], [22232, 0], [0, -15007], [0, 24171], [0, -20335], [8120, 0], [9996, 0], [-37744, 0], [0, -1273], [0, -17874], [0, 28049], [0, -2016], [-3990, 0], [0, 28832], [11322, 0], [0, -1974], [0, -4396], [0, 39929], [0, 1967], [-7882, 0], [0, 4179]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_99_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_99_4_d_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_4_d_b(:prec:=2)
    chi := MakeCharacter_99_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_d_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_4_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_99_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-3, 1]>],Snew);
    return Vf;
end function;