
// Make newform 99.9.c.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_99_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_99_c_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_9_c_a();".
// 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_9_c_a();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_99_c();"
function MakeCharacter_99_c()
    N := 99;
    order := 2;
    char_gens := [56, 46];
    v := [2, 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_c_Hecke();"
function MakeCharacter_99_c_Hecke(Kf)
    N := 99;
    order := 2;
    char_gens := [56, 46];
    char_values := [[1], [-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 := 9;
    raw_aps := [[0], [0], [-1151], [0], [-14641], [0], [0], [0], [531793], [0], [-1541233], [716447], [0], [0], [6080638], [15265438], [4101553], [0], [19806767], [-7043087], [0], [0], [0], [84100993], [-81155713], [0], [-3627838], [0], [0], [-101857247], [0], [0], [-363889247], [0], [0], [0], [-1205703073], [915081122], [0], [0], [-1217787887], [-2133265633], [327580753], [0], [0], [3135844802], [0], [4769506607], [0], [1323799487], [0], [0], [0], [2392024273], [-490675202], [0], [10285140958], [0], [0], [0], [0], [0], [0], [17481961918], [-3392091553], [-16649875967], [-14433190033], [0], [0], [0], [11058866113], [0], [-18367234033], [0], [31414911887], [-33498237167], [45786060193], [45726896162], [-45628828802], [0], [-18150491042], [-24380593438], [0], [69835918367], [0], [39207592273], [81058731073], [0], [0], [90859228847], [-80810161967], [0], [-27650152753], [0], [-105616178398], [0], [-111657987647], [131833528513], [0], [0], [0], [0], [0], [0], [0], [-21725112193], [-108008577122], [0], [-243784852802], [0], [0], [0], [58989120958], [289837535567], [138624088367], [264901289953], [108190430927], [82150241713], [-315667908287], [0], [-174258628993], [0], [0], [339818355358], [-398526352753], [0], [-492857674753], [432961440433], [48103779407], [0], [0], [0], [-603334355473], [645562562402], [0], [0], [249173298718], [0], [-735103677887], [0], [0], [0], [-301900203793], [0], [-444761282113], [-666474181007], [0], [0], [-274045979953], [-592718584322], [0], [770772426433], [-1115009652958], [0], [654958754402], [-876790346882], [0], [836301546238], [0], [0], [1606734411313], [0], [0], [-1751964834287], [1655433708193], [-471491735567], [-1638917153278], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_99_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_99_9_c_a();".
// 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_9_c_a(:prec:=1)
    chi := MakeCharacter_99_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 9));
    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_9_c_a();".
// 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_9_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_99_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,9,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;