
// Make newform 18.6.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_18_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_18_6_a_c();".
// 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_18_6_a_c();".
// 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_18_a();"
function MakeCharacter_18_a()
    N := 18;
    order := 1;
    char_gens := [11];
    v := [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_18_a_Hecke(Kf)
    return MakeCharacter_18_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 := 6;
    raw_aps := [[4], [0], [96], [-148], [-384], [-334], [-576], [-664], [3840], [-96], [-4564], [5798], [6720], [-14872], [19200], [-7776], [13056], [42782], [36656], [-64512], [-16810], [28076], [66432], [81792], [-29938], [-178656], [-115228], [76032], [-231118], [-142464], [-988], [-224256], [278976], [177200], [-236064], [-482836], [381086], [162920], [566016], [218208], [412416], [-25558], [-400128], [699650], [-406368], [-361996], [151856], [-1093324], [566400], [-587206], [-579456], [584448], [-414130], [1899648], [-447744], [67584], [-564192], [720308], [-141142], [-584448], [177056], [956832], [2882864], [2600448], [-2580790], [-2311008], [-637024], [3386654], [-2778240], [1555358], [-2117760], [-2174976], [1053356], [-677098], [-5107480], [-1632000], [4465632], [-611026], [6091584], [2891078], [2984064], [822074], [-6324480], [-851902], [-334732], [1762176], [-6518592], [-92074], [-257568], [3962276], [3489408], [-513024], [4144988], [-2758656], [660896], [944640], [7837728], [-3296448], [6502376], [9827138], [-3425800], [6433632], [-753024], [11148096], [191024], [10372178], [2970624], [7558272], [-6691584], [-3203590], [-13558492], [-10765402], [9335040], [9076640], [-11336740], [-15544896], [-8800264], [-10844928], [9887712], [14615040], [15779222], [6649394], [-4425504], [4678272], [-5541016], [6534432], [-3865414], [-4806144], [-19058980], [5696162], [18490160], [-9903360], [-9059140], [-11667670], [12739776], [10678274], [9186336], [-9282088], [-21792], [-12308544], [-23463592], [14420640], [34341884], [-21332736], [2637506], [10057728], [-23074762], [-35164608], [16602200], [29700864], [-11794258], [21037824], [21219176], [-22881792], [-9686920], [-9381120], [-42187012], [30455616], [14784698], [9919968], [-2916096], [-14086080], [15194852], [56121984], [-34562496], [-55638528], [-36002788], [23581094]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_18_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_18_6_a_c();".
// 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_18_6_a_c(:prec:=1)
    chi := MakeCharacter_18_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_18_6_a_c();".
// 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_18_6_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_18_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![-96, 1]>],Snew);
    return Vf;
end function;