
// Make newform 1764.4.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1764_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_1764_4_a_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_1764_4_a_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_1764_a();"
function MakeCharacter_1764_a()
    N := 1764;
    order := 1;
    char_gens := [883, 785, 1081];
    v := [1, 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_1764_a_Hecke(Kf)
    return MakeCharacter_1764_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 := 4;
    raw_aps := [[0], [0], [-20], [0], [-44], [44], [72], [-100], [120], [-218], [280], [-30], [120], [220], [88], [-110], [580], [-380], [-980], [112], [640], [-488], [660], [320], [-248], [220], [-1336], [-100], [682], [-370], [-160], [-300], [-1190], [-2220], [-2414], [2056], [-1060], [-1980], [488], [3476], [84], [-2180], [-696], [2690], [-5310], [-3080], [-4], [2416], [-4972], [4460], [-4550], [2112], [4840], [-4340], [-3520], [2560], [6820], [-3520], [-8690], [-3894], [6556], [-2484], [-308], [1760], [-24], [4890], [8108], [2990], [7860], [6060], [-3024], [-5896], [-2192], [-3410], [5916], [2040], [-11778], [-4708], [5874], [-14040], [-7940], [5214], [-10384], [-6520], [11520], [-13380], [-4098], [-16390], [9540], [-8920], [-8428], [-14520], [-10520], [5436], [18348], [-20944], [260], [14600], [-6820], [12166], [22660], [8490], [14300], [-18294], [-14388], [-25232], [16060], [-23856], [-3168], [12320], [8928], [-4510], [-5830], [24460], [15216], [-12478], [-15996], [-5880], [7510], [16508], [1220], [-13090], [15108], [-1540], [-5780], [10406], [28642], [7560], [20360], [-23276], [3172], [6600], [-16416], [36850], [-37400], [5720], [7356], [38572], [7812], [-45018], [-2740], [-1454], [37120], [7260], [11140], [8760], [-820], [-1320], [-42780], [-29880], [28170], [20800], [-20900], [11640], [-37580], [40832], [42768], [-8760], [-49632], [-36900], [33340], [-5610], [-25160], [1060], [38770], [-24296], [-1672], [-45628], [-2222], [46980], [-30756], [-11220], [-23904], [-13530], [-53480], [-5200], [-19036], [-21474], [38960], [-1580], [-19920], [-51700], [-1950], [50160], [22480], [26180], [-43450], [-49764], [3718], [-28360], [-14800], [20780], [-68884], [-36300], [-49420], [-44816], [-36146], [1690], [-70688], [32256], [53866], [82280], [-18700], [-21280], [43140], [32748], [-58040], [-21540], [-65626], [-58300], [-54450], [-72420], [49960], [-84348], [24640], [20120], [2200], [13320], [-51840], [93670], [31042], [25920], [-97602], [-25600], [84612], [-28366], [-55616], [75152], [-52272], [-14652], [-6390], [-19340], [-74888], [-14234], [19596], [36832], [-72680], [-10990], [40652], [-26480], [101180], [32380], [-34576], [-69514], [-36384], [-74040], [45760], [-54780], [49132], [-85000], [-29370], [-73040], [-9360], [-103040], [-50820], [26308], [-31438], [-20540], [58812], [-30712], [-8800], [45980], [-46300], [12276], [-53160], [73700], [-95862], [-97440], [131540], [-123970], [-85620], [-39980], [-33616], [-8096], [-63520], [-80464], [-64700], [-111674], [-24806], [87780], [126896], [145664], [138680], [92660], [80332], [116016], [-65230], [-48270], [50440], [23760], [75114], [-79772], [153450], [86020], [-25290], [-84700], [15040], [-14204], [66300], [17908], [-173624], [-137610], [-134168], [-42900], [57084], [59870], [-133580], [179060], [76824], [-111650], [-6000], [57198], [-35358], [-77740], [55760], [103040], [86620], [-79016], [-108416], [36338], [76340], [64310], [86220], [148120], [112870], [9480], [-163692], [129180], [240], [30910], [-111562], [-124410], [103560], [156956], [-131452], [-49468], [-6662], [-146320], [50960], [-123288], [76910], [-29430], [-158620], [-78816], [131110], [-99924], [-167700], [202540], [60720], [-116772], [119460], [132890], [-92246], [-40984], [13838], [-83600], [153000], [-117740], [-84370], [216520], [16670], [25440], [-176640], [29612], [-173540], [-105210], [119988], [-73480], [198506], [-213324], [123860], [68200], [-145838], [-65120], [108550], [-45980], [104384], [-138304], [18040], [8008], [148500], [225030], [-161280], [154270], [224180], [115776], [-137544], [82932], [-130540], [-261672], [65822], [199964], [55228], [44980], [226664], [206710], [-54120], [175560], [-40876], [-67122], [219860], [60350], [61160], [129272], [27060], [218120], [4150], [-63954], [110660], [-177060], [217184], [3630], [-143660], [-104852], [132870], [-275340], [270072], [-18680], [-78760], [53424], [-121178], [144100], [-167880], [63180], [288424], [23468], [-290100], [152394], [158740], [-13880]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1764_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1764_4_a_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_1764_4_a_a(:prec:=1)
    chi := MakeCharacter_1764_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_1764_4_a_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_1764_4_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1764_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![20, 1]>,<11,R![44, 1]>,<13,R![-44, 1]>],Snew);
    return Vf;
end function;