
// Make newform 1764.4.a.f in Magma, downloaded from the LMFDB on 24 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_f();".
// 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_f();".
// 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], [0], [0], [0], [-19], [0], [107], [0], [0], [-289], [323], [0], [71], [0], [0], [0], [182], [-127], [0], [-271], [-1387], [0], [0], [-1330], [0], [-1801], [0], [2213], [0], [2267], [0], [0], [467], [0], [1748], [-3850], [-3400], [0], [0], [0], [-4699], [0], [-3961], [0], [-5236], [6032], [-3220], [0], [2357], [0], [0], [-7378], [0], [0], [0], [0], [812], [9197], [0], [-9469], [0], [-3943], [0], [-9109], [0], [-10891], [12293], [0], [-11914], [0], [0], [9413], [-12601], [-6103], [0], [0], [-14257], [0], [-16543], [0], [-6679], [0], [16739], [14924], [0], [0], [6497], [0], [11969], [0], [0], [-6103], [0], [21743], [0], [0], [0], [23921], [1889], [1640], [0], [0], [0], [-23941], [-9613], [0], [0], [0], [17351], [6137], [17390], [0], [26693], [1892], [0], [19259], [0], [0], [0], [33731], [9899], [0], [0], [-20179], [0], [36146], [0], [38033], [-39331], [6047], [0], [-17569], [-41470], [0], [-34417], [0], [43400], [0], [0], [39368], [0], [-12220], [0], [30077], [0], [8279], [0], [31304], [0], [50150], [0], [52109], [0], [-49447], [0], [46817], [0], [40283], [0], [0], [0], [-3907], [0], [0], [0], [-14041], [-62893], [-59347], [0], [0], [-40138], [0], [-48259], [44207], [0], [57581], [0], [67340], [30203], [-67537], [0], [2141], [0], [0], [0], [56810], [42569], [21797], [0], [-76519], [0], [-37312], [0], [0], [0], [75419], [-56950], [0], [0], [0], [19241], [-83230], [-49714], [0], [0], [85484], [0], [0], [-41272], [-80137], [0], [48620], [0], [0], [-88489], [15860], [0], [0], [0], [-24751], [-101593], [0], [-77380], [0], [53534], [0], [0], [67517], [0], [-91279], [-69496], [13049], [0], [-63649], [0], [-36307], [0], [0], [0], [0], [84131], [-109261], [-49786], [0], [0], [39797], [0], [-118816], [0], [106037], [0], [0], [-126667], [0], [0], [116801], [-17803], [0], [4970], [127889], [0], [-131563], [97631], [0], [-139897], [0], [0], [142760], [0], [-107749], [92720], [93221], [-120763], [-71890], [127820], [0], [-33193], [-135001], [0], [0], [-96139], [0], [-44749], [-138040], [0], [161729], [0], [128483], [0], [0], [0], [0], [0], [-42589], [0], [-152101], [0], [0], [-113923], [75530], [0], [171236], [0], [161279], [-21223], [0], [164033], [0], [122039], [0], [0], [0], [-131560], [0], [45197], [0], [0], [-150211], [0], [2879], [194867], [0], [-150499], [0], [182789], [-97147], [199439], [0], [0], [141821], [0], [207107], [0], [-213571], [0], [6806], [0], [-207919], [-88237], [-172621], [0], [0], [-220681], [0], [0], [226529], [188480], [0], [0], [-222712], [70310], [0], [178220], [35027], [0], [0], [0], [0], [0], [7310], [0], [0], [0], [240623], [152570], [0], [102059], [-179062], [0], [-185113], [0], [0], [-117199], [-162163], [0], [0], [178850], [0], [248957], [0], [0], [-205900], [0], [-271096], [0], [193229], [219170], [277829], [0], [214073], [0], [0], [-164467], [0], [-254410], [206444], [0], [-63631], [0], [-112714], [0], [191900], [0], [0], [251099], [-41830], [0], [222731], [0], [108290], [0], [0], [235352], [40697], [0], [0], [-284293], [0], [0], [0], [281447], [0], [0], [277130], [0], [0], [0], [-323512], [0]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1764_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![0, 1]>,<13,R![19, 1]>],Snew);
    return Vf;
end function;