
// Make newform 1521.4.a.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1521_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_1521_4_a_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_1521_4_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1521_a();"
function MakeCharacter_1521_a()
    N := 1521;
    order := 1;
    char_gens := [677, 847];
    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;

function MakeCharacter_1521_a_Hecke(Kf)
    return MakeCharacter_1521_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 := [[-4], [0], [-17], [20], [32], [0], [13], [30], [-78], [-197], [-74], [-227], [165], [-156], [162], [-93], [864], [145], [862], [-654], [215], [-76], [-628], [266], [238], [819], [1638], [-522], [-1634], [-327], [-2158], [-730], [-1671], [912], [2115], [514], [2901], [2360], [-280], [-1326], [-4264], [-403], [1246], [267], [-1278], [4238], [3070], [-5378], [3974], [-6298], [-4030], [984], [943], [2730], [1885], [-4032], [-4006], [-4296], [-5551], [5557], [3120], [-8301], [8678], [-8658], [-5250], [-6413], [3488], [-1833], [-7230], [-5258], [-3163], [10068], [7438], [-9683], [-1062], [3532], [11063], [-5986], [-5935], [-15089], [10814], [-6535], [-1980], [-6929], [-4576], [8812], [-1918], [-11761], [-901], [1372], [6396], [-3270], [19920], [-6552], [1746], [-14692], [-8077], [-11247], [2732], [-18375], [-10346], [-345], [8580], [19682], [26624], [-14101], [-1408], [1241], [-11078], [-13817], [8270], [22273], [18989], [72], [-23380], [-6383], [-17104], [-6994], [5250], [4340], [-4179], [22867], [-5410], [13578], [12744], [-16406], [709], [7644], [-15808], [-2583], [4076], [34056], [-364], [-6914], [-13982], [-18066], [-14434], [-15398], [36842], [-41511], [23066], [-28838], [27456], [33572], [-45799], [-32286], [20937], [7189], [-32498], [-8428], [-6847], [-29731], [-23738], [-27588], [-37128], [-20516], [-21006], [-20427], [33191], [36422], [39630], [-57642], [2162], [19758], [12489], [28658], [-42794], [-52583], [10290], [26091], [-42774], [23173], [51766], [-9837], [-15366], [-23103], [-15138], [11199], [5668], [56214], [17414], [-69108], [52457], [-1434], [-54308], [-29133], [-69914], [-22486], [60899], [2580], [13951], [32054], [-72200], [-17726], [-27924], [8658], [15487], [26533], [-40931], [5250], [72663], [63182], [-66335], [33659], [80502], [-39977], [81124], [-46204], [24085], [-11960], [31187], [-55902], [-8266], [-18766], [23318], [20183], [71606], [-53471], [-8688], [26214], [-67351], [-71558], [-3631], [-42428], [-48984], [-15275], [-100479], [70900], [19006], [-50096], [30381], [-74020], [20570], [-55263], [-5822], [62370], [27554], [17387], [-70694], [-87388], [-20362], [-31538], [59828], [81501], [-99363], [-16536], [3946], [-70740], [-39724], [25662], [21626], [-54110], [-5844], [-27053], [1818], [-280], [-93442], [-10084], [46098], [25783], [41028], [6680], [60233], [14341], [-59391], [37078], [28570], [-19335], [53762], [36179], [9893], [24444], [92515], [49478], [-91674], [-147182], [87356], [-53735], [-90317], [-38454], [-126648], [-109102], [73136], [10937], [153056], [-112762], [81031], [-46089], [-129304], [-122759], [55638], [75684], [53361], [-112002], [-6134], [-9685], [-24438], [-949], [160444], [-90538], [9659], [76043], [111280], [93052], [125270], [70971], [-15860], [7009], [-103344], [273], [14584], [54414], [51101], [45580], [69478], [-64933], [-37314], [854], [149806], [-38103], [110162], [-43754], [147795], [74328], [-54438], [87531], [133634], [-107948], [81224], [115503], [22289], [49047], [9778], [130474], [-63176], [68090], [45325], [-133634], [-116281], [-108680], [44646], [-62779], [59143], [99164], [13594], [-4992], [-11555], [92476], [-722], [184106], [-194736], [3894], [-153885], [161018], [-99619], [-182015], [155980], [14868], [178113], [31898], [111297], [233805], [49554], [119544], [-197444], [-117674], [786], [-100808], [83187], [19758], [55770], [-156202], [-85425], [101610], [-62775], [210882], [-95472], [-91178], [-97879], [165243], [-157086], [67593], [101732], [235842], [153316], [64070], [13680], [134550], [89522], [18094], [211907], [79231], [25096], [180572], [-270836], [157198], [139364], [-188526], [-24308], [-141681], [225710], [205218], [-46988], [95049], [124351], [203894], [204910], [48265], [-82764], [89628], [100607], [29755], [150252], [-137462], [-89193], [-247821], [133090], [7178], [-28230], [-138164], [-145561], [-94311], [166552], [231034], [-75058], [-205729], [-126048], [-266751], [-220798], [-303630]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1521_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1521_4_a_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_1521_4_a_b(:prec:=1)
    chi := MakeCharacter_1521_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_1521_4_a_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_1521_4_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1521_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![4, 1]>,<5,R![17, 1]>,<7,R![-20, 1]>],Snew);
    return Vf;
end function;