
// Make newform 450.6.a.k in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_450_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_450_6_a_k();".
// 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_450_6_a_k();".
// 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_450_a();"
function MakeCharacter_450_a()
    N := 450;
    order := 1;
    char_gens := [101, 127];
    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_450_a_Hecke(Kf)
    return MakeCharacter_450_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], [0], [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], [36408062], [46657056], [54022656], [54021662], [4671744], [39368794], [12342716], [27492864], [-5982016], [-16969248], [-21002996], [7358990], [-7846700], [-30198912], [70222150], [-16379136], [-33886464], [21833184], [-33933890], [-28563872], [-25288570], [-64028160], [48979906], [22269696], [-53410624], [-41534880], [-61696512], [-87581184], [-3610558], [64863454], [-65015040], [32338944], [62286432], [-95650924], [-96020810], [-68878066], [-14396544], [-92788128], [69827636], [81826176], [70552704], [-53119144], [95460334], [-53211936], [-32805716], [-31266048], [-35986944], [98233610], [-33888428], [-77064000], [-1670400], [-26961504], [-48318310], [-95263660], [-18761664], [22157404], [88722048], [59161910], [75004800], [-49439232], [39509500], [-115611648], [25691086], [142890968], [51036740], [103822080], [-88122488], [44242944], [30808862], [-92772576], [-81922944], [39460608], [137770368], [52472000], [116859268], [-107356786], [73569792], [-14392320], [118499812], [16753920], [-111382624], [-77156352], [-13489442], [144777600], [35712768], [61032614], [105330912], [-92738304], [86327210], [22456048], [119217888], [137061862], [36446188], [23039232], [81418778], [-17333822], [-78149184], [202788032], [26382816], [-160370304], [-34209656], [-169050336], [-181853650], [-67463696], [-115163798], [33748028], [-1266254], [-212685428], [46964352], [181053374], [17961194], [268131072], [107493888], [115811540], [-218788608], [-86667898], [208676008], [223405056], [-67728494], [6273696], [-210649108], [-21071232], [35590368], [29336832], [-94929792], [-203522304], [959602], [-82885152], [-215205052], [249395424], [-288881280], [328893448], [-101603546], [-111208032], [-164043844], [16790016], [39905288], [202783138], [-44033664], [69968690], [273302016], [73276378], [-284577024], [-27661536], [-324909888], [308764792], [-389693952], [157188998], [-119510016], [-252503040], [-189273902], [68429184], [6159776], [-3163562], [-190458720], [269848564], [-296791104], [-150027010], [376332680], [-83889896], [128219136], [444433824], [-53294638], [-376024416], [226297364], [-174003456], [-48563152], [-281653632], [-229578946], [417495936], [-196212682], [-307842212], [-319374518], [-182178624], [225501216], [185980676], [149722080], [32515200], [-156324454], [-162726080], [66777600], [83561184], [400007384], [-349140566], [-49519200], [515017804], [-126684010], [-340655616], [-269052672], [-135714816], [-300723648], [67047936], [516029818], [188199552], [-217158912], [-464959104], [213665776], [-223059722], [-132818592], [316925044], [-106955782], [322557312], [-43628056], [-307368960], [-507808608], [24070652], [-399268802], [-460949376], [-134348544], [-681707234], [400650048], [-252938102], [621120864], [389107776], [-338983436], [581261568], [701448920], [637675776], [-230032228], [390548278], [-368649488], [118518528], [140384318], [-326947296], [580649472], [423827896], [215622912], [386014906], [-397149388], [100354368], [417297608], [187594080], [107555102], [-202381632], [-509209028], [411365568], [-572733792], [204596300], [394528558], [-135478464], [136059256], [-105961344], [-75484706], [-195279456], [-392109696], [518109512], [121580938], [-198181728], [205104384], [-678781292], [184446144], [-424660992], [-283949088], [684163258], [117307392], [-634790784], [-204911066], [153694560], [-67635840], [749319552], [469645400], [-571165440]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_450_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_450_6_a_k();".
// 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_450_6_a_k(:prec:=1)
    chi := MakeCharacter_450_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(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_450_6_a_k();".
// 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_450_6_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<7,R![-148, 1]>,<11,R![384, 1]>,<17,R![-576, 1]>],Snew);
    return Vf;
end function;