
// Make newform 240.8.a.h in Magma, downloaded from the LMFDB on 19 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_240_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_240_8_a_h();".
// 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_240_8_a_h();".
// 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_240_a();"
function MakeCharacter_240_a()
    N := 240;
    order := 1;
    char_gens := [31, 181, 161, 97];
    v := [1, 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_240_a_Hecke(Kf)
    return MakeCharacter_240_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]];
    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 := 8;
    raw_aps := [[0], [27], [-125], [-1380], [3304], [8506], [-9994], [-41236], [-84120], [132802], [55800], [228170], [-139670], [755492], [-836984], [1641650], [989656], [-1658162], [4523844], [389408], [5617330], [-3901080], [9394116], [2803746], [5099426], [15172266], [-4705268], [-2631204], [-4300594], [3982334], [-28017676], [8199192], [-1666458], [58745596], [-19369742], [53395208], [20452922], [733588], [-16892520], [11818554], [31374640], [-58355530], [-4061656], [-133220566], [13077138], [69850232], [-32853500], [-69519356], [230778860], [146157054], [311907438], [-227310296], [-198482654], [132535968], [-35864226], [479640488], [-70876414], [-407490256], [-79347702], [-87006846], [-277611876], [246490482], [-384964564], [-46443456], [210558002], [-960971366], [-399922812], [269184538], [821868004], [648353902], [652666326], [877430856], [-286988988], [-1770132542], [-1463113060], [990456408], [-796901166], [795584194], [-2016318342], [-54802966], [-110924536], [-2120637986], [2485326480], [29995642], [1048734680], [-204679116], [2639623418], [1837383370], [3576778602], [-1280677668], [4199840036], [-2482019880], [1080653044], [947786672], [2076919244], [-2732654896], [4524697154], [-279532670], [-1463762068], [-2721936690], [673048220], [-7199944134], [-5801136324], [-1670688894], [-5701386956], [2430634946], [3333782076], [3422803438], [928661192], [-6569738374], [3507897644], [3639701194], [7825585182], [-8351725844], [21681968], [-7853610750], [8818195092], [8719969024], [-665755238], [9995126680], [-2891603978], [-5832356070], [-12252424782], [9692933868], [-3332853076], [-2731988702], [344619886], [-3357493728], [13201695988], [4301789378], [18241424636], [236847232], [14432560904], [16493357962], [-2968454046], [8154002498], [-2780590398], [8209818596], [-54791606], [2049730330], [15947121708], [9659231274], [-3903976844], [13222440964], [12048232350], [-31060309000], [25215704442], [22531444814], [-4078626820], [11064318272], [36059458650], [-26295355606], [-2788729388], [18296966616], [1212451716], [20918089408], [18154477040], [-33922505414], [36402256938], [50711742490], [-23099415492], [-4163677114], [52046779324], [364572720], [-18178540474], [18577791360], [-18252005552], [-1245342558], [-45668224990], [53984362594], [-27751565496], [32607826438], [-43004334264], [-13448384478], [-37422643432], [-25517736414], [-55844036100], [-33353795934], [-31435140188], [-54517248890], [-57489586340], [81142941704], [46680310234], [-19657644506], [-50444380008], [18875394770], [-22806991974], [-68251267996], [-90764321398], [12682519816], [3082692298], [43278949876], [45907468692], [76331389226], [-39217289436], [-79594421786], [27089200786], [23417804394], [61086510414], [-48212011072], [-18953195622], [-52448361768], [-39548481102], [56702600146], [21048330072], [-103591659510], [21026287864], [60441851068], [127196466258], [79082835556], [-77938154998], [136344974242], [64656768540], [79782497524], [93486615936], [44792777738], [9512246708], [-87857832582], [-143268555328], [61609750698], [-97745982626], [-27035518232], [90437843490], [188863390316], [-84510987724], [-22065563410], [-87545748858], [-141620018656], [-99457488124], [58505238456], [121447379130], [-73719278956], [65797513744], [-100486190302], [-243524106884], [-148397444736], [-13693926318], [223361892322], [31401860904], [43072393392], [-72510881844], [136544617236], [90790441028], [-73996960962], [220365294582], [30703509376], [26203972132], [240022056160], [186687324436], [-112037305232], [180733720450], [13545573122], [23158431720], [51661113354], [-246025629878], [158361220248], [237425916742], [-225504039476], [-210084390942], [17471954738], [-161896176988], [-18689669620], [-333822071218], [-307561803246], [-10512839394], [-82485499588], [-202189285654], [400252038298], [244582607500], [428686727954], [-347633533066], [-248445797044], [-262174824478], [76583335128], [99139519194], [-278081712932], [54267027252], [74983230750], [-9271541094], [-380633799240], [1546792880], [-220666002664], [175333365920], [233175750366], [117345497076], [-370978741320], [173359728082], [366587239362], [418631004568], [283108623330], [-531809619254], [-354579588124], [-472898293218], [129759552456], [-54078174478], [-101963626470], [-219574994072], [471282220866], [541843984920], [365876019388], [390336455874], [18308465882], [436429902368], [703830462276], [195486189572], [-509791890574], [302521015836], [306956606726], [-190515079776], [528743042898], [491377767312], [-612976804614], [-398317664790], [-626780876372], [-227354481664], [526201604746], [-463949214672], [-96523022472], [-690400273830], [867168749922], [-366257818100], [736292421170], [-747272640334], [-93705223588], [-594310785282], [93279572386], [-725985987980], [440793011828], [-159664346472], [165898891634], [-92222062698], [129537727946], [-524504393720], [679820292884], [-515814352388], [-269897717332], [-28146752842], [-79539895706], [-792846077062], [-485701032396], [-823504206542], [-895751923730], [787417912554], [937312907448], [825875571194], [-985859023920], [656901189518], [558133162292], [666607436280], [-646986213662], [222087095364], [-1259804449302], [912777468850], [11074277684], [-804153003114], [-198368763426], [636458708624], [340774004376], [-726372732402], [445826039896], [719133254146], [-1053205479046], [-1071494865528], [-807862435784], [-344941418684], [508911004090], [1051924212426], [-611209903044], [-1050396519574], [-673625606064], [-600897540276], [822917946280], [-531740275254], [-1050305621064], [-7653440510], [1572327410224], [-480738413568], [-928346981614], [-110632647222], [-357829318166], [-163400715398], [908235760998], [-638757942628], [247200626086], [1096562832708], [20673495832], [-108518187464], [-312338572246], [1619299602852], [644312549856], [455928005634], [-4720553326], [747290156008], [-1562291253740], [-115099558328], [-1304809779718], [-1474919562800], [747434754234], [113988714260], [1316008657794], [810162737638], [830133757326], [-1986732518620], [-405603952010], [-198445358054], [1406706543680], [-968668890814], [1323708330106], [-152029910892], [-545734561224], [-551690459934], [1646900431554], [2164499011900], [-796382010516], [1169724976754], [1096241037794], [1211276632544], [1143373900316], [-1133032525410], [-2184082034776], [526789955578], [1661164648738], [-1083827788968], [-887001280064], [-670575666094], [-1649154244486], [-1029381975100], [-714054504630], [859264261204], [-2335019279384]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_240_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_240_8_a_h();".
// 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_240_8_a_h(:prec:=1)
    chi := MakeCharacter_240_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_240_8_a_h();".
// 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_240_8_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<7,R![1380, 1]>],Snew);
    return Vf;
end function;