
// Make newform 400.8.a.a in Magma, downloaded from the LMFDB on 06 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_400_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_400_8_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_400_8_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_400_a();"
function MakeCharacter_400_a()
    N := 400;
    order := 1;
    char_gens := [351, 101, 177];
    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_400_a_Hecke( : Kf := Kf)
    return MakeCharacter_400_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], [-87], [0], [-1366], [1083], [-5468], [-25269], [-33485], [5838], [125280], [73798], [395926], [-22683], [100148], [1145244], [354882], [-1098360], [-422998], [2558579], [2287428], [-6372443], [2019250], [7972983], [2185935], [5823646], [433062], [-10429492], [21316389], [16030430], [-905463], [16035194], [-3830772], [-9693429], [-44940605], [-49420800], [-20107802], [-78669434], [-69974917], [-83891796], [27366792], [79763415], [52871402], [103225338], [49965067], [33385626], [-92510660], [-56115347], [-77993572], [131342484], [228442340], [-377948598], [113186160], [-200235943], [462823173], [487751346], [-401821122], [455386320], [-508458602], [-646826864], [189546462], [463383803], [-123173358], [564202169], [-551416662], [171415702], [1033492596], [-24123257], [875079661], [-83244891], [397606670], [293709882], [171667050], [-955463356], [-344673458], [-1452725255], [-1878392802], [-1126315770], [1457334316], [1958492517], [-1255936675], [44654775], [-58312108], [-2158067862], [-2028669173], [2516405080], [-2567198457], [828488505], [-345538169], [481628172], [1183069868], [1011914004], [-3786731670], [-720259126], [1796669628], [-2545397660], [-1314007452], [-2503855830], [-1220282283], [1933184273], [-2549203888], [4582911689], [4942073076], [782258448], [1196276025], [-2410706492], [6453941341], [527193759], [8990563257], [-3407202390], [-4008079453], [3611514704], [-416268038], [232065666], [1509137620], [-11851212602], [-3894752118], [-687186652], [-2241571956], [-3445687518], [-7597527855], [-549900688], [-72947258], [3970034256], [-11894291247], [-2089482857], [12181797372], [3807262820], [-2744255130], [515574464], [-9167913788], [7951336900], [12033305358], [14665047088], [18675696496], [14550158517], [7455064685], [-18567068808], [-217239796], [-10402077714], [15401977170], [-9379140092], [-14918954718], [17290505228], [-192632721], [15519895940], [-10410809040], [15450604462], [-2503113189], [-20584244525], [19107304248], [-33306337124], [-9313125138], [4385142833], [-17876414436], [-2847419236], [38464295028], [-10737538190], [5344928670], [14812454461], [21873447372], [24516334104], [-13405631073], [-4647400816], [32485958283], [6065058111], [-21084216402], [-42647626382], [15922634716], [-54367870675], [38627389752], [-4156023615], [-37735608598], [30059736138], [-56882925803], [66024419080], [40958336895], [47022740653], [5403446262], [-58603048072], [-16614091030], [-1020822796], [25896400863], [30468586282], [-68315344239], [-69509009052], [-34792990620], [-30020869514], [23006180213], [-18900080335], [23517788448], [41571849937], [-11563140477], [46637702653], [-69579787338], [-98465451771], [14995860102], [-60968625853], [-61838107298], [-99594561339], [67933485438], [-51197152710], [98009030908], [-40222544], [79039959155], [-23637628380], [-130027206354], [3780841270], [84532288428], [-94548761970], [56159689288], [124200135646], [-137882397918], [65079492848], [-43094134671], [15731337270], [138298672457], [-58802107606], [-71660377083], [-184526457066], [-14834011098], [-70527689578], [134859756940], [144658138665], [-55961652202], [-89434448511], [110169093950], [173624642217], [145166668560], [32726850884], [86459121393], [116531235262], [-100348353335], [-66818478602], [171873930207], [-26388611272], [-34626461706], [234940491005], [141710240472], [32010917085], [-101927486172], [-163009308207], [106274944033], [4667604578], [-184126502140], [-172880121363], [-38490154560], [61218671774], [145638208428], [-65761278455], [-185044810632], [-229498832624], [316147363227], [-197961695946], [253767437315], [222227335752], [20442455985], [-67008658288], [68538912119], [232711503846], [30905470771], [262513566038], [-94736804661], [261791929400], [-42648686348], [165087696891], [31383583615], [-128831710650], [-221004972738], [115432702673], [-68601748338], [100903925492], [116680323359], [-113737564583], [-197300526440], [-199256748194], [282213960188], [-296157522831], [171367634120], [48274123022], [335194991703], [-220304447532], [249255317218], [364404092184], [-499196663308], [-537457508221], [-481177685592], [-392944619198], [-493131577344], [211852830880], [-365455437150], [134604064482], [190097864724], [-88433843403], [-106177737627], [259524177412], [-116263839210], [156864774598], [611225805192], [-634299233460], [256151237399], [133793417407], [-22096354374], [-94825665440], [-284178314067], [-449785082147], [399432086551], [264673054884], [-655364300050], [343557891750], [-745066265078], [-383660470722], [594500274330], [-94007102103], [466728030728], [56577673164], [-251553692890], [-462069430125], [-279734508792], [192012496177], [-555146829405], [49315141108], [164363417851], [588957730572], [52297033778], [-374451591258], [100722117602], [-554490889235], [243040591508], [-635501419806], [-355486501908], [-736867230508], [-771967494834], [-344186790110], [55286387373], [10500717943], [-754229414136], [-362137491400], [-455719986498], [632421691967], [43163089274], [-1623950408], [-955860734169], [-622112606250], [-199550208332], [519763718802], [-430713295620], [-233959229308], [-89127363181], [1028626473648], [-421164903654], [-143592281927], [-46898905394], [-898212595338], [-515315621812], [-61726050310], [-409643788803], [-842840095620], [-1198510297497], [-389829017934], [83500524348], [-450070923674], [-289894295613], [-988754410026], [500266496460], [1080045477119], [-269370947243], [-227591191224], [-942614651032], [755857862237], [1322184225228], [92871840595], [-1160914351332], [-1181909724420], [565294849888], [888847884706], [266318140635], [-1410646820082], [667138308037], [-591282016935], [1525954993486], [-1205865540318], [-1501831754073], [1077028844744], [-600677916999], [-462643377440], [-1360678690662], [-1511570850512], [-1858353556064], [-441643031152], [1619990342034], [830965248635], [496397263212], [717295141485], [-625423870441], [664059240318], [1174608529147], [-1078635584180], [-736723528215], [-424323951437], [1065879862842], [674118007880], [-928649395413], [-1425003951076], [1426490778111], [-1624501204560], [46644755128], [-558480962954], [309404802282], [967841943908], [699803772405], [-61877958143], [-792960838224], [1155719981943], [-375232513292], [-1236830576759], [-1007561262498], [-347999946870], [-1827430235926], [422571743151], [293242292718], [531299576160], [-836498931254], [1830879298074], [-603876949935], [-284363137613], [247186814016], [-1856269555032], [1820487251595], [-1367544808457], [-1603313337990]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_400_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_400_8_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_400_8_a_a(:prec:=1)
    chi := MakeCharacter_400_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_400_8_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_400_8_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<3,R![87, 1]>],Snew);
    return Vf;
end function;