
// Make newform 1080.2.d.g in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1080_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1080_d_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1080_2_d_g();".
// 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_1080_2_d_g();".
// 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
        poly := [45658, 157882, 302532, 244612, 78784, 37798, 49749, 23019, 2243, 1456, 1194, -96, -78, 36, -3, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2689937510010371245965572803175936, 2732541523404331265441186456675686, 4435294168143189478419454834972696, 1317727487635383788120858856767730, -127696115268596669962320832755376, 841460963211356679462474100466914, 405163675978233988011284373334759, -108042448863972106970412784976819, 32334063594810278586330766195352, 30709209012705539373498878120695, -12862260501825953682216710285180, 690872865962071542233180933875, 1270521640866234227739985065478, -567607760522674625597150073539, 100724542642732097318917507681, -2968246925305068366016861850], [52568564805374419618599547102869772, 99826922284602257453687426322992042, 45141980695973382302092265699617754, 45344381428787029708635032124128754, 23858007035730180850771394555342950, 3777963322588830713383769205094970, 8467394825371394503090403915954102, 5106555474537776169119004536951975, -714160107263881642262295627250559, 238764241238035190291337795854198, 355136783502262874112895101270311, -123794214856126853131814258112592, 1280995371795872811510501294005, 14115708744432730405169099899238, -5698040312704688625458600718907, 912048055583909114267368116599], [-1337216947772539479600848983644, -1928951608894628214963884861990, -3634579657946262309219440042838, -3153495124008224044780151788338, -271094238468154240554342214662, -284369681153041160135598123346, -778979448410751676709062383580, -152801224095641326968018331549, 38817024668411052751281577847, -40685642718566191529383659706, -9395358398162249739183319193, 6391954850963295425202433488, -1011247126758606846765662537, -391047289941977590220537926, 247541873575584587339885975, -46449794153507378462464313], [-16990181512359178068636832973313631, -37790026492467100033699517719869512, -34239618598105509035514735539030978, -8766523120552899761705807462372940, -2653892245875051729944810054208199, -7002054938507546354794938174196250, -3066101441992904461906948617949439, 112460540849969375003801107126426, -246655569135727184170919021724532, -185366960858756617517495506128185, 42511857392227576452026209731523, 4866844975149741053755465785922, -6464862877355428644800549389922, 1697786711553689228248940920153, 37263251707070194444744863223, -87860439233952427110875383394], [-3845233369531395357024450336700672, -26638241791253248321152919660354946, 27445211779793228070230591753899822, 37060205580310396209554068038099546, -4567354000268000403299377705361686, -817154759746499552597480606920214, 8850079069182169877354024787862828, 1658000774363623058550026565275425, -720008839227572037312814014258415, 488411285479145977958195793277498, 130730311624230541313015806067437, -84054762402289778006839537753592, 12930280575804544144208147028949, 5010504368748251349071998447510, -3031628099783860500899046757931, 536496212669605662673476107317], [65880008362435834886181250593669910, 15120231805568494292014688489771966, 61649576778676627296697081786586078, 118752405678956579179929249305407242, 20020407352584882924212303378445466, -1582419709853013369276506762342836, 23571699770791007139044388303758879, 7910146154395458408804969175067633, -2190048354033453433242699770361608, 1073426688804900417923904013442399, 549709591732674664049541903736694, -262639752234699500369196183809149, 26062463748198874069772966237588, 20188649462793866218084252073975, -9922501609277149191081771501907, 1678931869961246025751482656342], [-7120121894870179000886142728057188, 27119226155482261014122777726908648, 212736973128501478417582565566352002, 101455532695234330579648455701256366, -25090437022170849116478328135778602, 27237627010917239373486356055469288, 32918211766122447150143301884213953, -1054533793274871176390716695929090, -460678775942581977273304875462640, 2260578209286740445791742177475222, -236469348276488075538345678285800, -149542567276496964297616324748777, 77421827507796914931461143743850, -11788544643920816249790473945714, -2403344268270664082095633125341, 981733289547917239205031922273], [-32851625501020684039007608623164452, -125262376724566867740849963423298916, -252593482614583269343278949145479346, -96306096851138238841515994060985652, 2199535618314716984336541288078260, -38059966625057139546185394795765896, -29475107380374563927888171052871031, -1294217798191495103187598002166976, -413064522998180380861220400335713, -1715007642909326614286140833046751, 46607007193792557113868933579847, 142031045943419277209906176650229, -50280854846991367442576118955505, 518950110945043540986636785293, 4968738705497027910097754403238, -1293743257681942256442186325799], [758155685661014305806352162382534, 350327932206386288345113888784732, 6165283908794913503161657485950632, 5257400432853981328558449299067812, -302412293843340936235784256784052, 332199503775126359246903819246014, 1312356476908740198818910590863629, 198543627588457770517461457130486, -84450951113311548939038816986753, 77635620002219776839556714752383, 12380345160350258667539354240149, -11351830743787609314394630963035, 2510640089244908582995590437747, 338022536738320361410346741407, -341623495597449926990911857042, 68460096829407729493072618685], [-1385261056561774631404342637172970, -624434644366499204342386836185704, 4580785124927317668191068692291748, 4494433878046643427987644416961192, -27205067464592780445843442014512, 33473401889470466941629285783106, 1042749769941272139410207444162757, 266219488261185805938979561060616, -88287746232844903500229403060527, 53390695339553739670087002742331, 20683505336587692259796543586895, -11303975568665029687638292569291, 1488042873595497882371003908229, 740962837339389597865749684739, -409016750792810739016049720256, 70896363012619975352047365047], [-207794035139009378088396289841703886, -726884682171308571155783053543879340, -1036329442832847630470415460523284544, -401863543064934780565531559134870572, -67851985676704121023772583567798508, -173101196918431897285355745901015262, -127916290173464130300754420540957025, -13568216187504163801136339011810166, -1470617694604805867906139806085295, -7234739792925968930593802023929239, -143667817049424646174448098831829, 711372172741298485663046882993911, -220138357135127837547298790093483, -4718552561761306915881069203927, 24384290511076096526562020786398, -6149563994619954688638779707937], [-31587100994435148956841949945540210, -105304314957815300573758882340184806, -145171978655040313170872301388654991, -64761128639918113258547739261608700, -7985899677975973723172712922975579, -22480618898714890313138256946084163, -18741903802354980703715571404576057, -1930516773969094645935373235463383, -14185174417871716231293317069707, -1068994750098133024151149897842617, -29523529413906777493617000674543, 108171519253326925442652432374614, -32565341693744781589549986685301, -941078557932148630775019122912, 3599043381514092590775304833862, -884749222076827840163383480061], [-66185876186127129622090663942398286, -596064930865510949019247370622245756, -1036724982131751835039698759990491552, -481693103931873447446882666834166924, -28194944284812483239401157917335316, -155345327511197745613962098280501470, -145808451479287995280982367655085153, -16609911107011105456136005854111014, 1094983459797783593718319523936489, -8326297826207090073597894132202439, -503398277914670530340935274351237, 926797171801198127075784520724839, -252013692954719590152433256919851, -17107023363453311887525186709639, 31385661507909987353368135035934, -7269103474487362172933174332865], [-152095410191664098988657516871227004, -357672149290479265608983356352444654, -582201197180551186444520919564636482, -346112115238358927869461747300875766, -30405655107395500282521243569159062, -73467828060319886892184581181208762, -92810415687486762484021665828620090, -13703051666477086217585774710858421, 2693891094309741866477833200482489, -5266605292925521747930627375811942, -599047303927928447636178067892993, 683600955386312395547326219767976, -165894472843756913348070666740291, -16930496194389552475606824693290, 21869480600682083825586720360229, -4777405049874405220290293508725], [691319728804900074759618490740074, 2109648831841915726153072408406390, 3114340389660371141511675680175130, 1798355690683910481445547905696838, 240551486807570209228379215086046, 413736876395432004538516927161328, 479683665037345623980106225703263, 83329656222185135728278336221777, -12362514354697864859638372770130, 26000333648504466704692111919297, 3984836439443536634231966000260, -3602384666056073460301120040997, 752740054714243080564442352918, 145337649633553291461789413149, -128592701387822337905647833681, 26752646231523755733786132878]];
    Rf_basisdens := [1, 537937288243246638195923625151482, 29048613565135318462579875758180028, 1285393759243122194016543907172, 2420717797094609871881656313181669, 9682871188378439487526625252726676, 29048613565135318462579875758180028, 14524306782567659231289937879090014, 14524306782567659231289937879090014, 717249717657662184261231500201976, 717249717657662184261231500201976, 58097227130270636925159751516360056, 7262153391283829615644968939545007, 58097227130270636925159751516360056, 29048613565135318462579875758180028, 119541619609610364043538583366996];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1080_d();"
function MakeCharacter_1080_d()
    N := 1080;
    order := 2;
    char_gens := [271, 541, 1001, 217];
    v := [2, 1, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1080_d_Hecke();"
function MakeCharacter_1080_d_Hecke( : Kf := false)
    N := 1080;
    order := 2;
    char_gens := [271, 541, 1001, 217];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0, 0], [0, -1, 0, -1, -1, 0, 0, 1, 0, -1, 1, 0, 0, 0, 0, 0], [1, 0, 0, -1, 1, -2, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0], [0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 0], [0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 1, 0, 1, 0, -1, 0], [0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0], [-1, 0, -1, 2, 0, 0, 0, 0, 0, -1, 1, 0, -1, 0, 0, 0], [0, -2, 0, 0, -1, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0, -1], [0, -1, 0, 1, -1, 0, -1, 1, -1, 0, 0, -1, 0, -1, 0, 0], [0, -1, 0, -1, 0, 0, 1, -1, 0, 0, 1, 0, 0, 1, 0, -1], [1, 0, 0, -1, 0, -2, 0, 0, 0, -1, 1, 0, 0, 0, -2, 0], [6, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 2, 0, -1, 0, -1, 0, -2, 1, 0, 1, 0, -1], [-1, 0, -1, 1, 1, 2, 0, 0, 0, -2, 2, 0, 1, 0, -5, 0], [0, -2, 0, 0, -1, 0, -2, 0, 1, 0, 1, 1, 0, -2, 0, -1], [0, 1, 0, -1, 0, 0, 2, 1, -1, -1, 0, -1, 0, 2, 0, 1], [0, 0, 0, 0, 2, 0, -1, 0, -1, 1, -1, 1, 0, 1, 0, -1], [2, 0, -1, 3, 0, 0, 0, 0, 0, 3, -3, 0, -1, 0, 0, 0], [6, 0, 2, -1, 0, 0, 0, 0, 0, 3, -3, 0, 2, 0, 0, 0], [0, 3, 0, 3, 2, 0, -1, -1, 0, 2, -3, 0, 0, -1, 0, 1], [0, 0, 0, 0, -1, 0, 2, 0, -1, 1, 2, 1, 0, -2, 0, -1], [0, 0, 0, 0, -3, 0, 2, 0, 1, -1, 2, -1, 0, -2, 0, -1], [0, 0, 0, 0, -3, 0, 2, 0, 1, 0, 3, -1, 0, -2, 0, -1], [-3, 0, -2, -1, 0, 0, 0, 0, 0, 2, -2, 0, -2, 0, 0, 0], [0, 0, 1, 0, 3, 0, 0, 0, 0, 1, -1, 0, -1, 0, -1, 0], [-1, 0, 0, 1, 0, 2, 0, 0, 0, -1, 1, 0, 0, 0, -2, 0], [0, 0, 0, 0, 3, 0, -4, 0, 1, 3, 0, -1, 0, 4, 0, 1], [0, 0, 0, 0, 1, 0, -2, 0, 1, 1, 0, -1, 0, 2, 0, -1], [2, 0, -2, -2, 3, -4, 0, 0, 0, 0, 0, 0, 2, 0, -3, 0], [0, 0, -2, 0, 2, 0, 0, 0, 0, 2, -2, 0, 2, 0, 2, 0], [0, 0, 0, 0, -5, 0, 2, 0, 3, -2, 3, -3, 0, -2, 0, 2], [-1, 0, 0, -5, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0], [0, 1, 0, -3, 1, 0, 4, -1, 0, -1, 1, 0, 0, 4, 0, 0], [0, -2, 0, 0, -1, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0, -1], [3, 0, 1, -3, -3, -6, 0, 0, 0, 0, 0, 0, -1, 0, 3, 0], [-3, 0, 5, 6, 0, 0, 0, 0, 0, 1, -1, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 2, -1, -1, -2, 0, 2, 0, 2], [4, 0, -3, -4, 1, -8, 0, 0, 0, -1, 1, 0, 3, 0, -3, 0], [0, -2, 0, -4, -1, 0, 2, 0, 1, -2, 3, 1, 0, 2, 0, -1], [0, 0, 0, 0, -1, 0, -1, 0, 2, 1, 2, -2, 0, 1, 0, -2], [-6, 0, -3, 7, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0], [-4, 0, 0, -6, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0], [-3, 0, 2, 3, 3, 6, 0, 0, 0, -1, 1, 0, -2, 0, -5, 0], [0, 0, 0, 0, 3, 0, -2, 0, -1, 0, -3, 1, 0, 2, 0, 1], [-9, 0, 4, 4, 0, 0, 0, 0, 0, -1, 1, 0, 4, 0, 0, 0], [0, 0, 5, 0, -3, 0, 0, 0, 0, -1, 1, 0, -5, 0, 1, 0], [-2, 0, 0, 2, 0, 4, 0, 0, 0, 2, -2, 0, 0, 0, 4, 0], [0, 2, 0, -2, 0, 0, 2, 2, 0, -2, 0, 0, 0, 2, 0, 2], [-4, 0, -10, 5, 0, 0, 0, 0, 0, -2, 2, 0, -10, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0, -2, 0, -2, 2, 0, 0, 0, 4], [4, 0, 4, -4, 1, -8, 0, 0, 0, 2, -2, 0, -4, 0, 3, 0], [-3, 0, -6, 3, 0, 6, 0, 0, 0, -1, 1, 0, 6, 0, -2, 0], 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-18, 0, 0, 0, 0, 0, -2, 2, 0, 12, 0, 0, 0], [0, 0, 0, 0, -1, 0, 1, 0, 0, 3, 4, 0, 0, -1, 0, -8], [24, 0, 2, -6, 0, 0, 0, 0, 0, -4, 4, 0, 2, 0, 0, 0], [26, 0, -4, -6, 0, 0, 0, 0, 0, 2, -2, 0, -4, 0, 0, 0], [9, 0, -6, 16, 0, 0, 0, 0, 0, 14, -14, 0, -6, 0, 0, 0], [0, 4, 0, 2, 1, 0, 0, 2, -1, 0, -3, -1, 0, 0, 0, 3], [-8, 0, 2, 8, 10, 16, 0, 0, 0, 4, -4, 0, -2, 0, -2, 0], [0, 0, 0, 0, 8, 0, -7, 0, -1, 13, 5, 1, 0, 7, 0, -1], [0, 0, 0, 0, 12, 0, -10, 0, -2, 11, -1, 2, 0, 10, 0, 1], [0, 0, 0, 0, 0, 0, -2, 0, 2, -3, -3, -2, 0, 2, 0, 0], [-21, 0, 12, 1, 0, 0, 0, 0, 0, -2, 2, 0, 12, 0, 0, 0], [0, 6, 0, 0, 5, 0, 0, -4, 5, 2, -3, 5, 0, 0, 0, 1], [-1, 0, 4, 4, 0, 0, 0, 0, 0, 6, -6, 0, 4, 0, 0, 0], [0, 0, 0, 0, -2, 0, -4, 0, 6, 5, 7, -6, 0, 4, 0, -3], [0, 4, 0, 2, 5, 0, 0, -6, 3, 4, -3, 3, 0, 0, 0, -1], [-5, 0, -17, 5, -1, 10, 0, 0, 0, -2, 2, 0, 17, 0, -3, 0], [9, 0, -5, 0, 0, 0, 0, 0, 0, 6, -6, 0, -5, 0, 0, 0], [-12, 0, -2, 12, 0, 24, 0, 0, 0, 4, -4, 0, 2, 0, 8, 0], [14, 0, -15, 7, 0, 0, 0, 0, 0, -11, 11, 0, -15, 0, 0, 0], [0, -3, 0, 3, -3, 0, -5, 3, -1, 0, 0, -1, 0, -5, 0, 0], [0, 0, 0, 0, 10, 0, -6, 0, -4, 5, -5, 4, 0, 6, 0, -5], [12, 0, 4, -23, 0, 0, 0, 0, 0, 9, -9, 0, 4, 0, 0, 0], [2, 0, -3, -2, 9, -4, 0, 0, 0, 9, -9, 0, 3, 0, 9, 0], [2, 0, 4, -2, -11, -4, 0, 0, 0, 2, -2, 0, -4, 0, 15, 0], [0, 5, 0, 1, 1, 0, 6, 3, -6, -1, -3, -6, 0, 6, 0, 4], [0, 0, 0, 0, -2, 0, 0, 0, 2, 8, 10, -2, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, 0, 7], [0, 7, 0, -3, 4, 0, 6, -1, 1, -1, -2, 1, 0, 6, 0, 3], [6, 0, 0, -6, 3, -12, 0, 0, 0, 6, -6, 0, 0, 0, 9, 0], [-4, 0, 4, -8, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0], [0, 0, 0, 6, 3, 0, -6, -6, 3, 6, -3, 3, 0, -6, 0, -3], [0, 9, 0, 7, 3, 0, -3, 3, -1, 2, -8, -1, 0, -3, 0, 6], [-7, 0, 21, 7, -11, 14, 0, 0, 0, -2, 2, 0, -21, 0, 7, 0], [-34, 0, 8, 17, 0, 0, 0, 0, 0, -2, 2, 0, 8, 0, 0, 0], [27, 0, -1, 4, 0, 0, 0, 0, 0, -4, 4, 0, -1, 0, 0, 0], [0, 0, 0, 0, 6, 0, -4, 0, -2, -1, -7, 2, 0, 4, 0, -8], [0, -13, 0, -5, -5, 0, -2, -3, 2, -1, 9, 2, 0, -2, 0, -8], [27, 0, 2, -5, 0, 0, 0, 0, 0, -5, 5, 0, 2, 0, 0, 0], [14, 0, 2, -14, 1, -28, 0, 0, 0, 4, -4, 0, -2, 0, 7, 0], [0, 0, 0, 0, 0, 0, -2, 0, 2, 3, 3, -2, 0, 2, 0, 0], [-6, 0, -3, 6, -3, 12, 0, 0, 0, -9, 9, 0, 3, 0, -15, 0], [0, 0, 0, 0, 4, 0, 0, 0, 0, -2, 2, 0, 0, 0, -8, 0], [0, 8, 0, 4, 4, 0, 4, 0, -4, 2, -6, -4, 0, 4, 0, 4], [0, 0, 0, 0, -9, 0, 2, 0, 7, -9, 0, -7, 0, -2, 0, 3], [0, 0, 0, 0, 7, 0, -1, 0, -6, 13, 6, 6, 0, 1, 0, -4], [-2, 0, -4, 2, -10, 4, 0, 0, 0, 0, 0, 0, 4, 0, 10, 0], [-5, 0, -7, 5, -3, 10, 0, 0, 0, -14, 14, 0, 7, 0, -25, 0], [0, 0, 0, 4, -2, 0, -8, 4, 2, 0, -2, 2, 0, -8, 0, 2], [0, 6, 0, 0, 3, 0, 8, 0, -5, 0, -3, -5, 0, 8, 0, 3], [-4, 0, -7, 4, 7, 8, 0, 0, 0, -9, 9, 0, 7, 0, -25, 0], [-7, 0, -14, 2, 0, 0, 0, 0, 0, -12, 12, 0, -14, 0, 0, 0], [-21, 0, -1, 8, 0, 0, 0, 0, 0, -3, 3, 0, -1, 0, 0, 0], [0, 0, 0, 0, 5, 0, -8, 0, 3, 3, -2, -3, 0, 8, 0, 5], [5, 0, -11, -5, -7, -10, 0, 0, 0, -4, 4, 0, 11, 0, -1, 0], [0, -3, 0, 1, -6, 0, -9, 9, 2, -4, 1, 2, 0, -9, 0, 3], [-12, 0, 29, -1, 0, 0, 0, 0, 0, 10, -10, 0, 29, 0, 0, 0], [0, 0, 0, 0, 5, 0, -5, 0, 0, 1, -4, 0, 0, 5, 0, 10], [0, 0, 0, 0, -1, 0, 2, 0, -1, -8, -7, 1, 0, -2, 0, 2], [33, 0, 20, -10, 0, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0], [6, 0, 5, -6, -5, -12, 0, 0, 0, 1, -1, 0, -5, 0, 7, 0], [0, 10, 0, 12, 7, 0, -4, -4, -1, 8, -11, -1, 0, -4, 0, 3], [-2, 0, 18, 2, -3, 4, 0, 0, 0, 8, -8, 0, -18, 0, 19, 0], [-8, 0, 2, 8, 10, 16, 0, 0, 0, -2, 2, 0, -2, 0, -14, 0], [0, 0, 0, 0, -9, 0, 2, 0, 7, -10, -1, -7, 0, -2, 0, 5], [0, 6, 0, 6, 4, 0, 0, -2, -2, 4, -6, -2, 0, 0, 0, 2], [0, -6, 0, -2, -8, 0, -6, 10, 0, -6, 4, 0, 0, -6, 0, 2], [24, 0, -4, 10, 0, 0, 0, 0, 0, -2, 2, 0, -4, 0, 0, 0], [-3, 0, -3, 3, 5, 6, 0, 0, 0, 4, -4, 0, 3, 0, 3, 0], [0, 2, 0, 2, 2, 0, 4, -2, -4, 2, -2, -4, 0, 4, 0, 0], [0, 0, 0, 0, -4, 0, 7, 0, -3, -17, -13, 3, 0, -7, 0, -3], [-25, 0, -15, 6, 0, 0, 0, 0, 0, 1, -1, 0, -15, 0, 0, 0], [0, 0, 0, 0, 4, 0, 3, 0, -7, 5, 1, 7, 0, -3, 0, 7], [0, 0, 0, 0, 2, 0, 4, 0, -6, -8, -10, 6, 0, -4, 0, -1], [-6, 0, 0, -13, 0, 0, 0, 0, 0, -13, 13, 0, 0, 0, 0, 0], [-1, 0, 13, 1, -5, 2, 0, 0, 0, 6, -6, 0, -13, 0, 17, 0], [8, 0, 10, 1, 0, 0, 0, 0, 0, -11, 11, 0, 10, 0, 0, 0], [0, 0, 0, 0, -6, 0, 3, 0, 3, -16, -10, -3, 0, -3, 0, -1], [7, 0, -14, -7, 0, -14, 0, 0, 0, 3, -3, 0, 14, 0, 6, 0], [-16, 0, 1, -13, 0, 0, 0, 0, 0, 8, -8, 0, 1, 0, 0, 0], [-1, 0, 17, 1, -5, 2, 0, 0, 0, -6, 6, 0, -17, 0, -7, 0], [-7, 0, 17, 7, -11, 14, 0, 0, 0, -8, 8, 0, -17, 0, -5, 0], [0, 12, 0, -2, 9, 0, 6, -6, 5, 2, -5, 5, 0, 6, 0, 3], [0, 11, 0, 1, 8, 0, 2, -5, 5, 3, -6, 5, 0, 2, 0, 3], [0, 0, 0, 0, -2, 0, 4, 0, -2, -1, 1, 2, 0, -4, 0, 16], [-30, 0, 2, -10, 0, 0, 0, 0, 0, -13, 13, 0, 2, 0, 0, 0], [-31, 0, 15, -10, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0], [0, -12, 0, 6, -5, 0, -8, -2, -3, 4, 3, -3, 0, -8, 0, -7], [0, 0, 0, 0, -9, 0, 6, 0, 3, 2, 11, -3, 0, -6, 0, -3], [-12, 0, -16, 11, 0, 0, 0, 0, 0, 13, -13, 0, -16, 0, 0, 0], [0, 0, -2, 0, 16, 0, 0, 0, 0, 14, -14, 0, 2, 0, 12, 0], [0, 0, 0, 0, -3, 0, -2, 0, 5, -7, -4, -5, 0, 2, 0, 3], [0, 9, 0, 7, 10, 0, 0, -11, 3, 9, -8, 3, 0, 0, 0, -1], [0, 0, 0, 0, -15, 0, 6, 0, 9, -4, 11, -9, 0, -6, 0, -3], [-16, 0, 5, 13, 0, 0, 0, 0, 0, -8, 8, 0, 5, 0, 0, 0], [-12, 0, -14, 20, 0, 0, 0, 0, 0, 1, -1, 0, -14, 0, 0, 0], [0, 0, 0, 0, 6, 0, 2, 0, -8, 5, -1, 8, 0, -2, 0, 4], [0, 19, 0, 9, 11, 0, 3, -3, -1, 6, -14, -1, 0, 3, 0, 8], [0, 0, 0, 0, 10, 0, -9, 0, -1, 8, -2, 1, 0, 9, 0, 13], [27, 0, 5, 18, 0, 0, 0, 0, 0, 1, -1, 0, 5, 0, 0, 0], [-4, 0, -8, 10, 0, 0, 0, 0, 0, -1, 1, 0, -8, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1080_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1080_2_d_g();".
// 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_1080_2_d_g(:prec:=16)
    chi := MakeCharacter_1080_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1080_2_d_g();".
// 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_1080_2_d_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1080_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2556, 0, 2356, 0, 500, 0, 39, 0, 1]>,<11,R![284, 0, 2032, 0, 556, 0, 43, 0, 1]>,<13,R![101104, 0, -27228, 0, 2492, 0, -88, 0, 1]>,<53,R![999, -754, 206, -24, 1]>],Snew);
    return Vf;
end function;