
// Make newform 1250.4.a.n in Magma, downloaded from the LMFDB on 10 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1250_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_1250_4_a_n();".
// 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_1250_4_a_n();".
// 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 := [-45086320, 82757040, 78433160, -193725880, 14900696, 83435508, -11070230, -14035110, 1899751, 1143340, -151199, -47880, 6166, 990, -125, -8, 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], [-246199036512048478408023186494845472738280, 43592415860598142321513222204225446810600, 355215759083463672172019602978367449601780, 27418608768202620812488955377971677068480, -121258614206735892870779006788380048156050, -4955841075597817775053159224480902850320, 19451659006919605260059797040720617896381, -114037677041660259543130930625170867914, -1595083636278409773754961636429641821529, 63420377294206307807438746530988036812, 67580041586027164861044464273490798762, -4274381975720612921393707762243549644, -1409309606904377665774888046574651151, 113184032207733731831996632775585498, 11458748724136290787055843463663027, -1069464125100162760874994114577352], [674632020803992772218504312, -1259773156528429963097166440, -1719061490662910410639523920, 1729263928096266435801236540, 1051303528391285401287874544, -439351209129842597187757728, -204154993827830505465902881, 50965673955785044487274117, 17848101691824502199050053, -3257047745667554042215933, -778037562329546807375568, 117849012148856420223872, 16522397332288035810951, -2253306920357720796683, -136292957292407835199, 17625511305178629155], [-5875465188252648721302663964660351242440, 3767714476876716949514239361102560634480, 15938213770601248540294604489659606488640, -7585375717329464357595908655447989263060, -8406109603019161061693668500585386733812, 2144281857622254150127684086430238692676, 1536970569933869326235307696843767833455, -277654602707779377821114763426484669888, -130002191542975375440089716490418978137, 19510950085628416395379326066228067450, 5539140695557685454396207598590903032, -752093617162397955429422339591987704, -115470603717990592474040073582634173, 14872850481759576487145306919999340, 936957767754206142941135541010775, -117797041579734287440895572047534], [895468469026884389449772106930940331884440, -297247586617071041096258758758755888685640, -2112159137481212595891295728087402195797840, 685362164285964770753665649533483350470860, 833379201946614448660369661818106604143956, -168695612466641039489011209819440560359408, -122559411302551252018313978056095147131709, 17807664173781619460119334277404615963405, 8767649503815782672902613668267653604677, -1027834385026494736389817354366478121813, -329486716082356727657242345685927714604, 34608399309534444410636029948144632608, 6275619019104687322492917745129756523, -639900769524642961522527337653075747, -47904186671572308715300113259431023, 4982685750591786789420540408217515], [-30298046953018234914819829872853356997480, 5051702966234669902039097072552493439440, 50127245982626991632318381914707515531280, -12950366637676873484000678082590814172460, -25704721030214343509101461663434379219876, 3117121589851076412494584303812149962628, 4587294504796889492467285214504338122269, -382659571188430808465634401012949851980, -377164851650991885610116038376887521307, 28846099149373867460026963299918205458, 15623179167300487697763291778782489584, -1254072866913605887723363601555484368, -317916555323515174646584822828010823, 27901618554610719631015596693125432, 2532621941726194253851227693932293, -243688942197329839239135174807430], [1162311646692893325200389301419415007564040, 917545904204176276859874488607158780729760, -3140759437991286743189625583387944518989120, -156296119372914626715299557003579614995940, 1336547665174322064539739840326470007013976, -66963755033227134973100525069863257760048, -228633998056214944951598657559008341655489, 19757063889727468747118375502138135498910, 19095627173598718259154199969734847587607, -1956553779270407371427197809618790059108, -821466856065325636590873942833813020484, 90307268013095637001913987840291999228, 17439398471539341466549336915763550443, -1988327310336818964384480319401253422, -144490304525805199322801888494987153, 16904741661227646355570712424817700], [1957031126341062081492090286630266702546720, -992669034296528260044361276943041851199800, -4817883627459669250122348594440287099699920, 1827375591108314924658910994596388634612280, 2237423872273932721438408256620406109631920, -454285406251684883637491109947893360765300, -370237481517761012346766454484621696774674, 51826136680698784864809396445080031367251, 28744972414254899122925485456553413814236, -3323364043483193486869517480587909025293, -1135757884323345278853062382616101709708, 121854996368902900148777923836321913016, 22167523979020769730631628599004816114, -2353984505865169327361990743937211417, -169940386149267259545126861711518048, 18432688368518010852483775494574123], [-841174519412973735362248119832240466720440, 612439527586883938560374485525997006522120, 2771771121834210799110499407078878577538320, -1474066350035604143463718749805955281631100, -1397171421027228416224499336084520209113432, 442760636709979328847717005832223160454264, 249179439771144538748433099854510839424507, -58133613256575853920247264257020460007067, -20827238163453426287833689460305755508495, 3996591471027561178761373500226815883107, 880536324499863109186530747957669146552, -147728998048197103383532174909627035192, -18199904570360010210171847438447594397, 2777776247106471155492814574156896901, 146037738205990198117170871424288285, -20879704966685736566052415421346693], [1761163574756325500950439895217624629718280, -131413936629237220277990750974908752866200, -4614156562511199163409784827686348597164080, 1511284538647774990070915132722035532492220, 2096583018902809769555839464393003652804580, -544620716478810822991233843417872874866500, -372390093640421537607752415097905278011751, 78644219242145268883440007057137854538199, 31979244309405125138135795882085509695039, -5742793954819218225825882000126281485307, -1406815256527351741839715346041633882092, 223100277237153335173757476110460782284, 30396214200510478206287601335153413361, -4408720865655947845216526656660109433, -255272751156302106122134068901980477, 34929171208073888019493149619988877], [-1858685074638955818571232361878932573946240, 1252585537985058211839674569630384688258000, 4427202318110488530889390123520662771608260, -2370501644349463587074332962011885946793420, -2142983317602207115876138036009822476812256, 723314814285573230418745417906575496582320, 386144437356153357812038971792617331282607, -96185050962886909568431447859728465599336, -32931254679656953273168234538711819123589, 6650595042765878291941280145039183942546, 1425556883568307648540220028966074974084, -247261419219557634267202170605066591060, -30202355147084292941440505008703965989, 4694913550129491779499141963218542436, 248444101405326685017118438313257143, -35815184981891882761227650004703446], [-1307772143193149926963381518514215951457960, 613295475071331206817710152919317072418020, 3015605523971300520895524011080699123858600, -1404603563866747961237527342204540741479010, -1414593841594535854946848808472130024461603, 436467641746070811746995940045693643018074, 250989120547588084907701127320673860017081, -58442883048512383532920423419039637385021, -21180569350713094472461229385705151480652, 4061106325256855718194245381957259173187, 909219838757315495010405421670240226385, -151656878418434004679348961399688625430, -19132194823777704430178061382980430063, 2890540681193301857710130363346402943, 156513696548082265152109780467635092, -22114709330151248158344459391550063], [6397832932352110691876173384108245431900560, -3923615446733933605517111731124400938110600, -16536597715585073676720162237672423128093360, 8323525925059153685203668659135949261825120, 8119708722203691052159931094020964707680828, -2472494441874350637098754591724909489834820, -1440675210587403565650373775507300082839534, 320646188389041794565987152035757212950285, 120176777646940688082756490190911730360984, -21852436969228945655690573062948387471899, -5090257886055202447561375777749428111288, 808356151321676827190653755004607438512, 105838584730153748704345790090194374350, -15355244083472014736152337933148200567, -857433758016969263059435603390240620, 117451733351824900826577794999332069], [4321069197055685595484914883391676837240, -2434183417286781746537223003903405872580, -10104393664921602622631756712569099952940, 5097052558089603661903924996488609443760, 4788775418832348466564845170751188325952, -1504359024277127953673722031288245639471, -837474708866429616110420978383698808555, 193552392946919303454961281807152425073, 69393716300896638590668415572293020677, -13080816284782169841336063918622012550, -2930704290534293045011681345222040672, 479956856115963595285001345166690409, 60861487714685060880888478177945233, -9048634601740397219583929250667915, -492762129799335008194934801374075, 68749390374322572604944472837314], [-1795022179033208745354462447932113467621520, 804284562367384592076252412968337718130760, 4623429385048770416620563593083323091533540, -2163364443778120876821080892187183140820510, -2229464179084444811798841839734765173366278, 674857721727583896724659786267819584957167, 395721192050854595519331096380416488364376, -89963181270824855703272550020321927849217, -33180431176286993971632322940969342790262, 6223637373399386343977577275702337772950, 1413153851957748689423178263004117368906, -231618340280317797947180035635506420125, -29510497940296196484836484353301128620, 4401622810494876749034855441006605763, 239757645813782841775855601205742998, -33576259405212958085687963827304828], [-5603551945083809218303151957200584606774320, 3145130894857715235240929037929398404013480, 13343608268778718335039867723808765699699020, -6587146956166762919029142590651594130967080, -6396231542864710648600748068221821533488926, 1993840534346318482080822050223153638455646, 1133434714704299774602933550235841766549165, -262068517506575055063623805829048856628240, -94988770321618173139210128533680755492581, 17988846521454852077484957776254459941668, 4046995391170587659632163038577341129182, -666736646455866471750060220455538543222, -84591283965062382247769925899387858231, 12648698361735442142003281545130103900, 688155496548949169603813644441843631, -96468999845803454661929370354937712]];
    Rf_basisdens := [1, 5603291159985434938816495524295022572440, 57181453573239074761611984, 273331276096850484820316854843659637680, 11206582319970869877632991048590045144880, 273331276096850484820316854843659637680, 11206582319970869877632991048590045144880, 11206582319970869877632991048590045144880, 11206582319970869877632991048590045144880, 11206582319970869877632991048590045144880, 5603291159985434938816495524295022572440, 2801645579992717469408247762147511286220, 11206582319970869877632991048590045144880, 6212074456746601927734473973719537220, 2801645579992717469408247762147511286220, 5603291159985434938816495524295022572440];
    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_1250_a();"
function MakeCharacter_1250_a()
    N := 1250;
    order := 1;
    char_gens := [627];
    v := [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_1250_a_Hecke(Kf)
    return MakeCharacter_1250_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 := 4;
    raw_aps := [[2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 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], [3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [13, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0], [13, 0, 2, -1, -1, 0, 1, 0, -1, 1, 0, 1, 1, 0, 1, 0], [6, 1, 0, -3, -1, -1, -1, 0, 0, 1, 1, 0, 0, 0, -1, -1], [20, 0, -3, -2, 2, 2, 0, 0, 2, -1, -1, 1, -2, -1, 1, -1], [27, 3, 0, -5, -2, -2, 1, 1, 1, 1, -3, -1, 0, 2, 2, 1], [6, 1, -3, -5, 2, -2, 0, -1, -3, 1, 2, -1, -2, 2, 0, 0], [42, -6, -3, -2, -1, 0, 2, 2, -6, 3, -3, 1, -2, 1, 3, -5], [34, 0, 1, -6, 8, 1, 4, -5, 6, -6, 3, -1, 0, -3, 2, -3], [61, -2, -1, -2, 1, -1, -1, 2, 3, 5, 6, -5, 1, 0, 1, 3], [49, 4, -2, 6, -1, -3, -2, 4, 7, -5, -7, -2, -3, 0, -5, 4], [30, -3, 20, 10, 6, -3, 1, -8, 6, -6, 6, 2, 8, -1, 1, -3], [59, -5, 16, -15, -6, 0, -1, 7, 2, 0, -5, 4, -5, -6, -1, -5], [47, 1, -11, -6, 7, -4, 4, -1, -7, 2, -9, 1, -1, -5, -2, -3], [86, -4, -14, 16, -10, -5, -2, -6, -6, 5, 7, 3, 0, 0, 1, -2], [101, -8, -15, -28, -4, -5, -3, -5, -3, 10, 12, 1, 7, 1, -2, 3], [114, -10, 32, 50, -5, -9, 0, 0, -11, 9, 8, 1, 1, 2, -4, 3], [-1, -6, 35, -16, 0, -7, -2, 4, -15, 1, -2, 8, 1, 2, -3, -6], [157, 9, 12, 40, -5, 6, 0, -5, -2, 1, -7, 7, -6, 9, 7, 1], [188, -17, 28, -28, 2, 5, 3, -11, 6, -8, -7, 8, 11, -10, 11, 1], [248, -23, -32, 47, 5, -4, 5, -6, 5, -7, 20, -3, 11, 4, -7, 0], [264, -4, -41, 4, 1, 13, 10, 17, -12, 0, -4, -11, -19, -6, 5, 7], [301, -11, 28, 44, 8, 13, -11, -6, -5, -10, -3, 0, -13, 8, -8, -10], [69, -12, -40, 11, 0, -1, 6, -6, -22, -9, 3, 13, -4, -14, -1, 4], [103, -4, 28, -10, 16, 7, -13, -14, 11, -25, 33, -14, -1, -10, -11, 11], [197, -12, 83, 11, -14, 12, -4, -7, 2, 14, 18, 8, 12, 4, 10, 4], [242, 18, -90, 6, -8, -11, 15, 20, -4, -10, -23, -11, -16, -3, -2, -8], [284, -10, -78, 86, -8, 15, 6, -28, -18, -15, -18, 32, 14, 12, 35, -7], [329, -8, -86, 58, 1, 8, -1, 35, 21, -6, 28, -17, -5, -7, 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-2103, 97, -678, 405, 853, -159, -245, 416, -932], [19647, 982, -952, -1320, 2685, -476, -1243, -1187, 1809, -1496, -144, -61, -2409, -155, -847, -564], [-10151, 264, -4754, 1879, 1436, 154, 1954, -994, -624, -1979, -956, 300, 308, -384, -489, -1397], [-13376, -658, -3877, 5250, 1682, 846, -218, -2532, 2073, -1447, 55, 562, 653, 480, 1329, -69], [38719, 467, 6045, 28, 276, -503, -1021, -1554, -1480, 793, -622, 1009, -173, -1185, -10, -2319], [-26067, -1261, 7385, -4966, 863, 739, 2584, 186, -393, -408, -2671, 1128, 65, -623, 1743, -1166], [-6622, -1199, 2675, -4680, -2714, -1101, -1301, 457, -1318, 1112, 773, 670, 1575, 1830, 103, 605], [10830, 383, -1120, 4330, 586, -1193, 201, 136, -175, 359, -1378, -131, 434, -1211, -210, 1404], [12839, -611, -8346, -9411, -222, -1958, 176, -516, -437, 896, -1038, -391, -465, 978, 714, 1634], [71961, 1412, 6709, -3718, 443, -734, -55, -378, 488, -282, -2605, -1273, -1016, 479, -916, 533], [51017, -561, -4232, 949, 2299, 887, -433, -1067, 603, -523, 1230, -1495, 595, -780, 449, 915], [8056, -231, -1653, -1554, -1046, -1000, 187, -1363, -1729, 1730, 1477, 157, 1156, 554, -581, -566], [1461, 618, -6609, 1106, -1019, -546, -897, -534, 565, 736, -759, -218, 795, -686, -1810, -261], [-18205, -3195, 3716, -1800, 273, -1706, 351, 34, -401, 294, -917, 619, 1, -536, 1424, 835], [28821, -374, 5015, -4115, 1365, -216, -85, -3043, 422, -1217, 1420, 630, 2, -335, 1473, -1022], [4417, 684, -7751, -7275, -4670, 294, -1834, 1585, -910, 1576, 802, 744, 2676, 2, -212, 430], [40697, -921, -7361, 5268, -605, -1734, 893, 2214, 805, -24, -2568, 355, 585, 874, -57, 306], [-7592, -1221, 9318, -10536, 1736, 380, 1002, -1199, -997, -841, 1573, -372, -721, -571, 381, -703], [3870, 2671, -7363, 2327, -1154, -874, -847, -243, -808, 1557, -123, 2933, 563, -126, 742, -190], [-21823, -744, -532, 232, 181, -856, 300, -1226, -762, 518, -1849, 264, -1095, 659, 867, -2562], [50300, 617, -3339, 279, -98, 1398, 1591, 925, 1655, -1876, -803, -612, 802, -1358, 1361, 1056], [-1240, 859, 3464, -5190, -1669, -972, -1453, -2350, -1094, 869, 1146, 1026, 1308, -694, 975, -908], [-32320, -2291, -530, -676, 1962, 1110, 955, 1427, -25, -2576, -1393, -2145, -2216, 272, -1139, -754], [16489, 1773, -2867, 43, 2128, 664, 24, 79, 3777, -1786, -138, -702, -282, -1312, 823, -75], [40082, 517, 306, 563, -1179, -1384, 800, 1441, -1872, 1307, -41, -2197, 1068, 267, 337, 1705], [-31111, 1478, -161, -4531, -2011, -1083, -744, 843, -902, 2033, -250, 1239, 1433, 978, 578, 136], [-34955, 1524, -768, -3667, 502, -100, -43, -1785, -727, -251, -60, -438, -43, 741, 11, 1718], [56689, -129, -5309, -671, -1484, -1324, 308, 1018, -742, -571, -101, 1005, -1784, -8, -1307, -831], [25859, 2057, -376, 4179, 352, -116, 1241, 32, 453, -239, 174, -1789, 660, -2003, -650, 586], [50246, 610, -10566, 763, -191, 445, 76, 41, 819, 250, -1576, 263, -1041, 881, -1052, 73], [-9782, -1455, -1665, 2847, 2104, -46, 2091, -2420, -1553, -382, -670, 422, 684, -7, 601, 747], [-2199, 2035, -3452, 6083, -2545, 91, -2290, 1949, 791, 1800, 368, -2911, -1069, -261, -3038, 1351], [34114, 1265, 1966, -5446, -99, 1877, 422, -530, 1457, 213, 1397, 202, 1597, 906, 935, -124], [18963, -2033, 2780, 3824, 1145, 1562, -633, -3586, -1309, -533, 1753, 1562, 1637, -649, 838, -1469], [25645, -1727, -1807, -2117, 763, -497, 2101, -801, -4523, -1187, -2032, 390, -1573, -233, 423, 127], [-10924, 969, -1853, 616, 1410, -173, 937, 1991, 1154, 142, -771, 671, 192, 1160, -558, 260], [-34407, -1854, -4852, 2980, 3012, -1168, 1558, -394, -1170, -361, -2716, 413, -1586, -756, 1009, -2397], [26188, 2354, -1723, 6031, 1985, -880, 2578, -2223, 395, -56, -1060, 579, 2123, -668, 1370, 848], [36498, 426, 4079, -4202, -403, -3203, -745, -1767, -1698, 2528, -2577, 2089, 1968, 499, 1090, 152], [43250, 3153, 6539, 2902, 741, -1312, -261, -984, -1222, 126, 1521, 603, -884, -513, -1410, -245], [-29124, -1321, -3128, -6832, -1438, 950, -635, -1322, 1488, -870, -1900, 2573, -381, -323, 453, -1902], [81059, 453, 2849, 6105, -1158, -1685, 67, 331, -1409, -366, -1465, -1716, -1537, 567, -2022, 1262], [-13912, 2709, 4314, 5694, -406, -602, 1279, 2309, 995, 552, -795, -1375, 578, -498, -447, -40], [31726, -450, -4081, -6405, 373, 278, 1097, -846, 657, 1886, 823, 1870, 2485, 326, 1514, -507], [-2749, -848, -11868, -2659, -3240, -184, -300, 852, -3823, 5378, -130, 169, 1967, 2288, 1850, 705], [19960, 557, 2651, 4474, -2370, 1036, -1526, 922, 695, -1383, 148, 1027, -939, 142, -402, 172], [-10076, -919, 1070, 12769, 2798, 1481, 3423, -270, -904, -2208, -1524, -241, -2797, -823, 129, -2648], [-8335, -482, 428, -1646, -1528, -330, 299, 858, -505, 879, -1757, 1767, -65, -672, 1917, -363], [10092, 1160, -5959, 2146, -2811, 1110, -1592, -1258, -2935, 3234, 1301, 1470, 3888, 1947, 1839, 753], [-31389, 923, 2167, 8330, 1668, 1085, 345, -3116, 328, -2157, -1286, 3057, 115, 625, 1864, -993], [-9696, 1950, 1766, -1, 49, 1430, 774, 2328, 2754, -709, -105, -3422, -175, 493, -1942, 1045], [-32245, 278, 9516, -1866, -2969, 2038, -1929, 3243, 1986, -314, -542, -348, -136, 340, -2075, 814], [11842, 3328, 6676, 992, -2455, -792, -1155, 1647, 101, 3402, -2329, -664, 571, 1497, 474, 2771], [24762, -528, 6854, -6130, 3288, 98, -104, 249, 245, -775, 136, -1725, -1487, -771, 104, -1044], [5947, 2487, -8867, 15850, -446, -131, -157, 744, -1990, 405, -1314, -73, -2401, -53, -1148, -903], [-16564, -3696, -4656, 2228, 1990, 459, 251, -586, 4306, -1726, 4253, -1285, 1206, -458, -796, 365], [81308, -1852, -499, 2832, 728, 157, -616, -933, -1183, 915, 596, 1021, -141, -206, -616, 463], [8194, -374, -8873, -168, -2218, 1580, -926, 466, -140, 2361, 890, 155, 1414, -882, 1367, 131], [117098, 1003, -7568, 731, -1177, -133, 16, 887, 1662, 1106, -776, -318, 1372, 551, 1224, -17], [-6809, 1301, -13757, 5760, 240, -301, 2113, 1099, -1474, 1779, -669, -1121, -281, 932, 1794, 78], [78898, -1483, 4449, -674, -2137, 504, -953, 1714, -993, -1808, 475, 496, -357, 740, -1500, 265], [-33402, -4101, 855, 9938, 3696, -1503, 2360, -1459, 1839, -3507, -471, 1418, 1045, -342, -983, -1674], [1800, 2103, -359, -7423, -2704, -1372, -827, 770, -1002, 1500, -2519, -1139, 666, 1990, 988, 315], [24441, 3707, 14637, 5505, 545, 742, -909, 806, 1855, -1048, 1959, -1296, 187, -804, -1374, 377], [-30551, 1948, 6937, -4870, 2088, -505, -979, -282, 49, -1713, -945, 662, -2907, 1461, -1122, 558], [4083, 1539, 12673, 8983, -279, 1959, -1751, -989, -913, 287, 1706, 674, -309, -307, -827, -201], [-39590, 848, 4281, 2373, 2234, 646, -1325, -1304, 2264, -1171, 351, 120, -912, 412, -1111, -2314], [6297, -1359, -16231, -10222, -2044, 908, -860, 1566, -302, -409, -1063, 1501, -1296, -1889, -917, 1231], [-16820, -2811, 4897, -9194, 3925, 2108, -467, -606, 3315, -2054, 3379, -600, 27, -3504, 1038, -1067], [8028, -1193, 3307, 1554, -287, -1064, -789, -2714, -817, 874, -1833, 2054, -67, 224, 984, -113], [29331, -271, -3149, 9763, 2168, 2074, 286, 1406, 4814, -3632, 145, -2796, -1255, -797, -2875, 407], [-5352, -3294, -6316, 3918, 764, -2736, 3430, -1293, -3434, -1777, -439, 207, -1072, -1117, 361, -929], [15795, -316, -20734, 8834, 1787, 1517, -198, -980, -868, 145, 1157, -1709, -1298, 1313, 747, -926], [-329, -1386, -2002, 372, 113, 262, 1376, 105, -1716, 2084, 336, 694, -795, 220, -33, 1627], [14870, 1216, -86, -2991, 1429, -425, 1718, -1251, 785, 82, 1790, -1453, 3433, -871, 1360, 3817], [-14470, 1863, -379, 3000, 496, 328, -619, 127, -45, -701, -502, -1721, 531, -1057, -1451, -74], [-5724, -2423, 7965, -1557, -1858, 2123, -1316, -1210, -3824, 3618, 820, 1744, 308, 1107, 2474, -799], [-64519, 2153, -6021, -2100, 212, -326, -439, -1259, -173, 1482, -1179, 1405, 1210, 364, 37, -1824], [8648, -671, 13053, -2437, -2032, -1350, -757, -1802, -3643, 3360, 1012, 2188, 2362, 357, 851, -613], [-51116, -704, -777, 7744, 718, 1653, -418, -348, 2707, 8, 3140, 257, 2013, 346, 2824, 1169]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1250_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1250_4_a_n();".
// 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_1250_4_a_n(:prec:=16)
    chi := MakeCharacter_1250_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_1250_4_a_n();".
// 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_1250_4_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1250_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![3019248976, 20024894992, 25201310220, -27083271420, -3722923875, 5470481026, 23831962, -432158670, 17778370, 16524000, -1015858, -325606, 23530, 3170, -250, -12, 1]>],Snew);
    return Vf;
end function;