
// Make newform 1859.4.a.i in Magma, downloaded from the LMFDB on 09 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1859_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_1859_4_a_i();".
// 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_1859_4_a_i();".
// 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 := [2596992, 27375104, 37775816, -32766128, -42560950, 20197463, 15765187, -6021311, -2799440, 941923, 267574, -81750, -13998, 3949, 375, -99, -4, 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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-13, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10, -20, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4005776073270108633041280, -46459625359141949016244480, -55604933131807308939165272, 45976808470153155586111784, 56492005825843061507150314, -23335646798509388610951591, -17774686946610383545589102, 5505225111267290509992731, 2482441345101946425892713, -637097390027371710941150, -171164136459150704236180, 37643143106635550853998, 5833727324126602849724, -1089691170983042995419, -86276209144188559950, 12220861324930981551, 320755191500311577], [16500463079882723556101760, 63056700801129872480984704, 42403000983433033556775464, -71259894123286134590648504, -62426016709271278113819526, 36559198506852545757974505, 21260617620147652922491442, -8331921762884182908956669, -3050211553396792419142647, 931492962122600546754578, 212515466194045461357988, -53562836030236774388930, -7264914351002868800972, 1519209013952630432517, 107019848753033523354, -16780707919544935689, -386045941214303951], [946935233399516690044032, -10436961863781423151253632, -6716099813670673792069688, 14346725921463554772267176, 4110265225736896062855778, -8138969644016812017528963, -447842633376516758769758, 2040905566609312580237903, -104307193522747761536547, -249259690591585967807054, 25515832962439663658804, 15435666144060616993790, -1999857507681436152316, -464620731506214219951, 67527700045633927674, 5377304972460401163, -834355344101763163], [1857642607657073539939200, 13048986773425192113396800, 11478063945715241514885672, -10968382728342737928642472, -9922801033207085479851006, 3736749823076970811763313, 2978640746162419862606938, -633588781178425639716957, -430194131871573684828351, 57101739172667830425634, 33021141054423613533156, -2776257706572940996546, -1373270106845628091124, 68695718062154067565, 29116573867575102514, -677699904112737401, -246001677822851943], [-312466578555150408503424, 3338510421881759195977280, 8923827583809858356515048, -1308433526648827103033128, -8951220179730454768044110, -493666337619772718455719, 3375383848603830705202654, 320926527514169096349683, -611102833047427733796087, -56243761730486894837330, 58123134527729820553268, 4240037519706955497122, -2970824170558640012788, -144117636813427476807, 77085158542626105006, 1807282731806681235, -794945999869239535], [445812494432237802387840, 263461682627912690185664, -1278541649684379721427448, -1739525836522434296317624, 1178777557344139423194594, 910563429895306171658973, -490373581971114473381906, -206016269428743313291251, 97828074288180730269825, 24095649275694889714462, -10006999854303408125730, -1477492620570823726966, 539898869913703807450, 44632339955601634477, -14599544582538301484, -520614068795656775, 155486252094825627], [-461415185777467950047232, 1596507155218615036240384, 3137680051681207126664184, -305814498176742101214632, -1789792709700367081277082, -1142012868968443514824737, 719868763535121102545312, 463858933659058379880837, -165631634438522277703593, -68569101746942057247100, 19577823932393552550384, 4683435812869384390948, -1187252250085449887296, -149319496393491484627, 35161134110441480840, 1792552784613818351, -401684023324405389], [-7153964600185080668547456, 14463609179121479652373504, -40595470245958879516187224, -70151831340015541353681176, 50661168686216023509934634, 39356639855540743272837129, -19276961548576070622697078, -8923246732173942392520677, 3463513350492767939837433, 1005145130013904108707722, -331454329038764362490708, -58904936706635741320058, 17237980831271333470684, 1708714101000259789197, -457468265004854947110, -19284674488241722281, 4829276221031694793], [-5138725070565031353389952, -19010609048725512549715520, 4238097793521018709476680, 32174626963205953829221624, -4706082324872649705752926, -16401285927440756711216487, 2601771794010579321439772, 3679657207272629908937815, -623546378582022859241151, -414728277741166000467232, 72958850156528943501796, 24312102165820772638288, -4366138031426659256228, -704677004901730046361, 128039367790348748568, 7942108156575237441, -1453038091328142239], [6517735619048495503435392, 55469087455397896100328512, 105658747314575401830992392, -32152647990081680920035688, -105845741991384190307302358, 12095714859611388303880485, 34756651463343465655682494, -2472464321682009327529213, -5320684556007493253013747, 252696123365868605562142, 426437228613283366401560, -13187166866038469263726, -18409681132571854969384, 338203908811421232909, 404863097090957807754, -3385940474786076861, -3555216097341445111], [-57296721716171638867584, 959678662302754067843392, 2090299834438029090161736, 152978406136835152639192, -1611689526537873451363206, -444293491713271477435443, 515710907077155482290694, 142732466580791125364691, -86750295822367095797883, -19038545289400276194442, 8132893409488820723376, 1229771504057779472410, -422086926091684992160, -37916402873948071123, 11255478868614860042, 445526327537030795, -119629311245531367], [4236507095656427146743168, 55470839455481591137640896, 116323706342218721755151144, -24915531096114567203929256, -115221199883171356814809630, 4696803797837985984577905, 39223941287140969807033922, -115186227595703219180573, -6343325546399511195442239, -75957202429120851331942, 542929528061973571293940, 8921433310284870691846, -25229580738510485037860, -365226872150539899915, 601138302887419570842, 5076712012817397759, -5747872223055716855], [-21080850440440604229280896, -99324566713284533917102016, -96065753360577250702798648, 70481065908800217157672024, 105841107352771081370713226, -17473398284762458322961771, -38499356501626419899083930, 1378626587778159489856783, 6582150944399490450099237, 58623640258067664718886, -590746390376011896797012, -13003027896057641526566, 28610574430359347732356, 587779790581119353253, -707170283228976671250, -8426738008742616801, 6986541270776725381]];
    Rf_basisdens := [1, 1, 1, 1, 198841345374492269121024, 198841345374492269121024, 198841345374492269121024, 33140224229082044853504, 99420672687246134560512, 16570112114541022426752, 16570112114541022426752, 198841345374492269121024, 49710336343623067280256, 99420672687246134560512, 3012747657189276804864, 99420672687246134560512, 99420672687246134560512];
    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_1859_a();"
function MakeCharacter_1859_a()
    N := 1859;
    order := 1;
    char_gens := [508, 1354];
    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_1859_a_Hecke( : Kf := Kf)
    return MakeCharacter_1859_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 := [[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, -1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 1, 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], [-11, 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, 0, 0, 0, 0], [8, 3, 1, 1, 0, -1, 2, -1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [22, 2, 0, 0, -1, -1, -1, 2, 1, 1, 0, 0, 0, -1, -1, -1, 0], [-15, 9, 0, 2, -3, 1, 2, 0, 3, 0, 1, -1, 0, -2, 0, -2, -1], [3, -1, 5, 3, 1, 1, 3, 2, -7, 0, -1, 1, 2, -2, 1, 2, -1], [41, -4, 0, 0, 1, -1, 0, -2, -1, 1, -1, 0, 1, 1, -1, -1, -1], [60, 11, 1, -2, -2, 0, -3, -1, 0, 1, 0, 0, 2, 2, -2, 0, 3], [13, -15, -5, 2, 6, 0, 2, 5, -9, -1, -2, -1, 3, 0, 2, 0, 2], [-38, 6, -10, 1, 3, 1, -1, 0, 0, 1, -3, -4, -1, -2, -3, -1, 3], [43, -1, 18, 3, -3, 5, -1, 3, -11, -4, 1, -2, -1, -3, 2, -5, 0], [-54, 35, -4, 3, 9, -2, -5, -1, 28, -3, 1, -3, 0, 2, 2, -1, -2], [136, -9, 23, 8, -2, 5, 10, -6, -8, 0, -1, 3, 4, -5, 2, 3, -6], [28, 7, -2, -5, 8, -3, 4, -5, -13, 12, 2, 2, 2, -1, -1, 5, 0], [-12, -15, -14, 4, -5, -2, -1, -1, 0, -5, 4, -12, -8, -5, 2, -6, 0], [32, 21, -5, 4, 1, -5, -4, 4, -30, 10, -6, 13, 11, 1, 5, 12, -4], [52, -83, 29, 7, -12, -3, -6, 6, -2, 2, 0, -6, 2, -10, 2, -1, -8], [-64, 29, 7, -1, 11, -1, -19, 2, 40, 2, -5, 3, 0, -4, -1, -5, -13], [13, -9, 7, -16, 3, 6, 6, 2, -53, -9, 5, -14, -13, 0, -9, 3, 9], [244, 1, 16, -9, 3, -14, -9, 15, 4, 7, 2, -5, -1, 2, -2, -12, -5], [139, 112, -4, -7, 4, 1, 7, -8, -16, -7, -6, -7, -5, 0, -8, 15, 6], [16, -76, 2, 5, 3, 0, 24, 14, 50, -6, -12, 5, 15, 3, 13, 19, -2], [65, -37, -38, -15, 6, 2, -18, 2, -12, -11, 4, -3, -11, 10, -5, -10, -2], [54, 82, 58, 1, -14, -2, 6, -19, -55, -8, 11, -4, 1, -3, -2, -17, -19], [374, 69, 35, 10, -1, -25, -26, 2, -15, 12, 0, -17, -9, 11, -5, -7, -8], [-28, 60, 4, 22, -9, 18, 9, -19, 4, -30, 2, 6, 4, 6, 15, 8, -7], [58, 24, 1, -22, -6, -7, 43, -10, 84, -15, -1, -34, -23, -16, -11, -7, 4], [-143, 160, 22, -2, -3, 2, 37, -3, -8, 8, -5, 13, 10, -8, -11, 16, 3], [-37, -104, -8, 44, 33, -25, -34, -25, -32, 3, 3, -1, 0, -7, 5, 9, -22], [23, 11, -65, 7, -38, -3, -58, -21, 99, 8, -9, 1, 2, 8, 2, -5, -6], [352, 181, -18, 11, -27, 28, 55, -14, 5, -33, -6, 2, 0, -34, 24, -1, -26], [-340, -65, 20, 14, -37, 16, 31, -12, -5, 9, 4, -6, 10, 25, 0, -9, -12], [-142, 54, 63, -19, -21, -20, 56, 5, 63, -8, -1, 0, -1, 8, 16, 17, 2], [331, -42, 5, -46, 14, 11, 10, 15, -20, -7, -1, 5, -26, 24, -23, -51, 11], [140, -30, -39, -7, -5, 16, 43, 41, 59, -18, 14, -50, -4, -8, -8, -34, 9], [-303, -14, 50, -6, -34, 9, 44, -23, -39, -3, 22, 32, -6, 3, -2, 0, 17], [197, -139, 35, -21, 4, 6, 27, 0, -34, 29, 0, 39, 25, 35, 2, -1, 10], [309, -178, -121, 43, -15, -3, 53, 14, -5, -21, -9, 6, 43, 11, 10, 18, -19], [812, -56, 57, -5, -37, -4, -53, 4, 16, 2, -14, 4, 4, 22, -6, -19, -35], [340, -78, -5, 57, 50, 24, -18, -29, -108, 2, -27, 63, 52, -2, 43, 44, -8], [957, 161, -50, -4, 31, 0, -68, 19, 235, 1, -19, -2, -31, 3, -6, -29, -3], [-274, -136, 30, 0, -52, 2, 99, -77, 196, 20, 8, -14, -16, -32, -18, 44, 28], [-724, 192, -33, -15, -61, -11, 9, -20, -122, 15, 21, 6, 7, -1, -28, -38, 25], [1625, -129, -51, -3, 86, 0, 6, 28, -27, 17, -2, 7, 29, -43, 6, 60, -2], [586, 267, -96, -1, -33, -16, 17, 78, 17, 1, 30, -17, -7, -26, -4, -58, -31], [654, 272, 24, 44, -19, 3, 104, -13, -111, -29, 8, -22, 10, -49, -5, -25, 8], [1278, 222, 39, 36, -68, 0, 72, -10, 12, 26, 28, -12, -18, -15, -1, -16, -4], [209, -215, -9, -72, 48, -23, 45, -30, 132, 54, -19, 50, 7, 9, -54, 86, 3], [1001, -54, 68, -20, -38, -25, 76, 34, -153, 55, 7, 12, 27, 35, -10, -15, -30], [-517, -92, 130, 54, 73, -9, -66, -107, 348, 11, -35, -14, 25, -25, 3, 38, -21], [-59, 5, -116, 1, -105, -4, 121, 62, 172, 17, -8, 5, -23, 42, -34, -23, 3], [-1166, -230, 13, -31, 76, -6, 64, 67, -183, -8, 13, -54, -17, -6, -33, 42, 1], [928, -332, -45, -13, 91, -15, -84, -22, 65, 5, -1, -72, -11, -43, 22, -50, -33], [943, 276, 25, -71, -20, 4, -58, -35, -104, -14, 54, -24, -82, -6, -31, -34, 24], [437, -449, 103, -12, -18, 33, -5, -111, 195, 10, 33, -25, 32, -65, -12, 5, -12], [666, 24, -7, 53, 126, -16, 2, 42, -14, -38, -35, -17, 12, -86, 19, 91, -16], [-308, -177, -75, 20, -36, -20, 19, -48, 188, -25, -19, 48, 73, 52, 28, -11, -44], [-115, 93, 244, 124, -100, 15, 6, -6, -211, -32, -49, 43, 32, -14, 65, 44, -42], [331, 292, 16, -96, -157, -4, 113, -37, 288, 38, 1, 7, -22, 30, -47, -30, 11], [2, -212, 28, 139, -27, 15, 11, -50, -295, -5, -15, 36, 17, -18, 47, 75, -9], [230, -409, 221, 52, 69, 9, 3, -23, -59, 14, -39, 6, 39, -5, -45, 81, 52], [-98, -118, -172, 70, -32, -14, 82, -155, 515, -8, 36, 21, 13, 22, -40, 18, 19], [572, 487, 156, 172, -173, 35, -12, -127, 93, -18, -15, 18, -21, -62, 36, 19, 14], [1308, -738, 19, -132, -49, 41, 43, 90, -292, -23, 43, -6, -31, -46, 2, -9, 95], [3270, 267, 107, -8, -38, 23, 89, 109, -92, -60, -9, -113, -80, -17, -30, -76, 94], [2729, -1112, 262, 8, -23, -30, -117, -52, -337, 18, -40, 60, -6, 38, -3, -68, -19], [104, 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-707, 301], [59654, -15184, -5552, -1870, -146, 144, 4533, 2047, -2692, -128, 544, -1823, -1645, -806, -348, 1398, 1931], [13114, 21141, -3867, 1714, 275, 1064, -2158, -525, 6623, -1293, 1287, 307, -492, -438, 755, 1090, 6], [-90849, -528, 8193, -1469, 3304, -728, -3647, -452, -1754, 900, -891, -24, 61, 1078, -961, -884, 121], [12991, 941, -780, -2049, -875, -836, -1225, 4817, -5724, -773, 928, -642, -1718, 674, -1296, -127, 1200], [18892, -22779, 165, -1476, -1385, 637, 330, 2899, 5107, -106, -385, 110, -35, 2511, -27, -125, -126], [27000, 998, 620, 1398, 1258, 866, 968, -3006, 2660, -304, -1448, 1122, 938, -126, 564, 2116, 164], [-94461, -5315, 4571, 1251, 258, 352, -851, -1555, 3797, -433, 944, -1244, -488, -1915, 500, 113, -1441], [33665, -470, -2228, -1178, 1348, -814, 179, 2344, 3642, 1312, -20, -42, 826, 576, 596, 1187, 738], [1130, -2813, -618, -1596, 1687, 1442, -6597, -685, 5772, -1061, 574, -1066, -1172, -1419, 1074, 417, 814], [13076, 11073, 1987, 203, 320, 448, 436, -3806, -7523, -1091, 39, 1282, 1209, -1321, 1434, 1376, -450], [19292, 10777, -3680, -331, -145, -525, 2672, 2351, -7034, 1236, -126, -786, -36, -331, -1004, 193, 1601], [-1458, -2240, 53, -933, 497, -1892, 202, 2872, 5163, 654, -339, -619, 48, 36, -358, -1982, -253], [-35284, -10586, 4470, -280, -2317, 131, -272, 2720, -2464, -191, -392, -1858, -488, -471, -1637, -516, 478], [-35454, -137, 2780, 1522, 1004, 995, 1447, -1996, -1684, -1492, -63, -987, -634, 32, 939, -574, 568], [-19889, 9935, 2508, 457, -1490, -455, 1119, 2553, 5738, 2370, -335, 1029, 206, 1207, -779, -454, 1974], [4715, 4026, 5378, 0, 1841, 329, -8037, 1018, -4498, 663, -644, 253, -1331, 569, -527, -2153, 1317], [-43176, -467, -1327, 1138, -1620, 1030, 1749, 213, 3915, -1661, -386, 1347, -1253, -760, 1550, 671, 86], [10639, -10356, -7163, 1546, 2115, -645, -2912, 3032, -9146, -1151, 90, -773, -1153, -884, 626, -1865, -387], [696, -18165, -3762, -423, 148, -1025, -2540, 4000, 5461, 422, -755, 1461, 1186, 951, 841, 350, 96], [-17869, 2139, 3766, -804, 1154, 1284, 3834, -1866, -4218, -9, -538, 1075, 1027, -2049, 257, 3609, 166], [-16721, 3912, 2877, 2416, -1798, -921, 4679, -3498, 7906, 513, -1970, 798, -106, -516, -951, 703, 297], [55548, -11906, -728, -397, -2305, 200, -2571, -3392, 9991, 268, -580, 1326, 370, 331, 1853, 1135, -31], [51648, 5844, 72, -689, 38, 202, -4023, -2462, -3122, -1564, -8, -1495, -649, 1279, -336, -798, -515], [-47411, -6022, -5423, -1407, -1391, 1015, 1481, 788, 4552, -1549, -683, 505, -536, 597, -374, 1447, 2746], [-64397, -6679, 663, 776, 599, 371, 5116, -232, 6686, -424, -488, -2592, -336, -1133, -1345, -2362, 879], [-7324, -24899, -222, -1343, -3786, -530, 1647, -2384, 2804, -147, 606, -782, -806, 194, -945, -2138, 723], [-1886, 12548, -3345, 619, -2286, 485, -1773, 694, 8740, -3103, 1232, -3444, -2658, -1489, 513, -2614, -515], [14127, -4976, -2064, -1482, 199, 574, 5063, 145, 1843, 870, 684, 329, -917, -728, -567, -1144, -458], [66990, 1292, 3800, 191, -441, 348, -4134, -433, 102, -1430, 1372, -2426, -728, -1563, -329, -2633, -707], [51269, 2456, -26, -1004, 1767, -673, 2285, -1202, -2189, 1219, 427, 86, 737, 7, -993, 2431, -731], [-16231, -18275, 1577, 2956, -2365, 1943, 988, -1181, -13476, -1076, 72, -450, 318, -1979, 1721, -2236, -1791], [-47537, -4359, 4245, 269, 2452, -199, 1897, 997, 6579, -482, 304, -2123, -515, -360, -940, -1124, 553], [28834, -5334, 5677, -1081, 951, -308, 801, 3610, 6139, 354, 149, 1458, 385, 1078, -8, -1477, -554], [-3995, 53, -3010, 33, -690, -406, 4083, 3548, -2339, -471, 436, -99, -1193, 2032, 741, -481, -242], [15252, -3894, -4886, 1046, 2223, -48, 974, 892, 7116, -1728, -310, 253, 267, 1326, 231, -731, -1160], [3505, 935, 595, 664, 2545, -114, 5461, -2050, -4374, 1219, -533, 131, -930, -550, -953, 1468, 976], [-30892, 11144, 2602, -572, 2208, 1554, -1995, -369, 10004, -1912, 23, 71, 398, 2020, -540, -236, -420], [-34259, -2045, 1312, -382, 1248, -934, 6752, 2207, 188, 1925, -224, 388, -892, -2233, 485, -1758, -1293], [16784, -339, 399, -2190, 398, -4, 2040, 1945, -7867, -463, 333, 709, 544, 1714, 112, 2591, 745], [4537, 2148, 252, -1720, 1088, -69, -1766, -1386, 6956, 319, 2567, 1226, -97, 243, -286, -1144, -1600], [-27766, -14291, 7984, 2259, 838, -529, -1991, 298, -2534, 1022, 919, -176, 2365, -1227, 213, -235, -347], [15888, 3471, -7406, -3818, -296, 782, 5766, -73, -10822, -695, -12, 349, -889, 757, -1989, 577, 694], [-38697, -8956, 696, 1706, 253, 424, 3356, 715, -10183, -1324, -455, -1256, 227, -1576, 103, -659, -1468], [16251, -1772, -8430, 419, 1186, -77, -1595, 3206, 5154, 847, -663, 707, 34, 1304, 264, 314, -759], [-36960, 11736, -2467, 434, -52, -1459, 4213, 2873, 12898, -779, 405, -1300, 69, -844, -381, 1153, 1666], [-56570, 3327, 7938, 1820, -568, -340, 2701, 849, -4812, -109, -581, 970, 1181, -1097, 1717, -1023, 224], [-27265, 3383, 5609, 463, -2899, 1241, 1396, -2025, 21, -576, -144, -261, -899, 2546, 987, -531, -328], [17819, -4000, 3739, -198, 3266, -638, -1331, 2780, 7927, -2266, 924, -1989, -959, -1851, -1195, -108, -97], [33592, -23302, 2309, -1320, 396, 849, 452, 3918, 1063, -1033, -466, -3088, -678, 866, 171, -1107, 1797], [65951, -14876, 103, 434, 1879, 1142, 811, 789, 2223, 2498, -718, 1269, 1581, -1505, -922, 2224, 1174], [-13083, -539, -716, -442, 578, 395, 2672, 4331, 7174, -2228, -745, -1028, -267, -222, 489, -1190, 1235], [-29929, 1597, 2740, -1955, -2242, -569, 4718, 2340, 2191, 946, 291, -2111, -1656, -663, -1743, -1807, 1336], [-49176, -1588, 500, -4858, -1798, -756, -796, -289, -6197, 522, 1464, -270, -964, -1736, -1476, -1940, 1348], [8463, -2338, 890, -233, 696, -427, -1773, -4227, 4152, 87, 921, 1173, 706, 1944, 716, -261, -1241], [13691, 3700, 1647, -6, 1598, 547, 1993, -767, -8626, -1031, 0, -674, -534, -2916, 1267, 2374, 3], [-2381, 28505, 720, 272, 767, -1337, -812, -1373, 10275, 1154, -226, 729, 1021, 2590, -574, 1847, -582], [10300, 4815, 5038, 2986, -771, -375, 714, -2128, 10482, 2364, -333, 3038, 2059, -880, 2216, 1750, -1888], [50747, -11982, 5325, -274, -5519, 1117, 4142, 2114, -2843, -343, -155, -1006, 159, -10, 388, -1705, -731]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1859_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1859_4_a_i();".
// 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_1859_4_a_i(:prec:=17)
    chi := MakeCharacter_1859_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_1859_4_a_i();".
// 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_1859_4_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1859_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![2596992, 27375104, 37775816, -32766128, -42560950, 20197463, 15765187, -6021311, -2799440, 941923, 267574, -81750, -13998, 3949, 375, -99, -4, 1]>],Snew);
    return Vf;
end function;