
// Make newform 1339.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1339_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_1339_2_a_d();".
// 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_1339_2_a_d();".
// 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 := [-63, -135, 1062, 855, -6099, -108, 14037, -5183, -14841, 9364, 7199, -6826, -1079, 2337, -279, -351, 106, 13, -9, 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, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-367316328, -2239694676, 522952632, 16419032754, 2390504031, -38810104606, -2587790527, 43197066608, -2547741249, -25338218398, 4425829614, 7874399570, -2146010844, -1193838810, 439605017, 64448440, -35333866, 866534, 518861], [-52348346, -352301263, 73057679, 2631800879, 484643920, -5515646682, -1014594649, 5323234806, 912123174, -2876506439, -475558874, 988608896, 126785428, -224139921, -9244957, 30046194, -1918661, -1733205, 266872], [-256932042, -729624204, 991928916, 5929583586, 290078265, -16348498764, -2506879364, 20944458162, 1743397843, -14303955112, 90805985, 5484083056, -461151195, -1175948955, 170797761, 131218405, -25217420, -5921985, 1351897], [-176479527, -369112455, 2472441105, 4816071849, -13454741109, -10754227640, 29212585114, 5268963067, -28454019576, 3594678784, 13490967474, -4136363897, -3119479608, 1441567245, 287327512, -219039295, 5062687, 12425812, -1733783], [11552727, 80522568, 547247193, -337108071, -2943638814, -3234689276, 7551460911, 8856919174, -11949471883, -6745738768, 9809723083, 1163695290, -3796144359, 494081487, 628906693, -188001263, -26659212, 16696447, -1774029], [-72057150, -354761775, 959132472, 1341841695, -4541442723, -3874829504, 11072810483, 5325699157, -15359073416, -1694028624, 11125918067, -1667724457, -3878467977, 1273304997, 557989561, -292838892, -9319981, 22214635, -2967169], [396383271, 1767136542, -2877998025, -13208855523, 12278066430, 32040083114, -29224136445, -34577299945, 35953796293, 16410444493, -23045098663, -1724394570, 7497899679, -1087566852, -1104351670, 340341434, 41511942, -27847177, 2975640], [-594935301, -2240429361, 6285394443, 17502771159, -26415907989, -40591999526, 52932730804, 38723604598, -54462269205, -14778103997, 29906142930, -13650818, -8621118225, 1527145143, 1157175646, -377869468, -39139232, 28229263, -2977490], [179541513, 165420858, -2847026964, -1311577932, 15320518098, 2468241711, -33239325248, 2244994815, 34008428146, -7761853987, -17717082622, 6248040703, 4641677091, -2232289599, -503026266, 370547824, -4157897, -23242464, 3222355], [380675985, 2255746206, -3103113504, -18421513458, 15066984189, 43775529148, -36360422821, -43261149383, 42616920457, 17991728124, -25462459450, -1139127976, 7728033165, -1329885306, -1060064717, 365202966, 32898422, -28246403, 3242789], [854546172, 2826082806, -7161240999, -21633770148, 26954603601, 50255632128, -54269913053, -49254184047, 58206182989, 20223944111, -33486398635, -917532422, 10037043489, -1718293269, -1382033019, 460096249, 45831970, -35489841, 3916792], [-538160700, -1542363603, 6282432237, 11076515934, -27167273565, -23941531913, 54803623152, 18383790961, -55839515491, -1069577854, 29962810723, -5372183439, -8288751996, 2731646766, 1009915663, -521843402, -14137671, 35302225, -4506342], [-312204861, -1686760458, 3588750066, 14632675959, -17035177710, -36892221704, 40741389017, 37822024414, -49097203763, -14808048921, 30469391777, -895515472, -9524314713, 2193453195, 1312656961, -537568329, -33835177, 40706596, -4982707], [-661897668, -2222818413, 5448393864, 16743331674, -20129603628, -41201640961, 42992247621, 43796725379, -50947890482, -19202427860, 32287372634, 488579049, -10388315715, 2093135799, 1478773274, -563967892, -42791964, 44295845, -5349330], [507806415, 1811028675, -5996011284, -14013135555, 29484011664, 30962880442, -64012151559, -23988369473, 67636343651, 1580203424, -36950019419, 6828072783, 10301983554, -3488842920, -1253274389, 666919645, 15470457, -45112259, 5831874], [532661505, 1875025842, -6412415181, -14803792149, 31499657322, 32655029953, -67703514507, -25530330101, 71956966178, 1569302456, -39967012229, 7773012714, 11324010768, -4042041579, -1378655723, 785351959, 10811283, -53818736, 7258260]];
    Rf_basisdens := [1, 1, 1, 24433851, 8144617, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851, 24433851];
    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_1339_a();"
function MakeCharacter_1339_a()
    N := 1339;
    order := 1;
    char_gens := [1237, 417];
    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_1339_a_Hecke(Kf)
    return MakeCharacter_1339_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+1)]];
    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, -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, 0, 0, -1, 0, 0, 0, 0], [-1, 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, -1, 0, 0, 0, 1, 0, 0, 0, 0], [1, -1, -1, 0, 0, 1, 0, -1, 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], [-3, 3, 2, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 0, 0, 1, 0, 1], [-1, 2, 0, 0, 0, -1, 0, 0, 1, 0, 1, -1, 0, 0, 1, 0, 0, 0, 0], [0, -1, -1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -2, 0, 0, -1, 1, 0], [-1, -2, -1, 0, -1, -1, 1, 1, -1, -1, -2, -1, 1, -2, -2, 1, 0, 1, -1], [-1, 3, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 2, 1, -1, 0, 0, 1], [-1, 0, -1, -1, 1, 0, -2, -1, 0, 1, 0, 2, -1, 0, 2, -1, 2, -2, 1], [-2, -1, -1, -1, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, 0, 0, 0, 1, 0], [1, -1, -1, -1, -3, -1, 1, 1, -1, -2, -2, 1, -1, -2, -2, 0, 1, -1, 0], [-2, -1, 0, 0, 1, 1, 1, 0, 1, -1, -1, -2, 1, 0, 0, 1, -2, 2, -1], [0, -3, -1, -1, -1, 0, 0, 0, 2, -2, -1, 2, -1, -2, -1, 2, 1, -1, -1], [-5, 0, 0, -1, -1, -2, -1, 1, -2, 2, -1, 1, -2, -3, 1, 0, 1, 1, 0], [0, -2, -2, -1, 0, 3, -1, -2, 2, -2, -1, -1, 0, 0, -1, 0, -2, -1, -1], [0, 2, 2, 0, 0, -1, 0, 1, -1, 1, 3, 1, 1, 2, 0, -1, 2, 0, 0], [-2, -3, 1, -1, -1, -1, -1, 0, -1, 1, 1, 0, 1, 0, 0, 1, 1, -1, 1], [0, -1, -1, 1, -1, 2, 0, 0, 0, -4, -1, 1, 0, 2, 1, 2, 0, -1, -4], [-1, 1, 4, 1, 2, 1, -1, 2, 0, 1, 3, 2, 1, 3, 0, 0, 2, -3, -1], [-1, -1, -1, 0, -2, -1, 1, 1, -1, 0, -1, 3, 0, 0, -1, 1, 1, 1, -1], [-4, -1, 0, 2, 0, 0, 1, 2, 0, -1, -1, -1, 1, 1, 0, 1, 0, 0, -2], [-1, 0, -1, 1, -2, -1, 1, 1, 0, 1, -1, 1, -2, -2, 2, -1, 1, 1, 2], [-3, 3, 3, -1, 2, -1, 2, -1, 2, 1, 2, 1, -2, 3, 3, -2, 0, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 1, 0, 1, 2, 0, -1, 2, 0, 3, 0, -3, 1, -1, -1, -1, -2, 1], [0, -2, 1, 2, 1, 3, 1, -1, 0, 0, 0, -2, 1, 3, -2, 1, -2, 1, -1], [0, -1, -1, -3, -1, 1, 1, -1, 0, -1, -1, 2, -1, -1, 0, 1, 1, 0, 0], [3, 2, 1, 4, 2, 2, 0, 2, -1, 0, 3, -1, 1, 4, 2, -1, 0, -2, 0], [1, -4, -5, 0, -2, -3, 1, -3, 0, -1, -2, -3, 2, -6, -4, -2, -1, 3, -1], [-1, 5, 4, 4, 2, -1, 0, 1, -1, 1, 4, -2, 1, 6, 2, -1, 0, -1, -1], [3, -3, -5, -1, -3, 0, -1, 0, -1, -3, -3, 3, -2, -3, -1, -1, 2, -1, -1], [-6, 0, 2, 0, 3, -1, -2, -3, 0, 3, 2, -1, 3, 1, 2, -2, 0, 0, 1], [-3, -1, 2, -1, 1, 0, -3, 1, -2, 4, 0, -1, 0, 0, -1, 1, -1, -1, 4], [-5, 2, 4, 2, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, -1, -1, -1, 2], [1, -3, -2, -1, -3, 1, -2, 2, -2, -2, -3, 1, -1, -4, -1, 1, -1, -1, 0], [-5, 7, 3, 2, 4, 1, -1, 1, 4, 3, 6, 0, 0, 4, 4, -1, -1, -1, 2], [0, -2, -2, 0, -1, 0, 5, 1, 2, -5, -3, -1, 1, 2, -3, 2, -2, 0, -6], [0, -8, 0, 1, -1, 2, 1, -1, -3, 0, -1, -4, 6, 2, -5, 3, -2, 2, -5], [0, -5, 0, -4, 2, 0, -1, 1, -2, -1, -2, 2, 0, 0, -3, 1, 1, 0, -1], [6, 2, 1, 1, 2, 5, 3, -4, 2, -3, 2, 0, 0, 8, 1, -3, -3, -1, -2], [-1, 5, 4, 2, 0, -4, 0, 0, -4, 5, 4, -2, 3, 5, 3, -2, 4, 3, 1], [-4, 2, 1, -1, 2, 0, 0, -1, -1, 5, 0, 0, -1, -2, 0, -4, -2, 2, 5], [-4, 6, 3, -3, 4, -5, -3, 2, -1, 4, 3, 2, 0, -3, 1, 0, 4, -1, 7], [-1, -3, -1, 4, 1, 3, -1, 1, 1, -2, -1, -2, 1, 0, -1, 6, -5, 0, -3], [2, 7, -2, 1, 2, 2, 4, 0, 6, -2, 4, 0, -3, 3, 0, -3, -2, 1, 0], [0, 9, 9, 3, 6, 2, 1, 0, 2, 2, 10, -1, 5, 11, 2, -1, 0, -3, 1], [-3, -2, -4, -4, -1, 0, -1, -5, 0, 1, -5, -1, -4, -7, 0, -2, -4, 1, 3], [-3, 0, -3, -2, -2, -5, -3, -1, 0, 3, 1, -3, 1, -8, -1, 0, -1, 4, 5], [-5, -2, -1, -1, 0, -2, -1, 3, -2, -2, -3, 4, -3, -3, 2, 2, 2, -2, 1], [4, 0, -2, 1, -1, 2, 1, 0, 1, -3, -2, -1, 1, 1, -2, 0, 2, -1, -4], [-4, -5, -4, -3, -1, 0, -3, -1, -1, 1, -4, -1, -1, -5, -1, 1, -2, 4, -1], [1, 1, -4, 1, 0, -2, 1, -3, 4, 0, 0, -3, 2, -2, 0, -2, -1, 0, 0], [1, 0, 4, -1, 3, -1, -3, 0, -3, 6, 0, -2, 3, 1, 2, -1, -1, 0, 0], [-15, -1, 2, -4, -2, -2, -1, 3, -4, 0, -2, 2, -3, -4, -3, 3, -2, -1, 2], [-3, 3, 3, -1, 1, 1, 2, 3, 0, -2, 2, 3, 0, 5, 1, 3, 1, -2, -2], [10, 0, -3, 4, -2, 1, 4, -2, 2, -7, 1, -3, 2, 1, -3, 3, -1, 0, -5], [-10, 7, 4, 4, 3, 0, -1, 2, 3, 3, 4, -1, 4, 2, -1, 2, 0, 0, 2], [1, -1, 1, 1, -6, -1, 2, 2, -6, -1, -2, 1, 1, 3, -2, 1, 2, 0, -3], [-5, 6, 1, 1, 2, 6, 1, -1, 5, -2, 2, 0, -2, 5, -3, 0, -2, -4, -3], [5, -4, -7, -4, -2, 0, 1, -4, 3, -2, -2, 0, -1, -6, -6, 2, -2, 2, 0], [-5, -2, 2, 0, 1, 2, 3, 4, -1, 0, -1, 4, 0, 3, -3, 3, 1, -3, -2], [1, 5, 3, -1, 1, 0, 2, -1, -2, 1, 3, -2, 3, 3, -4, -2, 0, 3, 1], [-6, -3, -7, -6, -2, -7, -3, -5, 0, 4, -5, -1, -2, -12, -5, -3, 0, 4, 5], [0, -2, -3, -1, -3, -2, 3, 3, 0, -5, -5, -1, -1, -5, -3, 2, 1, -4, -3], [2, -12, -8, -7, -4, -2, -4, -4, -2, -3, -10, 0, -8, -14, -1, 0, -4, 3, 0], [8, 1, 3, 3, 2, 5, 2, -1, 1, -5, 4, 0, 0, 6, 0, 2, -3, -2, -2], [3, -4, -2, 1, 0, 6, 1, 4, 2, -7, 1, 2, -1, 2, -3, 3, -4, -3, -5], [-1, -2, -4, 4, -5, 2, 4, 3, 0, -4, -6, -3, -2, -2, 1, 0, -4, 1, -5], [0, -2, 0, 3, 0, 6, 3, 4, -2, -5, -1, 3, 1, 6, -2, -1, 0, -2, -4], [-2, -6, -5, -6, -1, -4, -2, -2, 0, 0, -9, 2, -2, -14, 0, 2, -1, 0, 3], [7, -5, -4, -4, -4, -6, -1, -2, -2, 2, -5, -2, -1, -8, 1, -1, 3, 1, -1], [3, 2, 4, 3, 5, -4, 1, -1, 1, 4, 7, -3, 4, 5, 1, -2, 0, 4, 1], [-7, 11, 0, 4, 3, -2, -3, 0, 0, 5, 6, -2, 2, 2, 2, -1, 6, -2, 3], [1, -6, -3, 5, 1, 0, -1, 0, -2, 2, 2, -4, 6, 3, -2, 3, -5, 4, -2], [-4, 0, -2, 1, 0, -1, -3, -2, -1, 5, 0, -9, 1, -6, -5, 3, -5, 4, 1], [-3, -3, -3, -4, -9, -5, 1, 1, -2, -1, -7, 7, -5, -8, 2, -2, 8, 1, 1], [1, -3, 0, 4, -1, 0, 0, 1, 0, 3, -2, -3, 0, -3, -1, 1, -5, 0, 0], [8, -14, -6, -4, 0, 2, -1, -2, 2, -4, -6, 0, -2, -6, -3, 2, -1, -2, -1], [-1, 6, 7, 0, 2, 1, 5, -5, -2, 5, 2, 1, 0, 6, 0, -6, 1, 5, 1], [2, -1, 2, -2, 3, 0, -4, -2, -3, 4, 0, 1, 1, -1, -2, 0, 5, -2, 1], [10, -5, -8, 2, -3, -1, 0, 4, -2, -2, -1, 2, 3, 2, -5, 4, 5, -3, -4], [-2, 0, -2, -8, 0, 2, -1, -4, 3, -5, -3, 1, -5, -5, -1, 1, 2, -4, 3], [-3, 1, -1, 0, -2, 1, 1, -2, 0, -1, -1, 0, 3, 1, -5, -1, 1, 2, 0], [-3, 6, -1, 4, -2, -4, 3, -2, 1, 4, 4, -2, 2, 3, 5, -5, 4, 2, -2], [4, -4, -1, 4, 2, -1, -2, -3, 1, 2, -1, -3, 1, 1, 7, -3, -4, 5, 0], [-14, -1, -4, -4, -1, -3, -3, -3, 0, 0, -3, 0, -2, -8, -2, 4, 0, -3, 6], [7, -3, 0, -1, -5, -2, 3, 1, -2, -3, -3, 2, 0, -2, 3, -3, 1, 3, -2], [2, -5, 1, -5, -3, 4, -2, -1, -3, -2, -7, 4, -7, 0, -2, 2, -4, -3, 2], [10, -6, -2, 0, -6, 0, 1, 1, -3, 4, -3, 2, -3, 0, 2, -5, 3, 2, -1], [7, -5, -3, 4, 0, 3, 0, -4, 5, -1, -1, -6, 4, 1, -1, 2, -4, 1, -4], [-4, 5, 6, 1, -1, -3, 1, -2, -3, 8, 4, -1, 0, 1, 4, 0, 2, 3, 5], [-9, 5, 1, -4, 1, -4, -4, 1, 2, 3, 0, 7, -2, -6, 3, 0, 4, -3, 9], [6, 3, 3, -3, 3, 0, -1, -2, 3, -1, -1, 4, -1, 4, 7, 0, 4, -1, -1], [-6, 0, 0, 0, -2, -3, -4, -2, -2, 2, 1, 2, -1, -1, 2, -2, 1, -3, 8], [-5, -4, 1, 1, -1, 2, -3, 2, -2, -3, -2, -3, -4, -2, -2, 4, -2, -2, -5], [1, -5, -6, -2, -1, 7, 0, 1, 0, -2, -6, 5, -2, -2, -8, 3, -1, -4, 2], [7, 11, 6, 7, 2, 3, 3, 0, 5, 2, 11, 0, 3, 12, 2, -2, -2, -2, 2], [1, 1, 5, 3, 1, 6, 3, 0, 4, -2, 3, -6, 0, 6, -2, 0, -6, 2, -3], [-13, 2, 6, 0, 3, 2, -2, 0, -2, 5, -1, -5, 1, 0, 2, 1, 0, -1, 2], [7, 5, 2, 1, 0, -1, 2, 1, 5, -2, 6, 0, 6, 7, -5, 1, 1, 1, -1], [-8, -2, 7, 2, -2, 2, -4, 8, -7, 3, -4, 5, -3, -4, 1, 5, 3, -4, 1], [-3, 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-4, 5, -4, 7, -4, 11, -3, -14, -7], [-8, -1, -11, -6, -11, -10, 14, 5, 1, -5, -18, -1, -9, -24, 0, 2, -1, 6, 5], [-11, -2, 1, 3, 6, -4, 6, -9, 2, 2, 3, -23, 15, -7, -6, 4, -16, 14, 8], [-1, -11, -24, -13, -19, -1, 2, -2, 2, -14, -20, 13, -17, -21, 6, 0, 8, -10, -7], [-7, -1, -7, -4, -2, -5, 2, -8, -2, -4, -3, -11, 3, -23, -8, 0, -15, 15, 5], [24, -3, -8, 16, -3, 8, -2, 7, -3, 1, 11, -2, 7, 21, -2, -2, 5, -1, -8], [-8, -8, 2, 0, 5, 5, 7, -2, 4, 0, 6, 0, 3, 13, -5, -1, -5, 2, 2], [0, 23, 9, 9, 0, 6, 12, 4, 12, -15, 19, 4, 17, 21, -12, 10, 8, -12, -3], [-26, 16, 27, 7, 14, 18, 4, 4, -2, 8, 14, 10, 0, 35, 16, -7, 4, -5, 0], [-2, 6, 0, 8, -7, 6, 9, -1, 14, -15, 3, -3, -1, 7, -2, 2, -16, 9, -14], [10, -20, -2, 10, -14, 9, 6, 7, -8, -15, -9, 8, 7, 21, -3, 8, 0, -7, -24], [24, -29, -17, -13, -8, -5, -11, -2, -1, -10, -17, 8, -13, -16, -3, 3, -3, 0, -3], [-17, 6, 7, 4, 2, 8, -3, -1, -1, 3, 2, 11, -4, 9, 9, -1, 13, -5, 8], [-16, 34, 12, 21, 18, 7, 5, -2, 18, 6, 22, -6, 6, 24, 26, -9, -15, 6, 8], [23, -2, -19, -10, -4, 2, 2, -5, 9, -15, -5, 12, -13, -2, 8, -10, -1, -4, -3], [15, 4, -6, -2, 4, -2, 7, -9, 14, -7, 5, -9, 4, 7, -6, 3, -1, 9, -14], [-10, 15, 3, 0, -2, 1, 3, 1, 9, 2, -3, 8, -13, 11, 13, -6, 3, 2, 5], [-3, 11, -4, 3, 6, -1, 9, -6, 13, 7, 9, -8, 8, 5, 11, -7, 5, -1, 4], [9, 33, 21, 11, 10, 3, 6, 3, 14, 3, 22, 8, -1, 18, 4, -8, 0, -6, 1], [-7, -17, -1, 1, -4, 23, -2, -5, 4, -10, -13, 3, -10, 5, 5, 1, -17, -7, -7], [1, -9, 4, -1, -9, -2, -7, 4, -8, -7, -16, -9, -4, -13, -8, 8, -11, 3, -3], [2, 13, -5, 5, 8, 2, -6, -7, 6, 11, 5, -3, -5, 6, 18, -9, -7, 9, 7], [15, -25, -3, -4, -13, 8, 8, 4, -4, -22, -9, 10, 3, 6, -12, 20, 1, -6, -20], [7, -23, -19, 0, -6, 1, -2, -10, -3, -3, -13, -6, 5, -7, 0, -1, 0, 9, -6], [-10, 21, 25, 14, 12, 7, 0, 4, 7, 16, 35, -9, 23, 27, -8, 6, 0, 1, 7], [1, 29, 10, 2, 8, -11, 7, -2, 16, -1, 18, -2, 5, 18, 11, 2, 3, -1, 0], [-12, 19, 4, 7, 6, 0, -7, 6, -10, 9, 16, -8, 10, 18, 3, 0, -4, -1, 7], [0, -22, -20, -12, -10, 5, -3, -2, -1, -17, -20, 5, -11, -27, 3, 1, -6, 2, -6], [-15, 24, 17, 6, 10, 3, 8, 3, 1, 4, 9, 2, -1, 21, 14, -15, 1, 8, -2], [22, -12, 0, 9, 1, 12, 2, -8, 0, -10, 4, -21, 11, 4, -21, 11, -15, 4, -7], [24, 7, 10, 7, 9, 2, 6, -7, 1, 7, 21, -2, 6, 23, 5, -3, 0, 6, -6], [32, -3, -3, 2, 6, 11, 9, -11, 19, -12, 0, -9, 6, 12, 1, -2, -13, -5, -6], [-16, -3, -2, -15, 4, -5, 0, -11, -2, -2, -14, -9, -4, -19, -6, 5, -16, 11, 1], [8, 15, 0, 8, -10, -7, 9, 1, 3, -4, 8, -9, 16, 6, -5, 0, -8, 4, -4], [-14, 3, -6, -8, -8, -13, -5, 1, -8, 6, -9, -3, -10, -24, 9, 2, -1, 5, 8], [-29, -12, -10, -7, -6, 4, -7, 4, -7, -6, -19, -4, -8, -11, -4, 5, -10, 1, -9], [-22, -10, -8, -7, -4, -9, -2, 2, -2, 3, -5, 4, 3, -24, -12, 10, 4, 9, -1], [7, -6, -13, -18, -3, -12, 2, -4, 1, 0, -10, 2, -9, -12, -9, -9, 5, 11, 5], [-3, -28, -12, -11, -12, -4, -2, 5, -2, -7, -17, -2, -8, -27, -8, 16, -7, 1, 5], [5, -3, -3, 5, -4, 8, 9, 4, -3, -8, -1, -10, 5, 5, -19, 7, -12, 6, -13], [-5, -1, 7, -6, 9, 5, -9, 0, -3, 5, -8, 3, -10, 1, 27, -10, 1, 6, 4], [13, -29, -7, -3, -6, 1, -1, 17, -8, -10, -17, 13, -9, -2, -8, 13, 8, -8, -6], [24, -14, -13, -9, -11, -11, 8, 1, 4, -7, -8, 7, 1, -9, -3, 1, 17, -1, -14], [-18, 24, 7, -13, -3, -10, -1, 8, -8, 3, -9, 15, -15, -9, 7, -8, 18, 0, 4], [2, -29, -15, -7, -10, 0, 1, -4, -3, -11, -13, 7, 0, -10, -9, -2, 7, 0, -2], [1, -2, -4, -10, 10, 10, 4, -11, 9, -8, -4, -7, -6, 5, 2, -2, -9, -1, -10], [14, -24, -13, 3, -4, 0, 10, -1, 6, -13, -7, 3, 0, -11, -10, 5, -5, 6, -9], [-10, 10, -1, -1, 7, 11, 0, -7, 2, 1, 7, 4, -3, 8, 3, -9, 6, 0, 0], [-3, -3, -7, -5, -5, -9, 3, -6, -6, 9, -6, -5, -3, -6, 0, 4, 12, 11, 2], [-17, -12, -18, -10, -7, -2, 2, -1, 8, -13, -18, 3, -14, -21, -3, 4, -12, 2, 3], [-12, -11, 6, 4, 0, 9, 7, 2, -6, -4, -1, -5, 17, 12, -9, 14, -2, 8, -13], [23, -13, -24, -8, -7, 4, -8, 5, 1, -12, -12, 6, -15, -19, -7, 1, 0, -5, -10], [-22, 30, 9, -3, 2, -7, -1, 7, 7, 6, 6, 14, -19, 8, 21, -16, 13, -5, 13], [-7, 9, 3, 2, 11, 0, -10, 3, 4, -4, 15, -4, 10, 4, 3, 9, -2, -8, -2], [13, -15, -4, 7, -4, -6, 5, -3, -6, 4, -5, -10, 4, -9, -16, 3, -1, 2, 2], [-20, 39, 16, 9, 5, -1, 6, 4, 1, 5, 20, -7, 1, 11, 9, -19, 6, 0, 9], [1, -8, 5, -1, 15, 9, -3, -6, -3, -1, 6, -1, 6, 5, 12, 1, 1, -2, 0], [11, -12, 8, -2, 3, 15, -8, 4, -5, -4, -3, 22, -7, 8, -14, 2, 5, -15, 6], [-23, 6, 6, -1, -11, -2, 4, 12, -6, -7, -6, 15, -8, -7, -2, 10, 7, -2, -3], [9, -24, -21, -2, -5, -4, 2, -6, -1, -3, -4, -5, 12, -15, -27, 11, -6, 5, 1], [-42, -2, 0, 1, -2, -9, -10, 6, -8, 8, 3, 6, 3, -9, 0, 6, 12, -3, 2], [-9, 11, -4, 6, -1, -15, -2, 0, -3, 15, -1, -11, 6, -10, 14, 0, 6, 3, 7], [-13, 21, 15, -2, 11, 13, -3, 0, -1, -1, 3, 2, -13, 12, 11, -7, -4, -3, 9], [0, -7, -21, -5, -9, -7, -10, -4, -12, 2, -6, 0, 3, -14, 3, 2, 9, 6, 4], [11, -31, -10, -17, -4, 15, -8, -8, -8, -12, -19, 4, -13, -15, -8, 10, -10, -8, -4], [1, 25, 5, 14, 5, 3, 8, -5, 1, 3, 15, -4, 4, 24, 20, -14, 5, 4, -6], [-12, 17, 25, 11, -1, 7, 3, 15, -15, 4, 3, 14, -2, 25, 7, -3, 17, -6, -9], [5, 14, 6, -6, 3, 0, 2, -2, 3, 10, 8, 3, -4, 7, 6, -1, 11, 0, 14], [7, 9, -9, 9, -12, -1, 1, 8, 6, 1, 10, 7, -7, 2, 5, -3, 4, -3, 7], [-27, 29, 13, -13, 8, -8, -6, -2, 2, 10, 6, 3, -16, -5, 17, -10, 17, 0, 14], [-23, 2, 20, -14, 3, -13, -15, 8, -23, 14, -10, 7, 2, -16, -1, -2, 3, 8, 12], [0, 19, -6, 1, 1, -2, 11, -10, 20, 3, 1, -15, 5, -9, -1, -6, -9, 9, 5], [-4, -2, 14, 5, 10, 9, -16, -2, -9, 16, -2, 6, 0, 18, 12, -7, 4, -7, -1], [-15, 4, 7, -1, -1, 15, 2, 16, 2, -10, -5, 6, -9, 3, -8, 9, -4, -8, -8], [0, -17, -13, 3, 1, 2, 8, -3, 12, -14, -13, -17, 4, -13, -12, 11, -22, 8, -11], [2, -5, 8, 17, 12, 1, 2, 0, 1, -1, 5, -17, 19, 7, 0, 11, -15, 5, -5], [1, -28, -2, -1, -5, 19, -7, -1, -7, -6, -11, 7, -9, -1, -5, 5, 0, -9, -8], [9, -8, -4, 5, 3, -14, -4, -9, 2, 3, 7, -10, 9, 3, -1, -2, -2, 5, -1], [-25, 11, 1, -26, 0, -14, -2, -7, 8, 9, -3, 21, -11, -18, 4, -6, 21, -5, 17], [4, -12, 3, 3, -8, -4, 1, -4, -8, 11, -5, 9, 2, 6, 15, -8, 10, 4, 5], [9, 21, 4, 5, 4, -10, 11, 3, 9, 2, 16, -2, -3, 6, -7, -2, 8, -6, -3], [-22, 27, 13, 0, 7, -8, 1, -2, 9, 7, 14, -12, 9, -3, 16, -3, 3, -3, 4], [-32, 35, 26, 8, 9, -3, -10, 3, -16, 22, 12, 0, 2, 19, 14, -10, 7, 2, 16], [4, 5, 22, 10, 8, 5, -3, 4, -2, 13, 23, -9, 8, 25, -5, 5, -3, -1, 2], [16, 7, -17, -3, -13, -12, 9, -2, 1, -3, -14, 5, -7, -7, 13, -17, 10, 10, -7], [-13, 2, -24, -4, -11, -11, -4, -10, 1, -4, -24, -9, -5, -23, 16, -3, 7, 8, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1339_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1339_2_a_d();".
// 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_1339_2_a_d(:prec:=19)
    chi := MakeCharacter_1339_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_1339_2_a_d();".
// 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_1339_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1339_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![63, -135, -1062, 855, 6099, -108, -14037, -5183, 14841, 9364, -7199, -6826, 1079, 2337, 279, -351, -106, 13, 9, 1]>,<3,R![-7, -78, 559, 1516, -4334, -7725, 12819, 15623, -17404, -14664, 11390, 7507, -3840, -2194, 677, 358, -59, -30, 2, 1]>],Snew);
    return Vf;
end function;