# q-expansion of newform 546.6.a.q, downloaded from the LMFDB on 30 May 2026.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 546
weight = 6
poly_data = [-46440, -2752, -1, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0], 1], [[0, 1, 0], 1], [[-1826, -27, 1], 7]]

hecke_ring_character_values = None
aps_data = [[-4, 0, 0], [9, 0, 0], [-6, 1, 0], [49, 0, 0], [62, -8, 1], [169, 0, 0], [570, 2, -6], [-82, -35, 3], [1010, -15, 13], [1188, 19, 9], [-374, 113, -53], [3362, 170, 78], [-1904, 18, -169], [-2190, 157, -7], [12268, 49, 146], [7144, 91, 143], [24434, 250, -35], [3558, -472, -184], [-6824, 690, 358], [22222, -746, -85], [-1600, -799, 261], [-11862, -127, -505], [37792, 461, -310], [-5422, -77, -446], [20888, -275, 213], [-53866, -564, -170], [9940, 2476, -746], [51408, 1228, 1272], [-24182, 1752, 862], [40788, -1667, 891], [-37004, -2280, -2066], [-232, -1596, 1366], [188204, -3416, -1025], [184148, 3206, 792], [114844, -474, -1327], [21228, -6748, 326], [28978, -7510, 1582], [310456, 1440, -1226], [214944, 6863, -1884], [329202, 8430, -3660], [229930, 6197, 5855], [130038, -948, -3484], [120616, -592, -4258], [516730, 4716, -4292], [643832, -8178, -365], [245480, -6828, 1500], [133142, -6059, 759], [281382, 15731, 8633], [-183206, 12198, -3019], [89446, -26908, 3304], [638824, -17669, 185], [470878, -3758, -9421], [623928, -15787, -2083], [587756, -11822, 340], [10218, 30600, -1578], [-498250, 12715, 2251], [444954, -7522, 5196], [-1348024, -18864, 4920], [-573668, -17319, 3529], [187856, 4134, 10189], [-1436316, 12736, -4744], [1147462, -18387, 5126], [-116394, -5869, -23497], [467528, 40914, 5068], [-2048054, 14756, -3980], [604268, 9472, 27061], [-892536, 22546, -4522], [-626032, -32347, -5967], [1628116, -13316, -6670], [1375672, 3545, 18115], [-14696, 12110, -26149], [-1348658, -48598, -5293], [-734136, 36762, 12740], [-1846410, -21768, -15436], [-2246124, -55086, 23048], [198886, 35858, 21461], [268142, -33664, -28532], [-40840, 37121, 25947], [-118992, -1964, -519], [1481600, -19631, 1797], [1611788, -87250, 19336], [-3175506, -25738, -27136], [553006, 748, 2567], [-1690818, 29758, -8602], [-2353632, -31378, -43780], [-3704890, -44443, -4427], [-252328, 64844, 5161], [-382238, 29650, -15812], [4582480, -107870, -10903], [2406092, 74550, 24366], [-1446732, 86510, -9660], [-21324, 60635, -43842], [569880, -19670, -25312], [-2586472, 66708, 1672], [-1627628, 12184, 27388], [399104, -188022, 5788], [-980830, 63753, 23344], [-1155370, 69078, 70312], [-3980108, -51926, 9712], [4041086, -94946, -12612], [-4808858, 216723, 21679], [3223704, 67820, 28239], [1322196, -13424, 11724], [407480, 188459, -32807], [677830, 30553, 58015], [2894750, -22568, 25986], [4578944, -87867, 6346], [3846342, -172891, -68280], [372794, 42505, -43595], [6238514, 130112, -70656], [393376, -55854, 42688], [-4404862, 83906, 45246], [-4639824, 35162, -7731], [5706612, 64400, 20924], [2441368, 85162, 14452], [-2333696, -78051, 33857], [7455972, 51532, 20276], [581212, 145762, -31246], [-6796214, -94168, 67538], [-11041514, -110493, -53587], [-1416800, -104659, 16459], [-554024, 111109, 49585], [-5148654, 234276, -7248], [-5822126, -158742, -60229], [-1034298, 63023, -129133], [-3386860, 116893, 134401], [217474, -52860, -16754], [-15834376, -94670, -31484], [609780, -385334, 21578], [-8233300, 17573, 108189], [5456416, 137450, -35078], [-3088810, -155298, 192151], [10818022, -349775, -21239], [-4882764, 100969, -34831], [-8022646, 142005, 55534], [4860824, 234431, 97113], [2937740, -122974, 10159], [-1296694, 133193, -68679], [-4762174, 318532, -152468], [-352820, -62719, -58909], [10837048, -177436, -62438], [-686204, -401170, -37963], [5209008, 245864, -133240], [-3679670, -460510, -116941], [2194726, -194356, -110948], [13743846, -38498, 193677], [2901220, 519059, 102433], [-6605622, 27404, 129498], [-14546040, -202662, 129518], [-2085454, 256174, -71879], [24452682, -237054, 44942], [8765242, -536096, -7270], [-14311368, -276668, -360310], [-14047088, -220436, -191872], [33424598, -90607, 102387], [-9425438, -512689, 71429], [21222228, -507316, 35246], [-28849278, -178007, -166746], [-33291122, -194450, -22190], [-5938526, -83489, -399784], [-12785046, 434380, 164415], [-3347084, 96449, 367559], [-4244152, -350206, 352524], [775544, 240798, -30518], [-6364932, 799486, -41109], [24505308, 267927, -270390], [11729376, -799636, -25528], [-36664662, 26702, -158362], [15735190, -474792, 43438], [-50936238, -120462, -79944], [27406946, 164620, 129727], [-43405726, -212720, -238542], [590158, 86683, 31931], [22592482, -249354, -151340], [-98908, 528140, 167190], [-23392910, -302982, 32810], [25613868, 242144, -34600], [13634572, -565506, 53885], [24006336, -192246, 262448], [-22327866, -117940, 377216], [19892064, 335372, -368572], [-5691984, 692142, 52002], [-42756554, 314144, -75260], [-22000690, 148769, 336424], [20106426, 750196, 176709], [-26647942, -522272, 152740], [3820152, -195893, -421585], [916846, -697547, 243391], [-9293858, 678680, -31586], [10957792, 714203, -178852], [-7020810, -120350, -371326], [7889262, 200984, 128889], [-24829582, 478757, -156315], [-35529210, -85099, -348726], [-29858626, 640623, -233951], [7743342, -386714, 174252], [17679290, -384704, -153564], [21988580, 175717, 364077], [-40281326, 652651, -114538], [-21786036, -551706, 9786], [-41258444, 1208792, -134935], [-1867500, 1419532, 80630], [4224564, -1393831, 296381], [-51395242, 191482, -477126], [-13557572, -429743, 138752], [-49665546, -883406, 188658], [9148370, -1468555, -595893], [21543782, -1282407, -34931], [1308672, -41136, 365193], [25740680, 104818, -82770], [38023428, -28565, -24793], [-3070586, -128894, 858242], [58034868, 108396, -451618], [-51133954, -143294, 27139], [-26182144, -1301291, -267956], [-2030968, -891533, 306141], [4944834, 553841, 533495], [8306010, 1124966, -251982], [29511670, -73310, 582491], [17832504, 1775328, -339189], [20210540, -1639043, -355191], [7159208, -1117296, -669444], [-35336400, 389440, 254925], [31675984, -261026, 802936], [83076652, -120478, -263020], [51389904, 137763, 524687], [-43183166, 209854, 39830], [-2653898, -2101777, 71807], [-24270748, 464174, -294990], [80343634, -1133524, -19333], [85817894, 470892, -575412], [43133168, 296072, 211914], [30776624, 1231028, -909720], [-27761710, 1149908, 858844], [-8735068, -700470, -116868], [4983772, 2238833, -344362], [-51154342, -194152, 34008], [23509696, -1804428, -470833], [-13694586, 471755, -158115], [93977340, 319170, -97650], [-8659782, 1809594, -390021], [47679568, 94886, -291410], [48053408, 1870112, -7440], [74623270, 548176, -523472], [-59691750, -631217, 596058], [-11073628, -2930540, 59614], [-25094366, -1994529, 1065319], [113507552, 372299, -424610], [-81548504, 1444180, 384970], [75426326, 1264143, -232025], [102972918, -268802, 644768], [33943912, 478198, -1222453], [-106826682, 1121872, -402585], [39558642, 1273248, -1062124], [25325854, -1348124, 422030], [-30740934, 2312526, -191985], [81958312, 551219, 788479], [-42451104, -2321804, 261578], [14729022, 2744422, -734382], [-156380060, 1361877, -29609], [28171906, 175841, -862421], [19130550, 907375, 1265961], [18334588, 664885, 636193], [-59921778, -352984, -313912], [-122317780, 505922, -80231], [-3354888, -2368746, -462130], [8953904, -1565650, 106531], [97033726, -547453, 62168], [-21554144, -510838, -1097906], [-84421686, -1776952, 304710], [-32516418, 2704, -368572], [-10780284, 190580, 933236], [174536938, -346676, -387620], [-25534694, -3140933, 1260007], [-131681730, 1665294, -137110], [44454488, 2052800, 1592628], [-23629586, -2463950, -1865551], [27204002, -1957210, -222438], [-140669482, 2183622, -98886], [-78953584, -727188, 36670], [-126003348, 1419794, -474438], [17874940, -2368342, -1278278], [31691264, -470816, -1552388], [8020072, 3586097, -380537], [-86710092, 342488, -1237132], [97583358, 279073, 835671], [-63308860, 4586547, -182745], [-18277912, -3122576, 554899], [-132369132, 580212, 884990], [7234934, -1175746, 876640], [-100135728, -419537, 1476471], [-126944444, -1982728, 1577576], [-18716460, 777602, 772251], [-36649544, -1399593, -1060858], [118622452, -856083, 687409], [-19297970, 5336422, -745162], [-91025912, -3068330, 1949692], [89231214, 1892470, -978474], [-20461028, 2150804, -1137580], [-98078118, -1053619, 1149593], [-177064214, 3556980, 145936], [-136373452, 726652, 555517], [946134, 5776441, -139459], [-26662758, 6057701, 70995], [-128099026, -2622637, 742887], [-78478866, -3850968, -404590], [-190035676, 1633176, -240350], [-202207006, -365534, 779922], [-35629994, -780688, -440401], [-49972800, -2122157, -1812205], [21854172, 1151004, -1354806], [109950360, 2425206, -945897], [-128271290, 1507648, 1205522], [-182020470, -956121, 1339613], [-149187530, -709562, 78443], [-3678602, 5048758, -1192622], [188639616, 1705755, 267186], [-3293586, -1742352, -13797], [-214910540, -1899929, -604457], [-24804728, -64871, 855365], [87158288, 213408, -592926], [-275159234, -1295802, 694510], [159784754, 2787032, -835628], [67737340, 2851676, 383722], [-207370736, -2500200, -418519], [-101801754, 2316914, 1611590], [-93975704, 417572, 1877686], [-90935050, -2551873, 433611], [60703280, -5110764, -790214], [25291034, -7485340, -1518962], [14155266, 3161264, 1601042], [57541844, -5805385, 749137], [-123942868, 2399698, -1576374], [-117260416, -2284323, 1569454], [27548292, 4407982, -1845844], [226411268, 606319, -1341656], [-25053086, 1290630, 568234], [-24755800, 3998130, -1872161], [-72743190, 7630628, 133892], [-180117256, -1686798, -1727256], [157315258, 2293286, -2201102], [61909566, -3374848, -773190], [-9937836, -4189418, -1838145], [88866064, -3194040, -197756], [-113540732, 3904916, 1708853], [-122541404, -4765148, -2968966], [-14099554, 7284404, 1328484], [-273598616, 2737468, 125230], [259826240, -356117, 401620], [131238146, -528140, 2859736], [-37545174, 6863501, 2660357], [169422662, -2098644, 586458], [117840912, -1561560, 235563], [27616356, 6407306, 482498], [-192497514, 1265008, -338716], [277179010, 1571552, -783022], [5331948, -814799, -918306], [127185348, -1804351, 882084], [86234590, 3925140, -530806], [276878054, 3527084, 86983], [139406102, -802752, -295140], [258470310, -1770104, 1051284], [12844162, 5913837, 246413], [-70781802, -7013490, 916653], [-147295668, 534882, -3259000], [-121719444, -1952209, 251843], [29620162, -3932387, -722800], [-92414776, -4874542, 3956460], [194391356, 1658815, 2200371], [220925120, 41628, -343160], [252562340, -1890626, -2431692], [-82950050, 9843110, -903817], [-182389576, -8693665, 3926191], [-188711896, -3568672, 1644024], [-101171936, -620145, 333019], [23180230, 4907130, -1353085], [18418888, 4730760, -1054534], [6965398, -9460548, 1470598], [197780146, 3316430, 2705126], [125961874, 3014536, -358304], [-358294362, 4099689, -714534], [104997040, 3819306, 536945], [144652704, 6253648, -2881036], [132586952, -3678308, -2216495], [-145813686, 263, 1947617], [193382800, 3369869, -1246432], [58337912, -10114220, 822540], [126946138, -5411130, 487126], [120929768, 6146092, 1242066], [285388844, -5181860, -458450], [250054806, 3185364, -3475210], [270097658, -6065133, 1259608], [-102299982, 9870758, -1055241], [-227482636, -5799760, -3841956], [23916262, 2151938, 504911], [500536824, 958981, -2054479], [-129970424, 4675192, 2709244], [90018654, -299180, 726612], [-211285894, 6256363, 24069], [-120424164, 2183936, 2617725], [-228045428, -843381, 1130809], [356650732, -771125, 480983], [-206151380, 14968794, 1139782], [-204378110, 7304171, 1413050], [-123140782, 1690893, -3447068], [268277888, -4638816, -344232], [164807022, 8536514, 5304812], [24421736, 7349352, -2120831], [-122130746, -1302789, -5388989], [-91410000, 6570129, -1099770], [87753982, -2562154, 1122586], [-271485276, -6647111, 1848405], [372511150, 86675, 3226049], [-357107948, -5346280, 182224], [257916916, -301289, 1046983], [-162904874, -2633046, -7041226], [-117488378, 10059370, -1309225], [-380082384, -106092, 1015892], [52209684, 11390, 3568821], [-162102356, -8777576, -2102158], [-129392740, 661455, -668339], [456277970, 5112800, -2389266], [-68629914, -243970, -2548917], [-221129544, -10762024, 1371834], [-403866026, 6985144, -2338814], [-452081354, 13046737, 601346], [9137574, -15433395, 770481], [138400400, 290812, 6599605], [-88622086, -9238113, -428979], [-283480788, -5432454, -1042494]]