# q-expansion of newform 252.8.a.e, downloaded from the LMFDB on 25 June 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 = 252 weight = 8 poly_data = [-882, -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], 1], [[-3, 6], 1]] hecke_ring_character_values = None aps_data = [[0, 0], [0, 0], [-21, -1], [343, 0], [-3714, 14], [5915, 3], [-7896, -106], [13307, 9], [-16320, 504], [79008, 154], [-90370, 882], [-22912, -2562], [160860, 2714], [511934, -2058], [-832986, -998], [205314, -1344], [-851067, 7263], [-273763, -4863], [-1295308, -16968], [-2064636, -2716], [-4004434, 1836], [1235228, 41076], [4950393, -17693], [-7711746, 12056], [-8688736, -47790], [-6686925, -29097], [1021202, 84342], [-17039184, 37100], [-5363716, 72282], [-16084284, -51814], [-5247220, -194124], [-1377579, -65577], [-7353654, 22428], [-33208987, 132183], [-41648610, -81564], [-10072612, -74676], [31642835, -295941], [-1240018, 345870], [-70175238, 7326], [-18285183, 267809], [-40213992, -278516], [41297627, 26907], [61541244, -126364], [-41644648, 140826], [36225486, -83804], [70509278, -267414], [-115452292, -192864], [212910236, -153372], [-196966203, -676857], [83046215, 854559], [189161742, 716184], [91440480, -44240], [-129524248, 140730], [94666131, 1009953], [237601014, -1066808], [240155940, -1273468], [133070721, 1868929], [-198794680, 191400], [335701598, 1632960], [334905702, 1099952], [77543795, -1408311], [155462223, -203877], [-100274965, -1674183], [-39466224, 2027048], [405799604, -704442], [-380674026, -2381932], [297931502, -373338], [1013665136, 635250], [-3029238, -1529934], [-121218265, -4286649], [954609726, -238112], [-1124382876, 3012996], [711709124, -51348], [-559193086, -450660], [-338587990, 3894618], [-570004554, -6528294], [226030440, -10813894], [166157159, -123945], [518953992, -1342250], [-803935552, 3945402], [-1394225133, 674825], [-154883674, 5058984], [-1246200024, 7173544], [1535484860, 6229446], [608238008, -2609136], [-1396203156, -11084192], [-532414746, -16698472], [-178457872, 2216634], [-353148411, 592797], [-1734187744, -7965552], [-48658407, 5013379], [-2537995614, 3173934], [-2063441056, -920808], [3025150356, 7173544], [-2689337164, 6274632], [-1226922228, -8712116], [3322871097, -3248519], [-200138988, -17556286], [-1394753017, 17412453], [-1057733278, -12543468], [917086346, -17014830], [-434448786, -26046356], [-3956875713, 91725], [-461471400, 25804142], [-1433355790, -10422510], [-314927158, 26422776], [4011215271, -25554435], [7019564034, 6887468], [873566784, 14690200], [-388427326, 43024296], [2389745120, -12959040], [-2801207896, -28660842], [7024795464, -10031378], [-3332371021, 12255417], [5172995252, 6544692], [1462161504, 50420398], [-2676989119, 45996771], [2130330930, -5852602], [-1410595932, -52515610], [-749305506, -64928010], [2715028841, -40097667], [6429790262, -47336436], [6672308013, 5392525], [-5475986292, 22056048], [14279735939, -11398431], [-2238333156, 54171614], [9429070328, -31226202], [5985329322, 58028950], [-4946173750, -53826354], [874857221, 98687289], [-5979306226, 24905454], [3642260124, 21234444], [-7499966296, -26967864], [-1347007408, -4955874], [11266700424, -33010594], [-4444330492, -40838226], [-11074164969, -21718413], [-11843876065, -28186299], [-6139436919, 43304905], [-7329829656, -70339570], [10090767575, 48489909], [-27044161542, -560952], [-7740238144, 19432392], [14776464816, 39218620], [8828641157, 110630361], [-3027431946, -63044190], [11521615847, 36860175], [-21229008864, 70630310], [-20286754915, -77370801], [-3604580268, -140308756], [-26590847800, 8520750], [-7919434782, 5104940], [5215013456, -116188212], [24434792718, 30833594], [5890157384, -17563140], [-12890969580, 117587708], [-33264357568, 10175592], [19928444256, -21374546], [-5067179026, -42365172], [-24849508509, -43736849], [-24541389450, -27995842], [10760807502, 172592168], [7706420216, -89583648], [43369383981, -65191801], [-17159523534, -126829444], [-2214845178, 193140258], [13601142536, 68636568], [23853254705, 57347109], [23924615432, 73839738], [15829006641, -88634311], [-6920742300, -151983384], [-26601561301, -194050173], [18991165092, 19722500], [-5928390046, 278457816], [-43179411058, 4241202], [4495816290, 312024788], [4301027336, -39891348], [2856124968, 71941338], [-12230237572, -153348996], [21718472909, -107722911], [35200355912, 122193624], [78816114951, -34291499], [-15530389060, -81743550], [10213946526, -324346328], [30161784708, 13004236], [-4899532155, -155465947], [-21411911698, -97641432], [57844977659, 48452793], [16963034582, 164797668], [7811952918, 130598074], [58472431178, 163625016], [38602936068, -199539816], [57626188688, -21513492], [42070864941, 219984133], [-30030170520, -66503444], [16287896394, 197689276], [-63823100176, 196929810], [-43311140560, -53837322], [-331818942, 367196316], [32045369664, -490321720], [-38325233574, -49669312], [42075545162, 238218774], [-21630627823, 50177109], [-14040023410, -30110220], [17983269297, 505049603], [-45572535309, 49662255], [59547195722, -160588698], [-73128776718, 354056178], [-10209727956, 28384202], [3021834641, -239396085], [10778528864, -375989502], [3634447467, -22651633], [52292877596, -23792244], [-17658816147, 658056663], [-7704525024, -458223416], [-2779353304, 660139842], [-37054498288, -333138624], [22087777908, 124731114], [-18417920352, 132409144], [-60980024508, -323196650], [-170143787644, -63673638], [-53332690432, 414900936], [5727226326, -29056664], [-76063757200, 566651232], [27123365829, -924788297], [100651535108, 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[-55372216123, -1594595631], [-78484491547, -1208220135], [175003683120, -382007394], [-97555889851, -1049746401], [-147180651426, 1279898172], [106773994446, -936457816], [171884078096, 912677892], [167526496620, -252411306], [-128298045505, 979881087], [99649675154, -154142310], [3063502694, -902413980], [-107172180460, 1736501004], [-52154591566, 1507170144], [272096590052, 162806916], [-115606477038, -878718134], [193074233906, 963785508], [223629343316, 86152374], [6966975393, -508623941], [320289790212, -717050180], [123883387784, 775453392], [-190780593402, 1043619522], [19015058153, -524942259], [-322758212161, -1025249043], [353747619588, -74645172], [-16897159126, 98502600], [115534533270, 1544708396], [-350494060966, 827128254], [-21033790722, -119708768], [5155937634, -328213480], [-49207079775, -2816023525], [53992052952, 1638374766], [-416974813143, -374676821], [-140609672146, -1104995976], [272179532961, 1954860145], [-257774673352, 1929340584], [-211206187227, 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