# q-expansion of newform 49.8.c.g, downloaded from the LMFDB on 01 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 = 49 weight = 8 poly_data = [810000, 216000, 149400, -22680, 9744, -378, 103, -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, 0, 0, 0, 0, 0], 1], [[21568329000, 7856666100, 13150689930, -241778964, 191205693, 6596632, 2089856, 2419], 3180326625], [[9362007000, 7051146300, -2766284640, 521375988, -35051121, 4986506, -221882, 51287], 25442613000], [[4975101000, -4535559000, 855847440, -161305998, 10026716, -1542751, 68647, -11627], 605776500], [[78343411500, 1424943000, 507479700, 1049560974, -17965948, 13007813, 215659, 141661], 4240435500], [[192094254000, 10330483500, 33045233760, -6228206292, 625838339, -59567354, 2650538, -832833], 8480871000], [[-1685376396000, 1662073688700, -289928802240, 54644382408, -4123329861, 522625796, -23255012, 5145317], 25442613000], [[-692595526500, -10997523000, -3916661700, -10024081914, 138658828, -100392593, -961699, -1093321], 4240435500]] hecke_ring_character_values = [[3, [0, 0, -1, 0, 0, 0, 0, 0]]] aps_data = [[-1, 0, 2, 1, -1, 0, 0, 0], [6, 0, -6, 1, 0, 1, 0, 0], [7, 3, 58, -7, 7, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0], [910, 0, -910, 196, 0, -30, -9, 0], [255, 37, -37, 0, -115, -37, 0, -37], [14087, 0, -14087, 251, 0, -243, 29, 0], [34, -316, 10578, -34, 34, 0, 11, -11], [-1741, 285, -32209, 1741, -1741, 0, 208, -208], [6135, -2055, 2055, 0, -2703, 2055, 0, -353], [101192, 0, -101192, -3328, 0, 1550, 527, 0], [-4592, -1396, 4343, 4592, -4592, 0, -10, 10], [-93987, -1905, 1905, 0, 19143, 1905, 0, 1513], [27080, 2988, -2988, 0, 27900, -2988, 0, 1212], [9580, -1206, 363334, -9580, 9580, 0, 2915, -2915], [148219, 0, -148219, 29326, 0, -4578, -4760, 0], [219962, 0, -219962, -78463, 0, 14649, -5428, 0], [-133928, 8660, 394077, 133928, -133928, 0, -2962, 2962], [18898, 0, -18898, 65381, 0, 42281, 7106, 0], [861522, 61410, -61410, 0, -27246, -61410, 0, -5466], [1607891, 0, -1607891, 158366, 0, -20854, 110, 0], [-80361, -81879, -905441, 80361, -80361, 0, -3864, 3864], [-2411958, -30186, 30186, 0, -22698, 30186, 0, -14502], [282892, 21924, 3888895, -282892, 282892, 0, -9032, 9032], [-738325, -25459, 25459, 0, -684635, 25459, 0, 1939], [5742869, 0, -5742869, -436645, 0, 11985, 11935, 0], [137571, -62535, 8254619, -137571, 137571, 0, 29142, -29142], [471074, -142200, 8101742, -471074, 471074, 0, 15965, -15965], [5113543, 0, -5113543, -317237, 0, -189119, -15617, 0], [3352173, -6309, 6309, 0, -574557, 6309, 0, 43717], [19171942, -128354, 128354, 0, -300178, 128354, 0, -43174], [1016965, 40455, 7790839, -1016965, 1016965, 0, -55370, 55370], [-20944667, 0, 20944667, 4621, 0, -474669, 70351, 0], [-2196850, -433630, 433630, 0, -8318, 433630, 0, 153742], [-534079, 370101, -8425498, 534079, -534079, 0, -108881, 108881], [7109956, 0, -7109956, 1330980, 0, 858870, 67905, 0], [-9321703, 0, 9321703, -404684, 0, -365528, 233314, 0], [-825427, -534821, -27150513, 825427, -825427, 0, -15932, 15932], [-44654553, 293583, -293583, 0, 2179407, -293583, 0, -357871], [-2779670, 613974, 12477595, 2779670, -2779670, 0, 47952, -47952], [-49422974, 0, 49422974, 4077580, 0, -283374, -172165, 0], [12654618, 1102960, -1102960, 0, 1046720, -1102960, 0, 102440], [-1030372, 252270, 45153818, 1030372, -1030372, 0, 192077, -192077], [-1239191, 0, 1239191, -3026589, 0, 128997, 405525, 0], [6406683, -630123, 630123, 0, 3174093, 630123, 0, -494189], [-30448804, 0, 30448804, -3044362, 0, 2864, -170107, 0], [-35507238, 1649630, -1649630, 0, 1722622, -1649630, 0, 239362], [53292378, -1115390, 1115390, 0, -4287982, 1115390, 0, 413318], [-82023298, 0, 82023298, 2387774, 0, -2659836, -182341, 0], [-10592619, -3308751, -108304924, 10592619, -10592619, 0, 39879, -39879], [1316531, -2692893, -2192566, -1316531, 1316531, 0, -673337, 673337], [-92465685, -1320645, 1320645, 0, -480453, 1320645, 0, -47883], [-13267369, 0, 13267369, -14264892, 0, 2464620, -282972, 0], [-244748247, 1628565, -1628565, 0, -15483627, -1628565, 0, -480397], [9656335, -4090281, -38762936, -9656335, 9656335, 0, -962865, 962865], [-120408596, 0, 120408596, -14641529, 0, -1545693, -1115130, 0], [34996673, 0, -34996673, 8339756, 0, 423744, 1341506, 0], [4603531, 4411145, 108505863, -4603531, 4603531, 0, 160274, -160274], [351067003, 0, -351067003, -2742334, 0, 1792586, -348364, 0], [-158997063, 7526895, -7526895, 0, -6943689, -7526895, 0, 1183121], [53056586, 0, -53056586, -2693576, 0, -893102, 787765, 0], [29816682, -5451096, 5451096, 0, -25967928, 5451096, 0, -65912], [-382065744, 5029012, -5029012, 0, 3778436, -5029012, 0, 1357844], [220386260, 0, -220386260, -1572901, 0, -2984805, 383384, 0], [-13759446, -2496222, -234626839, 13759446, -13759446, 0, 160542, -160542], [13634726, -3406518, -191878915, -13634726, 13634726, 0, -3189560, 3189560], [-37458993, -798555, 267758209, 37458993, -37458993, 0, -3756342, 3756342], [629936439, 8277533, -8277533, 0, -17977691, -8277533, 0, -320549], [-328077482, 0, 328077482, -8432369, 0, -1353609, 250196, 0], [455646105, 416935, -416935, 0, 51889007, -416935, 0, 1131857], [-320569129, 0, 320569129, -5712259, 0, -16097349, -1835445, 0], [13466402, 3371196, 603633416, -13466402, 13466402, 0, 3328503, -3328503], [458281472, 0, -458281472, 17681501, 0, -20951743, -4014166, 0], [-2061582, 16078494, 143426197, 2061582, -2061582, 0, 5466648, -5466648], [-242360795, -8225735, 8225735, 0, -37930375, 8225735, 0, 3504095], [69010978, 14734272, -45419168, -69010978, 69010978, 0, 2872217, -2872217], [1051634347, 0, -1051634347, -2860211, 0, -4236729, 1774969, 0], [-38165911, 17704333, -105317420, 38165911, -38165911, 0, -2681825, 2681825], [21296729, 9763425, -1025839288, -21296729, 21296729, 0, 1065841, -1065841], [-279476173, 0, 279476173, -22617393, 0, -12288351, 8168637, 0], [-836836680, -18567300, 18567300, 0, 16602636, 18567300, 0, 3398076], [307312037, -17773785, 17773785, 0, -67324833, 17773785, 0, 2263017], [-243366248, 0, 243366248, 27644074, 0, -1749720, 5349799, 0], [348974271, 32847793, -32847793, 0, -93990199, -32847793, 0, 475343], [-68293438, -16000616, 1031087980, 68293438, -68293438, 0, 517093, -517093], [-9254212, -23812926, -259260028, 9254212, -9254212, 0, 534103, -534103], [-10701225, 35519805, -35519805, 0, 65119941, -35519805, 0, 2158731], [71129438, -9773066, -307313337, -71129438, 71129438, 0, -4205630, 4205630], [2275013133, -20230725, 20230725, 0, -55659549, 20230725, 0, 283021], [-1443446376, 11343512, -11343512, 0, -77238536, -11343512, 0, -7614248], [58192003, 24575877, -1327760759, -58192003, 58192003, 0, -7412620, 7412620], [-295281040, 0, 295281040, 126375539, 0, 35377419, -4429916, 0], [-644409092, 0, 644409092, -104276677, 0, 35492879, -1591294, 0], [783315117, -36939855, 36939855, 0, -61983567, 36939855, 0, 10779623], [55152461, -23416949, 1808237847, -55152461, 55152461, 0, 3270124, -3270124], [-227991222, 945450, -945450, 0, 107883738, -945450, 0, 11417838], [71452975, 8388099, -1992339698, -71452975, 71452975, 0, 4421925, -4421925], [203866859, 0, -203866859, -62707060, 0, -60823620, 3684200, 0], [-223502246, -32677132, -404294394, 223502246, -223502246, 0, 10491887, -10491887], [-251687007, 27154125, 596042578, 251687007, -251687007, 0, -5366853, 5366853], [782678679, 8730083, -8730083, 0, 85861219, -8730083, 0, -12027419], [1436475223, 0, -1436475223, 208609915, 0, -20662383, 7670271, 0], [-201588634, 0, 201588634, 97066667, 0, -5907945, 368310, 0], [201103310, -18281034, -3551074417, -201103310, 201103310, 0, -2436310, 2436310], [-387148922, 0, 387148922, 142656888, 0, -12027330, 12013323, 0], [580544687, 0, -580544687, -142529048, 0, 22564696, 19880104, 0], [-1068092340, -4518024, 4518024, 0, 46298280, 4518024, 0, -1446256], [-99605219, 8239605, -1285065206, 99605219, -99605219, 0, 21521837, -21521837], [1339357084, 0, -1339357084, 84347437, 0, 5946621, 10700272, 0], [-1936693357, 37782845, -37782845, 0, 120085333, -37782845, 0, 11830243], [-33613098, -21282444, 679526492, 33613098, -33613098, 0, -20610915, 20610915], [-1488140381, 0, 1488140381, -182844949, 0, -12545143, -5095645, 0], [525764187, 91231881, -91231881, 0, 127824177, -91231881, 0, -11880609], [-455965846, 0, 455965846, -112771865, 0, -37832141, -21107210, 0], [5074328402, 22975410, -22975410, 0, 210710562, -22975410, 0, 18186102], [2214589921, 0, -2214589921, -89722940, 0, 92091420, -23454500, 0], [5654892304, 101241876, -101241876, 0, 100686084, -101241876, 0, -16888572], [4171162892, 0, -4171162892, -271114669, 0, -20430573, -6249176, 0], [276164546, 102622902, -2340339271, -276164546, 276164546, 0, -869732, 869732], [-463731090, -74672910, 74672910, 0, -336475854, 74672910, 0, 16096686], [-575934739, 0, 575934739, 199860568, 0, 144560380, 14560138, 0], [-1493112897, -4284379, 4284379, 0, -455012147, 4284379, 0, 2322307], [-30518039, 21386085, 8310600448, 30518039, -30518039, 0, 16382315, -16382315], [2879087974, 0, -2879087974, 44511250, 0, -141861372, -35877935, 0], [74884929, 73942515, 389634275, -74884929, 74884929, 0, -11792934, 11792934], [1921799802, -13601100, 13601100, 0, 309033588, 13601100, 0, -20132292], [505621696, -57771364, 3576109847, -505621696, 505621696, 0, -5761546, 5761546], [140137219, -105605631, 833464303, -140137219, 140137219, 0, -58446934, 58446934], [2210193106, -32220798, 32220798, 0, -276003534, 32220798, 0, 10548294], [241534934, 116755378, -595274009, -241534934, 241534934, 0, 38048692, -38048692], [1796520514, 0, -1796520514, -358109618, 0, -90925904, 179257, 0], [-3849252594, -44801106, 44801106, 0, 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0], [2752483200, -292449240, 292449240, 0, 242088072, 292449240, 0, -38553208], [-4589753993, 165922389, -165922389, 0, -343484163, -165922389, 0, 78582123], [7195474907, 0, -7195474907, -827363938, 0, 253551066, -27681306, 0], [818050116, -248720706, -10414686964, -818050116, 818050116, 0, 13515609, -13515609], [-534145417, -184289511, -9617274553, 534145417, -534145417, 0, -32700752, 32700752], [601236197, 165524833, 5688794214, -601236197, 601236197, 0, -16108745, 16108745], [-21225133482, 226378320, -226378320, 0, 408444960, -226378320, 0, 35040600], [16018427606, -33515526, 33515526, 0, 166120986, 33515526, 0, -2925018], [764013208, 143413182, 1055179246, -764013208, 764013208, 0, 90691875, -90691875], [-1271292722, 0, 1271292722, -7812525, 0, 54233667, -2762664, 0], [-12340058928, 11119680, -11119680, 0, -1329805152, -11119680, 0, -43006752], [456098718, -244078104, -19826121020, -456098718, 456098718, 0, 22823307, -22823307], [-771389006, 272677074, -1679772347, 771389006, 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