# q-expansion of newform 280.10.a.f, downloaded from the LMFDB on 17 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 = 280 weight = 10 poly_data = [112929763661700, -3369732326760, -43705600687, 1492027269, 1263838, -75886, -3, 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], 1], [[0, 1, 0, 0, 0, 0, 0], 1], [[-21692, 21, 1, 0, 0, 0, 0], 1], [[-3899423728410185100, 59737119899912100, 1518639093365525, -8732670272012, -80308488126, 187093904, 1154557], 66778542929520], [[26026916453976233820, -803324128579671780, -10701660215135249, 47207270330012, 618724990470, -684377024, -9058969], 467449800506640], [[9713053024036955295, -36915201141400530, -5657024991175939, 6464438882638, 313455738273, -154194511, -4393385], 29215612531665], [[-14930556238456921140, 90368777335138140, 7730175589822561, -10236188932728, -405866763290, 212175876, 5465461], 25969433361480]] hecke_ring_character_values = None aps_data = [[0, 0, 0, 0, 0, 0, 0], [2, -1, 0, 0, 0, 0, 0], [-625, 0, 0, 0, 0, 0, 0], [-2401, 0, 0, 0, 0, 0, 0], [1613, -82, 0, 0, 0, 0, -1], [8102, -140, -1, 0, 1, 1, 2], [16299, -34, 1, -3, 1, -4, 1], [5088, -6, -24, 4, -13, 0, -3], [37021, 1852, -47, 19, 6, -7, 3], [533793, -890, -37, -36, 35, 0, 80], [257639, 7365, -77, -67, 32, 12, -70], [1586741, 8790, -29, 16, 57, 77, -61], [6481926, -1868, 744, 196, -99, -18, 145], [1430458, 7429, 118, -295, -316, 17, -198], [-1774500, 81335, -93, 144, 439, -90, -577], [2360768, 113728, 1020, 35, -977, -46, 136], [-1348072, 166320, -714, 1168, 410, -48, 2058], [-8157503, 429253, 2511, -103, 1198, -358, -410], [-12928134, 138606, -4250, 1385, -1611, 1842, -580], [-52105249, 754486, -2771, -792, 4778, -1031, 4002], [-46558962, 986890, 230, 1915, -1753, -1304, -3172], [-23780519, 734253, -379, -3047, -4610, 3569, 715], [2463380, 628348, 8592, 357, -3249, 1966, -5540], [-38776252, 195694, 6784, -5372, 5563, -4062, 1509], [-73880272, 1845190, 2878, -3176, 504, 6515, -10119], [-344012035, 4588607, -6215, -4691, -17584, 5158, -9608], [115515027, -391637, 69220, -3227, 5758, -5737, 10617], [23810254, -3085760, -28674, -5205, 975, -14230, -29036], [-103542588, 1078266, 1590, 13765, 7920, 13467, 38658], [-304214684, 6962209, 60828, -10354, 18437, -4683, 21142], [-13766306, -10511636, -101588, -12394, -31952, -4538, 31662], [-464201868, -13786208, 13886, 9964, -19619, -11658, -87763], [-256371816, 3085937, 119004, 10496, 51457, -39921, -20038], [-1291088188, -10701566, -245286, 19888, 43496, -1600, 556], [-1128569501, -7196000, -143885, -58855, 13485, -40717, -130560], [-3361869674, -6064295, 62786, -46722, -21675, 32396, 61625], [-112711161, 4895686, 190731, 42903, -33039, 24913, 91276], [-3114709704, -7075132, -97438, 116843, 66479, 49828, 57252], [-2345192609, -9927199, -139294, -12651, -41568, -10491, 73833], [672384883, -8998654, -4128, 81645, -196460, 52944, 9716], [-4274377865, -27332540, 390975, 105172, -25124, -43529, -45650], [-4094768294, 12692570, -189336, -10346, 252768, -80032, 221138], [-12325839791, 15920527, 228135, -37823, -185452, -23023, -117827], [-5580782606, 87531463, 250782, -184092, 157173, -69189, 96316], [-4700899665, 40437428, 73353, -182572, -106855, 6219, 290651], [-11058814504, -43810324, 405330, 109050, -78094, -43288, 167396], [-8902716276, 20908092, 876031, 114579, -416225, -201067, -659701], [-18247238526, 165091701, -523851, -180172, -14967, 17698, 41031], [-9278780502, 15938267, -561194, -43858, -122282, 576748, -349928], [-1343732577, -72493575, 137031, -160339, 119426, 109200, 434550], [-3048179896, -24954700, -182688, 542848, -190032, -134892, 728534], [-15616878403, -82141511, -465415, 142919, 729810, -138671, -152095], [1020054698, -9787116, 1838536, -554034, -229439, 272036, 660593], [-511729956, 161069996, 642678, 366206, 638638, 394934, 1377638], [797556538, -103734832, 1733078, -1078419, 386865, 113614, 872298], [-23403367728, 147113188, 1179856, -425556, -321634, 323306, -1187568], [-3027094008, -302833730, 1261956, 770100, 481284, -176008, 804238], [1737103161, 42172909, 2276267, -457761, -1263156, 38774, -723256], [-33294171902, -74954790, 694554, 192299, -476437, 21096, -1424232], [19682968333, -170820027, -1404966, 1501667, 1164893, -315901, -679284], [-40334504330, 168134011, -3064190, 14292, -882492, -715206, -369246], [10242905485, 200561092, 2653220, 1115835, -1430424, 348352, 157832], [-33492067397, 255853673, -3815939, 823023, -356643, 625007, -1682450], [-63238948341, 243920491, -184169, -2181307, 1776394, 523624, -1335156], [61039448637, -246632532, -6408619, -1193895, 645713, -242754, 1180033], [-25027176208, -553315430, 3337898, -2482277, 2813897, -588490, 269330], [-12462079396, -785508654, 8506918, 1547173, 522031, -260536, 2398490], [103395287346, -317084972, 455638, 2381644, -1787446, -16800, -818034], [-7887533744, -368534859, 7673838, -3402289, 2501168, 502003, 4695912], [131403560489, -172643653, 3827301, 457313, -521194, 73130, -817714], [139389106736, 880405238, -1025950, 1664570, 3918288, -646281, 187599], [-48830569038, -1127950884, 10965438, -42669, -3499377, 2332708, -2394066], [-25132134961, 904113121, -7654274, 1653233, -5695492, -1696559, -3191427], [239932799723, 61804236, 13534187, 391474, -2095857, 2216071, -4513193], [-4866396951, -1507285058, -7705791, 3380084, 4136764, 436911, -3590588], [13129006321, 1487535138, 3118771, 157545, -3651799, -1278643, -3450106], [94714613280, 759693764, -9148874, -2935151, 5077984, -2965879, 5557276], [332604002343, -192777582, -12112226, 1047643, 1839270, 2466184, 2714096], [192882330364, 1063719591, -12766375, 3512698, 190048, 2067720, -2823176], [397900859374, -686309868, -11872254, -1322882, -3493542, 793232, 1254036], [41818228444, -1601611508, -6436466, 4524048, -11587178, -1262772, -8506242], [237591506570, 1090463596, 6428638, 2179411, 1390966, -2435821, 591672], [-40819352700, 1993004329, -17359864, 11550594, -17513507, -3280184, -4694857], [353229120171, 2992327032, 739457, -5854890, 3401886, -2993363, -2792438], [7762883024, -2557670048, 13015340, -5186642, 13615780, 3113318, -275776], [-232835329514, -1486731417, -29557586, -1719085, 8352618, 1914015, -11929084], [121908143340, -2533781377, -1866771, 2150036, -4715674, -809986, 3927962], [525167845788, -258472961, -23092626, -1356560, -10693203, -10310895, -6874220], [-36604032048, 5483431410, 22389844, 1941738, 7041246, 4481402, 2030012], [84289908691, 626430104, -289703, -2497077, 3541682, 3164833, 15109441], [59619412065, -3007258691, 51163891, 5930121, 13157411, -2191223, 44010], [-155074384387, -1186838793, 24758011, -24044511, 12910804, -7602626, -2348974], [266652780630, -75256992, 43848216, -7822518, -805822, 2143160, -11912070], [-289364937302, -5618965768, -11184473, -9722357, -7000549, -15144517, -21388145], [-167033692818, -8648132668, -24144385, 14742437, -2613317, 4592265, 21489555], [-145267892454, 6858557465, -28688373, 4000146, -19307247, 1070488, -14853937], [-84891182472, -9892235240, -31332292, -6663892, -12902214, -1560346, -27587854], [581173755866, 42072248, 62157250, 10403406, 8185355, 3556568, 6227815], [-425671915562, -1844368656, -9819754, -2084183, -6751519, -6522692, 33069886], [655847329495, -5184116432, -256089, -35323522, 9294203, 3322312, 24417880], [-237428791538, -7553248937, 1162672, 10525317, -34753584, -10518771, -33274206], [436345203974, -2837027908, 25889212, 14606229, 31542311, 1570316, -10845362], [-460475404800, -1731811646, 88338220, -22862407, 9131641, 17706380, 2094420], [-458363673341, -4737091914, 1456911, 10210201, 13192655, 1187965, 42893520], [-489733915915, -8775287450, -47234743, 541592, 9019294, -5542781, -6700930], [1096174086481, 3020763552, -59902489, 27089665, 8413011, -19518086, -59826221], [-291990532666, 4768199294, 55252906, -30466423, -13061341, 20429050, 22418902], [-993061599661, -18615893000, 64674591, 14648509, 23079169, -2085468, 19301257], [-925774535470, 12634614123, 11349286, 19794828, 16569703, -5432104, 37301945], [-84904764218, 2137597692, -14629806, 7536948, -12756386, 18198996, 54216402], [-1975118626534, 10165876837, -98007919, -1433252, -51142535, 16328688, -47541259], [-439479431492, -13479478582, 109831594, 17936829, 21057101, 8262690, 30957736], [-817914023080, -4980717608, 95122676, -27342978, 13405882, 3814890, 3787184], [-1679335331200, 6198273590, -22854862, 41534424, -11392073, -7463012, 39743445], [-792354173691, 2624860673, -20077069, -8298101, 16408518, 41982643, -34231905], [-2065257851951, 17827152838, -67831579, -18123309, -45248623, 27452191, -78668416], [-822693730492, 11122381373, 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