# q-expansion of newform 432.4.i.d, downloaded from the LMFDB on 11 October 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 = 432 weight = 4 poly_data = [48, 0, 49, 0, 13, 0, 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], 1], [[4, 17, 0, 9, 0, 1], 8], [[52, 77, 12, 21, 0, 1], 4], [[-26, 0, -6, 0, 0, 0], 1], [[92, 31, 54, -3, 6, -1], 4], [[-92, 31, -54, -3, -6, -1], 4]] hecke_ring_character_values = [[271, [1, 0, 0, 0, 0, 0]], [325, [1, 0, 0, 0, 0, 0]], [353, [-1, 1, 0, 0, 0, 0]]] aps_data = [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [-2, 2, 0, 0, 0, 1], [0, 2, 1, 0, 1, 0], [0, 17, 3, 0, -2, 0], [4, -4, -4, -4, 0, 5], [37, 0, 0, 6, -1, 1], [-5, 0, 0, 2, 7, -7], [70, -70, 3, 3, 0, -5], [0, -152, 0, 0, 5, 0], [-16, 16, -3, -3, 0, 15], [-16, 0, 0, 8, 10, -10], [-299, 299, 12, 12, 0, -14], [0, -43, -27, 0, 0, 0], [0, 174, -21, 0, 39, 0], [368, 0, 0, 0, 22, -22], [151, -151, 15, 15, 0, 34], [0, -134, -12, 0, -3, 0], [71, -71, -3, -3, 0, 24], [20, 0, 0, 12, 52, -52], [125, 0, 0, 30, -21, 21], [0, -184, 5, 0, 23, 0], [0, -204, 33, 0, -15, 0], [154, 0, 0, 60, 44, -44], [0, 31, -56, 0, 70, 0], [0, -274, -84, 0, -5, 0], [206, -206, 33, 33, 0, -39], [817, 0, 0, -18, -19, 19], [-340, 0, 0, -24, -66, 66], [204, -204, 30, 30, 0, -45], [-830, 0, 0, -36, -126, 126], [-1264, 1264, 9, 9, 0, -7], [0, 433, 156, 0, -46, 0], [-817, 817, -91, -91, 0, 8], [188, -188, -72, -72, 0, -85], [0, -10, -27, 0, 45, 0], [-938, 938, 60, 60, 0, -165], [1420, 0, 0, -80, 8, -8], [-1848, 1848, -141, -141, 0, 81], [0, -84, -36, 0, -159, 0], [1204, 0, 0, -72, -136, 136], [1982, 0, 0, -244, -152, 152], [0, -1166, 147, 0, -85, 0], [2287, -2287, 72, 72, 0, -54], [1360, 0, 0, -60, -178, 178], [670, 0, 0, -88, 142, -142], [644, -644, 283, 283, 0, -5], [0, 3026, 213, 0, -39, 0], [0, 1521, 39, 0, -186, 0], [-1280, 1280, -204, -204, 0, 129], [2051, 0, 0, 78, 205, -205], [-774, 774, -9, -9, 0, 135], [0, 2635, -4, 0, -166, 0], [-2369, 0, 0, -78, 203, -203], [-3837, 3837, 384, 384, 0, -276], [0, -1226, 3, 0, -97, 0], [3644, 0, 0, -96, -134, 134], [3556, 0, 0, 192, 132, -132], [0, -4376, 120, 0, 75, 0], [0, -4004, 66, 0, -451, 0], [2396, -2396, -213, -213, 0, 363], [-1408, 1408, -12, -12, 0, -145], [2005, 0, 0, -282, -303, 303], [842, -842, 435, 435, 0, -121], [0, 1447, -40, 0, 734, 0], [0, -1600, 264, 0, 301, 0], [0, 4892, -265, 0, 527, 0], [-3071, 3071, 496, 496, 0, -80], [3165, -3165, -81, -81, 0, 564], [0, 5002, -304, 0, 245, 0], [0, 2251, 180, 0, 212, 0], [2178, 0, 0, -12, -6, 6], [0, 224, -291, 0, -129, 0], [-4856, 4856, -256, -256, 0, 689], [5659, 0, 0, 334, -505, 505], [-6580, 6580, -57, -57, 0, -115], [0, 2588, -312, 0, 361, 0], [6104, 0, 0, 528, -438, 438], [1969, -1969, -120, -120, 0, 934], [10699, -10699, -152, -152, 0, 118], [2076, -2076, -219, -219, 0, -555], [0, 3742, 32, 0, -823, 0], [3038, 0, 0, 252, 862, -862], [-4273, 0, 0, 378, -27, 27], [0, 6356, 465, 0, -597, 0], [0, -1473, -93, 0, -492, 0], [-8175, 0, 0, -186, -405, 405], [0, 4723, -204, 0, 642, 0], [0, 4204, -336, 0, 281, 0], [-6808, 6808, -275, -275, 0, 967], [-5213, 0, 0, -414, -937, 937], [0, -3150, -681, 0, 1011, 0], [11722, 0, 0, -460, 82, -82], [5461, -5461, 939, 939, 0, 868], [5249, -5249, -459, -459, 0, -378], [11688, 0, 0, -372, 168, -168], [3214, -3214, 672, 672, 0, -71], [4525, 0, 0, -1014, -841, 841], [3196, 0, 0, -660, -456, 456], [-952, 0, 0, -292, 706, -706], [0, 7685, -387, 0, 288, 0], [-3290, 0, 0, -732, -76, 76], [9501, -9501, 51, 51, 0, -636], [0, -2119, -168, 0, -350, 0], [-5767, 5767, 29, 29, 0, -250], [125, 0, 0, -474, 627, -627], [0, 16627, -309, 0, -568, 0], [-7190, 0, 0, 372, -268, 268], [-512, 512, -261, -261, 0, -167], [0, -8147, 24, 0, -1092, 0], [2288, -2288, -55, -55, 0, -793], [2348, 0, 0, -360, 342, -342], [-1757, 1757, -1344, -1344, 0, 196], [0, -6391, 33, 0, -516, 0], [88, 0, 0, 1192, 1328, -1328], [0, -6527, -972, 0, 98, 0], [-5227, 5227, 1377, 1377, 0, -1782], [-9090, 0, 0, -996, 150, -150], [5180, -5180, 336, 336, 0, -445], [0, 17260, 765, 0, 161, 0], [334, -334, -24, -24, 0, -573], [0, 5296, 1330, 0, -1721, 0], [0, -9670, 324, 0, 1159, 0], [-4323, 0, 0, 6, 249, -249], [0, 8624, 1647, 0, -189, 0], [8162, 0, 0, -1452, -272, 272], [0, -11282, -668, 0, 241, 0], [-2124, 0, 0, 552, -1884, 1884], [0, -1006, 981, 0, 441, 0], [9376, -9376, 1428, 1428, 0, -39], [-4493, 0, 0, 582, -1113, 1113], [-20262, 20262, 1119, 1119, 0, -225], [-52, 52, -1327, -1327, 0, -1381], [9890, 0, 0, 24, 2388, -2388], [-16632, 16632, -66, -66, 0, 447], [4288, -4288, -406, -406, 0, 1457], [-330, 0, 0, -660, 1980, -1980], [12500, -12500, 471, 471, 0, 1155], [2202, -2202, 1704, 1704, 0, -1395], [-647, 0, 0, 150, -1621, 1621], [11077, 0, 0, -594, -1863, 1863], [0, -15544, 1380, 0, -371, 0], [-76, 76, 217, 217, 0, -629], [-7828, 0, 0, 2904, 2200, -2200], [-19930, 0, 0, 388, -280, 280], [0, -22200, 507, 0, -1575, 0], [0, 7156, -1020, 0, 2319, 0], [0, -16452, -966, 0, -147, 0], [-21103, 21103, -1339, -1339, 0, 1406], [32216, 0, 0, 456, -440, 440], [4282, -4282, -920, -920, 0, 511], [842, 0, 0, 1356, 2980, -2980], [-7799, 0, 0, 1142, 2773, -2773], [-13108, 13108, 1119, 1119, 0, -2587], [0, 2003, -1039, 0, 158, 0], [0, -25860, -789, 0, -3, 0], [-506, 0, 0, 924, 1758, -1758], [0, -2016, -3042, 0, 2025, 0], [-22354, 0, 0, -340, -1892, 1892], [-17662, 17662, 1860, 1860, 0, -1891], [0, 14751, 2811, 0, -924, 0], [-15903, 0, 0, -1722, -573, 573], [-20854, 20854, 1869, 1869, 0, -1083], [-9396, 0, 0, -1332, -2880, 2880], [15617, -15617, -1332, -1332, 0, 782], [0, -23804, 2079, 0, 1517, 0], [12592, 0, 0, -840, -2400, 2400], [0, -15050, 1164, 0, -3255, 0]]