# q-expansion of newform 58.3.c.a, downloaded from the LMFDB on 23 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 = 58 weight = 3 poly_data = [9, 0, 0, 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], 1], [[0, 1, 0, 0], 1], [[0, 0, 1, 0], 3], [[0, 0, 0, 1], 3]] hecke_ring_character_values = [[31, [0, 0, -1, 0]]] aps_data = [[1, 0, -1, 0], [1, 0, -1, 1], [0, -2, -1, -2], [2, 3, 0, -3], [-3, 0, 3, 7], [0, -4, 3, -4], [-8, 0, 8, 8], [0, 0, 0, -2], [4, 16, 0, -16], [-29, 0, 0, 0], [19, 0, -19, 11], [-8, 0, -8, 0], [-29, 20, -29, 0], [39, 0, -39, 17], [-7, -27, -7, 0], [-35, -20, 0, 20], [-16, 8, 0, -8], [-4, 0, 4, -20], [0, 8, 56, 8], [0, -5, -10, -5], [12, -60, 12, 0], [-33, 0, 33, -3], [28, 32, 0, -32], [51, 0, -51, -20], [69, 36, 69, 0], [120, 0, -120, 8], [122, -27, 0, 27], [2, -17, 0, 17], [0, -10, 25, -10], [39, -24, 39, 0], [-114, 0, 114, 16], [-34, 80, -34, 0], [-128, 0, 128, 8], [-44, -24, 0, 24], [0, -84, -11, -84], [0, -19, -126, -19], [-9, 52, -9, 0], [-95, 39, -95, 0], [0, 21, 238, 21], [0, 80, -96, 80], [0, 79, -14, 79], [-31, -122, 0, 122], [50, 0, -50, -32], [-55, 0, 55, -136], [80, 100, 0, -100], [122, 5, 0, -5], [-125, 57, -125, 0], [10, 45, 0, -45], [142, 33, 0, -33], [81, 60, 81, 0], [-15, 0, 0, 0], [-148, 80, 0, -80], [0, 72, -97, 72], [21, 0, -21, -15], [35, -130, 0, 130], [-35, 0, 35, -51], [-196, 4, -196, 0], [77, 163, 77, 0], [136, -76, 0, 76], [-183, 0, 0, 0], [0, -56, -300, -56], [-59, 0, 59, -104], [-37, 0, 37, 65], [86, 0, -86, 0], [-83, 18, 0, -18], [8, 88, 8, 0], [183, 81, 183, 0], [20, -276, 20, 0], [0, 57, 242, 57], [-77, -16, 0, 16], [0, -104, -152, -104], [59, 0, -59, 221], [278, 0, -278, 160], [-121, 26, 0, -26], [-150, 0, 150, -272], [0, -213, 42, -213], [32, 248, 32, 0], [-353, 118, 0, -118], [193, -36, 0, 36], [-163, 0, 163, 252], [0, -7, 262, -7], [36, -100, 36, 0], [-384, 16, 0, -16], [236, -108, 236, 0], [0, -32, -548, -32], [298, 48, 298, 0], [387, 0, -387, -68], [0, 92, -608, 92], [-119, -260, -119, 0], [0, 264, 20, 264], [93, 0, -93, -185], [-285, -83, -285, 0], [-166, 221, 0, -221], [177, 273, 177, 0], [0, 80, -240, 80], [-123, -5, -123, 0], [-245, -168, 0, 168], [0, -94, 709, -94], [-428, -32, 0, 32], [-180, 0, 180, -132], [-566, -159, 0, 159], [0, 84, 480, 84], [649, 71, 649, 0], [-177, -344, -177, 0], [-748, 120, 0, -120], [48, 288, 48, 0], [-120, -40, 0, 40], [0, 14, 285, 14], [-11, 0, 11, -349], [-49, 0, 49, 80], [-233, -195, -233, 0], [0, 158, 231, 158], [656, -8, 656, 0], [113, -249, 113, 0], [0, -3, -382, -3], [356, 0, -356, 340], [0, -23, 382, -23], [0, -16, -400, -16], [-75, -136, -75, 0], [585, 0, -585, 193], [656, -184, 0, 184], [0, -272, -39, -272], [-193, 300, -193, 0], [352, -232, 0, 232], [20, -328, 0, 328], [0, 308, -533, 308], [0, 272, 293, 272], [-366, 83, 0, -83], [222, 0, -222, -112], [0, 64, 0, 0], [-677, 0, 677, -1], [-128, -90, -128, 0], [162, -432, 162, 0], [508, 0, -508, 340], [64, 188, 0, -188], [336, -240, 336, 0], [-765, 0, 765, 144], [0, 47, -198, 47], [228, 0, -228, 580], [-864, 0, -864, 0], [0, -33, -462, -33], [0, -192, 317, -192], [578, 0, -578, 240], [33, -57, 33, 0], [-180, 0, 180, -12], [-741, -261, -741, 0], [-465, 124, -465, 0], [-723, 18, 0, -18], [411, 375, 411, 0], [0, 237, 762, 237], [-747, 52, 0, -52], [597, 0, -597, -428], [0, 201, 518, 201], [71, 0, -71, -149], [-546, 96, -546, 0], [-169, -35, -169, 0], [-98, -365, 0, 365], [-392, -176, 0, 176], [0, -168, 1112, -168], [0, -310, 65, -310], [-93, 0, 93, -7], [49, 476, 0, -476], [377, -35, 377, 0], [254, 0, -254, -624], [909, 46, 0, -46], [275, 779, 275, 0], [0, -101, 646, -101], [-196, 0, 196, 332]]