# q-expansion of newform 15.18.a.b, downloaded from the LMFDB on 25 May 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 = 15 weight = 18 poly_data = [-350700, -37234, 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], 1], [[24809, 58, -1], 41], [[-1514169, 16142, 61], 41]] hecke_ring_character_values = None aps_data = [[-147, 1, 0], [-6561, 0, 0], [-390625, 0, 0], [1669256, 45211, 87], [349664264, -860632, -3704], [-1031412738, -2473637, 22487], [-5063082238, -69116101, 190519], [-37344186648, -136142045, 299903], [261414923228, -293800139, -3041351], [466409552954, 3841743816, -34761368], [-4152889688724, -449335347, 22114289], [10694102158366, -11807887517, -366052481], [-34092395456254, 269256210, 538232506], [-47122334100140, -19121868936, 364522840], [-18103693835980, -128142786925, 712179247], [115337586824898, 470359907170, 2922218058], [692071536970488, 691987226536, 11552809608], [-530137346891330, 742088043234, 25940289226], [-805892575157548, 25167672328, -34538566872], [1672117299499872, -3169526320816, -35831147376], [-3596857625642662, 13506249418262, -82629513330], [-13555607537032780, -24736162128941, -45208212561], [-7265628966994956, 26213101161648, -156924270480], [-884931687901878, 45428892095982, -6084570810], [-26561857197009438, 75688754118848, 683972358848], [72690131939773154, -96631230917828, 1021975678060], [-69928410910343240, 22251984100315, -752992433129], [152113128697875940, 20855378372774, -966722985634], [3286022475940662, -73729427573896, -2957578065448], [28410354334085594, -606587837851393, 286444215499], 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