# q-expansion of newform 3024.2.cx.c, downloaded from the LMFDB on 19 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 R = PolynomialRing(QQ, "x") f = R(poly_data) K = NumberField(f, "a") betas = [K.gens()[0]**i for i in range(len(poly_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 = 3024 weight = 2 poly_data = [1, -1, 1] # The basis for the coefficient ring is just the power basis # in the root of the defining polynomial above. hecke_ring_character_values = [[1135, [-1, 0]], [757, [1, 0]], [785, [-1, 1]], [2593, [-1, 0]]] aps_data = [[0, 0], [0, 0], [-1, -1], [1, 2], [-2, 1], [-3, -3], [-4, 8], [4, 0], [-3, -3], [0, -9], [7, -7], [10, 0], [3, 3], [10, -5], [0, 3], [6, 0], [-3, 3], [2, -1], [-5, -5], [2, -4], [4, -8], [-6, 3], [0, -9], [-4, 8], [-6, 3], [22, -11], [-5, 5], [10, -20], [2, 0], [15, -15], [10, -20], [21, -21], [0, 3], [-5, 5], [3, -3], [26, -13], [1, 1], [-2, 4], [-3, 3], [6, -3], [2, -4], [-8, 16], [14, -7], [5, -5], [-6, 0], [4, 0], [3, 3], [0, -19], [0, 3], [1, 1], [6, 0], [-11, -11], [2, -1], [-12, 0], [-9, -9], [22, -11], [0, 0], [8, 0], [0, -19], [0, 27], [-13, 13], [-9, -9], [4, 0], [9, -9], [2, -1], [0, -9], [-30, 15], [-23, 23], [5, 5], [-38, 19], [22, -11], [14, -28], [0, 17], [13, -13], [10, -20], [-3, 3], [0, 3], [8, -16], [-21, 21], [17, 17], [9, -9], [0, 17], [-2, 4], [4, -8], [0, -7], [-10, 5], [-6, 0], [0, 13], [14, -7], [3, 3], [-36, 0], [0, -9], [-6, 12], [9, 9], [15, 15], [-36, 0], [11, 11], [16, -32], [8, 0], [-2, 0], [2, -1], [-42, 0], [9, -9], [0, 39], [11, 11], [20, -40], [0, -9], [-20, 40], [1, 1], [-38, 19], [-13, 13], [2, 0], [3, -3], [0, -35], [-22, 44], [0, 39], [23, -23], [24, 0], [-9, 9], [30, -15], [1, 1], [0, -43], [-26, 13], [-30, 60], [0, -23], [-42, 0], [0, -11], [24, 0], [0, 25], [13, 13], [-14, 28], [25, 25], [7, 7], [38, 0], [-5, -5], [-15, -15], [24, -48], [-5, 5], [3, 3], [-6, 0], [-52, 0], [0, -9], [-21, -21], [26, -52], [-24, 48], [0, -45], [-62, 31], [6, -3], [23, -23], [-18, 36], [13, -13], [4, -8], [-18, 36], [-15, 15], [42, -21], [6, -3], [18, -36], [22, -11], [-20, 40], [7, 7], [-50, 25], [42, 0], [19, 19], [48, 0], [-33, 33], [0, 15], [10, -20], [42, -21], [-2, 0], [-5, -5], [22, -11], [-3, -3], [-3, -3], [0, 53], [-13, 13], [15, 15], [26, -13], [30, 0], [32, 0], [-22, 11], [50, -25], [0, 39], [41, -41], [-12, 24], [-23, -23], [-2, 1], [-34, 0], [0, 1], [-31, 31], [48, 0], [-12, 24], [30, -15], [38, -76], [-42, 21], [18, -36], [23, 23], [53, -53], [0, -43], [-50, 25], [-24, 0], [63, -63], [0, -43], [-15, -15], [50, -25], [-60, 0], [16, -32], [-32, 0], [1, 1], [0, 39], [-13, 13], [38, 0], [27, 27], [-46, 23], [0, -69], [69, -69], [-46, 23], [-5, -5], [38, -19], [30, -60], [-21, 21], [-35, 35], [19, -19], [-45, 45], [34, -68], [-27, 27], [0, 29], [-34, 17], [-26, 52], [0, 1], [-66, 33], [25, -25], [28, 0], [-13, -13], [63, -63], [0, -19], [0, -69], [21, 21], [6, 0], [-23, -23], [0, 0], [-10, 5], [20, 0], [59, -59], [34, 0], [35, 35], [0, 51], [32, 0], [-39, 39], [19, 19], [-2, 4], [-35, 35], [-4, 8], [-35, -35], [34, -17], [-26, 13], [14, -28], [10, 0], [0, 53], [-28, 56], [-4, 8], [-54, 27], [-10, 5], [-7, -7], [-40, 80], [11, 11], [0, 37], [-18, 0], [-2, 1], [-29, -29], [27, -27], [-11, -11], [38, -76], [-62, 31], [-1, -1], [-19, -19], [-16, 0], [-7, -7], [0, 17], [46, 0], [0, 3], [-51, 51], [23, 23], [0, -33], [-86, 43], [-77, 77], [26, -52], [38, 0], [-33, 33], [0, 37], [16, -32], [0, -33], [-24, 0], [-57, 57], [57, -57], [0, -83], [-49, -49], [0, -55], [30, -15], [-12, 0], [0, -71], [-15, -15], [-18, 0], [2, -4]]