# q-expansion of newform 8670.2.a.b, downloaded from the LMFDB on 27 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 = 8670 weight = 2 poly_data = [0, 1] # The basis for the coefficient ring is just the power basis # in the root of the defining polynomial above. hecke_ring_character_values = None aps_data = [[-1], [-1], [-1], [-1], [2], [1], [0], [-1], [-4], [8], [7], [3], [2], [-9], [8], [-2], [-6], [5], [-11], [8], [-2], [0], [2], [4], [3], [2], [3], [0], [10], [0], [-1], [-10], [12], [-17], [10], [1], [-13], [16], [12], [-8], [0], [-25], [-26], [9], [10], [4], [13], [-8], [-20], [6], [-6], [24], [10], [20], [0], [-28], [24], [15], [17], [-20], [-29], [16], [8], [20], [1], [-8], [13], [-10], [32], [22], [26], [0], [13], [-22], [-25], [8], [30], [18], [18], [-11], [-18], [27], [-10], [-23], [-3], [18], [24], [-2], [20], [-5], [-18], [-4], [-25], [12], [13], [38], [-16], [0], [4], [18], [-1], [4], [18], [18], [-5], [-15], [32], [6], [6], [15], [-17], [-41], [48], [20], [-25], [6], [-23], [14], [-42], [32], [-43], [49], [-12], [42], [7], [0], [43], [-8], [-1], [33], [21], [10], [45], [53], [54], [50], [-36], [9], [46], [6], [-25], [30], [41], [50], [23], [-30], [-13], [-10], [55], [-10], [-22], [54], [-28], [6], [31], [-52], [16], [18], [29], [10], [-28], [-6], [8], [-14], [-6], [-4], [56], [7], [21], [6], [-6], [-31], [-6], [-53], [-33], [24], [17], [-18], [-40], [-51], [52], [44], [-15], [8], [6], [-30], [37], [-52], [-39], [30], [59], [-20], [-25], [44], [-52], [60], [-37], [62], [30], [-58], [-60], [12], [-19], [-11], [-26], [56], [-45], [-62], [24], [69], [-7], [0], [24], [-18], [-34], [-9], [-17], [24], [8], [-48], [-41], [-40], [62], [-43], [44], [49], [-14], [0], [63], [-70], [29], [-56], [43], [36], [27], [24], [-30], [-66], [4], [-48], [10], [41], [-40], [70], [12], [-72], [-25], [-48], [-65], [34], [-11], [20], [-52], [53], [48], [30], [-17], [-25], [-2], [53], [8], [-20], [-25], [55], [68], [20], [12], [-60], [17], [-78], [18], [-31], [19], [-19], [30], [-5], [0], [-31], [-25], [-38], [-10], [-56], [30], [1], [-12], [28], [-26], [4], [43], [38], [10], [18], [-66], [-38], [-50], [58], [-28], [16], [-48], [85], [66], [-56], [7], [-76], [-67], [2], [26], [-85], [-26], [-77], [-42], [-4], [44], [-32], [42], [-7], [30], [2], [31], [78], [-13], [-70], [-62], [-49], [-12], [62], [-52], [76], [46], [-78], [-5], [-26], [-88], [-14], [19], [-2], [-51], [6], [53], [24], [22], [68], [42], [48], [58], [48], [-26], [17], [64], [-8], [60], [83], [6], [55], [-11], [-32], [-18], [-12], [-10], [16], [-14], [-76], [82], [-42], [-16], [6], [-8], [24], [51], [-78], [65], [-50], [20], [0], [-67], [-74], [18], [61], [52], [7], [0], [-4], [-11], [54], [-37], [40], [37], [73], [-48], [-46], [-103], [-32], [-42], [64], [-12], [59], [85], [-56], [-77], [-30], [85], [-70], [96], [-4], [6], [-63], [49], [-10], [60], [60], [66], [-88], [-42], [-61], [-17], [34], [78], [43], [46], [36], [-20], [43], [-36], [60], [-87], [76], [-78], [-28], [39], [-26], [5], [-12], [21], [-22], [59], [64], [-14], [-75], [28], [36], [-42], [-36], [-75], [4], [65], [-72], [-5], [-36], [-89], [-77], [-112], [-30], [22], [-68], [3], [40], [-9], [-6], [113], [72], [-59], [71], [-68], [-55], [-61], [15], [17], [-98], [62], [-59], [4], [-8], [22], [46], [18], [33], [84], [-5], [-14], [34], [-42], [50], [7], [-100], [-16], [62], [26], [114], [-109], [-48], [82], [24], [-37], [-66], [16], [70], [-36], [-36], [84], [12], [-100], [-29], [60], [-100], [-85], [-104], [-86], [-88], [10], [-44], [-44], [42], [-62]]