
# q-expansion of newform 1870.2.a.f, downloaded from the LMFDB on 31 May 2024.

# 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 = 1870
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], [0], [1], [-3], [1], [3], [4], [-3], [-7], [8], [-4], [-2], [-9], [-3], [-11], [-1], [-12], [-7], [7], [0], [-6], [-1], [-9], [4], [16], [0], [-1], [-13], [-1], [-8], [2], [-14], [-20], [22], [-2], [20], [-14], [8], [-12], [18], [18], [18], [20], [-9], [10], [9], [-3], [-26], [-3], [-24], [-22], [-20], [-24], [1], [-27], [-10], [-4], [5], [17], [19], [30], [0], [-26], [-6], [-17], [15], [35], [20], [-14], [18], [-4], [22], [-4], [17], [38], [-2], [-2], [31], [14], [-12], [16], [32], [-16], [-18], [10], [-8], [38], [23], [-14], [9], [18], [-27], [14], [-20], [-22], [12], [4], [6], [3], [-31], [44], [-31], [10], [-12], [-2], [24], [24], [-2], [46], [-23], [-33], [0], [0], [12], [-20], [-19], [0], [29], [14], [19], [12], [21], [-10], [-6], [11], [-11], [15], [-34], [-11], [-48], [-5], [11], [-30], [-5], [-26], [-37], [-30], [-10], [-18], [-45], [-44], [-8], [-2], [43], [-34], [13], [-31], [4], [-22], [-12], [14], [20], [-33], [-20], [-44], [-40], [16], [45], [3], [38], [32], [1], [22], [24], [25], [-18], [-16], [54], [-9], [52], [-33], [-58], [26], [6], [50], [-41], [29], [14], [39], [10], [-32], [-64], [28], [34], [10], [24], [4], [-64], [-46], [1], [1], [10], [25], [-46], [-30], [-18], [-45], [11], [-6], [59], [-2], [7], [20], [2], [4], [-48], [6], [67], [13], [26], [-26], [0], [-25], [-70], [-53], [-57], [38], [-5], [68], [-55], [34], [-12], [58], [-12], [-69], [-43], [-15], [-8], [47], [-14], [8], [35], [62], [-4], [-26], [-32], [64], [38], [-25], [52], [20], [30], [-29], [21], [58], [-8], [-40], [-77], [25], [60], [-67], [26], [41], [45], [-53], [-28], [-4], [-76], [44], [-42], [-46], [6], [3], [4], [42], [75], [37], [2], [-15], [20], [-38], [32], [-2], [-33], [38], [-24], [-27], [-57], [-72], [-59], [10], [58], [-68], [42], [15], [-32], [-56], [-19], [75], [9], [-18], [-72], [-10], [35], [20], [-69], [-80], [-4], [-56], [15], [77], [11], [40], [-70], [54], [-35], [12], [39], [28], [70], [22], [-54], [-65], [33], [-73], [-32], [-26], [41], [50], [-65], [38], [-16], [-51], [69], [-58], [-9], [-24], [4], [-74], [58], [-8], [30], [-74], [-36], [-80], [-78], [-64], [8], [-9], [48], [37], [-58], [6], [-18], [22], [46], [15], [-52], [76], [-2], [-60], [44], [-44], [6], [-36], [-40], [70], [57], [8], [36], [84], [0], [-58], [84], [2], [5], [-68], [-69], [60], [54], [12], [-22], [74], [-10], [49], [-50], [33], [74], [-31], [0], [38], [15], [50], [6], [-10], [-94], [-23], [12], [-8], [76], [69], [-42], [-2], [-70], [-70], [86], [4], [58], [-94], [42], [43], [-91], [84], [64], [-45], [-30], [22], [-14], [-77], [21], [-76], [-8], [4], [-57], [-40], [-18], [-7], [-73], [42], [-4], [86], [73], [1], [-11], [84], [-44], [49], [-80]]
