
# q-expansion of newform 3850.2.a.f, downloaded from the LMFDB on 20 April 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 = 3850
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], [0], [0], [1], [-1], [-2], [-2], [0], [8], [-2], [-8], [2], [10], [-4], [-8], [-6], [0], [10], [12], [16], [14], [0], [0], [-6], [-10], [18], [0], [4], [-2], [-2], [-8], [0], [6], [-16], [14], [0], [14], [20], [-16], [14], [4], [-22], [8], [-2], [18], [16], [4], [-16], [-16], [-22], [6], [-8], [18], [24], [30], [24], [18], [16], [-30], [-22], [24], [30], [16], [8], [14], [18], [4], [-18], [28], [2], [-18], [32], [-16], [-14], [4], [-8], [-18], [-18], [18], [10], [16], [22], [-8], [-26], [40], [-12], [-30], [22], [-30], [8], [-24], [8], [40], [12], [-28], [24], [2], [34], [-8], [38], [36], [10], [-8], [-6], [-44], [-2], [-24], [-42], [0], [-46], [16], [-22], [6], [-16], [0], [-30], [32], [0], [26], [28], [2], [14], [-26], [-12], [24], [30], [30], [-8], [-32], [22], [12], [0], [-32], [-14], [-6], [34], [-18], [-32], [-2], [42], [-40], [30], [-32], [52], [-38], [0], [-26], [22], [-56], [-40], [2], [-30], [-4], [32], [4], [-48], [24], [-6], [-18], [-38], [52], [22], [-32], [-16], [-2], [0], [-8], [30], [-14], [-26], [60], [-46], [40], [-58], [8], [-38], [44], [14], [24], [-38], [0], [8], [2], [14], [-64], [42], [-22], [-24], [26], [-8], [-34], [-28], [20], [-14], [-4], [6], [50], [-38], [-42], [24], [-26], [-56], [-34], [-22], [0], [6], [56], [36], [-6], [8], [46], [-14], [40], [-32], [-24], [-14], [-32], [18], [48], [10], [-42], [40], [-30], [-16], [0], [-26], [-10], [0], [-40], [20], [10], [56], [-32], [42], [0], [0], [42], [18], [60], [-40], [36], [-8], [-24], [14], [-42], [0], [32], [-56], [36], [-48], [-54], [26], [-16], [10], [-26], [36], [-34], [32], [-2], [-26], [56], [20], [18], [22], [6], [0], [30], [10], [68], [10], [82], [68], [6], [-40], [-58], [-8], [12], [-10], [58], [0], [48], [32], [-48], [-62], [8], [-48], [14], [-46], [48], [42], [-18], [-8], [6], [-40], [74], [2], [0], [86], [-64], [-72], [-26], [-46], [80], [36], [20], [-50], [44], [58], [72], [18], [0], [-34], [50], [84], [-72], [18], [-80], [48], [78], [-30], [56], [22], [10], [-40], [-42], [-54], [44], [48], [24], [-6], [-42], [58], [8], [-56], [-20], [0], [22], [54], [90], [-72], [-38], [6], [26], [-32], [-54], [-84], [-70], [92], [48], [30], [-24], [22], [-2], [-8], [-18], [-66], [-8], [40], [62], [24], [82], [34], [-16], [60], [40], [6], [14], [32], [26], [36], [56], [32], [-82], [56], [-30], [72], [56], [-26], [-46], [-90], [-94], [-58], [40], [-2], [88], [-88], [8], [78], [-76], [-8], [-14], [-34], [28], [64], [-24], [54], [-40], [-46], [-4], [-58], [-54], [-66], [-48], [78], [42], [16], [2], [18], [-32], [-32], [6], [90], [12], [52], [-58], [66], [-16], [-80], [-10], [-56], [70], [30], [8], [16], [-98], [6], [36], [-54], [8], [-80]]
