
# q-expansion of newform 3850.2.a.m, downloaded from the LMFDB on 25 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], [-2], [0], [-1], [-1], [-2], [-2], [6], [-6], [4], [0], [-8], [0], [-4], [4], [12], [0], [2], [8], [-12], [6], [10], [12], [14], [-4], [10], [8], [-20], [4], [6], [-8], [-6], [22], [-6], [4], [-6], [10], [24], [0], [18], [-4], [2], [8], [-2], [2], [16], [-20], [-16], [24], [6], [26], [6], [-8], [0], [28], [-8], [14], [28], [22], [-2], [-32], [6], [12], [16], [12], [0], [12], [2], [-20], [6], [-20], [-6], [0], [26], [4], [-16], [18], [-14], [-34], [0], [-16], [-26], [-22], [-16], [12], [-40], [14], [-22], [-26], [10], [-38], [8], [18], [12], [-28], [-16], [2], [-10], [-4], [-16], [-20], [18], [24], [26], [-40], [-4], [2], [-26], [44], [44], [-16], [26], [42], [-32], [-8], [-30], [-34], [0], [36], [8], [42], [42], [-38], [4], [12], [-8], [-30], [48], [44], [2], [-36], [-48], [4], [-52], [-24], [-28], [6], [48], [-30], [54], [-30], [-8], [-14], [-12], [-26], [-40], [-22], [30], [48], [-30], [-14], [-14], [-28], [-48], [4], [-12], [26], [30], [-6], [18], [52], [-18], [56], [0], [54], [56], [16], [42], [-10], [18], [40], [2], [6], [10], [-8], [18], [4], [30], [40], [-30], [62], [10], [28], [-30], [-42], [6], [26], [22], [34], [48], [-56], [4], [-4], [-6], [-36], [-56], [38], [-44], [34], [-24], [-56], [64], [14], [16], [44], [14], [-40], [4], [42], [-60], [-18], [-2], [-22], [66], [56], [30], [28], [64], [10], [36], [4], [-36], [62], [-10], [56], [-40], [64], [20], [-48], [-32], [8], [-46], [-30], [18], [10], [0], [-10], [6], [12], [72], [-36], [-38], [36], [-10], [-62], [-12], [-32], [-72], [-4], [-8], [-58], [60], [-10], [34], [46], [36], [-22], [-28], [70], [-22], [80], [12], [-46], [-38], [60], [-80], [-6], [-42], [44], [-10], [-62], [0], [0], [14], [18], [-68], [72], [-68], [26], [-18], [-56], [60], [24], [82], [-48], [-72], [18], [62], [32], [0], [22], [-42], [10], [78], [54], [-42], [48], [26], [22], [-68], [66], [2], [22], [12], [-12], [-50], [-44], [14], [28], [26], [48], [-2], [18], [-4], [-84], [40], [-28], [54], [-40], [46], [-82], [-10], [-30], [-38], [30], [-2], [52], [34], [56], [-30], [-16], [-24], [-28], [-76], [32], [30], [2], [74], [-58], [-56], [28], [-66], [-50], [8], [64], [60], [86], [-96], [20], [30], [30], [-78], [-6], [-88], [76], [46], [56], [66], [-42], [14], [-2], [-72], [-66], [4], [38], [-78], [-10], [60], [14], [-84], [28], [-12], [84], [-4], [-24], [28], [50], [-78], [20], [66], [-18], [-6], [20], [26], [54], [4], [0], [-82], [-52], [48], [-2], [-18], [-44], [18], [48], [22], [-28], [-50], [-4], [-80], [-78], [-78], [88], [56], [-66], [76], [8], [-82], [22], [0], [-22], [78], [-8], [72], [26], [18], [18], [24], [60], [0], [6], [38], [2], [82], [4], [-42], [104], [30], [-72], [80]]
