
# q-expansion of newform 2850.2.a.m, downloaded from the LMFDB on 24 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 = 2850
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], [0], [0], [4], [-2], [-2], [-1], [-4], [6], [4], [6], [10], [4], [12], [-6], [-12], [-2], [-4], [8], [6], [-4], [12], [10], [-2], [10], [4], [4], [-6], [14], [4], [-12], [-10], [-20], [18], [-12], [2], [4], [0], [10], [20], [18], [20], [-2], [-2], [0], [-4], [-12], [20], [-10], [-18], [12], [26], [-20], [-2], [4], [-2], [8], [2], [-6], [28], [2], [-4], [-28], [6], [26], [28], [-26], [12], [-10], [-26], [12], [-24], [22], [-20], [0], [-22], [18], [-14], [18], [12], [26], [8], [-34], [12], [-4], [-30], [6], [-38], [-8], [28], [20], [20], [-36], [-36], [36], [-18], [-6], [-28], [-10], [-20], [-42], [-4], [-38], [-44], [30], [28], [-10], [16], [-38], [-4], [-22], [14], [4], [-8], [18], [4], [-36], [6], [-12], [10], [14], [42], [-36], [-4], [18], [-10], [28], [-40], [-46], [52], [-16], [-4], [-38], [18], [-46], [-30], [52], [-54], [34], [-20], [-22], [-8], [-36], [10], [0], [-14], [-26], [36], [-8], [46], [-30], [-20], [48], [-28], [24], [8], [-6], [-10], [30], [-52], [-42], [32], [-36], [-2], [-56], [4], [-38], [-6], [-34], [44], [-54], [-12], [22], [60], [10], [12], [-62], [28], [62], [-24], [28], [14], [6], [12], [2], [-58], [20], [-14], [52], [-42], [-44], [-44], [6], [-60], [54], [-30], [50], [-42], [60], [-42], [12], [-26], [-14], [4], [-2], [-16], [36], [-54], [52], [62], [-70], [-16], [-12], [64], [18], [-32], [50], [-64], [14], [50], [28], [-14], [-48], [60], [54], [-26], [24], [36], [-12], [66], [60], [60], [58], [76], [-12], [34], [-42], [-36], [-8], [44], [-28], [56], [-38], [14], [-12], [-64], [-20], [-12], [36], [-62], [18], [36], [-70], [6], [76], [-58], [-12], [-38], [6], [68], [60], [30], [6], [-82], [44], [54], [10], [28], [-74], [42], [-28], [-10], [-56], [70], [40], [36], [-38], [-22], [12], [16], [16], [20], [2], [-52], [20], [30], [-78], [-16], [66], [58], [-52], [54], [-20], [6], [18], [-20], [-18], [20], [52], [6], [-46], [-16], [-20], [-28], [-50], [76], [26], [4], [34], [-12], [18], [18], [-44], [4], [-54], [-44], [-48], [-18], [2], [-92], [54], [-74], [-12], [22], [-38], [-60], [44], [0], [-50], [-42], [2], [0], [-76], [-20], [-12], [82], [-66], [26], [32], [-58], [38], [-82], [-68], [-14], [20], [62], [36], [40], [54], [4], [86], [-38], [4], [-22], [-42], [60], [28], [-18], [-8], [-38], [58], [72], [44], [-84], [62], [30], [-4], [50], [-12], [-12], [68], [46], [-80], [-38], [-60], [100], [46], [42], [-58], [6], [-10], [32], [6], [44], [-16], [0], [-62], [-12], [88], [18], [90], [60], [-44], [-24], [6], [-44], [-6], [-4], [58], [90], [14], [4], [-90], [6], [56], [26], [18], [-28], [84], [-26], [6], [4], [-44], [-74], [-78], [-40], [68], [-98], [0], [-90], [38], [28], [44], [-42], [42], [-4], [74], [-4], [60]]
