
# q-expansion of newform 2800.2.a.m, 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 = 2800
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 = [[0], [0], [0], [-1], [-4], [6], [-2], [0], [0], [6], [-8], [10], [2], [4], [8], [2], [8], [-14], [-12], [16], [-2], [8], [8], [10], [-2], [-6], [16], [12], [6], [-2], [-8], [16], [6], [-16], [6], [-8], [-10], [-4], [0], [22], [12], [-14], [-24], [-2], [-14], [16], [-4], [16], [8], [-14], [6], [-16], [10], [0], [22], [8], [-6], [0], [18], [26], [32], [-10], [-8], [24], [22], [-22], [-4], [14], [4], [10], [6], [-8], [16], [-14], [12], [24], [14], [-10], [-14], [-30], [-24], [-10], [0], [-2], [8], [-20], [-30], [-10], [10], [-16], [-40], [-24], [32], [-12], [12], [-24], [10], [2], [16], [-18], [12], [-22], [-16], [-6], [12], [-10], [8], [6], [0], [42], [-32], [-6], [22], [-8], [16], [2], [-32], [16], [-14], [-28], [-38], [14], [-18], [-4], [32], [22], [-2], [-24], [0], [14], [-28], [48], [16], [-22], [18], [10], [-42], [-8], [6], [-6], [-8], [6], [-8], [44], [-46], [16], [46], [54], [48], [-32], [-14], [2], [-12], [-16], [-52], [0], [16], [58], [-50], [2], [44], [54], [16], [24], [30], [-16], [-40], [22], [34], [14], [52], [26], [-48], [6], [40], [-38], [20], [30], [-40], [34], [-40], [24], [58], [-50], [24], [18], [-22], [8], [58], [24], [6], [-20], [-52], [26], [-12], [-18], [34], [-46], [-42], [56], [-50], [40], [6], [18], [24], [46], [-56], [-36], [-6], [-32], [-18], [-54], [64], [32], [24], [-46], [16], [42], [-56], [-30], [-34], [40], [-30], [-24], [-40], [62], [62], [16], [-8], [-20], [50], [-24], [8], [-54], [64], [48], [10], [34], [4], [-40], [28], [-8], [72], [46], [54], [0], [32], [0], [-20], [-16], [26], [18], [24], [42], [30], [-4], [62], [-8], [6], [-42], [-72], [28], [-38], [46], [-18], [-8], [-18], [26], [-68], [-54], [-6], [12], [46], [-48], [70], [56], [68], [14], [-6], [-64], [16], [-32], [48], [42], [-16], [-48], [-34], [-30], [16], [18], [-18], [-40], [-58], [40], [-6], [42], [-64], [-34], [48], [32], [-42], [-62], [8], [60], [28], [78], [-76], [18], [-24], [-14], [-48], [-18], [-14], [44], [-72], [-70], [-72], [72], [-42], [50], [-40], [-10], [50], [0], [54], [-22], [-44], [48], [-8], [-54], [62], [-30], [56], [-48], [4], [-48], [22], [54], [10], [-88], [2], [-58], [-30], [-80], [18], [-28], [-30], [52], [0], [6], [-56], [6], [78], [-40], [-90], [-18], [24], [8], [30], [-96], [34], [-22], [-88], [-28], [40], [86], [86], [64], [10], [60], [-16], [-80], [-90], [-8], [-70], [16], [0], [70], [-6], [-42], [10], [86], [72], [-66], [-72], [-24], [72], [22], [-4], [88], [-46], [-58], [-76], [48], [-40], [-10], [-88], [82], [-12], [30], [-78], [-18], [-80], [6], [-14], [-16], [58], [-30], [0], [-72], [-26], [90], [36], [-4], [-26], [10], [24], [96], [30], [8], [-74], [-74], [-48], [-80], [6], [14], [-52], [90], [80], [64]]
