
# q-expansion of newform 2898.2.a.f, downloaded from the LMFDB on 19 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 = 2898
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], [2], [-1], [-4], [2], [-6], [0], [-1], [2], [4], [6], [6], [12], [12], [-6], [4], [-10], [4], [16], [2], [8], [16], [-6], [-2], [14], [0], [12], [-2], [6], [-8], [-4], [-10], [4], [-6], [0], [-18], [-4], [-20], [6], [-4], [-2], [-8], [18], [18], [0], [28], [-4], [24], [-18], [-10], [-8], [-10], [-24], [6], [24], [6], [20], [-10], [22], [24], [34], [-20], [-12], [14], [10], [28], [-22], [12], [-6], [14], [16], [-24], [22], [20], [0], [-38], [34], [-18], [26], [-16], [-26], [-16], [-2], [20], [-20], [30], [-6], [22], [-32], [-8], [-40], [8], [12], [-36], [-40], [6], [26], [-8], [-18], [-28], [-14], [0], [-42], [-20], [10], [12], [6], [-40], [2], [20], [-10], [6], [32], [16], [30], [24], [-36], [34], [12], [-18], [34], [34], [-36], [-20], [34], [6], [-28], [16], [14], [-4], [0], [0], [38], [-18], [-10], [26], [-40], [34], [6], [28], [-30], [32], [-20], [18], [24], [-38], [22], [-12], [8], [-26], [-6], [-36], [44], [-28], [24], [-16], [46], [30], [-38], [36], [-34], [24], [24], [-18], [24], [-56], [-6], [18], [-26], [28], [-22], [32], [18], [60], [-38], [-36], [-6], [24], [-34], [8], [-48], [-50], [22], [16], [10], [46], [24], [-6], [-4], [2], [-36], [28], [-18], [44], [-6], [-22], [62], [-38], [44], [-46], [20], [-6], [38], [0], [-18], [48], [-28], [-42], [4], [-62], [-2], [56], [48], [-44], [-2], [-32], [-42], [-64], [18], [46], [40], [-2], [24], [60], [-34], [42], [-24], [16], [-28], [-26], [40], [-40], [-42], [-56], [8], [22], [10], [-44], [-60], [-20], [44], [44], [-2], [54], [20], [20], [-32], [44], [-72], [-34], [-62], [40], [30], [-10], [76], [38], [64], [22], [58], [16], [-20], [10], [22], [-2], [-56], [-46], [-54], [-20], [26], [-38], [12], [22], [-64], [58], [-52], [-60], [-82], [-22], [40], [28], [-72], [-24], [14], [-44], [56], [34], [42], [52], [14], [34], [8], [22], [-56], [46], [-38], [-64], [-26], [-4], [-28], [-26], [-46], [80], [52], [-52], [-78], [68], [-26], [24], [-26], [-12], [-54], [-82], [-68], [24], [86], [68], [-24], [14], [30], [64], [-62], [-10], [64], [-66], [-26], [-12], [-76], [8], [-6], [-18], [18], [20], [44], [28], [-52], [-66], [66], [58], [0], [70], [70], [54], [56], [34], [-84], [-70], [60], [-80], [-78], [-44], [-54], [-22], [64], [22], [-58], [48], [-80], [-66], [72], [-26], [22], [-80], [36], [-28], [74], [-34], [-40], [-6], [36], [-28], [80], [-54], [16], [-2], [-28], [-80], [-74], [-62], [-78], [18], [-82], [16], [-66], [56], [20], [88], [26], [-4], [16], [-54], [54], [4], [76], [-48], [66], [16], [74], [-76], [-6], [74], [14], [-72], [-86], [-26], [20], [-82], [-66], [-48], [-60], [82], [58], [-92], [28], [34], [-42], [-80], [20], [50], [-56], [-30], [-10], [-48], [36], [10], [-58], [52], [-42], [84], [84]]
