
# q-expansion of newform 1368.2.a.f, downloaded from the LMFDB on 23 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 = 1368
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], [0], [2], [-2], [4], [1], [2], [6], [4], [2], [-2], [4], [10], [6], [4], [2], [4], [4], [-6], [8], [-2], [6], [-10], [4], [0], [12], [-2], [6], [12], [-6], [4], [12], [-20], [-8], [2], [-4], [-8], [10], [-12], [-2], [-10], [-6], [-8], [0], [12], [16], [24], [2], [-16], [6], [-18], [-18], [-22], [-10], [-14], [-16], [18], [-6], [-12], [-14], [-24], [-30], [-2], [18], [8], [-18], [-22], [22], [-4], [14], [-8], [2], [0], [-36], [16], [38], [-10], [-26], [-6], [22], [12], [2], [-16], [18], [34], [-14], [-28], [-8], [6], [30], [-28], [-30], [4], [30], [-14], [-18], [4], [14], [-32], [12], [16], [42], [-4], [22], [22], [8], [-16], [2], [-12], [26], [36], [4], [-32], [30], [20], [18], [-12], [40], [10], [-6], [30], [36], [-36], [20], [-26], [-30], [-32], [-18], [-44], [44], [40], [26], [-52], [-10], [26], [-28], [42], [48], [24], [-4], [16], [12], [30], [4], [6], [46], [20], [-36], [-22], [-52], [-44], [24], [52], [40], [-16], [-8], [-38], [-30], [-38], [-6], [-32], [0], [-18], [-8], [20], [26], [-10], [36], [28], [-38], [-46], [-10], [36], [-16], [52], [-60], [-44], [14], [16], [-48], [-50], [-14], [-14], [-44], [38], [4], [-2], [2], [-34], [-6], [-8], [-42], [50], [6], [2], [-66], [0], [-42], [6], [-4], [-50], [-58], [-54], [12], [40], [-36], [-36], [8], [22], [24], [48], [-4], [-4], [-46], [32], [38], [20], [-40], [-66], [32], [-50], [-16], [8], [-6], [-38], [20], [-52], [-62], [-30], [-8], [36], [14], [20], [-6], [50], [-24], [42], [-36], [-44], [-20], [-40], [-54], [18], [-18], [32], [-24], [56], [18], [-14], [-48], [50], [-26], [64], [66], [-42], [-16], [-18], [38], [64], [72], [-42], [-6], [72], [4], [54], [-40], [64], [0], [-22], [-20], [6], [64], [22], [48], [-10], [10], [10], [-42], [60], [-72], [-2], [10], [-20], [-14], [30], [58], [-56], [42], [20], [-34], [42], [36], [-46], [48], [64], [-48], [0], [-84], [70], [30], [-56], [84], [12], [38], [48], [-74], [74], [-42], [-34], [-16], [-58], [-72], [18], [78], [-14], [0], [26], [60], [-20], [58], [22], [16], [4], [6], [80], [28], [12], [28], [50], [46], [88], [42], [-60], [-2], [-54], [-18], [22], [-80], [-62], [20], [-2], [-20], [14], [48], [42], [36], [-64], [-60], [-68], [-18], [8], [-60], [-70], [-34], [-42], [-42], [36], [36], [62], [-20], [-12], [-84], [-68], [2], [-68], [-52], [-38], [-18], [-16], [-22], [78], [-56], [-2], [-32], [42], [34], [-32], [-26], [-18], [64], [8], [24], [12], [-68], [32], [50], [4], [40], [-50], [-34], [18], [-68], [-12], [94], [44], [-54], [92], [-16], [-22], [24], [40], [-22], [26], [-8], [46], [-14], [-16], [-54], [2], [-72], [-48], [-68], [58], [-60], [48], [-100], [-52], [28], [-82], [-46], [-14], [-64], [50], [42], [-68], [24], [28], [2]]
