
# q-expansion of newform 1122.2.a.f, 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 = 1122
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], [-2], [0], [1], [-4], [-1], [-4], [-4], [0], [2], [0], [6], [-12], [-10], [-4], [4], [-2], [-12], [8], [-10], [8], [4], [-6], [-2], [-2], [8], [20], [-2], [6], [22], [-8], [6], [-20], [-10], [22], [-14], [20], [2], [8], [16], [4], [6], [-22], [-4], [-14], [4], [12], [-16], [-14], [-6], [0], [-6], [-16], [-14], [12], [14], [-6], [10], [30], [-28], [6], [8], [20], [-10], [-18], [-4], [2], [-4], [-28], [-30], [-20], [26], [-12], [-20], [-6], [24], [-28], [26], [14], [20], [-22], [-10], [-2], [28], [-24], [-14], [-26], [18], [-8], [-28], [10], [14], [12], [-24], [-6], [-40], [-30], [24], [18], [28], [18], [20], [18], [-28], [22], [8], [-14], [-18], [-26], [-16], [12], [-6], [40], [-4], [-2], [16], [18], [-6], [-20], [-18], [10], [-4], [-44], [44], [-30], [-48], [-16], [-28], [-28], [-16], [-30], [-10], [22], [-14], [54], [0], [28], [12], [14], [-20], [-36], [-6], [-32], [30], [4], [26], [-10], [-4], [-54], [46], [-14], [-12], [54], [-36], [8], [26], [46], [-14], [-12], [-12], [18], [42], [-28], [-2], [0], [-2], [18], [14], [-50], [60], [30], [-2], [-38], [48], [10], [52], [62], [-34], [32], [20], [40], [16], [38], [-22], [-48], [46], [-28], [-26], [46], [34], [36], [20], [-8], [-64], [-18], [-30], [-36], [42], [32], [24], [4], [-38], [42], [36], [-68], [16], [-36], [-22], [-60], [-2], [28], [-66], [12], [-66], [6], [18], [26], [-48], [-64], [-52], [48], [6], [34], [-28], [-44], [-50], [24], [22], [-24], [-14], [-4], [-42], [-18], [28], [0], [-34], [24], [36], [50], [44], [-52], [-24], [18], [74], [50], [-18], [-60], [0], [20], [52], [-22], [30], [-54], [14], [-76], [72], [60], [-66], [38], [-44], [-76], [-6], [-42], [-14], [-28], [-56], [18], [-20], [42], [0], [-28], [-42], [10], [-62], [8], [-32], [0], [-38], [36], [-60], [-26], [-66], [28], [-44], [-18], [-34], [36], [-12], [-18], [54], [60], [-6], [-12], [42], [60], [0], [60], [-12], [68], [-82], [38], [8], [4], [-64], [-18], [-48], [-16], [26], [-48], [-78], [10], [-2], [52], [-48], [10], [68], [-58], [34], [-62], [-44], [-42], [38], [-40], [38], [-50], [-84], [16], [-66], [60], [26], [42], [16], [60], [68], [-36], [54], [26], [-38], [-86], [-50], [-90], [40], [20], [36], [-60], [-28], [-76], [10], [-78], [4], [78], [-80], [64], [52], [22], [62], [24], [94], [54], [14], [-78], [-6], [-8], [68], [46], [52], [86], [-10], [52], [0], [-74], [-42], [-82], [44], [-68], [-18], [-38], [-22], [-10], [-42], [30], [92], [18], [-28], [60], [80], [14], [-44], [10], [26], [-66], [84], [-20], [-66], [-10], [18], [70], [-16], [14], [30], [-66], [-2], [22], [-18], [20], [-10], [46], [-4], [84], [82], [54], [16], [44], [-70], [-18], [50], [46], [10], [56], [-96], [2], [8], [-44], [54], [88], [36], [18], [36], [-30]]
