
# q-expansion of newform 1344.2.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 = 1344
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], [1], [-2], [1], [0], [-6], [-2], [-4], [-4], [10], [-8], [-6], [-2], [4], [8], [10], [-12], [2], [-12], [-12], [-14], [-8], [-12], [-2], [10], [6], [0], [0], [2], [18], [8], [4], [2], [12], [-22], [-8], [18], [20], [0], [-2], [16], [-14], [-20], [18], [-6], [8], [-20], [0], [-20], [-14], [2], [20], [10], [20], [6], [12], [6], [8], [-22], [18], [4], [6], [-4], [0], [-22], [18], [4], [2], [-24], [-14], [30], [-12], [-16], [10], [-12], [0], [2], [2], [-6], [18], [-20], [26], [12], [-14], [-24], [-8], [34], [10], [-18], [-40], [-36], [-8], [-8], [24], [-4], [40], [-42], [6], [-20], [2], [20], [2], [-4], [-30], [-20], [18], [-28], [-18], [-44], [10], [16], [26], [-30], [12], [-16], [10], [-4], [-48], [10], [-8], [10], [-14], [6], [24], [-44], [-22], [-22], [0], [-16], [26], [20], [20], [32], [26], [-18], [34], [-10], [-28], [-18], [-6], [-36], [-6], [-16], [0], [-38], [0], [26], [-18], [20], [20], [-14], [-2], [-4], [-48], [-44], [-20], [-8], [-26], [-54], [-42], [56], [58], [-48], [-4], [-6], [-24], [0], [26], [-30], [38], [32], [18], [-60], [10], [48], [-10], [-20], [-30], [16], [50], [-56], [-44], [-22], [22], [36], [-42], [-62], [52], [42], [-48], [-22], [48], [28], [30], [8], [6], [-62], [-14], [22], [64], [34], [0], [-14], [-38], [60], [30], [-16], [-24], [-70], [-28], [18], [30], [40], [-36], [32], [-6], [48], [14], [60], [-22], [42], [64], [-46], [0], [68], [10], [-66], [20], [64], [-24], [2], [-20], [-32], [10], [-28], [-48], [26], [-6], [8], [-72], [64], [-20], [8], [-46], [6], [-32], [-56], [-20], [-36], [20], [-62], [62], [-44], [18], [14], [-24], [-38], [52], [54], [42], [56], [0], [66], [2], [30], [4], [2], [70], [12], [-30], [-46], [-52], [2], [0], [42], [32], [-80], [2], [-70], [60], [0], [-64], [72], [-6], [76], [12], [-78], [-22], [0], [6], [-30], [20], [50], [20], [-14], [14], [48], [46], [-60], [-44], [74], [42], [24], [72], [12], [-46], [-72], [26], [4], [-38], [-16], [-14], [-38], [-4], [44], [-14], [20], [-44], [-86], [-6], [28], [-22], [78], [-32], [26], [-70], [68], [-76], [-36], [18], [18], [-6], [40], [-52], [-20], [4], [2], [70], [2], [-8], [-54], [18], [-66], [56], [58], [-72], [-46], [-28], [-32], [22], [-20], [58], [-30], [0], [42], [-42], [-48], [28], [18], [36], [-38], [-66], [12], [0], [-60], [-70], [54], [0], [10], [64], [52], [12], [26], [-80], [-30], [20], [-12], [50], [62], [-14], [62], [26], [40], [18], [4], [72], [-64], [-62], [76], [-8], [-94], [62], [0], [12], [52], [-22], [64], [22], [92], [-30], [18], [2], [72], [-18], [46], [56], [82], [34], [-60], [-84], [-30], [-78], [-56], [92], [-54], [-42], [4], [16], [-18], [16], [90], [10], [12], [-36], [42], [-18], [32], [-30], [-20], [16]]
