
# q-expansion of newform 3360.2.a.m, downloaded from the LMFDB on 17 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 = 3360
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], [1], [1], [6], [4], [6], [6], [-4], [-2], [-2], [6], [6], [-12], [0], [-4], [-4], [-10], [-12], [2], [16], [4], [-12], [-6], [-12], [-6], [-8], [4], [18], [12], [8], [0], [16], [6], [2], [-16], [-24], [-4], [0], [-22], [6], [-14], [22], [-22], [0], [-26], [-4], [-4], [-4], [22], [-12], [18], [-2], [20], [26], [-16], [-14], [-14], [-2], [-2], [-4], [18], [-8], [16], [-4], [12], [28], [26], [-16], [6], [-18], [6], [-36], [-6], [8], [32], [-14], [-4], [-6], [26], [-16], [6], [18], [12], [-10], [32], [18], [14], [-42], [-40], [12], [24], [8], [22], [-16], [-24], [10], [-10], [16], [-34], [-20], [-36], [-4], [34], [-40], [-20], [12], [-34], [38], [-18], [12], [-34], [28], [-34], [4], [-42], [-24], [16], [-8], [42], [14], [-26], [-6], [-48], [-14], [-42], [6], [12], [-20], [-44], [-16], [-16], [36], [-38], [26], [-34], [42], [32], [-30], [46], [-10], [6], [16], [-28], [2], [20], [44], [26], [50], [-44], [-2], [42], [52], [32], [52], [6], [4], [-14], [40], [30], [40], [4], [8], [60], [-12], [-16], [4], [-36], [62], [-6], [6], [-46], [34], [26], [10], [-26], [28], [-54], [-44], [26], [40], [12], [-38], [-42], [32], [42], [54], [-56], [-18], [-36], [12], [-4], [32], [-18], [-48], [-2], [46], [50], [-6], [40], [-58], [-2], [40], [14], [-44], [42], [-30], [-52], [14], [10], [-26], [18], [24], [-12], [44], [-10], [24], [18], [-24], [32], [54], [-34], [-18], [64], [-28], [-22], [-54], [-26], [-64], [6], [50], [-58], [32], [-50], [-48], [-48], [10], [24], [-66], [4], [-12], [-2], [64], [-30], [-30], [-76], [12], [-8], [-20], [36], [54], [18], [-52], [-6], [-14], [-34], [-22], [-4], [66], [20], [-24], [8], [-58], [56], [-2], [50], [62], [38], [44], [-16], [-22], [60], [4], [-4], [-52], [-20], [-12], [-50], [-22], [-20], [-48], [52], [0], [-6], [32], [6], [70], [44], [14], [18], [-50], [12], [20], [40], [10], [6], [-62], [78], [68], [8], [16], [-8], [0], [80], [-32], [18], [12], [-34], [90], [10], [-40], [38], [18], [-36], [-52], [-34], [-40], [-26], [-52], [-14], [-46], [-10], [14], [-40], [16], [6], [40], [-4], [84], [-4], [-54], [-80], [-38], [44], [-76], [-84], [-50], [2], [6], [52], [82], [20], [42], [8], [-24], [10], [-26], [-12], [-20], [-78], [14], [38], [38], [64], [90], [-66], [-36], [40], [52], [-72], [-46], [14], [48], [-6], [32], [-34], [-10], [-32], [58], [-90], [10], [-64], [-54], [-94], [70], [-16], [6], [-76], [-34], [56], [66], [4], [0], [8], [14], [-48], [100], [4], [-52], [-40], [-10], [-74], [-14], [-64], [90], [-22], [-70], [-30], [24], [-42], [-38], [44], [88], [26], [-58], [34], [-90], [26], [4], [-84], [16], [12], [-104], [60], [-14], [-30], [-6], [60], [-10], [8], [-46], [-36], [32], [-28], [76], [-46], [20], [-62], [26], [-60]]
