
# q-expansion of newform 3360.2.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 = 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], [4], [2], [-2], [0], [4], [2], [-8], [10], [2], [-4], [-4], [10], [-4], [-2], [-4], [0], [-6], [-8], [4], [-6], [10], [18], [16], [4], [-2], [6], [-8], [12], [-2], [-8], [18], [-8], [18], [20], [4], [18], [12], [6], [0], [26], [-6], [8], [-8], [0], [4], [-2], [6], [24], [2], [-20], [-10], [4], [-6], [16], [-22], [18], [-28], [26], [-28], [0], [10], [2], [0], [-6], [12], [14], [-18], [16], [-8], [18], [-16], [4], [-30], [26], [10], [-38], [-12], [30], [-8], [-14], [24], [36], [-6], [10], [34], [24], [12], [-24], [0], [28], [8], [-44], [-22], [-22], [-28], [-18], [-28], [-30], [44], [-30], [-32], [10], [-36], [6], [-48], [26], [0], [26], [22], [-8], [16], [2], [4], [12], [34], [-12], [-42], [-6], [-6], [12], [-8], [18], [-34], [48], [-16], [26], [8], [-4], [0], [-38], [-6], [-30], [42], [-20], [2], [-38], [-16], [-38], [-56], [12], [-18], [16], [-30], [54], [8], [36], [10], [34], [52], [20], [-44], [8], [40], [-30], [2], [2], [28], [-34], [56], [12], [14], [-60], [40], [-22], [-14], [-22], [-36], [-42], [-24], [-30], [32], [-22], [-8], [-30], [8], [-26], [-64], [12], [58], [-2], [-36], [-6], [10], [-60], [10], [0], [2], [-12], [8], [-14], [52], [30], [66], [26], [6], [-28], [-54], [-56], [-38], [50], [-36], [34], [-16], [36], [18], [56], [10], [-22], [16], [-12], [8], [58], [-40], [18], [-60], [-14], [-34], [8], [18], [24], [28], [-50], [-34], [-16], [0], [52], [42], [72], [40], [2], [-44], [-28], [66], [-54], [44], [-64], [-60], [-24], [24], [-34], [14], [40], [8], [-12], [72], [-12], [34], [58], [-68], [-22], [34], [-76], [22], [-20], [10], [42], [0], [-20], [46], [10], [70], [-32], [26], [66], [-12], [-54], [6], [20], [-38], [-8], [-14], [64], [12], [-2], [10], [-68], [-68], [8], [-52], [-2], [4], [-48], [-46], [-70], [-8], [42], [10], [-36], [-42], [-44], [-14], [34], [40], [42], [-20], [-28], [10], [66], [-16], [-4], [-32], [-62], [68], [22], [-40], [-30], [-36], [42], [-38], [76], [4], [-22], [-52], [48], [-46], [82], [72], [18], [-70], [40], [-90], [34], [80], [-4], [52], [26], [70], [-78], [0], [-28], [-8], [-52], [-58], [-66], [-86], [56], [2], [-42], [50], [24], [2], [-84], [-2], [-12], [0], [-22], [-16], [74], [18], [0], [-58], [54], [48], [36], [-90], [36], [34], [-78], [-12], [12], [92], [10], [2], [-16], [10], [-76], [56], [-52], [-22], [0], [34], [-20], [-88], [-38], [34], [66], [-46], [-34], [-88], [-2], [56], [-36], [64], [-46], [92], [68], [-46], [58], [12], [-52], [-24], [-86], [-56], [-30], [-64], [2], [-90], [-90], [-72], [-34], [34], [88], [98], [-6], [4], [-44], [-6], [-22], [76], [-80], [-46], [-54], [-96], [-72], [6], [-84], [-70], [-22], [-12], [-12], [-54], [50], [92], [10], [-24], [0]]
