
# q-expansion of newform 6160.2.a.m, 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 = 6160
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], [2], [1], [-1], [-1], [-4], [0], [4], [-6], [-6], [4], [2], [-6], [4], [-6], [-6], [0], [-10], [-2], [-12], [8], [16], [12], [-18], [-10], [-6], [10], [0], [-10], [-18], [4], [12], [6], [-20], [6], [-8], [14], [-2], [12], [-24], [0], [-22], [0], [20], [12], [-20], [4], [-26], [-12], [-10], [0], [0], [26], [12], [-6], [-12], [18], [16], [8], [6], [4], [-12], [16], [24], [26], [6], [4], [20], [-24], [-10], [18], [-24], [10], [-4], [-20], [18], [-30], [14], [30], [-10], [-12], [38], [24], [-34], [-8], [6], [18], [8], [-30], [10], [18], [0], [-2], [36], [-20], [-24], [-18], [6], [-8], [-10], [28], [-12], [-24], [-6], [4], [-10], [42], [0], [-12], [26], [-44], [20], [-18], [-20], [-20], [18], [10], [6], [-6], [12], [2], [-4], [36], [18], [4], [-6], [26], [0], [-2], [32], [28], [36], [-20], [38], [6], [26], [-30], [-32], [-18], [6], [-20], [-30], [-50], [-48], [-34], [-36], [-16], [-48], [4], [6], [-4], [-18], [-2], [12], [46], [-24], [40], [-18], [20], [18], [54], [-24], [4], [-48], [-42], [-30], [-32], [32], [-58], [-6], [60], [50], [-24], [-40], [-32], [-6], [52], [-42], [16], [-46], [58], [12], [-34], [-36], [-54], [42], [20], [-50], [26], [0], [38], [36], [-56], [42], [60], [-54], [-10], [26], [60], [12], [-54], [-8], [50], [26], [60], [-18], [52], [0], [-18], [-32], [-40], [54], [46], [-6], [-24], [14], [-32], [54], [-54], [66], [-22], [64], [-6], [-2], [-24], [-10], [-18], [-48], [-44], [-60], [38], [52], [-32], [30], [-2], [-48], [-22], [-24], [36], [36], [6], [-20], [10], [2], [12], [48], [58], [36], [-68], [12], [56], [6], [-30], [-58], [0], [-60], [50], [28], [54], [20], [76], [60], [-10], [44], [-6], [40], [-78], [18], [76], [24], [-34], [-2], [14], [40], [-4], [-2], [30], [-22], [-82], [36], [84], [-56], [-48], [-46], [4], [-48], [-58], [24], [-56], [6], [-30], [6], [24], [-12], [-16], [-6], [64], [-30], [-36], [16], [-4], [60], [16], [66], [-20], [-46], [30], [-82], [-60], [20], [24], [54], [-54], [-86], [-36], [-22], [84], [48], [62], [-30], [4], [-82], [78], [-62], [-72], [50], [-8], [10], [48], [-36], [50], [-90], [88], [-48], [28], [78], [-70], [-36], [26], [-8], [38], [-78], [-6], [64], [42], [-84], [26], [10], [0], [66], [4], [-70], [6], [-80], [2], [48], [-60], [-12], [-72], [30], [8], [-18], [6], [-60], [-2], [-82], [0], [-44], [-34], [24], [-44], [-12], [-66], [-80], [38], [12], [24], [92], [-42], [20], [30], [90], [-20], [12], [-68], [66], [52], [-58], [-32], [54], [50], [-54], [-12], [-38], [-12], [32], [16], [-54], [-56], [-78], [50], [54], [-32], [-6], [-6], [-56], [-10], [42], [-50], [0], [-28], [48], [30], [-44], [-100], [-78], [-48], [70], [42], [72], [-66], [-76], [-78], [-60], [-70], [66], [-42], [-30], [-20], [0]]
