
# q-expansion of newform 6080.2.a.f, 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 = 6080
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], [2], [0], [-6], [2], [1], [-2], [2], [4], [10], [-10], [-6], [-6], [-6], [4], [-2], [2], [12], [-6], [8], [2], [2], [-18], [10], [-6], [18], [-10], [-10], [6], [-16], [-6], [-12], [-10], [12], [-18], [-6], [14], [-6], [12], [10], [20], [22], [-2], [8], [-8], [2], [-14], [18], [-22], [-24], [14], [-16], [-18], [-26], [-26], [-28], [14], [22], [-22], [2], [-22], [12], [10], [-30], [-16], [-2], [6], [26], [-6], [-24], [26], [-6], [-20], [-6], [6], [-10], [18], [-10], [-4], [-10], [-36], [-18], [-8], [-30], [-34], [-38], [34], [-26], [-10], [24], [-26], [-32], [36], [-26], [-30], [42], [-26], [34], [2], [-26], [38], [22], [-40], [-38], [-18], [2], [48], [-10], [22], [22], [34], [-4], [-12], [30], [26], [18], [-18], [-36], [-26], [-10], [2], [30], [8], [30], [-14], [8], [2], [6], [28], [10], [4], [-10], [-6], [-14], [42], [-14], [-38], [-6], [40], [-10], [38], [18], [-42], [0], [-2], [-42], [4], [-54], [26], [14], [34], [22], [10], [52], [32], [54], [-46], [-14], [22], [14], [18], [0], [6], [18], [-28], [-2], [-6], [-58], [12], [-30], [60], [-14], [-32], [-6], [40], [-6], [-6], [-50], [-62], [8], [2], [22], [-18], [50], [2], [-42], [54], [60], [-34], [18], [-32], [62], [54], [-10], [-50], [-10], [2], [30], [2], [28], [-62], [58], [-60], [-18], [0], [62], [10], [0], [42], [-42], [-26], [-38], [32], [6], [-14], [50], [-58], [30], [22], [-32], [42], [70], [2], [22], [-10], [-32], [-2], [40], [14], [20], [-20], [54], [-22], [-30], [22], [22], [60], [-60], [78], [32], [-74], [34], [30], [-24], [18], [-64], [-44], [14], [-34], [18], [-46], [-58], [22], [4], [-70], [2], [42], [-62], [34], [-6], [-30], [-46], [-22], [-44], [50], [58], [22], [-66], [50], [-46], [-78], [-32], [54], [70], [-18], [-30], [6], [0], [10], [12], [66], [-70], [-2], [12], [26], [-22], [32], [-50], [14], [-10], [-66], [56], [26], [46], [36], [62], [-4], [30], [58], [58], [-72], [-2], [48], [-50], [-14], [2], [-24], [30], [70], [18], [18], [-26], [74], [-62], [-12], [-60], [58], [6], [-16], [26], [78], [-46], [26], [-66], [52], [30], [-82], [78], [-18], [-6], [56], [42], [-80], [86], [-26], [-66], [-42], [10], [-30], [-30], [-2], [-36], [-14], [-60], [-58], [34], [-60], [70], [56], [-2], [14], [-14], [74], [-10], [-24], [56], [74], [26], [78], [-6], [22], [-60], [94], [54], [22], [-30], [54], [-32], [76], [-34], [14], [20], [38], [-60], [76], [34], [-90], [-34], [-30], [82], [26], [-30], [-60], [66], [-60], [70], [74], [-42], [-2], [10], [-12], [-10], [76], [-42], [-24], [2], [-40], [-42], [-10], [-6], [38], [-42], [-66], [20], [-50], [2], [94], [-36], [38], [14], [-50], [-64], [-54], [2], [80], [-42], [-22], [42], [38], [-30], [58], [52], [38], [-54], [6], [-2], [40], [-24]]
