
# q-expansion of newform 6080.2.a.m, downloaded from the LMFDB on 25 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], [0], [1], [-2], [4], [4], [6], [-1], [-2], [6], [-8], [-4], [6], [6], [6], [-8], [12], [-6], [0], [0], [-10], [-8], [-14], [14], [16], [14], [-20], [4], [2], [0], [12], [8], [6], [8], [10], [-16], [18], [14], [16], [8], [-4], [10], [12], [16], [22], [-24], [-4], [4], [4], [6], [-14], [16], [6], [0], [-20], [14], [-14], [-24], [26], [30], [-22], [24], [-4], [-8], [-22], [12], [-20], [24], [6], [-14], [-2], [4], [-10], [8], [-12], [32], [-26], [10], [-30], [10], [4], [-10], [24], [-16], [8], [-30], [2], [10], [18], [22], [14], [-12], [-32], [24], [-24], [6], [-26], [-30], [36], [-34], [-20], [18], [4], [-30], [-36], [2], [6], [42], [16], [30], [16], [2], [-10], [48], [-40], [14], [34], [-6], [10], [36], [-42], [28], [0], [44], [36], [18], [10], [-12], [14], [-42], [-44], [24], [40], [-42], [-10], [-2], [-36], [-12], [-16], [10], [-28], [22], [50], [-24], [-14], [-40], [-18], [-52], [28], [-40], [24], [10], [6], [-4], [-44], [24], [-32], [-42], [38], [18], [-18], [60], [-14], [-4], [4], [0], [-48], [-22], [-10], [10], [60], [42], [8], [-46], [-24], [-42], [16], [-18], [-40], [34], [-14], [-36], [-44], [20], [-26], [-6], [32], [-52], [-14], [-36], [-52], [2], [20], [-34], [-30], [-64], [-66], [14], [-6], [26], [22], [48], [-28], [-2], [-40], [58], [4], [-64], [34], [20], [34], [-30], [6], [16], [0], [-50], [-58], [-38], [68], [-34], [62], [64], [-50], [38], [-12], [6], [-44], [0], [28], [8], [14], [44], [32], [22], [66], [30], [-14], [-58], [48], [56], [52], [-64], [-30], [6], [24], [-48], [-26], [-68], [-52], [66], [62], [50], [-6], [-38], [-18], [-4], [74], [56], [72], [-10], [-4], [-8], [38], [56], [-78], [-28], [-42], [-50], [8], [-50], [-38], [72], [-42], [20], [40], [10], [-70], [-42], [-66], [52], [-32], [-24], [-70], [-30], [70], [8], [-38], [76], [-64], [-22], [-18], [74], [0], [44], [-24], [-46], [-64], [-58], [52], [34], [78], [0], [-36], [-8], [-56], [-16], [-72], [30], [0], [14], [-38], [54], [-30], [-28], [-22], [-34], [-64], [0], [-46], [-18], [-28], [22], [-42], [-56], [62], [62], [84], [60], [68], [-30], [10], [32], [-20], [-66], [-44], [-54], [-14], [-44], [22], [2], [-8], [-58], [-54], [-40], [24], [-36], [38], [8], [16], [-30], [-36], [76], [-6], [-88], [-50], [-4], [0], [0], [-26], [88], [-38], [70], [32], [60], [-66], [40], [-54], [80], [38], [12], [28], [-38], [10], [-68], [-30], [84], [36], [10], [-10], [-8], [10], [-6], [30], [42], [-28], [44], [-72], [42], [-66], [48], [-70], [40], [36], [-58], [8], [28], [8], [-34], [4], [-42], [-62], [-30], [8], [-52], [-14], [52], [-6], [-54], [4], [28], [68], [46], [96], [52], [-34], [-46], [-8], [68], [-2], [24], [10], [-24], [-2], [12], [16], [-64], [100], [-102], [16], [12]]
