
# q-expansion of newform 6720.2.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 = 6720
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], [-2], [4], [-2], [6], [-4], [10], [2], [2], [-10], [-4], [-8], [-4], [-4], [2], [-4], [6], [0], [-4], [4], [-6], [4], [6], [8], [-20], [14], [12], [8], [0], [16], [22], [6], [-16], [0], [20], [8], [6], [-18], [6], [18], [-22], [24], [10], [-20], [-12], [-20], [2], [-12], [6], [14], [4], [2], [8], [30], [14], [10], [30], [-4], [-2], [8], [0], [28], [28], [12], [-6], [24], [34], [6], [2], [-28], [-18], [8], [8], [6], [-20], [10], [10], [0], [-6], [-10], [28], [-6], [-24], [-30], [-2], [26], [-24], [12], [24], [24], [-2], [-16], [-16], [-26], [-10], [32], [18], [20], [-36], [-4], [18], [40], [28], [12], [6], [-30], [14], [36], [-38], [-20], [46], [44], [38], [40], [-24], [-16], [-30], [42], [6], [6], [-24], [34], [18], [10], [36], [-44], [-44], [-16], [40], [12], [-34], [42], [-18], [-42], [-32], [46], [-2], [38], [-46], [-16], [-36], [6], [-52], [-36], [18], [50], [-12], [-22], [10], [-20], [40], [-20], [34], [44], [-30], [8], [34], [16], [-28], [40], [12], [-28], [24], [-52], [28], [-2], [-26], [30], [-42], [6], [-54], [-26], [-10], [28], [-18], [-20], [62], [-8], [12], [14], [46], [24], [-42], [50], [24], [-50], [-12], [-52], [36], [-64], [2], [-8], [6], [-18], [-26], [-30], [16], [-30], [50], [-16], [14], [-60], [-10], [46], [36], [-18], [-22], [22], [-2], [8], [36], [-44], [-26], [40], [-30], [48], [8], [-38], [-62], [30], [-48], [52], [38], [-46], [-14], [48], [-50], [54], [6], [-16], [14], [64], [-56], [-6], [-48], [-58], [-68], [12], [-18], [0], [46], [-22], [-4], [52], [24], [60], [36], [2], [34], [-52], [26], [14], [-10], [-10], [28], [62], [-60], [-72], [16], [2], [-16], [-74], [34], [10], [6], [36], [-24], [30], [20], [-44], [-28], [-68], [68], [-36], [2], [10], [76], [56], [-20], [-8], [62], [16], [66], [-10], [28], [-46], [-30], [26], [12], [-12], [40], [14], [-22], [-82], [-30], [-44], [-88], [64], [-48], [16], [56], [-80], [-62], [68], [-22], [-50], [-18], [48], [-30], [-62], [-44], [-4], [-34], [8], [-78], [-52], [82], [18], [-42], [-62], [8], [48], [-74], [56], [44], [-28], [28], [86], [24], [6], [-4], [-44], [-4], [34], [-6], [-26], [28], [-74], [-12], [70], [72], [64], [66], [50], [-36], [84], [-2], [14], [38], [18], [0], [-74], [-42], [-12], [24], [36], [-48], [-10], [-18], [24], [18], [-64], [46], [58], [64], [58], [-82], [74], [-56], [-50], [30], [2], [-80], [82], [52], [94], [72], [-2], [-76], [64], [40], [-34], [24], [-4], [4], [20], [16], [86], [-70], [42], [-16], [-18], [-38], [-10], [-30], [-40], [-62], [30], [12], [-24], [2], [26], [14], [34], [-54], [-44], [-20], [-48], [28], [64], [-68], [2], [78], [30], [-12], [30], [0], [-26], [76], [24], [-12], [-84], [-2], [92], [34], [42], [-36], [-6], [-54], [-108], [-64], [-72], [-18], [10], [62], [-84]]
