
# q-expansion of newform 6720.2.a.m, downloaded from the LMFDB on 24 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], [-4], [2], [2], [-4], [0], [10], [0], [-6], [-6], [-4], [8], [-6], [-4], [10], [4], [16], [-14], [-8], [-4], [10], [10], [10], [-8], [-4], [-14], [-6], [-16], [-12], [18], [4], [2], [-16], [-14], [4], [0], [18], [4], [2], [-24], [2], [-6], [-24], [-4], [16], [-4], [2], [18], [8], [-14], [-20], [2], [0], [18], [0], [10], [-22], [20], [-6], [-4], [-24], [-30], [-14], [-12], [18], [-36], [26], [-14], [-32], [-32], [10], [20], [-24], [-30], [18], [-30], [-22], [4], [-22], [8], [-38], [8], [28], [-14], [-6], [2], [32], [-20], [0], [40], [-4], [-36], [0], [34], [-6], [4], [-46], [-28], [18], [-4], [-22], [20], [-38], [36], [34], [-32], [10], [-32], [-6], [18], [20], [16], [-30], [-4], [16], [-14], [-28], [-46], [18], [-6], [-20], [-36], [10], [-38], [0], [24], [34], [-4], [16], [-40], [-22], [42], [2], [-6], [44], [-30], [42], [-44], [2], [24], [-52], [42], [56], [10], [26], [52], [8], [18], [2], [20], [32], [-52], [8], [32], [50], [50], [18], [-12], [18], [-56], [44], [-54], [48], [-8], [-22], [-14], [10], [-4], [10], [-32], [-22], [16], [10], [-12], [-30], [24], [-22], [48], [36], [-38], [26], [24], [58], [34], [-36], [42], [48], [26], [-52], [12], [18], [-44], [-38], [18], [2], [-30], [0], [58], [-32], [26], [-46], [-36], [-30], [16], [52], [-38], [52], [-62], [42], [40], [-12], [-24], [-22], [32], [-30], [0], [34], [-6], [40], [66], [32], [60], [-6], [-6], [24], [-72], [-20], [-46], [-20], [56], [-70], [36], [-72], [66], [-6], [-20], [-40], [36], [-44], [24], [-46], [-46], [-24], [-48], [-12], [4], [56], [-62], [18], [48], [-54], [-46], [4], [10], [52], [26], [2], [-32], [36], [-46], [2], [50], [28], [-6], [42], [44], [-70], [26], [52], [82], [24], [-22], [8], [44], [66], [74], [-12], [24], [-16], [-48], [18], [52], [8], [-62], [74], [40], [18], [-38], [-36], [-30], [12], [18], [-14], [48], [-22], [-36], [44], [34], [-78], [-24], [36], [-60], [34], [60], [-54], [0], [-70], [24], [-78], [-14], [20], [-32], [-70], [-60], [-72], [58], [18], [-68], [-22], [18], [16], [-78], [-78], [-20], [36], [8], [-54], [-62], [-30], [48], [28], [4], [20], [50], [-62], [-70], [-64], [-54], [50], [90], [64], [18], [36], [-30], [12], [32], [42], [-84], [-70], [10], [-32], [-22], [74], [64], [-20], [42], [64], [58], [-38], [-24], [60], [-20], [-6], [-78], [8], [-38], [20], [-92], [72], [50], [24], [-30], [68], [-8], [-6], [66], [18], [-46], [66], [40], [-6], [-68], [0], [-24], [10], [60], [56], [2], [-6], [28], [-68], [-32], [58], [16], [26], [-28], [-30], [-38], [-6], [32], [-6], [-70], [56], [50], [66], [60], [-76], [58], [-38], [44], [28], [-22], [-14], [-24], [8], [-94], [-48], [10], [10], [-40], [-36], [50], [-14], [36], [-6], [-92], [-24], [-22], [-60], [-12], [-8], [66], [82], [-22], [-38], [44]]
