
# q-expansion of newform 2240.2.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 = 2240
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], [1], [0], [2], [2], [-8], [8], [-6], [0], [2], [-6], [-8], [-8], [2], [-8], [2], [-8], [0], [10], [-16], [-16], [10], [-14], [-6], [8], [-8], [10], [-14], [-8], [8], [10], [8], [18], [-16], [-14], [-8], [-8], [2], [0], [-6], [0], [18], [18], [-16], [16], [-8], [0], [-22], [-6], [0], [-14], [24], [18], [24], [-30], [-16], [2], [10], [-16], [-6], [0], [16], [-6], [26], [32], [-14], [8], [18], [18], [32], [-24], [34], [16], [8], [18], [2], [-30], [10], [-24], [2], [16], [-30], [-32], [8], [-14], [10], [2], [-8], [-32], [16], [24], [16], [-32], [24], [18], [-38], [-16], [10], [-24], [42], [0], [-6], [32], [18], [-16], [34], [-16], [10], [8], [2], [-6], [24], [32], [18], [32], [24], [-6], [-48], [10], [-14], [10], [-24], [40], [-38], [-14], [48], [-8], [2], [-16], [40], [16], [50], [10], [-30], [-6], [-32], [-46], [10], [-56], [18], [-40], [24], [2], [16], [-6], [26], [-24], [-40], [-38], [18], [40], [-40], [-40], [-16], [-48], [-30], [-54], [50], [8], [-6], [-40], [-24], [-30], [-56], [0], [-38], [-30], [-6], [0], [-46], [32], [26], [16], [42], [-48], [18], [-8], [18], [24], [-8], [-30], [-22], [-56], [-38], [26], [48], [42], [0], [-14], [-24], [16], [50], [24], [-54], [50], [-54], [-46], [-40], [58], [-32], [42], [66], [56], [-30], [-16], [-8], [-6], [-40], [34], [-22], [-56], [0], [0], [-38], [40], [-46], [-24], [-22], [50], [-64], [-14], [-24], [-32], [2], [-70], [64], [72], [-64], [26], [-24], [0], [-6], [-16], [8], [-46], [2], [32], [48], [-8], [-24], [56], [-70], [66], [-16], [72], [8], [16], [-72], [58], [-30], [-24], [26], [-46], [-64], [-30], [16], [-6], [-6], [40], [72], [26], [2], [18], [40], [-6], [74], [-24], [34], [2], [-24], [-38], [64], [-62], [56], [40], [42], [58], [8], [24], [-64], [-40], [-38], [-64], [-48], [34], [34], [-80], [66], [-6], [32], [74], [40], [74], [50], [48], [26], [72], [80], [-22], [-6], [-16], [24], [-32], [50], [40], [50], [32], [-14], [24], [66], [-78], [56], [40], [-38], [72], [-32], [-46], [-46], [56], [90], [-62], [56], [-22], [-14], [48], [-16], [-8], [-30], [-70], [10], [48], [-32], [-16], [0], [-38], [-78], [-6], [-40], [-30], [-6], [74], [-64], [-86], [16], [74], [8], [-48], [42], [-40], [-54], [-70], [-56], [-46], [26], [48], [8], [2], [-88], [82], [-38], [24], [64], [32], [-70], [2], [40], [58], [48], [-24], [56], [-46], [-16], [-6], [24], [-48], [82], [18], [-70], [50], [74], [72], [-94], [40], [24], [-32], [-22], [-88], [-72], [2], [-38], [64], [80], [-80], [26], [80], [90], [80], [-30], [-30], [-46], [8], [-6], [-86], [-32], [10], [-30], [32], [8], [34], [18], [-104], [-80], [58], [66], [16], [-8], [2], [88], [-6], [-102], [-24], [-24], [74], [-46], [-72], [-54], [-40], [48]]
