
# q-expansion of newform 2940.2.a.f, 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 = 2940
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], [0], [6], [4], [-6], [-2], [0], [6], [10], [2], [6], [-4], [0], [-12], [0], [-14], [-4], [6], [4], [-16], [12], [-6], [16], [-6], [16], [0], [2], [0], [8], [12], [-12], [22], [18], [-4], [-8], [20], [0], [-6], [6], [-2], [18], [26], [-24], [10], [20], [4], [-12], [-14], [-24], [6], [10], [-24], [-18], [-12], [-6], [-2], [26], [6], [4], [18], [16], [24], [-8], [-12], [-4], [-22], [12], [10], [-30], [-30], [4], [14], [-16], [0], [18], [4], [-6], [22], [-12], [-10], [-18], [-8], [10], [12], [-6], [-34], [-18], [-16], [12], [-24], [32], [-18], [-40], [0], [-30], [6], [-32], [38], [-28], [12], [12], [-30], [32], [-32], [-36], [-6], [-30], [10], [4], [2], [0], [-2], [-16], [30], [-32], [-48], [48], [42], [10], [14], [18], [12], [34], [-42], [38], [36], [28], [-20], [8], [-36], [32], [-10], [-30], [10], [18], [16], [-30], [30], [46], [6], [-16], [-48], [-26], [12], [4], [-42], [-14], [24], [-22], [-30], [-52], [0], [20], [-6], [-16], [-6], [-20], [-18], [-36], [48], [56], [0], [48], [0], [-40], [28], [14], [18], [-18], [-50], [54], [-46], [-50], [30], [20], [-30], [-44], [22], [8], [0], [14], [-54], [-12], [-54], [50], [-8], [-34], [-12], [16], [24], [-16], [54], [-12], [-54], [-34], [62], [-18], [-48], [-18], [-38], [40], [10], [48], [-30], [-50], [24], [-42], [-14], [38], [-54], [-16], [12], [12], [-14], [8], [18], [60], [-24], [38], [58], [6], [-16], [-12], [-22], [-66], [42], [-8], [-42], [-58], [-74], [20], [6], [-8], [0], [-74], [-24], [-42], [-60], [24], [-26], [-32], [-22], [6], [36], [28], [-36], [-52], [24], [-22], [18], [-24], [-26], [42], [-18], [-22], [4], [42], [16], [8], [-60], [34], [16], [66], [58], [6], [54], [-4], [24], [22], [-52], [40], [-40], [16], [28], [48], [-34], [-70], [-24], [48], [-16], [24], [46], [64], [18], [38], [-36], [-14], [-54], [54], [-84], [72], [60], [-10], [-18], [-50], [6], [0], [-56], [76], [48], [-4], [36], [32], [74], [-48], [-86], [30], [-10], [48], [30], [-54], [44], [-48], [82], [60], [66], [-80], [-30], [34], [62], [54], [-16], [-60], [-38], [-16], [-44], [-48], [36], [-46], [72], [46], [12], [-52], [-12], [-10], [-66], [-38], [52], [86], [48], [-30], [44], [-72], [90], [34], [-52], [-36], [18], [-50], [-34], [-66], [-80], [50], [66], [12], [-36], [0], [-12], [38], [54], [12], [-6], [-32], [-82], [30], [-40], [26], [-18], [-38], [-36], [-6], [-74], [98], [12], [-30], [-8], [78], [-44], [-78], [-48], [-16], [-12], [22], [72], [80], [76], [44], [0], [-10], [78], [90], [-8], [-18], [2], [-2], [42], [-40], [-42], [94], [-24], [56], [-18], [6], [22], [2], [-6], [4], [0], [4], [-12], [-12], [20], [-22], [-54], [-90], [28], [-78], [-96], [90], [28], [-36], [-72], [-56], [66], [-24], [-6], [-38], [60]]
