
# q-expansion of newform 4046.2.a.f, downloaded from the LMFDB on 23 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 = 4046
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 = [[-1], [2], [0], [-1], [0], [-4], [0], [2], [0], [6], [4], [-2], [-6], [8], [-12], [6], [-6], [-8], [-4], [0], [-2], [-8], [-6], [-6], [10], [0], [-4], [-12], [-2], [-6], [-16], [-18], [18], [-14], [-18], [8], [-4], [16], [12], [12], [-12], [-20], [24], [-14], [18], [-20], [4], [8], [-18], [-4], [6], [24], [10], [-18], [18], [0], [12], [-16], [10], [-6], [22], [24], [2], [24], [10], [-6], [8], [-14], [24], [-28], [18], [-24], [-8], [14], [16], [36], [18], [-20], [18], [14], [-6], [-10], [-24], [-34], [-8], [-12], [-18], [-10], [12], [32], [-6], [36], [16], [-12], [4], [0], [36], [-6], [2], [-38], [-8], [6], [30], [6], [-32], [2], [-42], [-6], [-24], [-26], [-32], [2], [-6], [-26], [-16], [18], [-14], [-12], [-18], [-24], [-40], [-26], [12], [12], [46], [18], [46], [-12], [44], [-40], [-16], [-24], [40], [2], [-18], [14], [24], [22], [-12], [-6], [-2], [-6], [40], [36], [56], [-12], [-44], [18], [14], [-24], [22], [54], [20], [36], [-44], [-48], [56], [-6], [2], [24], [-24], [-54], [32], [-6], [-6], [36], [16], [-8], [34], [36], [36], [-4], [0], [26], [-4], [30], [-44], [30], [-16], [8], [8], [-30], [22], [-6], [48], [36], [34], [-46], [-50], [12], [-2], [60], [20], [60], [12], [-66], [-14], [-26], [6], [-24], [-30], [28], [-40], [26], [18], [24], [20], [0], [-30], [-58], [34], [-48], [-8], [42], [24], [-62], [32], [-18], [24], [-30], [-22], [40], [18], [-56], [-30], [26], [-42], [-24], [-64], [48], [50], [-38], [32], [-6], [-46], [0], [58], [-30], [12], [0], [72], [2], [32], [-34], [-42], [12], [16], [-66], [-40], [0], [-58], [-66], [48], [-26], [-72], [-60], [-74], [10], [0], [-34], [-8], [48], [-32], [-56], [42], [-10], [54], [-6], [4], [-6], [76], [56], [-46], [32], [50], [-16], [72], [-10], [-34], [42], [-24], [40], [-12], [8], [-50], [48], [-2], [-78], [44], [18], [18], [-42], [54], [30], [-62], [-84], [-64], [-60], [-30], [-82], [14], [-78], [16], [48], [40], [70], [-72], [-32], [0], [38], [-24], [30], [-42], [32], [0], [-70], [30], [-48], [-26], [-30], [10], [-50], [-72], [56], [-54], [-34], [88], [46], [48], [54], [10], [78], [28], [54], [-92], [-18], [2], [-18], [22], [20], [-82], [-42], [12], [-64], [-30], [72], [-20], [80], [24], [24], [14], [70], [-66], [28], [14], [18], [-36], [-42], [-18], [48], [-62], [30], [-48], [-60], [2], [-10], [-84], [-16], [-38], [36], [34], [0], [30], [-40], [-38], [78], [-72], [-70], [-42], [-82], [-36], [6], [-56], [-18], [-14], [-24], [8], [32], [-20], [-12], [-62], [48], [60], [-10], [24], [94], [-52], [-42], [4], [66], [40], [-78], [32], [18], [60], [-92], [62], [30], [86], [66], [-26], [42], [0], [-44], [-22], [-12], [24], [-44], [-54], [0], [18], [-56], [96], [6], [-38], [-96], [-24], [54], [38], [-36]]
