
# q-expansion of newform 483.2.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 = 483
weight = 2
poly_data = [-3, -1, 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], [-3, 1], [-1, 0], [-5, 0], [0, 1], [-1, -2], [3, -2], [1, 0], [-3, 4], [3, 0], [-9, 0], [-3, -4], [-6, 5], [-4, -2], [-1, -5], [-3, 3], [-8, 3], [-5, -3], [-6, 3], [7, -4], [-1, 0], [1, 2], [4, 3], [-13, -2], [7, 1], [-4, 0], [-4, 5], [3, -3], [3, 1], [4, -1], [15, -2], [3, 6], [9, 3], [4, 2], [12, 2], [10, -6], [7, -9], [-11, 6], [-9, -4], [-5, 1], [7, -6], [0, -2], [-14, 10], [-14, 3], [1, 5], [-9, -8], [13, 3], [5, -11], [-9, 5], [11, -11], [-21, 3], [-11, 10], [10, 6], [4, 10], [15, -2], [-6, 3], [-1, 2], [-10, -1], [14, 6], [15, 1], [12, -16], [9, 4], [2, -5], [-8, 4], [-19, -5], [-10, 12], [-18, 1], [-13, 12], [-17, 3], [-11, 0], [-1, -15], [10, 5], [7, 4], [-4, -8], [-9, 6], [-9, -12], [-32, 2], [5, -8], [1, 10], [-1, 7], [12, -1], [12, -9], [2, -10], [11, 4], [4, -16], [-15, -9], [-27, 7], [-13, 9], [-5, -6], [19, 2], [-9, -4], [13, -6], [12, -13], [25, 3], [-3, 9], [8, -18], [-6, 12], [18, -8], [8, -6], [13, -9], [-18, 10], [-18, -11], [20, -4], [1, 4], [-18, -8], [0, -17], [14, 10], [-28, -5], [11, 13], [-36, 5], [-11, 6], [-6, 9], [26, -7], [-35, 10], [24, -15], [-14, 25], [-9, 11], [19, 5], [15, 12], [7, 6], [1, -12], [-2, 17], [17, -8], [-20, -5], [9, -15], [-22, -11], [5, 4], [13, -26], [-22, -2], [-25, -8], [19, 5], [11, -1], [-31, 12], [12, 0], [14, -18], [-5, 16], [-7, 17], [-16, -10], [-4, 13], [-15, 2], [-24, 2], [-5, 5], [-3, -3], [11, -18], [36, 1], [-26, -14], [-18, -14], [34, 4], [-38, 14], [-24, 4], [-30, -6], [-30, 5], [18, -13], [31, -17], [8, -14], [-15, 2], [15, 11], [-23, 8], [36, -16], [14, -12], [-26, -11], [-14, 24], [-42, 3], [-8, 17], [-9, -6], [24, 5], [23, -10]]
