
# q-expansion of newform 8034.2.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 = 8034
weight = 2
poly_data = [-16, -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 = [[-1, 0], [-1, 0], [0, 0], [1, 0], [3, 0], [1, 0], [2, -1], [0, 0], [-3, 1], [0, 0], [4, 0], [-3, -1], [-1, 1], [-1, 1], [-5, 1], [2, -1], [0, -2], [-7, 1], [-8, -1], [-8, 2], [-6, 3], [0, -2], [0, 2], [-4, -2], [0, -4], [-5, 2], [1, 0], [4, 1], [-15, 1], [5, -1], [15, 0], [-12, 0], [9, -3], [-12, 1], [9, 0], [-12, 0], [8, 1], [3, 3], [2, -1], [21, 0], [-7, -2], [-2, -1], [4, -1], [-9, 3], [-4, 2], [-5, 0], [-3, -1], [-1, 4], [-12, 3], [4, -6], [13, 1], [8, -1], [5, 0], [-4, 4], [2, 2], [18, 1], [-18, -2], [-18, 0], [-20, 2], [6, 0], [15, -3], [10, -2], [-20, -1], [-3, 3], [-3, -3], [-18, -3], [6, -7], [-11, 2], [21, -1], [-2, 8], [10, -7], [-6, 3], [-10, 0], [9, 5], [6, 5], [3, -9], [-11, -4], [18, 4], [25, 1], [-4, 4], [9, -8], [-24, -3], [6, 2], [16, 0], [1, 0], [0, 8], [6, 5], [-1, -9], [19, -1], [-28, -2], [-25, 2], [-1, -5], [-6, -6], [-8, -5], [-18, 6], [11, -7], [9, 2], [-15, 3], [2, -9], [30, 1], [6, -6], [-2, 10], [-36, 2], [-9, -5], [-8, 0], [-3, 0], [0, 2], [18, -3], [6, 3], [-18, 6], [-42, 0], [-4, 3], [0, -2], [-4, 0], [-11, 8], [-32, 5], [-23, 3], [-15, 3], [-21, 3], [0, 5], [-6, -4], [-15, -2], [14, 2], [-24, -1], [8, 4], [-10, 2], [28, -6], [-18, -1], [-16, 3], [-33, -1], [1, -7], [8, 2], [-22, 6], [27, -3], [2, -8], [11, 6], [-11, -2], [-3, 3], [-34, 2], [-2, 10], [-6, -6], [-7, -10], [-19, -5], [5, -1], [12, -10], [26, 2], [-8, 9], [12, -3], [23, -9], [13, -5], [-27, 3], [-38, 2], [-2, -3], [-1, 11], [6, 3], [-2, -2], [-29, -6], [-19, -1], [4, 8], [6, -5], [7, -7], [2, -5], [12, -8], [0, -4], [17, 1], [-18, 3], [-12, 10], [-14, 0], [-32, -4], [19, 4], [6, 8], [-11, -7], [0, 4], [23, -6], [-12, 6], [16, -11], [16, -12], [4, -8], [-12, 0], [20, -3], [-48, 0], [3, -6], [5, -9], [8, -2], [-8, 14], [8, -2], [8, 1], [31, -1], [42, 1], [-4, 5], [6, 12], [12, -2], [8, -12], [-18, 12], [20, -10], [-8, -7], [48, 0], [-10, 1], [-16, 2], [6, 3], [-22, 2], [-38, 6], [28, -1], [-2, 11], [-8, -4], [44, 2], [-6, 16], [34, 4], [56, 2], [-42, 6], [19, 2], [45, 1], [-9, 9], [12, 4], [-42, -3], [10, 0], [26, -12], [15, -11], [-48, 0], [59, -1], [4, -3], [5, 3], [-28, 5], [-24, 12], [8, -14], [-38, -1], [-27, -7], [-19, 11], [-16, -3], [6, 0], [48, -6], [-18, 11], [-16, -3], [17, 5], [30, 7], [-9, 13], [9, 0], [30, -2], [-52, -2], [6, -15], [-2, 2], [12, 0], [-3, -14], [50, 0], [-14, 10], [-28, 14], [30, 0], [38, 8], [-9, -5], [27, -5], [40, 5], [15, -7], [-56, -4], [70, 0], [-2, -8], [35, 2], [13, 3], [60, 4], [14, -16], [2, 0], [-12, -6], [63, 0], [39, 3], [-2, 9], [1, 7], [-35, 9], [-30, -9], [28, -11], [34, -6], [-6, -11], [43, 3], [26, 7], [52, 6], [-34, 0], [39, 6], [45, 0], [-8, -2], [2, 3], [-18, -10], [-19, -1], [-39, -9], [-4, -6], [0, 5], [-46, 4], [-27, 7], [32, 10], [-2, -15], [15, -12], [8, -11], [42, 6], [18, 0], [-34, 4], [43, 5], [-4, 7], [19, -15], [7, 7], [4, 2], [40, 10], [40, 10], [12, 0], [-33, -2], [12, -4], [-36, 3], [-11, -1], [-15, 15], [-2, -10], [6, -8], [-35, -1], [-30, 3], [-30, -4], [-6, 6], [22, 5], [18, -6], [-12, 12], [-51, 3], [-39, 2], [-13, 14], [-16, 2], [78, -3], [-27, 15], [-23, -3], [69, -2], [-16, 4], [16, -9], [26, -14], [-24, -6], [-33, 9], [-56, 5], [-10, -7], [-42, 10], [36, 9], [58, -8], [-34, -6], [-20, 8], [-16, 12], [36, -2], [50, 4], [-22, -4], [14, -12], [57, 6], [42, 4], [10, 2], [36, 10], [-8, -1], [6, -6], [-32, -2], [12, 6], [5, 9], [29, -1], [-26, 10], [21, -21], [-70, 4], [-4, 8], [46, -4], [30, -16], [-39, 15], [-26, 2], [-84, -3], [-15, -15], [-17, -5], [30, 4], [16, -5], [68, -4], [41, -4], [-60, -4], [9, 0], [74, 2], [-12, -18], [-27, 9], [40, -5], [-12, -12], [0, -12], [-63, 1], [26, 6], [-10, 0], [-20, 10], [-47, 1], [-84, 0], [-28, 17], [-18, 0], [42, 13], [-1, 1], [-36, 3], [21, -7], [68, 0], [49, -11], [-28, 0], [-30, 12], [-34, -3], [-52, -10], [51, -6], [-22, -7], [-12, -9], [80, 3], [-14, -2], [58, 2], [-3, -13], [70, 2], [36, 6], [1, -11], [-14, -12], [-75, 5], [14, 0], [-6, 19], [-13, 2], [-40, 0], [42, -1], [-50, -5], [8, -8], [-22, 14], [6, 5], [0, 6], [-6, 10], [-13, 1], [-2, 15], [31, -2], [-12, -13], [-75, 6], [15, 9], [-5, 1], [-16, 10], [-4, 7], [-81, 5], [29, 10], [-45, -9], [-14, 2], [27, 0], [-66, 5], [46, 6], [42, -15]]
