
# q-expansion of newform 8043.2.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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 8043
weight = 2
poly_data = [1, -3, -1, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0], 1], [[-2, 0, 1], 1], [[2, 2, -1], 1]]

hecke_ring_character_values = None
aps_data = [[0, -1, 0], [1, 0, 0], [-1, 1, 0], [-1, 0, 0], [1, 0, 1], [5, -1, 0], [-4, 1, 1], [0, 2, 2], [2, 3, -1], [2, 1, 3], [-6, -1, -1], [10, 0, 0], [5, 1, -2], [2, 2, 0], [2, 0, 2], [4, 1, 1], [4, 2, 2], [-7, -3, 2], [-10, 4, 2], [2, -3, -3], [2, -4, 0], [-5, 2, 1], [8, -4, 0], [1, -3, -2], [7, -1, -2], [4, -5, -1], [0, 0, 0], [-3, 2, 3], [6, -4, -4], [10, -7, -1], [-9, -4, -1], [0, 0, 4], [6, -4, 0], [6, -3, 1], [-2, -3, 3], [-21, 0, -1], [1, 9, -2], [5, -2, -5], [0, -4, 0], [2, 3, 1], [-1, 0, -5], [7, 3, 2], [7, 6, 1], [-6, -5, 5], [8, -3, 1], [4, 1, -5], [-11, -2, -5], [-6, 5, 1], [2, 5, -3], [2, -6, 2], [0, -1, 3], [3, 0, -5], [17, -3, 2], [-4, -6, -2], [-3, 3, 0], [6, -3, 1], [11, -7, -4], [24, -3, -1], [-18, 0, 0], [4, -3, 1], [0, 9, -5], [8, 7, -5], [-10, 5, -3], [-4, -2, 2], [2, -10, -6], [18, 1, 3], [-6, -4, -6], [20, 4, -2], [-5, -2, -11], [-5, 3, -2], [0, 1, 1], [11, -2, 5], [-2, 3, 11], [-14, -4, 0], [1, -4, -7], [-1, 0, 0], [-14, 16, 4], [10, -6, 6], [18, -3, -5], [1, 7, -4], [16, -4, -8], [18, -4, 0], [-8, -5, 1], [2, -4, 0], [-6, -1, 7], [-7, 10, -1], [10, 1, -5], [10, -4, -4], [7, -15, 0], [-9, 6, 1], [18, 7, -5], [4, 0, -4], [7, 2, -3], [-16, 3, -3], [14, -6, 0], [-2, -16, 2], [0, -3, -3], [-6, 5, -9], [-10, -13, 3], [-4, 2, -8], [9, 4, -3], [14, -4, -4], [20, -2, -2], [-6, 0, 8], [-9, -2, -3], [-35, -1, 0], [-10, 1, 5], [-17, -9, 6], [-16, -3, 7], [15, 1, -8], [-14, 3, -5], [6, 8, -8], [-32, 5, 1], [-12, 11, 1], [-8, -4, -8], [-6, 6, -6], [-22, 5, 5], [16, 2, 2], [2, 9, -13], [-2, -7, -3], [0, 2, -4], [-36, 2, 4], [16, 3, -5], [-4, 9, -5], [24, -3, 3], [-6, -7, 7], [-22, -4, 0], [-12, 2, 2], [-18, -3, 5], [-2, -4, -12], [11, -2, 13], [-19, 2, -5], [-10, 4, -10], [-10, 12, 12], [4, -3, -7], [-8, -2, 12], [30, 1, -1], [-2, -19, 1], [-6, 13, 11], [18, 5, 3], [-14, 21, 5], [-26, -3, -5], [8, 6, -2], [-9, 8, -1], [30, 2, -6], [20, 10, -6], [-2, -2, 10], [7, -3, -12], [14, -7, 5], [3, 8, -5], [-10, -6, 6], [3, -9, -6], [15, -4, 3], [-20, -2, -10], [-11, 2, 11], [11, 0, 7], [-2, -16, 6], [-9, 7, 14], [24, 4, 6], [-10, -3, 15], [-15, -4, 5], [-22, -9, 9], [2, 4, 18], [34, -3, -3], [-8, 11, -9], [2, -7, -3], [22, -12, -2], [11, -9, 2], [30, 4, -8], [1, 1, 10], [11, 4, -13], [-15, 9, -2], [-24, -7, -9], [4, 6, -8], [14, -1, -9], [-1, -17, -2], [-23, -18, 11], [-2, -4, -12], [4, -1, 1], [-14, -6, -2], [23, -28, -5], [2, 12, -10], [-2, 4, -8], [56, -1, -1], [25, -8, -11], [-2, 7, 9], [-16, 4, 14], [4, -14, -2], [6, 15, -3], [-6, 13, 9], [-20, 8, 2], [16, 5, -1], [-3, -14, 7], [-32, -11, 5], [6, 1, -3], [-13, -3, -8], [-8, -10, 4], [2, 12, 0], [-28, -3, 1], [16, 2, -6], [-16, 15, 3], [-10, 7, 7], [17, -21, -8], [16, 6, 16], [-18, -9, 7], [16, 13, -7], [4, 10, 10], [-4, -17, 1], [6, 19, -3], [-44, 22, 10], [-12, 4, -2], [24, -1, -9], [25, -6, 11], [0, -16, -12], [-8, 10, 10], [-18, 20, 12], [-29, 4, 7], [29, -7, -2], [-13, 2, 13], [-26, -6, -2], [18, 15, -3], [-2, 1, -15], [-2, 4, -16], [16, 10, -14], [-24, -8, 12], [-46, 0, -8], [-18, 5, 15], [13, 4, 5], [0, -6, -10], [-19, -6, -1], [-2, 9, -1], [-20, 13, 11], [18, -16, -2], [-14, -5, -11], [22, 7, -5], [42, -11, -7], [7, 23, -2], [2, 5, 7], [-10, 7, 7], [-22, 6, 8], [9, -14, -13], [-12, -15, 19], [14, -5, 3], [22, -1, 17], [-40, 1, 1], [38, 7, -5], [-6, -7, -11], [14, 0, -6], [7, 8, 11], [-56, 7, -3], [-6, -19, 7], [-14, 1, -9], [24, -31, -5], [21, 1, 2], [0, 9, -3], [6, -1, -5], [-2, -6, -6], [-10, -5, 15], [5, -7, 14], [-33, 29, 4], [-3, -28, 5], [-7, 8, 17], [4, -2, -20], [34, 4, -12], [18, 15, -11], [2, -13, -9], [0, -3, -3], [44, -9, 7], [51, 0, -1], [-26, -12, 12], [25, 7, 4], [-17, 10, 13], [-15, 27, 8], [-40, 30, 10], [21, -5, 4], [-14, 11, 3], [45, 2, 3], [-16, -4, 10], [42, -20, -8], [10, -11, 13], [-20, 8, 4], [11, -20, -9], [58, -16, -2], [41, 9, -2], [-6, 3, -17], [50, 15, -5], [6, -26, 6], [-20, 1, 5], [-6, -3, 9], [-12, -25, 11], [20, 15, 11], [-8, -18, -2], [-4, 9, 17], [-26, 3, -13], [58, 3, 1], [48, -3, -11], [70, -1, 3], [36, -21, 3], [-14, -3, 5], [16, -4, 0], [-47, 1, -10], [30, -15, -1], [24, 30, -18], [-15, 20, -3], [2, -22, -12], [-42, 7, -3], [-8, 19, 9], [28, -16, -10], [2, -5, 3], [22, 5, 27], [48, -2, -6], [-22, -31, 11], [-20, -3, -19], [-58, 12, -2], [-24, -9, -3], [30, -12, -12], [4, 4, 12], [-20, -1, -23], [-1, 21, -20], [-12, -25, -5], [0, 22, 14], [28, -10, 8], [38, 13, 3], [-42, 34, 12], [2, -2, 10], [-5, -13, 6], [-31, 28, 1], [-18, 19, -5], [36, -7, 7], [2, -3, -1], [-6, 17, 3], [46, 2, -2], [40, 14, -6], [-38, 20, -6], [-26, -6, 16], [20, -20, 8], [2, -10, -2], [55, -31, -8], [-29, -11, -8], [14, -23, 9], [-42, 0, -8], [8, 25, -15], [21, 9, 6], [-49, 4, 3], [26, -5, -3], [-21, -8, 7], [-20, 14, 4], [-4, -10, 14], [-40, 14, 18], [-47, 27, 4], [26, -9, -5], [54, -2, -6], [-8, -7, -31], [-48, -3, -5], [36, 2, 20], [3, 7, 10], [-60, 10, 10], [32, -28, -16], [34, 2, -10], [1, 14, 27], [-10, 0, -8], [-18, 37, 3], [-26, -1, 15], [28, -9, 9], [6, -17, -1], [-6, 32, 0], [-61, -1, 2], [16, -18, 6], [-2, -1, -27], [35, -16, -17], [-6, -15, -11], [59, -6, 9], [6, 10, -10], [-12, 7, -23], [16, 24, -18], [-10, 12, -22], [-26, 1, -3], [-54, -4, 4], [1, 17, -6], [27, -19, 8], [22, -13, -15], [50, -39, -17], [-3, 6, 11], [-58, 5, 11], [-16, -2, -26], [-32, 0, 0], [6, -20, -6], [-13, 7, 2], [-10, -18, -12], [78, -3, -3], [6, 16, -12], [-6, 21, -5], [0, 7, 5], [10, -21, -13], [23, -10, 5], [26, -12, -4], [-4, 6, -10], [-56, 23, 3], [-6, -32, 18], [58, 1, -9], [-18, 18, 6], [-50, 12, 16], [-50, -4, 10], [-8, 13, -3], [28, 15, 3], [8, -14, 22], [-6, -25, 13], [10, -17, -3], [26, 19, 7], [-56, 4, -8], [14, 24, -8], [-38, 9, 3], [36, -3, 11], [37, 22, -17], [-36, -8, 14], [-55, 13, -10], [-3, 0, -7], [26, -11, -7], [24, 29, -23], [8, -38, 2], [58, 5, 7], [19, -3, -24], [-11, 10, 11], [70, 5, -7], [-36, -16, -2], [38, 21, -5], [-9, 36, -1], [18, 5, 27], [28, 9, 11], [56, -8, 4]]
