
# q-expansion of newform 1521.2.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 = 1521
weight = 2
poly_data = [-4, -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], [0, 0], [-2, 1], [1, 1], [-2, 0], [0, 0], [0, -1], [-4, 2], [-2, 0], [-2, 3], [-1, 1], [-6, 1], [0, 1], [3, -1], [2, -4], [-4, -3], [-6, -2], [7, 2], [-3, 1], [14, 0], [7, -2], [7, 1], [-4, -2], [8, 2], [7, -1], [6, -1], [-5, -1], [2, -4], [-5, -5], [2, 5], [7, 1], [2, 6], [-4, 1], [-7, 7], [-4, -1], [0, -6], [11, -6], [-3, -5], [4, -4], [-14, 4], [8, 2], [2, 3], [6, -2], [1, -8], [4, -6], [5, -11], [15, -3], [-8, 0], [4, -2], [10, -4], [-26, 0], [-16, 6], [10, -5], [0, 12], [-12, 11], [0, 6], [12, -6], [19, -7], [10, 3], [10, -5], [-9, 3], [22, 1], [-23, 5], [16, -2], [-9, 3], [0, -9], [11, 5], [-3, 2], [-12, 10], [1, 5], [10, 3], [0, 6], [-3, 9], [19, -6], [-19, 3], [-4, 12], [-20, 9], [-11, 9], [16, 1], [3, 6], [28, -4], [-25, 4], [-8, 2], [15, 4], [-9, 3], [16, -12], [-2, 4], [3, 10], [-24, -5], [19, 3], [26, 0], [4, -4], [-4, 2], [30, -4], [8, 8], [22, -4], [-6, 5], [20, 7], [12, -14], [-1, 12], [-7, -7], [-32, 7], [14, -6], [-10, -12], [4, 6], [22, 3], [30, -6], [6, 15], [26, -10], [-18, -7], [-16, 0], [13, -14], [2, -5], [-9, 13], [17, -1], [2, -11], [5, -17], [22, 10], [-20, 2], [-4, -4], [-15, -6], [-17, 4], [0, -10], [-10, 18], [31, -13], [-4, -6], [-5, -8], [14, -6], [-9, 19], [-33, -8], [-4, 4], [-22, -6], [16, -18], [30, 0], [0, -6], [18, 0], [-18, 4], [-27, 11], [16, 10], [-4, -13], [-1, 1], [-6, -8], [8, 0], [4, 12], [1, -8], [6, -10], [1, 2], [-34, 1], [23, -1], [18, -8], [-22, 1], [20, 7], [1, -5], [-34, -6], [28, 0], [-20, 12], [24, 8], [0, 5], [-6, 1], [18, -8], [-20, 12], [-6, 16], [4, 2], [-12, -16], [44, -11], [-2, -10], [-16, 22], [-5, 10], [-25, 15], [6, -17], [28, -6], [26, -7], [-2, 22], [-25, -10], [40, 8], [46, -7], [5, -15], [-30, 13], [17, 3], [31, -5], [24, -22], [42, -2], [29, 8], [-38, 12], [30, -4], [10, -25], [17, -7], [4, -24], [-53, 2], [20, -2], [-27, -10], [0, 18], [-21, 11], [-42, -8], [-34, 0], [0, -10], [21, 8], [-43, 8], [-22, -1], [34, 2], [44, -5], [25, 3], [23, -12], [-15, -6], [22, -16], [-20, -11], [57, -3], [-36, 0], [-16, 29], [4, 20], [-17, 4], [-20, -14], [-41, -9], [-22, 0], [-24, 8], [-46, 3], [41, -13], [22, 9], [-42, -10], [-8, -6], [29, -8], [1, 15], [8, -23], [33, -1], [20, -6], [-61, 6], [2, 9], [-20, 24], [59, 5], [0, -20], [29, 8], [7, -15], [24, -4], [6, 9], [-25, 5], [-2, 14], [-13, 19], [-18, -11], [-52, 12], [26, -20], [-40, 6], [0, 10], [0, -2], [5, 26], [46, -17], [12, -8], [-45, -3], [18, -22], [-3, -3], [-32, -6], [23, -1], [26, -12], [-14, -2], [-19, 14], [10, -8], [-18, -20], [-2, 8], [-8, -26], [-28, 10], [2, 11], [28, 16], [16, 4], [-33, -8], [19, -10], [36, 9], [19, 7], [-30, 12], [-46, -13], [-20, 2], [-36, -3], [-37, -10], [3, -19], [18, 11], [44, -24], [-27, 29], [0, -14], [-54, 2], [-38, -11], [29, 4], [12, -22], [54, -4], [-24, 0], [10, 18], [5, -26], [28, 10], [10, -24], [9, -22], [2, -19], [15, -17], [54, -11], [-40, -2], [-54, 12], [-8, -11], [-10, 18], [43, -9], [12, 5], [-24, 0], [18, -9], [2, -30], [-27, -11], [-21, 28], [38, -15], [0, 10], [14, -2], [-21, -1], [-9, 18], [-58, 4], [18, 11], [-30, 10], [-18, -1], [64, -6], [-10, 12], [28, -33], [-7, 29], [-6, 24], [-3, -26], [20, -30], [2, 20], [-9, -1], [30, 9], [-7, 9], [45, 5], [8, 1], [35, -9], [54, -4], [-23, 18], [15, -29], [36, 10], [-42, -8], [-54, 3], [34, 9], [-18, 15], [-7, -23], [-76, 10], [3, 23], [-22, -16], [-39, 6], [-10, -12], [-38, 3], [33, -9], [38, -12], [-32, 19], [-42, 21], [-5, -17], [22, 12], [50, -16], [25, -24], [-32, -16], [30, -4], [-36, 10], [-47, -3], [-22, 16], [4, 7], [43, 7], [62, -1], [-4, -29], [40, 2], [4, 24], [-8, -25], [28, -6], [62, -3], [12, -5], [6, 2], [38, 6], [25, -37], [-17, 27], [0, -6], [3, 17], [50, -3], [-32, -20], [-19, 5], [14, -12], [36, 5], [40, 2], [34, -29], [-40, 22], [52, -14], [45, -35], [10, 17], [-39, 14], [-32, 10], [-54, -9], [8, 6], [-46, 0], [63, -23], [36, 18], [8, -6], [-7, 9], [53, -9], [66, 10], [74, -9], [-2, -15], [64, -2], [0, -6], [32, 16], [41, 1], [-64, -6], [-62, 24], [28, 0], [24, -13], [-43, -11], [-22, -4], [59, 1], [30, -1], [28, -29], [41, 1], [31, -13], [28, -31], [73, -7], [-44, 28], [-33, -4], [16, 17], [34, -12], [64, -20], [33, -38], [-16, 5], [-56, 12], [-11, 13], [4, -30], [24, -4], [42, -25], [-27, 14], [48, -18], [-84, -6], [-63, -11], [-22, -15], [-38, 34], [20, -31], [43, -1], [46, 6]]
