
# q-expansion of newform 5472.2.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 = 5472
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 = [[0], [0], [1], [-1], [-3], [-4], [3], [1], [8], [0], [2], [-8], [0], [11], [7], [-2], [-6], [-1], [-10], [-2], [5], [-2], [0], [-6], [-12], [-14], [-14], [-10], [-2], [-12], [16], [-13], [-5], [-5], [15], [-18], [2], [-4], [-12], [-18], [16], [6], [-19], [-14], [6], [7], [-22], [28], [16], [-11], [29], [5], [4], [11], [20], [11], [18], [24], [-15], [2], [15], [16], [-30], [15], [-26], [10], [-26], [-8], [-25], [-7], [-26], [-27], [4], [-24], [-10], [18], [-19], [-23], [30], [-22], [-36], [30], [28], [-28], [-28], [-9], [12], [-1], [-33], [17], [-33], [32], [38], [-4], [29], [24], [-28], [-42], [-26], [19], [-4], [7], [-4], [24], [40], [15], [3], [-6], [30], [26], [8], [9], [27], [4], [-25], [-28], [-29], [-27], [-25], [-30], [48], [-10], [4], [-16], [45], [50], [-26], [41], [-7], [42], [23], [-6], [-20], [-41], [-45], [-29], [46], [-44], [22], [29], [16], [-9], [-9], [30], [20], [-22], [2], [26], [-5], [4], [-46], [7], [7], [-18], [-30], [54], [-16], [10], [13], [-18], [4], [-18], [16], [52], [2], [-36], [-8], [61], [-18], [-21], [-12], [14], [-28], [-50], [28], [53], [-20], [23], [-8], [35], [17], [-12], [14], [46], [41], [-5], [6], [50], [60], [-12], [32], [21], [10], [-44], [-28], [24], [45], [-14], [-54], [-29], [-12], [-48], [58], [40], [-20], [-45], [-47], [12], [-57], [-62], [-26], [38], [-29], [24], [-16], [8], [0], [-4], [-42], [-39], [-62], [40], [28], [-19], [18], [-17], [-46], [54], [30], [-15], [11], [-74], [42], [18], [33], [57], [73], [45], [36], [-38], [2], [25], [16], [38], [0], [27], [57], [-18], [40], [-45], [-34], [29], [12], [-6], [-3], [-36], [-43], [-8], [-12], [-38], [-76], [-6], [35], [-62], [33], [46], [6], [-3], [34], [21], [-50], [52], [33], [-45], [-82], [-80], [41], [14], [-70], [-19], [-74], [48], [-37], [2], [-77], [45], [19], [64], [-41], [80], [2], [13], [16], [70], [-58], [30], [-48], [81], [-54], [-35], [-39], [34], [79], [-70], [25], [14], [28], [76], [49], [34], [-59], [-66], [-44], [16], [40], [-2], [63], [34], [55], [39], [40], [5], [-54], [-22], [-58], [22], [32], [-4], [-24], [-45], [-41], [-90], [8], [77], [63], [-13], [34], [-64], [-1], [44], [-30], [-49], [36], [10], [-54], [44], [-78], [60], [-16], [-25], [-88], [68], [21], [54], [-66], [44], [24], [16], [-69], [-48], [-55], [-22], [12], [30], [21], [44], [-54], [-94], [-70], [-92], [40], [-81], [16], [-29], [25], [44], [0], [-69], [29], [-20], [-74], [5], [33], [-11], [28], [80], [-44], [6], [69], [-100], [46], [-70], [68], [65], [58], [-18], [34], [-44], [-28], [-18], [-50], [73], [48], [-74], [74], [-31], [5], [32], [-48], [15], [-86], [-35], [46], [-47], [-6], [42], [-100], [10], [-42], [-34], [46], [-8], [63], [-12], [-4], [84], [48], [25], [67], [-101]]
