
# q-expansion of newform 5776.2.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 = 5776
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], [-1], [-1], [0], [4], [1], [3], [0], [-5], [-7], [4], [-10], [5], [5], [7], [-11], [3], [11], [-3], [11], [15], [-13], [0], [-3], [5], [-1], [0], [20], [-3], [-14], [-3], [-15], [-5], [-9], [3], [-16], [7], [4], [-15], [-15], [-12], [5], [-16], [-15], [2], [7], [-9], [-25], [-20], [2], [-21], [-12], [-19], [31], [-23], [9], [-27], [-31], [10], [-7], [9], [30], [27], [20], [11], [-15], [20], [5], [5], [10], [30], [-15], [-25], [10], [-12], [-29], [3], [-25], [-19], [17], [-12], [1], [21], [25], [-13], [-25], [-22], [-22], [11], [20], [20], [23], [-8], [1], [5], [-21], [-15], [-26], [-19], [-1], [25], [-5], [36], [-22], [4], [-10], [-15], [-5], [45], [26], [32], [-29], [-45], [20], [-41], [-39], [-19], [-32], [-30], [5], [-31], [10], [-10], [16], [-36], [35], [3], [-7], [-7], [-30], [13], [-25], [29], [-17], [30], [7], [21], [52], [-18], [-18], [21], [-9], [31], [-7], [-30], [-25], [39], [-15], [-13], [-36], [5], [6], [-25], [25], [35], [-24], [40], [23], [23], [-19], [-27], [25], [45], [35], [10], [-45], [-5], [-53], [53], [51], [-12], [-35], [-39], [14], [45], [15], [19], [-45], [40], [35], [23], [4], [41], [5], [24], [-42], [25], [31], [10], [-24], [-19], [55], [4], [-51], [-25], [9], [-25], [15], [-10], [24], [21], [1], [5], [61], [25], [-37], [7], [-51], [-57], [-40], [35], [-49], [-60], [-3], [24], [45], [33], [-30], [-32], [-45], [-39], [-40], [-23], [25], [-13], [-45], [10], [-9], [35], [-36], [39], [17], [-24], [50], [64], [-63], [-18], [50], [13], [9], [31], [20], [35], [-11], [9], [0], [-37], [53], [-37], [75], [-38], [71], [40], [25], [-25], [23], [-9], [20], [33], [7], [1], [15], [63], [21], [27], [28], [18], [-6], [-55], [75], [-42], [28], [-45], [-32], [-23], [-75], [-12], [65], [5], [-69], [60], [-4], [-57], [30], [-7], [75], [30], [-27], [-67], [-38], [-66], [72], [45], [44], [1], [-22], [-55], [35], [-29], [-12], [-5], [73], [-37], [-4], [69], [-23], [45], [13], [-45], [74], [44], [-33], [-31], [36], [-15], [-78], [3], [-89], [35], [-50], [-5], [-45], [9], [-19], [-45], [-39], [25], [-20], [-65], [-49], [31], [-3], [37], [-4], [75], [43], [30], [30], [86], [32], [29], [15], [33], [-40], [-11], [-53], [-21], [57], [11], [18], [-11], [-83], [-49], [24], [-23], [-46], [61], [81], [-5], [-75], [-17], [-49], [-75], [80], [25], [25], [78], [39], [-79], [-57], [12], [1], [-43], [-35], [50], [-97], [-56], [95], [-33], [33], [-66], [14], [47], [15], [-72], [-25], [-16], [-5], [11], [-20], [-95], [1], [12], [35], [21], [25], [67], [82], [55], [-9], [-95], [-29], [72], [5], [9], [25], [-65], [34], [-35], [40], [101], [15], [16], [64], [58], [66], [-87], [-40], [27], [-35], [33], [73], [-80], [13], [74], [18], [-76], [43], [76], [13]]
